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http://arxiv.org/abs/2307.02344v1 | 20230705145733 | Applying the Resonance Method to $\textrm{Re}\left(e^{-iθ}\logζ(σ+it)\right)$ | [
"Mikko Jaskari"
] | math.NT | [
"math.NT"
] |
We apply the resonance method to Montgomery's convolution formula for (e^-iθlogζ(σ+it)) in the strip 1/2 < σ < 0.88. This gives new insight into maximal values of (e^-iθlogζ(σ+it)) for t ∈ [T^β,T] for all β∈ (0,1) and real θ.
Quantum Limits of Position and Polarizability Estimation in the Optical Near Field
Stefan Nimmrichter
==================================================================================
§ INTRODUCTION
The Riemann zeta function is a famously important function in number theory. All the non-trivial zeros of the zeta function are in the critical strip 0 ≤(s) ≤ 1 and they are related to the distribution of the prime numbers. This is one of the reasons why it is important to study the behavior of the zeta function inside the critical strip.
In 1977 H. L. Montgomery <cit.> proved that if we let σ∈ (1/2,1) and T > T_0(σ), then for any real θ we have[We denote log_j as the jth iterated logarithm and for instance log_2T=loglogT.]
max_T^(σ-1/2)/3≤ t ≤ T( e^-iθlogζ(σ+it) ) ≥1/20( σ-1/2)^1/2(logT)^1-σ/(log_2T)^σ.
Moreover, assuming the Riemann hypothesis, Montgomery <cit.> obtains
max_T^1/6≤ t ≤ T( e^-iθlogζ(σ+it) ) ≥1/20(logT)^1-σ/(log_2T)^σ
for any real θ. In Montgomery's method the lower bound for t weakens notably without the use of the Riemann hypothesis when σ→ 1/2^+. C. Aistleitner improved the lower bound of the extreme value in <cit.> in the case θ≡ 0 (mod 2π) by showing that for fixed σ∈ (1/2,1) and sufficiently large T we have
max_ 0 ≤ t ≤ T |ζ(σ+it)| ≥exp( 0.18(2σ-1)^1-σ(logT)^1-σ/(loglogT)^σ).
Our goal is to prove a Montgomery type result with better lower bound for t when σ is close to 1/2.
Fix σ∈ (1/2,0.88), β∈ (0,1) and 0 < κ < min(σ-1/2,1-β). Then there exists a positive constant c independent from any of the chosen parameters σ, β or κ such that for any θ∈ and sufficiently large T we have
max_t ∈ [T^β, T]( e^-iθlogζ(σ+it) ) ≥ cκ^1-σ/√(|log(2σ-1)|)(logT)^1-σ/(log_2T)^σ.
Under the Riemann hypothesis we could choose κ < 1 - β and would not require κ < σ - 1/2. Due to condition κ < σ - 1/2, our unconditional result in the case t ∈ [0,T] is weaker than Montgomery's. Taking θ = π we obtain the following corollary.
Fix σ∈ (1/2,0.88), β∈ (0,1) and 0 < κ < min(σ-1/2,1-β). Then there exists a positive constant c independent from any of the chosen parameters σ, β or κ such that for sufficiently large T we have
max_t ∈ [T^β, T] -log|ζ(σ+it)| ≥ cκ^1-σ/√(|log(2σ-1)|)(logT)^1-σ/(log_2T)^σ.
Result like corollary <ref> can then be converted into estimate of the upper bound of the minimum of |ζ(σ+it)| for t ∈ [T^β, T].
We will prove Theorem <ref> by means of resonance method introduced by K. Soundararajan <cit.>. Our work is heavily inspired by the works of A. Bondarenko and K. Seip <cit.>, <cit.> and <cit.> and also by the work of K. Mahatab and A. Chirre <cit.>.
In <cit.> Bondarenko and Seip proved that there exists a positive and continous function ν(σ) on (1/2,1) bounded from below by 1/(2-2σ), with the asymptotic behavior
ν(σ) = (1-σ)^-1 + O(|log(1-σ)|), σ→ 1^-
(1/√(2) + o(1))√(|log(2σ-1)|), σ→ 1/2^+
and such that the following holds. If T is sufficiently large, then for 1/2 + 1/log_2T≤σ≤ 3/4,
max_t ∈ [√(T),T] |ζ(σ+it)| ≥exp( ν(σ)(logT)^1-σ/(log_2T)^σ)
and for 3/4 ≤σ≤ 1-1/log_2T,
max_t ∈ [T/2,T] |ζ(σ+it)| ≥log_2Texp( c+ν(σ)(logT)^1-σ/(log_2T)^σ)
with c an absolute constant independent of T.
This theorem already improves both Aistleitner's and Montgomery's theorems and is seemingly much stronger than Theorem <ref> in the specific case θ≡ 0 (mod 2π). The method Mahatab and Chirre used in <cit.> to prove under Riemann hypothesis that for each fixed β∈ (0,1) there exists a positive constant c > 0 such that for sufficiently large T we have
max_ t∈ [T^β,T]±ζ( 1/2 + it ) ≥ c√(logTlog_3T/log_2T).
would also give stronger result to the case θ≡±π/2 (mod 2π). However, these proofs do not directly generalize to the case θ∈ and particularly θ≡π (mod 2π).
Our strategy is to use Montgomery's convolution formula and apply resonance method by choosing a similar resonator as Bondarenko and Seip with some modifications. It is worth to mention that the result of Bondarenko and Seip has been recently improved by Zikang Dong and Bin Wei <cit.> by a factor 2^σ as σ→ 1/2^+. Their result gives that if we let 1/2 < σ < 1 and fix β∈ (0,1) we have that
max_t ∈ [T^β, T]log|ζ(σ+it)| ≥ c_β(σ)(logT)^1-σ/(log_2T)^σ
holds for a function c_β(σ) which has the asymptotic behavior
c_β(σ) = (√(2) +o(1))(1-β)^1-σ√(|log(σ-1/2)|), as σ→ 1/2^+.
§ THE CONVOLUTION FORMULA
We first introduce the needed convolution formula. The following lemma is <cit.>.
Let σ∈ (1/2,1) and t ≥ 15. Suppose that ζ(σ_0+iu) ≠0 for any σ_0 ∈ [σ,1) and any u ∈ such that |u-t| ≤ 2(logt)^2. Then, for any ψ > 0 and any real H,
2/π∫_-(logt)^2^(logt)^2logζ(σ+i(t+u))( sinψ u/u)^2 e^iHudu
= ∑_n=1^∞Λ(n)max(0,ψ-|H-logn|)/n^σ+itlogn + O( e^|H|+2ψ/(logt)^2).
Here Λ is the von Mangoldt function. We now argue as Montgomery did in <cit.>. Let ψ = 1/2 and take successively H = H_1,H_2,H_3 where H_1 := -logx, H_2 := 0, H_3 := logx where 1 ≤ x ≤ (logt)^2. Now the main term on the right hand side of (<ref>) vanishes when H ∈{H_1, H_2}. Nevertheless, multiplying (<ref>) for H_1, H_2 and H_3 by 1/2e^-iθ, 1 and 1/2e^iθ respectively and adding them together we get
2/π∫_-(logt)^2^(logt)^2logζ(σ+i(t+u))( sin(u/2)/u)^2 (1+cos(θ+ulogx))du
= 1/2 e^iθ∑_e^-1/2x≤ n ≤ e^1/2xΛ(n)/n^σ+itlogn( 1/2 - | logn/x| ) + O( x/(logt)^2).
§ PROOF OF THEOREM <REF> BY THE RESONANCE METHOD
The resonance method introduced by Soundararajan <cit.> is based on evaluation of the ratio of two moments. The numerator moment is the integral of the investigated function multiplied by a chosen non-negative resonator over a chosen interval and the denominator moment is the integral of the resonator over the same interval. The ratio then gives a lower bound for the investigated function on that chosen interval.
In order to construct the resonator needed in the resonance method we will first define various sets motivated by <cit.> and <cit.>. Fix σ∈ (1/2,0.88), β∈ (0,1) and 0 < κ < min(σ-1/2,1-β). Let us denote N = ⌊ T^κ⌋.
Define the following sets
* P is the set of primes in the interval (elogNlog_2N,e^2logNlog_2N].
* Let a > 1 be fixed, let
M := { m ∈ : m has at least alogN/|log(2σ-1)| prime factors in P }
and let M' ⊂ M contain all the integers in M that have prime factors only in P.
Let f be the multiplicative function that is supported on the square-free numbers with all prime factors in P and
f(p):= 1/√(|log(2σ-1)|), p ∈ P.
Define
ℳ := supp(f) ∖ M
which in other words means that
ℳ = { m ∈ : m is square-free, all prime factors of m are in P
and m has at most alogN/|log(2σ-1)| prime factors }.
Now let 𝒥 be defined as
𝒥 := { j ∈ : [(1+T^-1)^j, (1+T^-1)^j+1) ∩ℳ≠∅}
and
ℳ' := { m_j : j ∈𝒥, m_j = min{[(1+T^-1)^j, (1+T^-1)^j+1) ∩ℳ}.
Let γ be fixed so that 0 < γ < 1. Let L be the set of integers that have at most γlogN/|log(2σ-1)| prime factors in P. Let L' ⊂ L contain integers in L that have prime factors only in P. Now set
ℒ := ℳ∖ L.
Hence ℒ is the set of integers in ℳ that have at least γlogN/|log(2σ-1)| prime factors.
We now define the resonator we need.
Let r : ℳ' → be defined as
r(m_j) := ( ∑_(1-T^-1)^j-1≤ n ≤ (1+T^-1)^j+2
n ∈ℳ f(n)^2 )^1/2, for every j ∈𝒥
and the resonator R : → as
R(t) := ∑_m ∈ℳ'r(m)/m^it.
It is crucial that |ℳ| ≤ N for large N and this is essentially shown in <cit.>. Note that the sets we use are subsets of the sets used in <cit.> and we only consider the case k=1.
We will now move forward to the definition of the moments. First we denote
Φ(t) := e^-t^2/2
and let Φ̂ be the Fourier transform of Φ defined as
Φ̂(y) := ∫_-∞^∞Φ(t)e^-itydt = √(2π)Φ(y).
Since in Lemma <ref> we need to assume that there are no zeros of Riemann the zeta function on the right side of the contour we have to bypass all those cases where such zeros exists. We do so by defining the following indicator function.
Denote by ρ non-trivial zeros of the ζ-function and let I(σ,t) be an indicator function defined as
I(σ,t) = 1, if there is no zero ρ such that (ρ) ≥σ and |t-(ρ)| ≤ (logt)^2
0, otherwise.
Using the indicator function we define moments as follows.
Let σ∈ (1/2,0.88) be fixed. We define two moments M_1(R,T) and M_2(R,T) as
M_1(R,T) := ∫_T^β^TlogT( ∫_-(logt)^2^(logt)^2 K(u)du )|R(t)|^2Φ( t/T)dt,
M_2(R,T) := ∫_T^β^TlogT( ∫_-(logt)^2^(logt)^2 e^-iθlogζ(σ+i(t+u)) K(u)du )|R(t)|^2Φ( t/T)I(σ,t)dt,
where
K(u) = ( sin(u/2)/u)^2 (1+cos(θ+ulog(elogNlog_2N)))
+( sin(u/2)/u)^2 (1+cos(θ+ulog(e^3/2logNlog_2N)))
+( sin(u/2)/u)^2 (1+cos(θ+ulog(e^2logNlog_2N))).
Now
max_t ∈ [T^β,TlogT](e^-iθlogζ(σ+it)) ≥(M_2(R,T))/M_1(R,T).
We begin with the use of Lemma <ref> and (<ref>) to obtain
|I(σ,t)( ∫_-(logt)^2^(logt)^2 e^-iθlogζ(σ+i(t+u)) K(u)du )|
≥|I(σ,t)(π/8∑_n ∈ PΛ(n)/n^σ+itlogn + O( log_2T/logT)) |
≥| I(σ,t)(π/8∑_n ∈ PΛ(n)/n^σ+itlogn +o(1)) |
for t ∈ [T^β, TlogT]. We will first focus on the main term of (M_2(R,T)). Similarly to <cit.> we obtain
(∫_T^β^TlogTπ/8∑_n ∈ PΛ(n)/n^σ+itlogn|R(t)|^2Φ( t/T)dt )
= π/8∑_n ∈ PΛ(n)/n^σlogn( ∫_T^β^TlogT n^-it|R(t)|^2Φ( t/T) dt )
= π/8∑_m,v ∈ℳ'∑_p ∈ Pr(m)r(v)/p^σ( ∫_T^β^TlogTΦ( t/T)e^-itlog(mp/v) dt).
We first focus on evalution of
( ∫_T^β^TlogTΦ( t/T)e^-itlog(mp/v) dt).
Combining (<ref>) with the fact that Φ(t) is an even and real function we get
( ∫_0^∞Φ( t/T)e^-itlog(mp/v) dt) = 1/2∫_-∞^∞Φ( t/T)e^-itlog(mp/v)dt
= T√(2π)/2Φ( Tlogmp/v).
By the trivial estimate |Φ(u)e^-iy| ≤ 1 we have
| ( ∫_0^T^βΦ( t/T)e^-itlog(mp/v) dt) | ≤ T^β.
For T > 193 we have by rapid decay of Φ(t) as t →∞
|( ∫_TlogT^∞Φ( t/T)e^-itlog(mp/v) dt)| ≤∫_TlogT^∞1/t^2 dt= o(1) as T →∞.
We may then conclude by (<ref>), (<ref>) and (<ref>) that since β < 1 we have
( ∫_T^β^TlogTΦ( t/T)e^-itlog(mp/v) dt) = T√(2π)/2Φ( Tlogmp/v) + O(T^β).
In order to evaluate M_2(R,T) we have to remove all t ∈ [T^β,TlogT] with I(σ,t)=0.
Let N(σ,T) denote the number of zeros ρ of the ζ-function for which (ρ) ≥σ and 0 ≤(ρ) ≤ T. Now, for T ≥ 10 and 1/2 ≤σ≤ 1,
N(σ,T) ≪ T^3/2-σ(logT)^5.
See <cit.>.
We note that for each zero ρ with (ρ) ∈ [T^β,TlogT] there exists at most one interval where I(σ,t)=0 and the length of such interval is ≪ (logT)^2. By Lemma <ref> and similar estimate as done for (<ref>) we may conclude that
( ∫_T^β^TlogTΦ( t/T)e^-itlog(mp/v) (1-I(σ,t))dt) ≪ T^3/2-σ(logT)^9,
Hence
( ∫_T^β^TlogTΦ( t/T)e^-itlog(mp/v) I(σ,t)dt)
= T√(2π)/2Φ( Tlogmp/v) + O(T^3/2-σ(logT)^9 + T^β).
In order to evaluate the contribution of the error terms we have to estimate the size of the resonator.
For any real t we have
|R(t)|^2 ≤ 3T^κ∑_l∈ℳf(l)^2.
We follow <cit.>. Using |R(t)| ≤ R(0) we begin with
R(0)^2 = ∑_m,n ∈ℳ' r(m)r(n) ≤ |ℳ'|∑_m ∈ℳ' r(m)^2
where we used the inequality ab ≤ (a^2 +b^2)/2. Recall from the beginning of this section and notes after Definition <ref> that there are at most N = ⌊ T^κ⌋ elements in ℳ'. Now by the definition of the set ℳ' and the function r
∑_m ∈ℳ' r(m)^2 ≤ 3∑_l ∈ℳ f(l)^2.
We obtain the desired result by combining the upper bound of |ℳ'|, (<ref>) and (<ref>).
Combining the definition of M_2(R,T), (<ref>), (<ref>), (<ref>), (<ref>) and Lemma <ref> we obtain
(M_2(R,T)) ≥ Tπ√(2π)/16∑_m,v ∈ℳ'∑_p ∈ Pr(m)r(v)/p^σ(Φ( Tlogmp/v) )
+ O((T^3/2+κ-σ(logT)^9 + T^β+κ)∑_l∈ℳf(l)^2).
The next two lemmas allow us to lower bound the main term on the right hand side of (<ref>).
We have
∑_m,v ∈ℳ'∑_p ∈ Pr(m)r(v)/p^σΦ( Tlogmp/v) ≥∑_v ∈ℳ f(v)^2 ∑_p|v1/f(p)p^σ.
We follow <cit.>. We consider all triples m', v' ∈ℳ' and p ∈ P such that |pm'/v'-1| ≤3/T. We use the notation
J(m') := [(1+T^-1)^j, (1+T^-1)^j+1),
where j is the unique integer such that (1+T^-1)^j≤ m < (1+T^-1)^j+1. By the definition of r(m') and the Cauchy-Schwarz inequality we have for any p ∈ P and any m',v' ∈ℳ'
∑_m,v ∈ℳ
mp=v
m ∈ J(m'), v ∈ J(v') f(m)f(v) ≤(∑_m,v ∈ℳ
mp=v
m ∈ J(m'), v ∈ J(v') f(m)^2)^1/2(∑_m,v ∈ℳ
mp=v
m ∈ J(m'), v ∈ J(v') f(v)^2)^1/2
≤( ∑_m ∈ J(m') f(m)^2 )^1/2( ∑_v ∈ J(v') f(v)^2 )^1/2
≤ r(m')r(v')
and hence, by the definition of ℳ', that, for any p ∈ P,
∑_m,v ∈ℳ
mp=v f(m)f(v) ≤∑_m', v' ∈ℳ'
|pm'/v'-1| ≤3/T r(m')r(v').
Now
∑_m,v ∈ℳ'∑_p ∈ Pr(m)r(v)/p^σΦ( Tlogmp/v) ≥∑_p ∈ P∑_m,v ∈ℳ
mp=vf(m)f(v)/p^σ = ∑_v ∈ℳ f(v)^2 ∑_p|v1/f(p)p^σ.
In the last step we used multiplicativity of f. This completes the proof.
For fixed 0 < γ <1 we have
∑_v ∈ℳ f(v)^2 ∑_p|v1/f(p)p^σ≥γ∑_v ∈ℒf(v)^2 e^-2σ/√(|log(2σ-1)|)(logN)^1-σ/(log_2N)^σ
We follow similar ideas as in <cit.>. We recall the definition of the set ℒ from Definition <ref>.
We have
∑_v ∈ℳf(v)^2∑_p|v1/f(p)p^σ≥∑_v ∈ℒf(v)^2∑_p|v1/f(p)p^σ.
Now
∑_v ∈ℒf(v)^2∑_p|v1/f(p)p^σ≥∑_v ∈ℒf(v)^2 γlogN/|log(2σ-1)|min_p ∈ P1/f(p)p^σ.
We obtain the desired result by noting that
min_p ∈ P1/f(p)p^σ≥√(|log(2σ-1)|)(e^2logNlog_2N)^-σ
and combining (<ref>) and (<ref>).
Now by combining (<ref>) and Lemmas <ref> and <ref> we obtain the following lemma.
Fix σ∈ (1/2,0.88), β∈ (0,1) and κ < min(σ-1/2, 1-β). Then there exists a positive constant c_1 independent from σ, β and κ such that
(M_2(R,T)) ≥ c_1T∑_v ∈ℒ f(v)^21/√(|log(2σ-1)|)(logN)^1-σ/(log_2N)^σ
+O((T^3/2+κ-σ(logT)^9 + T^β+κ)∑_v ∈ℳf(v)^2).
Here the restriction κ < min(σ-1/2,1-β) guarantees that the error term is acceptable. We now move forward to the evaluation of the denominator moment M_1(R,T).
There exists a positive constant c_2 such that
M_1(R,T) ≤ c_2 T∑_n ∈ℳf(n)^2.
Since K(u) ≥ 0 for all u note that as in <cit.> we have
∫_-∞^∞K(u)du ≪ 1
since K(u) is a sum of three distinct kernels introduced in Montgomery's paper. Next we proceed as in <cit.> and note that according to Fourier transform (<ref>) we have
∫_-∞^∞ |R(t)|^2Φ( t/T)dt = √(2π)T∑_m,n∈ℳ'r(m)r(n)Φ( Tlogm/n).
By the definitions of the set ℳ' and the function r, we find that
∑_m∈ℳ' r(m)^2 ≤ 3∑_n∈ℳf(n)^2.
We note that
∑_m,n∈ℳ'r(m)r(n)Φ(Tlogm/n) ≤∑_m ∈ℳ' r(m)^2 + ∑_m,n ∈ℳ'
m≠nr(m)r(n)Φ(Tlogm/n).
We recall the definition of the set 𝒥 from Definition <ref>. To deal with the off-diagonal terms we find, that
∑_m,n ∈ℳ'
m≠nr(m)r(n)Φ(Tlogm/n)
≤∑_j,l ∈𝒥
j≠lr(m_j)r(n_l)Φ(T(|j-l|-1)log(1+T^-1))
≪∑_j,l ∈𝒥
j≠lr(m_j)r(n_l)Φ(|j-l|-1 )
Applying the inequality ab ≤ (a^2 + b^2)/2 we have
∑_j,l ∈𝒥
j≠lr(m_j)r(n_l)Φ(|j-l|-1 ) ≤∑_j,l ∈𝒥
j≠lr(m_j)^2Φ(|j-l|-1 ).
Hence
∑_m,n ∈ℳ'
m≠nr(m)r(n)Φ(Tlogm/n) ≪∑_j ∈𝒥r(m_j)^2.
Hence combining (<ref>), (<ref>), (<ref>), (<ref>) and (<ref>) we see that
∫_-∞^∞( ∫_-∞^∞K(u)du )|R(t)|^2Φ( t/T)dt ≪ T∑_n ∈ℳf(n)^2
and the result follows from the definition of M_1(R,T).
Now according to (<ref>) and Lemmas <ref> and <ref> there is a need to evaluate the ratio
∑_v ∈ℒf(v)^2/∑_n∈ℳf(n)^2
and this is done in our next lemmas.
Let a>1 be fixed and M is defined as in Definition <ref>. Then
1/∑_j ∈ f(j)^2∑_v ∈ M f(v)^2 = o(1) as N →∞.
See <cit.>.
If γ < 1 is fixed and ℒ is defined as in Definition <ref>. Then
1/∑_j ∈ f(j)^2∑_v ∉ℒ f(v)^2 = o(1) as N →∞.
We follow the ideas in <cit.>. We note that
ℒ = supp(f) ∖ (M ∪ L)
and by Lemma <ref> it is sufficient to show
1/∑_j ∈ f(j)^2∑_v ∈ L f(v)^2 = o(1) as N →∞.
We recall the definition of L' from Definition <ref> and use the fact that f is multiplicative to obtain
1/∑_j ∈ f(j)^2∑_v ∈ L f(v)^2 = 1/∏_p ∈ P (1+f(p)^2)∑_v ∈ L'f(v)^2
where, for any b ∈(0,1),
∑_v ∈ L' f(v)^2 ≤ b^-γlogN/|log(2σ-1)|∏_p ∈ P(1+bf(p)^2).
Recall from Definition <ref> that f(p)=1/√(|log(2σ-1)|). Define
C := inf_σ∈ (1/2,0.88)1/1+f(p)^2 = 0.21533….
For b that is chosen close enough to 1, we have
1/∏_p ∈ P (1+f(p)^2)∑_v ∈ L'f(v)^2 ≤ b^-γlogN/|log(2σ-1)|exp( C∑_p ∈ P (b-1)f(p)^2 ),
due to the Taylor expansion of
log( 1+bf(p)^2/1+f(p)^2)=log(1-(1-b)f(p)^2/1+f(p)^2).
The cardinality of P is at most e^2logN. By the prime number theorem
∑_p ∈ P f(p)^2 = (1+o(1))logN/|log(2σ-1)|(e^2 -e).
Hence we obtain
1/∑_j ∈ f(j)^2∑_v ∈ L f(v)^2
≤exp(( C(e^2-e)(b-1)-γlogb+o(1) )logN/|log(2σ-1)|).
If we choose b and γ close to 1 we may note that
C(e^2-e)(b-1) - γlogb < 0.
This completes the proof.
Since ℒ = supp(f) ∖ (M ∪ L) and ℳ = supp(f) ∖ M we have by Lemmas <ref> and <ref>
∑_v ∈ℒf(v)^2/∑_n∈ℳf(n)^2 = 1-o(1), as N →∞.
It follows from combining (<ref>), (<ref>) and Lemmas <ref> and <ref> that
max_t ∈ [T^β,TlogT](e^-iθlogζ(σ+it))
≥ cκ^1-σ/√(|log(2σ-1)|)(logT)^1-σ/(log_2T)^σ+O(T^1/2+κ-σ(logT)^9 + T^β+κ-1).
for some positive c. Theorem <ref> then follows by noting that the desired restriction T^β≤ t ≤ T is obtained by trivial adjustment, applying (<ref>) for T/logT in place of T and β' ∈ (β,1-κ) in place of β.
§ ACKNOWLEDGEMENTS
I was partially funded by UTUGS graduate school and completed this work while working in Academy of Finland projects no. 333707 and 346307. I would also like to thank Kaisa Matomäki for supervising my work and Kamalakshya Mahatab for giving ideas for this project.
plain
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http://arxiv.org/abs/2307.02799v1 | 20230706061757 | Few-Shot Personalized Saliency Prediction Using Tensor Regression for Preserving Structural Global Information | [
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Online Linear Regression Based on Weighted Average
Mohammad Abu-Shaira1Greg Speegle2
August 1, 2023
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This paper presents a few-shot personalized saliency prediction using tensor-to-matrix regression for preserving the structural global information of personalized saliency maps (PSMs). In contrast to a general saliency map, a PSM has been great potential since its map indicates the person-specific visual attention that is useful for obtaining individual visual preferences from heterogeneity of gazed areas. The PSM prediction is needed for acquiring the PSM for the unseen image, but its prediction is still a challenging task due to the complexity of individual gaze patterns.
For recognizing individual gaze patterns from the limited amount of eye-tracking data, the previous methods adopt the similarity of gaze tendency between persons. However, in the previous methods, the PSMs are vectorized for the prediction model. In this way, the structural global information of the PSMs corresponding to the image is ignored. For automatically revealing the relationship between PSMs, we focus on the tensor-based regression model that can preserve the structural information of PSMs, and realize the improvement of the prediction accuracy. In the experimental results, we confirm the proposed method including the tensor-based regression outperforms the comparative methods.
Salinecy prediction, personalized saliency map, tensor regression, person similarity, adaptive image selection.
Online Linear Regression Based on Weighted Average
Mohammad Abu-Shaira1Greg Speegle2
August 1, 2023
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§ INTRODUCTION
Humans can selectively obtain vital information from the abundant visual information in the complex real-world owing to their visual system. Traditionally, many researchers have tried to introduce such human mechanisms into image-processing models <cit.>. Specifically, a saliency map, which represents the
salient parts more noticeable than the neighbor parts, is predicted for reproducing the human instinctive visual perception <cit.>. Such a saliency map is predicted for each image without personalization. However, the different persons actually focus on different areas even when they gaze at the same scene, that is, individual differences exist <cit.>. To model the individual visual attention, the personzalization of the salinacy map has been addressed over the past few years <cit.>. For distinguishing between a traditional saliency map and its personalisation, we call a universal saliency map (USM) and a personalized saliency map (PSM), respectively. While the USM omits the differences between individuals, the PSM is predicted for each person. Since the personalized visual preferences can be reflected by differences between PSMs <cit.>, such individuality can be useful for many situations (e.g., personalized video summarization <cit.>). Here, for obtaining the PSM for unseen images in advance, it is required to predict the PSM from the individual gaze tendency.
To capture the individual gaze tendency, the relationship between the visual stimuli, e.g., images, and its individual PSM should be analyzed from eye-tracking data obtained from each person in the past. Then the gaze patterns emerging in images are quite complex and individually different, and those characteristics lead to the difficulty of the PSM prediction. For extracting the gaze patterns and tendencies, several researchers have collected eye-tracking data for thousands of images <cit.>. Moreover, in these researches, the simultaneous prediction of PSMs for several persons has been tried by using a multi-task convolutional neural network (multi-task CNN) <cit.> to compensate for the lack of data <cit.>. In <cit.>, personalized information has been introduced into the PSM prediction model of several persons as the same training data. The prediction models adopted in these researches are based on deep learning that requires a massive amount of training data for each person. Actually, the large-scale PSM dataset is openly available, but the acquisition of a large amount of individual eye-tracking data can be a significant burden and time-consuming for persons in the application. In this way, it is desired that the PSM prediction with the limited amount of training eye-tracking data.
To tackle the challenging task to predict the PSM from the limited amount of data, the way to use the gaze data obtained from persons that have a similar gaze tendency to the target person can be an effective strategy. For determining whether the person has a similar gaze tendency to the target person, several pairs of eye-tracking data for the same images are needed. Thus, such pairs cannot be acquired in large quantities, and the selection of images to acquire eye-tracking data is an important process. In <cit.>, images that induce scattering of gazes are selected by using adaptive image selection (AIS) for efficiently and steadily obtaining the similarity of gaze tendencies between the target and other persons (called training persons in this paper). Here, we assume that enough amount of eye-tracking data is acquired from training persons, and this is not arbitrary as there is a large-scale PSM dataset. Under this assumption, we can use the PSMs of training persons for any images since their PSMs can be predicted by using the previous researches <cit.>. In this research, the similarity of gaze tendencies between training persons and the target person is used for the simple method of taking a weighted average of PSMs obtained from training persons to predict PSMs of the target person. On the other hand, in the several researches, learning-based methods are used to predict the PSM of the target person from the PSMs of the training persons. Concretely, in <cit.>, the collaborative multi-output Gaussian process regression (CoMOGP) <cit.> is used for predicting the PSM. However, the methods using the general regression or its varieties need the vector format as inputs, and the structural global information of PSMs cannot be effectively used. Such information is the important clue for detecting salient areas in the human visual system <cit.>, and thus, the improvement of prediction performance is expected by constructing the prediction method preserving the structural information.
We propose an inter-person gaze similarity based on tensor regression for few-shot PSM prediction in this paper.
In the proposed method, we construct the tensor-to-matrix regression model <cit.> that can predict PSMs of the target person from the PSMs of training persons. Here, note that the PSMs of training persons are maps predicted by the deep learning-based model. Then the tensor-to-matrix regression model can treat the multi-array tensor format as its inputs and outputs. Thus, this regression model can reveal the transformation coefficient tensor from the input tensor to the output matrix, that is, the PSM. In this way, we can preserve the structural global information without vectorizing. Our contribution is that we construct the novel PSM prediction method using tensor-to-matrix regression for preserving the structural global informatin and reveal that the proposed PSM prediction model is effective by experimentation with the open dataset.
§ PROPOSED FEW-SHOT PSM PREDICTION
Our few-shot PSM prediction consists of three phases and the whole flow is shown in Fig. <ref>. We assume that there are the P training persons with a vast amount of eye-tracking data and a target person with a limited amount of eye-tracking data. In practice, the assumption that training persons exist is pragmatic since the large-scale dataset is available<cit.>.
First the multi-task CNN <cit.> is trained to predict the PSMs of the training persons by referring to the previous study <cit.>.
Next, we choose the common images that the target person gazes at based on the AIS scheme <cit.>. The common images are chosen such that they bring more diverse gaze patterns to persons. Finally, the proposed method predicts the PSM by using tensor-to-matrix regression <cit.> with the PSMs of training persons.
§.§ Multi-Task CNN for Training Persons
For predicting the PSMs of the P training persons, the multi-task CNN <cit.> is adopted by referring to the previous study <cit.> in the proposed method.
Concretely, we prepare the training images X_n ∈ℝ^d_1 × d_2 × d_3 (n=1,2,…,N; N being the number of training images) and its USM U(X_n)∈ℝ^d_1 × d_2, where d_1 × d_2 and d_3 are the size of the image the color channel, respectively. For effectively obtaining the predicted PSMs of training persons, the previous study <cit.> adopts the specific approach, that is, predicting the difference map M(X)_p ∈ℝ^d_1 × d_2 (p=1,2,…,P) between the USM and PSM as M(X)_p = S (X)_p - U(X),
where S (X)_p is the PSM of pth training person based on eye-tracking data for the image X. Next, to simultaneously predict PSMs of training persons, we construct the multi-task CNN consisting of one image encoder and P PSM decoders, and optimize its trainable parameters by minimizing the following objective function:
∑_p=1^P∑_n=1^N∑_l=1^L||M̂_l(X_n)_p-M(X_n)_p||^2_F,
where M̂_l(X_n)_p (l=1,2,…,L; L being the number of convolution layers in one decoder) is a predicted difference map calculated from lth layer, and ||·||_F^2 represents the Frobenius norm.
Given the test image X_tst, the predicted PSM of the pth person is calculated as Ŝ(X_tst)_p = M̂_L(X_tst)_p + U(X_tst).
The multi-task CNN can predict the PSMs of the training persons, simultaneously, and consider the relationship of PSMs between the training persons.
§.§ Adaptive Image Selection for PSM Prediction
We need to acquire eye-tracking data for capturing the gaze tendency of the target person.
We choose the limited number of images from N training images used in the previous subsection for obtaining the similarity of the tendency between the target and training persons. For effectively analyzing such similarity, the I common images that bring more diverse gaze patterns to persons are chosen by using the AIS scheme <cit.>. Concretely, the AIS scheme pays attention to the variety of the common images and variation in PSMs obtained from the training persons. For simultaneously considering these factors, the AIS scheme uses the variation in PSMs for objects in each image. First, we calculate the PSMs and their variance for each object B_n,j (j=1,2,…,J; J being the number of object categories in the training images) in training images X_n. To detect the object bounding box, we apply the novel object detection method <cit.> to the training images and obtain the rectangle whose size is d^h_n,j× d^w_n,j for jth object in ith image. The PSM variance q_n,j for object B_n,j is calculated as follows:
q_n,j = 1/d^h_n,jd^w_n,jP∑_p=1^P ||S̅ (B_n,j)_p⊙S̅ (B_n,j)_p||_F^1,
S̅ (B_n,j)_p = S (B_n,j)_p-1/P∑_p=1^P S (B_n,j)_p,
where S(B_n,j)_p represents the PSM for object B_n,j of person p, and ⊙ is the operator of the Hadamard product. Then we set q_n,j=0 when X_n not including jth object and set the largest q_n,j when the image X_n including several mth objects. Then we obtain the sum of q_n,j for nth image by q̅_n = ∑_j=1^J q_n,j.
Finally, by using q̅_n, we choose top I images as common images under the constraint to maximize the number of object categories in common images. In this way, the chosen common images have multiple object categories and objects in common images have the high PSM variance. Therefore, by using the AIS scheme, we can choose the common images with the consideration of the variety and the PSM variation.
§.§ PSM Prediction via Tensor-to-Matrix Regression
This subsection shows the tensor-to-matrix regression model for few-shot PSM prediction. The PSMs predicted in Sec. <ref> are used to predict the PSM of the target person in the proposed method. That is, we need to treat the several PSMs as input, and the input tensor 𝒮(X_i) ∈ℝ^P × d_1 × d_2 (i=1,2,…,I) corresponding to the image X_i chosen in Sec. <ref> is constructed as follows:
𝒮(X_i)= [Ŝ(X_i)_1,Ŝ(X_i)_2,…,Ŝ(X_i)_P].
Moreover, we prepare the supervised PSM S(X_i)_p^tst of the target person p^tst for the input tensor 𝒮(X_i). Here, we assume that the target person gazes at only the common images in Sec. <ref>, and we can obtain the supervised PSM S(X_i)_p^tst. In the tensor-to-matrix regression scenario, the weight tensor 𝒲∈ℝ^P × d_1 × d_2 × d_1 × d_2 is used for predicting the PSM of the newly given image as follows:
S_TReg(X_tst)_p^tst=⟨𝒮(X_tst),𝒲⟩_3,
where ⟨·,·⟩_Q represents the tensor product and Q is the number of input arrays.
For optimising the weight tensor 𝒲, we minimize the sum of squared error with L_2 regularization as follows:
min_rank(𝒲)≤ R∑_i=1^I||S(X_i)_p^tst-⟨𝒮(X_i),𝒲⟩_3||_F^2 + λ||𝒲||_F^2.
Note that it is difficult to solve this minimization problem due to the inputs and outputs being the multi-array. Thus, by referring to <cit.>, we assume that 𝒲 has the reduced PARAFAC/CANDECOMP rank such that rank(𝒲)≤ R, and solve Eq. (<ref>) under this constraint.
In this way, by using the tensor-to-matrix regression model, the proposed method can preserve the structural information without vectorizing the input tensor and the output matrix.
§ EXPERIMENTS
§.§ Dataset
In this subsection, we explain the settings of the dataset. The PSM dataset <cit.> that is the open large-scale dataset, was used in our experiment. For details, the PSM dataset consists of 1,600 images with corresponding eye-tracking data obtained from 30 participants. Experimental participants had normal or corrected visual acuity and gazed at one image for three seconds under the free viewing condition. For evaluating the predicted PSMs, we constructed the PSMs of each participant for all images from eye-tracking data as the ground truth (GT) map based on the previous work <cit.>. As the USM used in the proposed method, we adopted the mean PSMs of the training persons since we reduce the influence of USM prediction errors. In the proposed method, we needed the training images with eye-tracking data for training multi-task CNN and common images chosen from training images for training the tensor-to-matrix regression model. Thus, 1,100 images were randomly selected for training and the rest 500 images were used as test images in this experiment. Moreover, I common images were chosen from training images based on the AIS scheme. In addition, we randomly selected 20 participants from the PSM dataset as training persons and the rest 10 persons were treated as the target persons. Although eye-tracking data of the target persons were available, we only used eye-tracking data of the target persons for common images since we assume that the target persons gaze at common images for the PSM prediction.
§.§ Experimental Settings
This subsection describes the parameter settings of the proposed method and evaluation settings using compared methods. We optimized the multi-task CNN used in Sec. <ref> and the tensor-to-matrix regression model used in Sec. <ref>, separately. Concretely, we used the stochastic gradient descent <cit.> by referring to <cit.>, and then the number of layers L, momentum, batch size, epoch, and learning rate were set to 3, 0.9, 9, 1000, and 3.0× 10^-5, respectively. On the other hand, the tensor-to-matrix regression model was optimized by simply differentiating weight parameters with the tensor unfolding.
Moreover, we set I=100, and R ∈{5,10,…,50}, λ∈{0.01,0.1,…,10000}.
To objectively evaluate the proposed method, we adopted several USM and PSM prediction methods as compared methods. Concretely, we adopted the following USM prediction methods; Signature<cit.>, GBVS<cit.>, Itti<cit.>, SalGAN<cit.> and Contextual <cit.>. Then Signature, GBVS, and Itti were the computational models that predict USM only from the input image without training. SalGAN and Contextual were deep learning-based models trained by using the SALICON dataset <cit.> that is a large-scale dataset without considering personalization. In addition, we adopted the following two few-shot PSM prediction (FPSP) methods by using only common images and their eye-tracking data as baselines.
Baseline1: PSM prediction using visual similarity between the target and common images <cit.>.
Baseline2: PSM prediction based on Baseline1 and the USM prediction method <cit.>.
Moreover, we compared with following three PSM prediction methods that are the similar setting to the proposed method.
Similarity-based FPSP: FPSP based on the similarity of gaze tendency similar to the proposed method <cit.>, but this method used simply weighted average of the predicted PSMs of training persons.
CoMOGP-based FPSP: FPSP based on CoMOGP <cit.> instead of Sec.<ref>.
Object-based Gaze Similarity (OGS)-based FPSP: FPSP using the object similarity between the target and common images <cit.>.
As the evaluation metrics, we adopted Kullback-Leibler divergence (KLdiv) and cross correlation (CC) between the predicted PSM and the GT map from the literature <cit.>. Specifically, KLdiv was used for evaluating the similarity of the distribution, that is, the structural similarity, while CC was used for evaluating the pixel-based similarity. By using these two metrics, we can evaluate the proposed and compared methods from the perspectives of both global and local similarities between predicted PSMs and its GTs.
§.§ Results and Discussion
Figure <ref> shows the predicted results, and Table <ref> shows the quantitative evaluation results. From Fig. <ref>, the PSMs predicted by the proposed method have a distribution close to GTs, and thus, we confirm the effectiveness of preserving the structural global information. Besides, from Table <ref>, we compare the proposed and compared methods. In the evaluation metric “KLdiv”, our method outperforms all compared methods, and thus, we confirm that the tensor-to-matrix regression is effective to PSM prediction with considering the structural global information.
Concretely, it is confirmed the effectiveness of the PSM prediction since our method outperforms the USM prediction methods with the state-of-the-art (SOTA) USM prediction method, Contextual <cit.>. Moreover, by comparing our method with other PSM prediction methods, the effectiveness of focusing on structural global information is confirmed. The “KLdiv” of OGS-based FPSP <cit.> is similar to our method, but this compared method cannot have high generalization performances since this method just retrieves the similar object in the target image from common images for considering structural information. While, our regression-based FPSP can have high generalization performances since our method captures the relationships of gaze tendencies between training and target persons in the training process.
While, the evaluation metric “CC” of our method is comparative to the SOTA method, but not the best. This reason can be considered that the local information cannot be considered when the low rank approximation of the weight tensor is conducted in Sec. <ref>. Then “CC” is the pixel-based evaluation, while “KLdiv” is the distribution-based evaluation, and thus, the proposed method succeeds in preserving structural global information owing to the high “KLdiv” value.
Therefore, we indicate theat the proposed method is effective for PSM prediction preserving the structural global information.
In addition to the comparative experiments, we confirm the changes in the values of the evaluation metrics in response to changes in hyperparameters of the tensor-to-matrix regression. Figure <ref> shows the values of the evaluation metrics in response to R and λ. From this figure, we can verify that R becomes larger, the prediction performance also becomes better, while λ=1000 is the best performance regardless of R. Then we consider that higher R is better, but there is a risk of increased computational complexity. Moreover, λ needs not be so high since λ is the hyperparameter of the regularization. In this way, we show the desirable hyperparameters of tensor-to-matrix regression for few-shot PSM prediction.
§ CONCLUSIONS
This paper has presented a few-shot PSM prediction using tensor-to-matrix regression for preserving the structural global information of PSMs. By treating the input and output PSMs without the vectorization, the proposed method can preserve the structural information. The experiment on the open dataset shows the effectiveness of the tensor-to-matrix regression for PSM prediction.
IEEEbib
|
http://arxiv.org/abs/2307.00818v1 | 20230703075729 | Motion-X: A Large-scale 3D Expressive Whole-body Human Motion Dataset | [
"Jing Lin",
"Ailing Zeng",
"Shunlin Lu",
"Yuanhao Cai",
"Ruimao Zhang",
"Haoqian Wang",
"Lei Zhang"
] | cs.CV | [
"cs.CV"
] |
[-25]A Data-driven Under Frequency Load Shedding Scheme in Power Systems
Qianni Cao, Student Member, IEEE,
Chen Shen, Senior Member, IEEE
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In this paper, we present Motion-X, a large-scale 3D expressive whole-body motion dataset.
Existing motion datasets predominantly contain body-only poses, lacking facial expressions, hand gestures, and fine-grained pose descriptions.
Moreover, they are primarily collected from limited laboratory scenes with textual descriptions manually labeled, which greatly limits their scalability.
To overcome these limitations, we develop a whole-body motion and text annotation pipeline, which can automatically annotate motion from either single- or multi-view videos
and provide comprehensive semantic labels for each video and fine-grained whole-body pose descriptions for each frame.
This pipeline is of high precision, cost-effective, and scalable for further research.
Based on it, we construct Motion-X, which comprises 13.7M precise 3D whole-body pose annotations (i.e., SMPL-X) covering 96K motion sequences from massive scenes.
Besides, Motion-X provides 13.7M frame-level whole-body pose descriptions and 96K sequence-level semantic labels.
Comprehensive experiments demonstrate the accuracy of the annotation pipeline and the significant benefit of Motion-X in enhancing expressive, diverse, and natural motion generation, as well as 3D whole-body human mesh recovery.
§ INTRODUCTION
Human motion generation aims to automatically synthesize natural human movements. It has wide applications in robotics, animation, games, and generative creation. Given a text description or audio command, motion generation can be controllable to obtain the desired human motion sequence. Text-conditioned motion generation has garnered increasing attention in recent years since it behaves in a more natural interactive way <cit.>.
Although existing text-motion datasets <cit.> have greatly facilitated the development of motion generation <cit.>, their scale, diversity, and expressive capability remain unsatisfactory.
Imagine generating “a man is playing the piano happily", as depicted in Fig. <ref>(a), the motion from existing dataset <cit.> only includes the body movements, without finger movements or facial expressions.
The missing hand gestures and facial expressions severely hinder the high level of expressiveness and realism of the motion. Additionally, certain specialized motions, such as high-level skiing, aerial work, and riding are challenging to be captured in indoor scenes.
To sum up, existing datasets suffer from four main limitations: 1) body-only motions without facial expressions and hand poses; 2) insufficient diversity and quantity, only covering indoor scenes; 3) lacking diverse and long-term motion sequences; and 4) manual text labels that are unscalable, unprofessional and labor-intensive. These limitations hinder existing generation methods to synthesize expressive whole-body motion with diverse action types. Therefore, how to collect large-scale whole-body motion and text annotations from multi-scenarios are critical in addressing the data scarcity issue.
Compared to indoor marker-based mocap systems, markerless vision-based motion capture methods <cit.> become promising to capture large-scale motions from massive videos. Meanwhile, human motion can be regarded as a sequence of kinematic structures, which can be automatically translated into pose scripts using rule-based techniques <cit.>. More importantly, although markerless capture (e.g., pseudo labels) is not as precise as marker-based methods, collecting massive and informative motions, especially local motions, could still be beneficial <cit.>. Besides, text-driven motion generation task requires semantically corresponding motion labels instead of vertex-corresponding mesh labels, and thus have a higher tolerance of motion capture error.
Bearing these considerations in mind, we design a scalable and systematic pipeline for motion and text annotation in both multi-view and single-view videos.
Firstly, we gather and filter massive video recordings from a variety of scenes with challenging, high-quality, multi-style motions and sequence-level semantic labels.
Subsequently, we estimate and optimize the parameters of the SMPL-X model <cit.> for the whole-body motion annotation. Due to the depth ambiguity and various challenges in different scenes, existing monocular estimation models typically fail to yield satisfactory results. To address this issue, we systematically design a high-performance framework incorporating several innovative techniques, including a hierarchical approach for whole-body keypoint estimation, a score-guided adaptive temporal smoothing and optimization scheme, and a learning-based 3D human model fitting process. By integrating these techniques, we can accurately and efficiently capture the ultimate 3D motions.
Finally, we design an automatic algorithm to caption frame-level descriptions of whole-body poses. We obtain the body and hand scripts by calculating spatial relations among body parts and hand fingers based on the SMPL-X parameters and extract the facial expression with an emotion classifier. We then aggregate the low-level pose information and translate it into textual pose descriptions.
Based on the pipeline, we collect a large-scale whole-body expressive motion dataset named , which includes 13.7M frames and 96K sequences with precise 3D whole-body motion annotations, pose descriptions, and semantic labels. To compile this dataset, we collect massive videos from the Internet, with a particular focus on game and animation motions, professional performance, and diverse outdoor actions. Additionally, we incorporated data from eight existing action datasets <cit.>.
Using , we build a benchmark for evaluating several state-of-the-art (SOTA) motion generation methods. Comprehensive experiments demonstrate the benefits of for diverse, expressive, and realistic motion generation (shown in Fig. <ref> (b)). Furthermore, we validate the versatility and quality of on the whole-body mesh recovery task.
Our contributions can be summarized as follows:
* We propose a large-scale expressive motion dataset with precise 3D whole-body motions and corresponding sequence-level and frame-level text descriptions.
* We elaborately design a automatic motion and text annotation pipeline, enabling efficient capture of high-quality human text-motion data at scale.
* Comprehensive experiments demonstrate the accuracy of the motion annotation pipeline and the benefits of in 3D whole-body motion generation and mesh recovery tasks.
§ PRELIMINARY AND RELATED WORK
In this section, we focus on introducing existing datasets for human motion generation. For more details about the motion generation methods, please refer to the appendix.
Benchmarks annotated with sequential human motion and text are mainly collected for three tasks: action recognition <cit.>, human object interaction <cit.>, and motion generation <cit.>.
Specifically, KIT Motion-Language Dataset <cit.> is the first public dataset with human motion and language descriptions, enabling multi-modality motion generation <cit.>. Although several indoor human motion capture (mocap) datasets have been developed <cit.>, they are scattered. AMASS <cit.> is noteworthy as it collects and unifies 15 different optical marker-based mocap datasets to build a large-scale motion dataset through a common framework and parameterization via SMPL <cit.>. This great milestone benefits motion modeling and its downstream tasks. Additionally, BABEL <cit.> and HumanML3D <cit.> contribute to the language labels through crowdsourced data collection. BABEL proposes either sequence labels or sub-sequence labels for a sequential motion, while HumanML3D collects three text descriptions for each motion clip from different workers.
Thanks to these text-motion datasets, various motion generation methods have rapidly developed and shown advantages in diverse, realistic, and fine-grained motion generation <cit.>.
However, existing text-motion datasets have several limitations, including the absence of facial expressions and hand gestures, insufficient data quantity, limited diversity of motions and scenes, coarse-grained and ambiguous descriptions, and the lack of long sequence motions. To bridge these gaps, we develop a large-scale whole-body expressive motion dataset with comprehensive sequence- and frame-level text labels. We aim to address these limitations and open up new possibilities for future research. We provide quantitative comparisons of and existing datasets in Tab. <ref>.
§ MOTION-X DATASET
§.§ Overview
As shown in Tab. <ref>, we collect from eight public datasets and online videos and provide the following motion and text annotations:
13.7M 3D whole-body SMPL-X annotation,
96K sequence-level semantic descriptions (e.g., walking with waving hand and laughing),
and frame-level whole-body pose descriptions. Notably, original sub-datasets lack either whole-body motion or text labels and we unify them with our annotation pipeline.
All annotations are manually checked to guarantee quality. In Fig. <ref>, we show the averaged temporal standard deviation of body, hand, and face keypoints of each sub-dataset , highlighting the diversity of hand movements and facial expressions, which fills in the gaps of previous body-only motion data.
§.§ Data Collection
As illustrated in Fig. <ref>, the overall data collection pipeline involves six key steps: 1) designing and sourcing motion text prompts via large language model (LLM) <cit.>, 2) collecting videos, 3) preprocessing candidate videos through human detection and video transition detection, 4) capturing whole-body motion (Sec. <ref>), 5) captioning sequence-level semantic label and frame-level whole-body pose description(Sec. <ref>), and 6) performing the manual inspection.
We gather 81K motion sequences from existing datasets using our proposed unified annotation framework, including the multi-view datasets (NTU-RGBD 120 <cit.>, AIST <cit.>), human-scene-interaction datasets (Egobody <cit.> and GRAB <cit.>), single-view action recognition datasets (HAA500 <cit.>, HuMMan <cit.>), and body-only motion capture dataset (AMASS <cit.>). For these datasets, steps 1 and 2 are skipped. Notably, only the mocap datasets (AMASS, Egobody, and GRAB) provide SMPL-X labels with body and hand pose, thus we annotate the SMPL-X label for the other 51.6K motion sequences. For the mocap data, which contains the body and roughly static hand motions, we skip step 4 and fill in the facial expression with a data augmentation mechanism. The facial expressions are collected from existing facial datasets BAUM <cit.> via a face capture and animation model EMOCA <cit.>.
Details about the processing of each sub-dataset are available in the appendix.
To improve the richness, we collect 15K monocular videos from online sources, covering various real-life scenes as depicted in Fig. <ref>.
Since human motions and actions are context-dependent and vary with the scenario in which they occur,
we design action categories as motion prompts based on the scenario and function of the action via LLM. To ensure comprehensive coverage of human actions, our dataset includes both general and domain-specific scenes. The general scenes encompass daily actions (e.g., brushing hair, wearing glasses, and applying creams), sports activities (e.g., high knee, kick legs, push-ups), various musical instrument playing, and outdoor scenes (e.g., BMX riding, CPR, building snowman). The inclusion of general scenes helps bridge the gap between existing data and real-life scenarios.
In addition, we incorporate domain-specific scenes that require high professional skills, such as dance, Kung Fu, Tai Chi, performing arts, Olympic events, entertainment shows, games, and animation motions. Based on the prompts describing the above scenes, we run the data collection pipeline to gather the necessary data for our dataset.
§ AUTOMATIC ANNOTATION PIPELINE
§.§ Universal Whole-body Motion Annotation
Overview.
To efficiently capture a large volume of potential motions from massive videos, we propose an annotation pipeline for high-quality whole-body motion capture with
three novel techniques:
(i) hierarchical whole-body keypoint estimation;
(ii) score-guided adaptive temporal smoothing for jitter motion refinement;
and (iii) learning-based 3D human model fitting for accurate motion capture.
2D Keypoint Estimation. 2D Whole-body keypoint estimation poses a challenge due to the small size of the hands and face regions.
Although recent approaches have utilized separate networks to decode features of different body parts <cit.>,
they often struggle with hand-missing detection and are prone to errors due to occlusion or interaction.
To overcome these limitations, we customize a novel hierarchical keypoint annotation method, depicted in the blue box of Fig. <ref>.
We train a ViT-WholeBody based on a ViT-based model <cit.> on the COCO-Wholebody dataset <cit.> to estimate initial whole-body
keypoints 𝐊^2D∈ℝ^133×2 with confidence scores.
Leveraging the ViT model's ability to model semantic relations between full-body parts,
we enhance hand and face detection robustness even under severe occlusion.
Subsequently, we obtain the hand and face bounding boxes based on the keypoints, and refine the boxes using the BodyHands detector <cit.> through an IoU matching operation.
Finally, we feed the cropped body, hand, and face regions into three separately pre-trained ViT networks to estimate body, hand and face keypoints, which are used to update 𝐊^2D.
Score-guided Adaptive Smoothing. To address the jitter resulting from per-frame pose estimation
in challenging scenarios such as heavy occlusion, truncation, and motion blur, while preserving motion details,
we introduce a novel score-guided adaptive smoothing technique into the traditional Savitzky-Golay filter <cit.>.
The filter is applied to a sequence of 2D keypoints of a motion:
𝐊̅_i^2D = ∑_j=-w^w c_j 𝐊^2D_i+j,
where 𝐊_i^2D is the original keypoints of the i_th frame,
𝐊̅_i^2D is the smoothed keypoints,
w corresponds to half-width of filter window size,
and c_j are the filter coefficients.
Different from existing smoothing methods with a fixed window size <cit.>, we leverage the confidence scores of the keypoints to adaptively adjust the window size to balance between smoothness and motion details.
Using a larger window size for keypoints with lower confidence scores can mitigate the impact of outliers.
3D Keypoint Annotation. Precise 3D keypoint can boost the estimation of SMPL-X. We utilize novel information from large-scale pre-trained models. Accordingly, for single-view videos, we adopt a pretrained model <cit.>, which is trained on massive 3D datasets, to estimate precise 3D keypoints.
For multi-view videos, we utilize bundle adjustment to calibrate and refine the camera parameters, and then triangulate the 3D keypoints 𝐊̅^3D based on the multi-view 2D keypoints. To enhance stability, we adopt temporal smoothing and enforce 3D bone length constraints during triangulation.
Local Pose Optimization. After obtaining the keypoints, we perform local pose optimization to register each frame's whole-body model SMPL-X <cit.>. Traditional optimization-based methods <cit.> are often time-consuming and may yield unsatisfactory results as they ignore image clues and motion prior. We propose a progressive learning-based human mesh fitting method to address these limitations. Initially, we predict the SMPL-X parameter Θ
with the SOTA whole-body mesh recovery method OSX <cit.> and face reconstruction model EMOCA <cit.>.
And then, through iterative optimization of the network parameters, we fit the human model parameters Θ̂ to the target 2D and 3D joint positions by minimizing the following functions, achieving an improved alignment accuracy:
L_joint = ‖𝐊̂^3D-𝐊̅^3D‖_1 + ‖𝐊̂^2D-𝐊̅^2D‖_1
+ ‖Θ̂ - Θ‖_1.
Here, 𝐊̂^3D represents the predicted 3D joint positions obtained by applying a linear regressor to a 3D mesh generated by the SMPL-X model. 𝐊̂^2D is derived by performing a perspective projection of the 3D keypoints. The last term of the loss function provides explicit supervision based on the initial parameter, serving as a 3D motion prior.
To alleviate potential biophysical artifacts, such as interpenetration and foot skating, we incorporate a set of physical optimization constraints:
L = λ_jointL_joint + λ_smoothL_smooth + λ_penL_pen + λ_phyL_phy.
Here, λ are weighting factors of each loss function and L_smooth is a first-order smoothness term:
L_smooth = ∑_t ‖Θ̂_2:t-Θ̂_1:t-1‖_1 + ∑_t ‖𝐊̂^3D_2:t-𝐊̂^3D_1:t-1‖_1,
where Θ̂_i and 𝐊̂^3D_i represent the SMPL-X parameters and joints of the i-th frame, respectively. To alleviate mesh interpenetration, we utilize a collision penalizer <cit.>, denoted as L_pen. Additionally, we incorporate the physical loss L_phy based on PhysCap <cit.> to prevent implausible poses.
Global Motion Optimization. To improve the consistency and realism of the estimated global trajectory,
we perform a global motion optimization based on GLAMR <cit.> to simultaneously refine the global motions and camera poses to align with video evidence, such as 2D keypoints:
L_g = λ_2DL_2D + λ_trajL_traj + λ_camL_cam + λ_regL_reg,
where L_2D represents the 2D keypoint distance loss, L_traj quantifies the difference between the optimized global trajectory and the trajectory estimated by Kama <cit.>. L_reg enforces regularization on the global trajectory, and L_cam applies a smoothness constraint on the camera parameters.
Human Verification. To ensure quality, we manually checked the annotation by removing the motions that do not align with the video evidence or exhibit obvious biophysical artifacts.
§.§ Obtaining Whole-body Motion Descriptions
Sequence motion labels.
The videos in were collected from online sources and existing datasets.
For action-related datasets <cit.>, we use the action labels as one of the sequence semantic labels. Meanwhile, we input the videos into Video-LLaMA <cit.> and filter the human action descriptions as supplemental texts.
When videos contain semantic subtitles, EasyOCR automatically extracts semantic information. For online videos, we also use the search queries generated from LLM <cit.> as semantic labels.
Videos without available semantic information, such as EgoBody <cit.>, are manually labeled using the VGG Image Annotator (VIA) <cit.>.
For the face database BAUM <cit.>, we use the facial expression labels provided by the original creator.
Whole-body pose descriptions. The generation of fine-grained pose descriptions for each pose involves three distinct parts: face, body, and hand, as shown in Fig. <ref>(a).
Facial expression labeling uses the emotion recognition model EMOCA <cit.> pretrained on AffectNet <cit.> to classify the emotion.
Body-specific descriptions utilizes the captioning process from PoseScript <cit.>, which generates synthetic low-level descriptions in natural language based on given 3D keypoints. The unit of this information is called posecodes, such as `the knees are completely bent'. A set of generic rules based on fine-grained categorical relations of the different body parts are used to select and aggregate the low-level pose information. The aggregated posecodes are then used to produce textual descriptions in natural language using linguistic aggregation principles.
Hand gesture descriptions extends the pre-defined posecodes from body parts to fine-grained hand gestures.
We define six elementary finger poses via finger curvature degrees and distances between fingers to generate descriptions, such as `bent' and `spread apart'.
We calculate the angle of each finger joint based on the 3D hand keypoints and determine the corresponding margins.
For instance, if the angle between 𝐕⃗(𝐊_wrist, 𝐊_fingertip) and 𝐕⃗(𝐊_fingertip, 𝐊_fingeroot) falls between 120 and 160 degrees, the finger posture is labeled as `slightly bent'.
We show an example of the annotated text labels in Fig. <ref>(b).
Summary. Based on the above annotations, we bulid , which has 96K clips with 13.7M SMPL-X poses and the corresponding pose and semantic text labels.
§ EXPERIMENT
In this section, we first validate the accuracy of our motion annotation pipeline on the 2D keypoints and 3D SMPL-X datasets.
Then, we build a text-driven whole-body motion generation benchmark on .
Finally, we show the effectiveness of in whole-body human mesh recovery.
§.§ Evaluation of the Motion Annotation Pipeline
2D Keypoints Annotation. We evaluate the proposed 2D keypoint annotation method on the COCO-WholeBody <cit.> dataset,
and compare the evaluation result with four SOTA keypoints estimation methods <cit.>.
We use the same input image size of 256× 192 for all the methods to ensure a fair comparison.
From Tab. <ref>(a), our annotation pipeline significantly surpasses existing methods by over 15% average precision. Additionally, we provide qualitative comparisons in Fig. <ref>(a), illustrating the robust and superior performance of our method, especially in challenging and occluded scenarios.
3D SMPL-X Annotation.
We evaluate our learning-based fitting method on the EHF <cit.> dataset and compare it with four open-sourced human mesh recovery methods. Following previous works,
we employ mean per-vertex error (MPVPE), Procrusters aligned mean per-vertex error (PA-MPVPE), and Procrusters aligned mean per-joint error (PA-MPJPE) as evaluation metrics (in mm).
Results in Tab. <ref>(b) demonstrate the superiority of our progressive fitting method (over 30% error reduction).
Specifically, PA-MPVPE is only 19.71 mm when using ground-truth 3D keypoints as supervision.
Fig. <ref>(b) shows the annotated mesh from front and side view, indicating reliable 3D SMPL-X annotations with reduced depth ambiguity. More results are presented in Appendix due to page limits.
§.§ Impact on Text-driven Whole-body Motion Generation
Experiment Setup. We randomly split into the train (80%), val (5%), and test (15%) sets. SMPL-X is adopted as the motion representation for expressive motion generation.
Evaluation metrics.
We adopt the same evaluation metrics as <cit.>, including Frechet Inception Distance (FID), Multimodality, Diversity, R-Precision, and Multimodal Distance.
Due to the page limit, we leave more details about experimental setups and evaluation metrics in the appendix.
Benchmarking Motion-X. We train and evaluate four diffusion-based motion generation methods, including MDM <cit.>, MLD <cit.>, MotionDiffuse <cit.> and T2M-GPT <cit.> on our dataset. Since previous datasets only have sequence-level motion descriptions, we keep similar settings for minimal model adaptation and take the semantic label as text input. The evaluation is conducted with 20 runs (except for Multimodality with 5 runs) under a 95% confidence interval. From Tab. <ref>, MotionDiffuse demonstrates a superior performance across most metrics. However, it scores the lowest in Multimodality, indicating that it generates less varied motion. Notably, T2M-GPT achieves comparable performance on our dataset while maintaining high diversity, indicating our large-scale dataset's promising prospects to enhance the GPT-based method's efficacy. MDM gets the highest Multimodality score with the lowest precision, indicating the generation of noisy and jittery motions. The highest Top-1 precision is 55.9%, showing the challenges of .
MLD adopts the latent space design, making it fast while maintaining competent results. Therefore, we use MLD to conduct the following experiments to compare with the existing largest motion dataset HumanML3D and ablation studies.
r0.50
< g r a p h i c s >
Visual comparisons of motions generated by MLD <cit.> trained on HumanML3D (in purple) or (in blue). Please zoom in for a detailed comparison. The model trained with can generate more accurate and semantic-corresponded motions.
Comparison with HumanML3D. To validate the richness, expressiveness, and effectiveness of our dataset, we conduct a comparative analysis between and HumanML3D, which is the largest existing dataset with text-motion labels. We replace the original vector-format poses of HumanML3D with the corresponding SMPL-X parameters from AMASS <cit.>, and randomly extract facial expressions from BAUM <cit.> to fill in the face parameters. We train MLD separately on the training sets of and HumanML3D, then evaluate both models on the two test sets. The results in Tab. <ref> reveal some valuable insights. Firstly, exhibits greater diversity (13.174) than HumanML3D (9.837), as evidenced by the real (GT) row. This indicates a wider range of motion types captured by . Secondly, the model trained on HumanML3D can not achieve a satisfactory result on the test set, while the model trained on performs well on the HumanML3D test set, even better than the intra-data training. These gaps arise from HumanML3D's limited action categories captured in a laboratory environment, whereas encompasses diverse motion types from massive outdoor and indoor scenes. For a more intuitive comparison, we provide the visual results of the generated motion in Fig. <ref>, where we can clearly see that the model trained on excels at synthesizing semantically corresponding motions given text inputs.
These results prove the significant advantages of in enhancing expressive, diverse, and natural motion generation.
Ablation study of text labels.
In addition to sequence-level semantic labels, the text labels in also include frame-level pose descriptions, which is an important characteristic of our dataset. To assess the effectiveness of pose description, we conducted an ablation study on the text labels. The baseline model solely utilizes the semantic label as the text input. Since there is no method to use these labels, we simply sample a single sentence from the pose descriptions randomly, concatenate it with the semantic label, and feed the combined input into the CLIP text encoder. Interestingly, from Tab. <ref>, adding additional face and body pose texts brings consistent improvements,
and combining whole-body pose descriptions results in a noteworthy 38% reduction in FID.
These results validate that the proposed whole-body pose description contributes to generating more accurate and realistic human motions. More effective methods to utilize these labels can be explored in the future.
§.§ Impact on Whole-body Human Mesh Recovery
As discovered in this benchmark <cit.>, the performance of mesh recovery methods can be significantly improved by utilizing high-quality pseudo-SMPL labels. provides a large volume of RGB images and well-annotated SMPL-X labels.
To verify its usefulness in the 3D whole-body mesh recovery task, we take Hand4Whole <cit.> as an example and evaluate MPVPE on the widely-used AGORA val <cit.> and EHF <cit.> datasets. For the baseline model, we train it on the commonly used COCO <cit.>, Human3.6M <cit.>, and MPII <cit.> datasets. We then train another model by incorporating an additional 10% of the single-view data sampled from while keeping the other setting the same. As shown in Tab. <ref>, the model trained with shows a significant decrease of 7.8% in MPVPE on EHF and AGORA compared to the baseline model. The gains come from the increase in diverse appearances and poses in , indicating the effectiveness and accuracy of the motion annotations in and its ability to benefit the 3D reconstruction task.
§ CONCLUSION
In this paper, we present Motion-X, a comprehensive and large-scale 3D expressive whole-body human motion dataset. It addresses the limitations of existing mocap datasets, which primarily focus on indoor body-only motions with limited action types. The dataset consists of 127.1 hours of whole-body motions and corresponding text labels.
To build the dataset, we develop a systematic annotation pipeline to annotate 96K 3D whole-body motions, sequence-level motion semantic labels, and 13.7M frame-level whole-body pose descriptions.
Comprehensive experiments demonstrate the accuracy of the motion annotation pipeline and the significant benefit of in enhancing expressive, diverse, and natural motion generation, as well as 3D whole-body human mesh recovery.
Limitation and future work.
While this paper provides preliminary investigations into diffusion-based models, the field of relevant models is still limited. Additionally, the evaluation metrics used may not fully reflect the true results. Therefore, there is a need for further development of the motion generation models and evaluation metrics, which we leave as future work.
As a large-scale dataset with multiple modalities, e.g., motion, text, video, and audio, Motion-X holds great potential for advancing downstream tasks, such as motion prior learning, understanding, and multi-modality pre-training. Finally, our large-scale dataset and scalable annotation pipeline open up possibilities for combining this task with large language model (LLM) to achieve an exciting motion generation result in the future.
With Motion-X, we hope to benefit and facilitate further research in relevant fields.
§ APPENDIX
§ MOTION-X: ADDITIONAL DETAILS
In this section, we provide more details about that are not included in the main paper due to space limitations, including statistic analyses, data processing, and motion augmentation mechanism.
§.§ Statistic Analyses
Fig. <ref>(a) shows each sub-dataset standard deviation of body, hand, and face joints. Our dataset has a large diversity of the hand and face joints, filling the gap of the previous body-only dataset in terms of expressiveness. Besides, as shown in Fig. <ref>(b), provides a large volume of long motion (>240 frames), which will be beneficial for long-term motion generation.
§.§ Processing of Each Sub-dataset
We gather 96K motion sequences from eight existing datasets and a large volume of online videos with the proposed annotation pipeline. As shown in Tab. <ref>, due to the lack of comprehensive annotations from their original datasets, we provide well-annotated whole-body motion, comprehensive semantic labels, and whole-body pose descriptions for all datasets. Here we introduce more details about each sub-dataset's data processing.
AMASS <cit.> is the existing largest-scale motion capture dataset, which provides body motions and almost static hand motions. To fill in the face parameters, we extract facial expressions from the facial dataset BAUM <cit.> with the SOTA facial reconstruction method EMOCA <cit.> and perform a data augmentation (in Sec. <ref>). For the text labels, we utilize the semantic labels from HumanML3D <cit.> and annotate the pose description with our whole-body pose captioning module.
NTU120 <cit.> is a widely used action recognition dataset. It provides the body keypoints, SMPL <cit.> parameters, action labels, and multi-view videos. Notably, we do not use the original body keypoints and SMPL parameters because of the insufficient quality. Instead, we annotate the SMPL-X format pseudo labels with the proposed motion annotation pipeline, which can generate high-quality whole-body motions. To obtain the semantic labels, we use the provided action labels and expand them with the large language model (LLM) <cit.>. The pose descriptions are annotated with our whole-body pose captioning method.
AIST++ <cit.> is a large-scale dance dataset with 3D body keypoints, multi-view videos and dance genres label. Like NTU120, we do not use the original body-only motion data and annotate the whole-body motion via our motion annotation pipeline. We obtain the semantic labels by expanding the dance genres label and providing frame-level pose descriptions.
HAA500 <cit.> is a large-scale human-centric atomic action dataset with manually annotated labels and videos for action recognition. It contains 500 classes with fine-grained atomic action labels, covering sports, playing musical instruments, and daily actions. However, it does not have the motion labels. We annotate the 3D whole-body motion with our pipeline. We use the provided atomic action label as semantic labels. Besides, we input the video into video-LLaMA <cit.> and filter the human action descriptions as supplemental texts. Pose description is generated by our automatic pose annotation method.
HuMMan <cit.> is a human dataset with multi-modality data, including multi-view videos, keypoints, SMPL parameters, action labels, etc. It does not provide whole-body pose labels. We estimate the SMPL-X parameters with our annotation pipeline. Besides, we expand the action label with LLM into semantic labels and use the proposed captioning pipeline to obtain pose descriptions.
GRAB <cit.> is human grasping dataset with body and hand motion. Meanwhile, it provides text descriptions of each grasping motion without corresponding videos. Therefore, similar to AMASS, we extract facial expressions from BAUM to fill in the facial expression. We use the provided text description as semantic labels and annotate pose descriptions based on the SMPL-X parameters.
EgoBody <cit.> is a large-scale dataset capturing ground-truth 3D human motions in social interactions scenes. It provides high-quality body and hand motion annotations, lacking facial expression. Thus, we perform a motion augmentation to obtain expressive whole-body motions. Since EgoBody does not provide text information, we manually label the semantic description using the VGG Image Annotation (VIA) <cit.> and annotate the pose description with the automatic pose captioning pipeline.
BAUM <cit.> is a facial dataset with 1.4K audio-visual clips and 13 emotions. We annotate facial expressions from BAUM with the SOTA face reconstruction method EMOCA.
Online Videos. To improve the richness, we collect 15K monocular videos from online sources, covering various real-life scenes. We design action categories as motion prompts and input them into LLM. Then, we collect videos from online sources based on the answer of LLM, after which we filter the candidate videos by transition detection and annotate the whole-body motion, semantic label and pose description for the selected videos.
§.§ Motion Augmentation Mechanism
Lower-body Motion Augmentation. contains some upper-body videos collected from online videos, like the videos in UBody <cit.>, where the lower-body part is invisible. Estimating accurate lower-body motions and global trajectories for these videos is challenging. Thanks to the precise low-body motions provided in AMASS, we can simply perform a lower-body motion augmentation for these sequences, i.e., selecting the closest motion from AMASS based on the SMPL-X parameters and replacing the lower-body motion with it. Meanwhile, we incorporate relevant keywords (e.g., sitting, standing, walking) in the text descriptions. Fig. <ref>(a) depicts three plausible lower-body augmentations for the motion sequence with the semantic label "a person is playing the guitar happily."
Facial Expression Augmentation. As shown in Tab. <ref>, the motion capture datasets AMASS, GRAB, and EgoBody do not provide facial expressions. Thus, we perform a facial expression augmentation for these motions by randomly selecting a facial expression sequence from the BAUM <cit.> dataset to fill the void and incorporating emotion labels (e.g., happy, sad, and surprise) in the semantic description. We perform interpolation for the selected sequence to ensure the same length as the original motion. An example of face expression augmentation is illustrated in Fig. <ref>(b).
§ MORE ANNOTATION VISUAL RESULTS
In this section, we present some visual results of the 2D keypoints, SMPL-X parameters, and motion sequences to show the effectiveness of our proposed motion annotation pipeline.
§.§ 2D Keypoints
As the main paper claims, we propose a hierarchical Transformer-based model for 2D keypoints estimation. To demonstrate the superiority of our method, we compare it with two widely used methods, Openpose <cit.> and MediaPipe <cit.>. We use the PyTorch implementation of Openpose and only estimate the body and hand keypoints as it does not provide the face estimator. As shown in Fig. <ref>, Openpose and MediaPipe can not achieve accurate results in some challenging poses. Besides, there exists severe missing detection of hands for Openpose and MediaPipe. In contrast, our method performs significantly better, especially the hand keypoint localization.
§.§ SMPL-X Parameters
To register accurate SMPL-X parameters, we elaborately design a learning-based fitting method with several training loss functions. We compare our method with two SOTA learning-based methods, Hand4Whole <cit.> and OSX <cit.>. As shown in Fig. <ref>, our method achieves a much better alignment result than the other models, especially on some difficult poses, which benefits from the iterative fitting process. Notably, Hand4Whole <cit.> and OSX <cit.> can only estimate the local positions without optimized global positions, which will suffer from unstable and jittery global estimation.
Furthermore, we compare with the widely used fitting method SMPLify-X <cit.>, using their officially released codes, in Fig. <ref>. Our method is more robust than SMPLify-X and can obtain better results about physically plausible poses, especially in challenging scenes (e.g., hard poses, low-resolution inputs, heavy occlusions). The results from the side view demonstrate that our method can properly deal with depth ambiguity and avoid the lean issue.
§.§ Motion Sequences
To highlight the expressiveness and diversity of our proposed motions, we illustrate examples of the same semantic label, like dance ballet, with six motion styles in Fig. <ref>. This one-to-many (text-to-motion) information can benefit the diversity of motion generation.
Then, we demonstrate more motion visualization in Fig. <ref> and <ref> for different motion scenes. These motions show different facial expressions, hand poses, and body motions.
§ EXPERIMENT
§.§ Experiment Setup
Motion Representation. To capture the 3D expressive whole-body motion, we use SMPL-X <cit.> as our motion representation. A pose state is formulated as:
𝐱 = {θ_b, θ_h, θ_f, ψ, 𝐫}.
Here, θ_b∈ℝ^22×3 and θ_h∈ℝ^30×3 denote the 3D body rotations and hand rotations. θ_f∈ℝ^3 and ψ∈ℝ^50 are the yaw pose and facial expression. 𝐫∈ℝ^3 is the global translation.
Evaluation Metrics. We adopt the same evaluation metrics as <cit.>, including Frechet Inception Distance (FID), multimodality, diversity, R-precision, and multimodal distance. We pretrain a motion feature extractor and a text feature extractor for the new motion presentation with contrastive loss to map the text and motion into feature space and then evaluate the distance between the text-motion pairs. For each generated motion, its ground-truth text description and 31 mismatched text description randomly selected from the test set compose a description pool. We rank the Euclidean distances between the generated motion and each text in the pool and then calculate the average accuracy at the top-k positions to derive R-precision. Multimodal distance is computed as the Euclidean distance between the feature vectors of generated motion and its corresponding text description in the test set. Additionally, We include the average temporal standard deviation as a supplementary metric to evaluate the diversity and temporal variation of whole-body motion.
Computational Costs. We use 8 NVIDIA A100 GPUs for motion annotation and 4 GPUs for motion generation experiments. It takes about 72 hours to annotate 1M frames with our annotation pipeline.
§.§ More Ablation Study
More Comparison with HumanML3D.
Previous motion generation datasets are limited in expressing rich hand and face motions, as they only contain body and minimal hand movements. To demonstrate the expressiveness of our dataset, we conduct a comparison between HumanML3D and on face, hand, and body, separately. Specifically, we train MLD <cit.> on each dataset and evaluate the diversity of generated motions and ground-truth motions by computing the average temporal standard deviation of the SMPL-X parameters and joint positions.
The SMPL-X parameters include body poses, hand poses, and facial expressions. Body, hand, and face joint positions are represented as root-relative, wrist-relative, and neck-relative, respectively. We randomly choose 300 generated samples from the validation set and repeat the experiment 10 times to report the average results. As shown in Tab. <ref>, the generated and ground-truth motions in exhibit a higher deviation, especially in hand and face parameters, indicating significant hand and face movements over time. These results demonstrate that the model trained with can generate more diverse facial expressions and hand motions, demonstrating the ability of our whole-body motion to capture fine-grained hand and face movements and expressive actions.
§ RELATED WORK
In this part, we introduce relevant methods for human motion generation.
According to different inputs, producing human motions can be divided into two categories: the general motion synthesis from scratch <cit.> and the controllable motion generation from given text, audio, and music as conditions <cit.>.
Motion synthesis encompasses several tasks, such as motion prediction, completion, and interpolation <cit.>, developed over several decades in computer vision and graphics. These tasks tend to utilize nearby frames with spatio-temporal correlations to infer estimated frames in a deterministic manner <cit.>.
On the other hand, motion generation is a more challenging task that aims to synthesize long-term, diverse, natural human motions.
Many generative models, like GANs, VAEs, and recent diffusion models, have been explored <cit.>.
This work mainly discusses text-conditioned motion generation. This field has evolved from inputting action classes <cit.> to sentence descriptions <cit.>, and generating motions from 2D to 3D keypoints, to the emerging parametric model (e.g., SMPL <cit.>). These models have become expressive and comprehensive toward real-world scenarios thanks to the development of related benchmarks.
Recently, diffusion model-based methods have rapidly developed and shown advantages in diverse, realistic, and fine-grained motion generation <cit.>. Some concurrent works <cit.> introduce novel diffusion model-based motion generation framework to achieve state-of-the-art (SOTA) quality. For example, MLD <cit.> presents a motion latent-based diffusion model with a representative motion variational autoencoder, showing its efficiency.
§ LIMITATION AND BROADER IMPACT
§.§ Limitation
There are two main limitations of our work. (i) The motion quality of our markless motion annotation pipeline is inevitably inferior to the multi-view mark-based motion capture system. However, as the quantitative and qualitative results demonstrate, our method can perform much better than existing markless methods, thanks to large-scale models pre-trained on massive 2D and 3D keypoints datasets and our elaborately designed fitting pipeline. Besides, a 30 mm PA-MPVPE error would be acceptable for the text-driven motion generation task since the target is to synthesize natural and realistic motions that are semantically consistent with the text input. Furthermore,
the experiment on the mesh recovery task has demonstrated that our dataset can also benefit the human reconstruction task, which requires a higher annotation quality.
Accordingly, a better motion annotation will be beneficial, and we will leave it as our future work.
(ii) During our experiment, we find out that existing evaluation metrics are not always consistent with visual results. Besides, SMPL-X parameters may not be the best motion representation for expressive whole-body motion representation. Thus, there is a need for further research on the evaluation metric, motion representation, and model designs for the expressive motion generation task. Since the main task of our work is to build a high-quality dataset, we leave them as future work.
§.§ Broader Impact
A large-scale 3D human motion dataset would have numerous applications and boost novel research topics in various fields, such as animation, games, virtual reality, and human-computer interaction. Until now, human motion datasets have had no negative social impact yet. Our proposed will strictly follow the license of previous datasets, and would not present any negative foreseeable societal consequence, either.
§ LICENSE
All data is distributed under the CC BY-NC-SA (Attribution-NonCommercial-ShareAlike) license. Detailed license and instructions can be found on the page <https://motion-x-dataset.github.io>. Further, we will provide a GitHub repository to solicit possible annotation errors from data users.
For the sub-datasets, we would ask the user to read the original license of each original dataset, and we would only provide our annotated result to the user with the approvals from the original Institution. Here, we provide a brief license of the used assets:
* HumanML3D dataset <cit.> originates from the HumanAct12 <cit.> and AMASS <cit.> datasets, which are both released for academic research only and it is free to researchers from educational or research institutes for non-commercial purposes.
* BAUM dataset <cit.> is CC-BY 4.0 licensed.
* HAA500 dataset <cit.> is MIT licensed.
* NTU120 dataset <cit.> is released for academic research only and is free to researchers from educational or research institutes for non-commercial purposes.
* HuMMan dataset <cit.> is under S-Lab License v1.0.
* AIST dataset <cit.> is CC-BY 4.0 licensed.
* GRAB dataset <cit.> is released for academic research only and is free to researchers from educational or research institutes for non-commercial purposes.
* EgoBody <cit.> is under CC-BY-NC-SA 4.0 license.
* Other data is under CC BY-SA 4.0 license.
* SMPLify-X <cit.> codes are released for academic research only and are free to researchers from educational or research institutes for non-commercial purposes.
* Codes for preprocessing and training are under MIT LICENSE.
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|
http://arxiv.org/abs/2307.03118v1 | 20230706164358 | Quantum Solutions to the Privacy vs. Utility Tradeoff | [
"Sagnik Chatterjee",
"Vyacheslav Kungurtsev"
] | quant-ph | [
"quant-ph",
"cs.CR",
"cs.LG"
] |
==
quote
1]Sagnik Chatterjee
2]Vyacheslav Kungurtsev
[1]Indraprastha Institute of Information Technology, Delhi (IIITD)
[2]Czech Technical University, Prague
affilsepx:
[ ]Email ids
affilsepx,
[1][email protected]
[2][email protected]
Quantum Solutions to the Privacy vs. Utility Tradeoff
[
=====================================================
In this work, we propose a novel architecture (and several variants thereof) based on quantum cryptographic primitives with provable privacy and security guarantees regarding membership inference attacks on generative models. Our architecture can be used on top of any existing classical or quantum generative models. We argue that the use of quantum gates associated with unitary operators provide inherent advantages compared to standard Differential Privacy based techniques for establishing guaranteed security from all polynomial-time adversaries.
§ INTRODUCTION
The privacy versus accuracy tradeoff in machine learning models has been a central challenge for algorithmic development and understanding. While large-scale generative models such as DALL-E 2 <cit.>, Imagen <cit.>, and Stable Diffusion <cit.> have advanced state-of-the-art in terms of the quality of synthetic data beyond previous expectations; their application presents a significant security risk in terms of privacy violation. The leading methodologies for generative modeling include generative adversarial networks (GANs) <cit.>, variational autoencoders (VAEs) <cit.>, and diffusion models <cit.>. Whereas their intention is to generate data that samples from the population distribution, in practice, all of
these generative model techniques have been shown to “memorize” <cit.> and regenerate the training data <cit.>, which makes them vulnerable to various adversarial attacks such membership-inference attacks (MIA) <cit.> where an adversary can infer if a given sample was used for training. Such attacks can lead to colossal privacy breaches <cit.>, which must be addressed for safe and reliable operation.
To tackle membership-inference attacks and, more generally, to ensure the privacy of training data, various sophisticated classical techniques <cit.> have been proposed with differential privacy (DP) based guarantees for diffusion models. However, it has been shown that there is a class of readily implementable adversarial attacks that can recreate training data samples for models satisfying DP-based guarantees <cit.>. In many domains of interest, if even a small fraction of the training dataset can be learned by an adversary, then the model cannot be considered private.
Key Idea: While there are many types of adversarial attacks one can consider, we focus on providing privacy and security guarantees against non-malicious adversarial (NMA) attacks, a broad class that includes MIA. It can be observed from <cit.> that realistic NMA attacks can be modeled by providing adversarial access to the discriminator. Ensuring proper defense against such attacks can be formulated in terms of an interactive game <cit.> between
a challenger and an adversary where the adversary should not be able to determine whether a particular example belongs to the training set even when it is given access to the learning model and the distribution over the data[MIA can also be viewed through the lense of a hypothesis test <cit.>.].
A novel reinterpretation to this work is that this game can be considered a cryptographic protocol, where we treat the training data as our message and the generated data as a cipher. The goal would be to ensure that the adversary should not be able to glean any meaningful information regarding the training data (our message) given the generated data (our cipher) and black-box access to the training model. This interpretation and its associated analysis uses tools of security guarantees such as CPA-security or CCA-security. These, in turn, can be considered to both subsume the weaker, more standard notions of privacy in ML, and provide essential guarantees qualitatively more significant than simply ensuring the privacy of the training data[Privacy alone does not imply security. For example, one-time pads (OTP) are perfectly private but not secure.].
Our contribution: Despite encryption being an extremely natural way to ensure security and privacy, classication encryption techniques are not particularly amenable to obtaining guaranteed accuracy or convergence of training. Therefore, incorporation of classical encryption techniques weights heavily towards complete privacy and no accuracy in the standard privacy-accuracy tradeoff balance.
In this work, we describe a quantum framework for tackling membership-inference attacks based on cryptographic primitives. Working in the quantum regime presents a number of unique computational advantages. Beyond the security advantages which we detail below, under standard complexity-theoretic assumptions, there may exist quantum-only distributions that classical generative models may not be able to generate <cit.>. Another motivation to use quantum generative models is the increase in stability and a significant reduction in the number of parameters over their classical counterparts <cit.>.
We also remark that our framework is essentially a generalization of classical diffusion models. In the quantum framework, our forward process does not rely on the same asymptotic guarantees which classical diffusion models <cit.> rely on, thereby hinting at possible sampling speedups. A classical solution <cit.> in which non-Gaussian multimodal distributions were used to model the denoising step, could fall prey to the instability issues described in <cit.> since there are absolutely no theoretical guarantees regarding the intermediate distributions. In our case, the discriminator needs to distinguish between two quantum states which have identical support. Therefore, the usage of a wide variety of divergences and distances in the discriminator is mathematically advantageous.
§ BACKGROUND
An n-qubit quantum state is a unit vector on a 2^n-dimensional complex Hilbert space. The uniform distribution of random quantum states on the 2^n-dimensional complex Hilbert space is referred to as the Haar measure over quantum states. Note that the Haar measure over quantum states is a continuous distribution even when the underlying support is finite. A simple example of a Haar random state is the ensemble
|ψ⟩_r=2^-n/2∑_y∈𝔽_2^n(-1)^r(y)|y⟩
where r is a truly random function. Since there are 2^2^n possible choices for r, |ψ⟩_r cannot be generated by polynomial-sized circuits.
Let 𝒦={𝒦_n}_n∈ℕ be a sequence of keys from which samples can be efficiently drawn. A quantum-secure pseudorandom function (QPRF) is a family of efficiently computable keyed functions QPRF={QPRF_k}_k∈ℕ:𝒦_k×{0,1}^n{0,1}^n <cit.> which can be used for the construction of Pseudorandom states. A Pseudorandom quantum state (PRS) <cit.> is a quantum state which is information-theoretically indistinguishable from a Haar-random state. In PRS, we replace the truly random function r with an efficiently constructible QPRF {f_k} indexed by secret key k, such that given query access to r and f_k:{0,1}^n→{0,1}^n, no efficient polynomially bounded non-uniform quantum adversary A_Q can distinguish between a PRS and a Haar random quantum state.
Therefore, any scheme which involves the construction of a PRS is by definition IND-CPA secure <cit.>. The existence of QPRF assuming one-way functions exist was proved in <cit.>. One straightforward candidate for a QPRF function is the family of pseudorandom random permutations (QPRP) <cit.>.
A Pseudorandom Unitary (PRU) <cit.> is a family of unitary operators which is efficiently constructible and indistinguishable from the Haar measure on the unitary group. Quantum Trapdoor functions (QTFs) <cit.> are a class of efficiently constructible unitaries associated with different secret keys. One can use them to form PRS from a classical message m and a secret key k. We show how to interpret the QTF construction as a phase-encoding+PRS construction in <ref>. Even though QTFs are unitaries, they are hard to invert without knowing the trapdoor k of the underlying QPRP.
§ TECHNICAL OVERVIEW
In this section, we outline three constructions that are provably secure against NMA attacks. We denote the real data distribution as P_d and the generated data distribution P_g. The crux of the idea is as follows:
Given x∼ P_d and x^'∼ P_g and a discrimination function 𝔻, we want to construct an IND-CPA (or IND-CCA) secure cryptographic scheme with an encryption function g s.t.
𝔻(x,x^')=𝔻(g(x),g(x^'))
For IND-CPA (and IND-CCA) secure cryptographic schemes, the encrypted tuple g(x),g(x^') must appear pseudorandom w.r.t. to the input tuple x,x^'. It is unclear how to construct classical functions which simultaneously perform IND-CPA secure encryption but also preserve the the discriminative labels in the encrypted tuple.
A straightforward answer lies in the realm of quantum operators and states. If there exist IND-CPA (or IND-CCA) secure cryptographic schemes with a unitary(quantum) encryption operator, then by the distance-preserving property of unitary operators, all notions of quantum state discrimination are preserved. Formally stated, if U_g is the unitary corresponding to the encryption g, and 𝔻 denotes the Fidelity measure, then for all pairs of quantum states |x⟩ and |x^'⟩, we have
𝔻(|g(x)⟩,|g(x^')⟩)=𝔻(U_g|x⟩,U_g|x^'⟩)=⟨x|U_g^† U_g|x^'⟩=⟨x||x^'⟩=𝔻(x,x^')
The above argument holds for all distinguishability measures on quantum states as well.
Even though 𝔻(|g(x)⟩,|g(x^')⟩)=𝔻(x,x^'), the discriminator only has access to the encrypted tuple of states (|g(x)⟩,|g(x^')⟩). Therefore, using the rules of CPA security, there is no polynomially bounded adversary (classical or quantum) who can perform MIA even with non-malicious access to the Generator.
We now investigate three different constructions that satisfy <ref>.
Phase Encoded PRS Construction. First, we recall the QTF construction <cit.> briefly.
A QTF is defined as the tuple (GenTR, GenEV, Eval, Invert): GenTR(1^n)tr. GenEV(tr)|eval⟩=|PRS(tr)⟩. Eval(|eval⟩,x)|ϕ⟩=Z^x|eval⟩. Invert(tr,|ϕ⟩)x.
The classical trapdoor tr is used to construct a quantum public key |eval⟩ which is a PRS. |eval⟩ can now be used to encode a classical string x using a Z-twirl operator: Z^x:=⊗_i=1^n Z^x_i. By properties of QPRF, the state Z^x|PRS(k)⟩ is also a PRS. We can now give the first construction.
Given a pair of inputs x∼ P_d and x^'∼ P_g,
* Create phase encoded quantum states |ϕ_x⟩=Z^x|+⟩^⊗ n and |ϕ_x^'⟩=Z^x^'|+⟩^⊗ n.
* Pick a classical trapdoor key k and generate unitary access to a QPRF {f_k}.
* Use the phase kickback trick on U_f_k once with with |ϕ_x⟩ and once with |ϕ_x^'⟩ to obtain the pair |PRS(x,k)⟩ and |PRS(x^',k)⟩.
* Train the Discriminator and Generator on the PRS tuple.
Firstly we note that |PRS(x,k)⟩ and |PRS(x^',k)⟩ are actually pseudorandom states. The proof follows directly from QTF construction <cit.> and the fact that pseudorandom states are invariant under unitary operation. Secondly, we highlight the fact that, unlike any classical crytographic map, 𝔻(|PRS(x,k)⟩,|PRS(x^',k)⟩)=𝔻(|x⟩,|x^'⟩). Therefore <ref> allows us to train the discriminator properly and securely against adversaries wanting to perform MIA.
Parameterized Phase Encoded PRS Construction. One semantic drawback of <ref> is the fact that we encode the phases uniformly across all features (bits of the string in this case). In practice, we might choose to create quantum state encodings of classical feature vectors based on some principles of optimal coding. To incorporate this option, we make use of the Parameterized Z-twirl operator: RZ(θ)^x:=⊗_i=1^n RZ(θ_i)^x_i. Since RZ(θ)^x is also a diagonal unitary, it commutes with the QTF construction as well.
Given a pair of inputs x∼ P_d and x^'∼ P_g, and a set of feature weights θ,
* Create parameterized phase encoded quantum states |ϕ_x,θ⟩=RZ(θ)^x|+⟩^⊗ n and |ϕ_x^',θ⟩=RZ(θ)^x^'|+⟩^⊗ n.
* Generate unitary access to a QPRF {f_k} and pick a classical trapdoor key k.
* Use the phase kickback trick on U_f_k once with with |ϕ_x⟩ and once with |ϕ_x^'⟩ to obtain the pair |PRS(x,k,θ)⟩ and |PRS(x^',k,θ)⟩.
* Train the Discriminator and Generator on the PRS tuple.
Basis Encoded PRS Construction. Instead of strictly working with phase-encodings, we may also be interested in basis-encoded quantum states. In order to construct PRS from basis-encoded states, we have to use a construction similar to <cit.>. We assume oracle access to a PRU U, a secret key k=k_1 k_2… k_T, where each k_i∈[4], and access to n-qubit Pauli operators 𝒪={I_n,X_n,Y_n,Z_n}. Given any initial state |ϕ_0⟩, the following construction was proven to yield a PRS, provided that the adversary only has black-box access to the PRU U.
|PRS(x,k,T)⟩=U𝒪_k_TU𝒪_k_T-1…𝒪_k_1U|ϕ_x⟩
Given a pair of inputs x∼ P_d and x^'∼ P_g, and a parameter T=n,
* Create basis encoded quantum states |ϕ_x⟩ and |ϕ_x^'⟩.
* Generate a classical trapdoor key k, and obtain unitary access to a PRU U.
* Construct the PRS tuple (|PRS(x,k,T)⟩,|PRS(x^',k,T)⟩) as in <ref>.
* Train the Discriminator and Generator on the PRS tuple.
§ DISCUSSION
In this work, we discuss three novel constructions for preventing MIA by leveraging the properties of quantum operators and using privacy and security guarantees from cryptography. Classical analogues of our approach would be difficult to construct, as we discussed. However, since MIA at its heart can be distilled down to a cryptographic game, we believe that there may be Zero-Knowledge-Proof based constructions, or attribute-preserving encryption schemes that could allow our framework to extend to classical techniques.
` |
http://arxiv.org/abs/2307.02393v1 | 20230705160705 | PSR J0026-1955: A curious case of evolutionary subpulse drifting and nulling | [
"Parul Janagal",
"Samuel J. McSweeney",
"Manoneeta Chakraborty",
"N. D. Ramesh Bhat"
] | astro-ph.HE | [
"astro-ph.HE"
] |
firstpage–lastpage
Collision integral with momentum-dependent potentials and its impact on pion production in heavy-ion collisions
Akira Ono
August 1, 2023
===============================================================================================================
PSR J0026–1955 was independently discovered by the Murchison Widefield Array (MWA) recently. The pulsar exhibits subpulse drifting, where the radio emission from a pulsar appears to drift in spin phase within the main pulse profile, and nulling, where the emission ceases briefly. The pulsar showcases a curious case of drift rate evolution as it exhibits rapid changes between the drift modes and a gradual evolution in the drift rate within a mode. Here we report new analysis and results from observations of J0026–1955 made with the upgraded Giant Meterwave Radio Telescope (uGMRT) at 300-500 MHz. We identify two distinct subpulse drifting modes: A and B, with mode A sub-categorised into A0, A1, and A2, depending upon the drift rate evolutionary behaviour. Additionally, the pulsar exhibits short and long nulls, with an estimated overall nulling fraction of ∼58%, which is lower than the previously reported value. Our results also provide evidence of subpulse memory across nulls and a consistent behaviour where mode A2 is often followed by a null. We investigate the drift rate modulations of J0026–1955 and put forward two different models to explain the observed drifting behaviour. We suggest that either a change in polar gap screening or a slow relaxation in the spark configuration could possibly drive the evolution in drift rates. J0026–1955 belongs to a rare subset of pulsars which exhibit subpulse drifting, nulling, mode changing, and drift rate evolution. It is, therefore, an ideal test bed for carousel models and to uncover the intricacies of pulsar emission physics.
stars: neutron - pulsars: general - pulsars: individual (PSR J0026-1955)
§ INTRODUCTION
Radio pulsars are rotating neutron stars with highly coherent radiation emanating from the vicinity of magnetic poles, which cross our line of sight once every pulsar rotation <cit.>. They possess a large mass (∼1 to ∼2 M_⊙) confined within a small radius (≲ 10 km), with strong gravitational (∼10^11 times stronger than the Earth's surface gravitational field) and magnetic fields <cit.>. Pulsars are the sites of some of the highest energy physical processes, making them powerful astrophysical laboratories, owing to such extreme environments of very strong gravitational and magnetic fields surrounding them. However, even though more than 3000 pulsars are known to date, a definitive exposition of the processes by which pulsars emit beams of radio waves is still non-existent in the literature <cit.>.
Radio emission from pulsars exhibits a variety of phenomena, which modulate their pulse-to-pulse emission, observable in the form of subpulse drifting, nulling, mode changing, etc., thereby providing a range of avenues to understand the complex physical processes that cause the emission. In many cases, individual pulses from a pulsar show substructure with one or more distinct components called subpulses. <cit.> observed the systematic `marching' of these subpulses with phase within the on-pulse window, leading to diagonal drifting structures in a pulse stack, called driftbands (pulse number vs rotation phase), as commonly seen in many pulsars <cit.>. For such a pulse stack, the drift rate is then defined as the reciprocal of the slope of the driftbands (^∘/P_1, where P_1 is the pulsar rotation period).
Theoretical models explaining subpulse drifting were suggested early on after the discovery of subpulse drifting in pulsars <cit.>. The most well-developed model at the time was able to explain the subpulse drifting phenomenon exhibited by pulsars studied then, most with stable drift rates, such as B0809+74 and B0943+10 <cit.>. The original model proposed by <cit.> associated drifting subpulses with a rotating `carousel' of a discrete number of sparks (electrical discharges) in regions of charge depletion just above the neutron star surface near the magnetic poles. This carousel of sparks circulates around the magnetic axis due to an E × B drift, and the electron-positron pairs produced in the discharges are ultimately responsible for the observed radio emission. The rotation rate of the carousel (P_4) around the magnetic axis is generally different from the pulsar period.
Two characteristic features of subpulse drifting pulsars are their drift rates and P_3. The drift rate is defined as D = Δϕ per pulse period (^∘/P_1), where Δϕ is the longitude shift in degrees during one pulse period P_1. A positive value indicates a drift from early to later longitudes, while a negative value corresponds to a drift from later to earlier longitudes. In a pulse stack, the vertical separation between driftbands at a given longitude is P_3 (typically expressed in units of the pulsar rotation period, P_1), which is a measure of time after which a subpulse will return at a particular phase. The caveat here is that the pulsar rotation only permits observation of the subpulse positions once every pulse. A specific subpulse in one pulse cannot be unambiguously identified in the next due to the difficulty in resolving the presence of aliasing, making it generally difficult to evaluate the true carousel speed. That is to say, if aliasing is present, the observed drift rate is related to the beating frequency between P_1 and P_4. Another consequence of aliasing is that the drift rate can appear to vary even if P_4 stays constant as long as the beamlet configuration changes. Thus, if multiple drift rates are present in a given pulsar, it does not necessarily mean that the rotation speed of the carousel has changed; it may be that the number of beamlets has changed instead <cit.>.
Even in the simplest case of a constant P_4 and a fixed number of beamlets, the apparent drift rate (i.e. the slope of the driftbands) is not a steady function of rotation longitude. This is a purely geometric phenomenon related to the projection of the beamlets' motion onto the line of sight trajectory, as explained in <cit.>. This results in the driftbands themselves appearing curved, referred to as “geometric curvature”. Geometric curvature is always present but, similar to the polarisation position angle (PPA) of the rotating vector model <cit.>, will only be visible if the pulse window is sufficiently wide for a given pulsar's particular viewing geometry. Geometric curvature is also similar to the PPA in that it is symmetric about the fiducial point, giving the driftbands a characteristic `S'-shape, with an excess (or deficit) of the drift rate appearing in the peripheral part of the pulse window.
The carousel model satisfactorily explains the subpulse drifting nature of some pulsars that show stable drift rates, citing the theoretical stability of electric and magnetic fields at the spark locations <cit.>. Furthermore, multi-frequency observations of a large fraction of the pulsar population have brought forth a variety of such atypical pulsars <cit.>. These studies show that a substantial fraction (∼50%) of known pulsars exhibit subpulse drifting. However, explaining the drifting behaviour in pulsars that exhibit anything more complicated than a single stable drift rate requires modifications or extensions to the basic carousel model. Such pulsars present ideal test beds to modify the classical carousel model. Several extensions have been proposed over the years to account for the observed complicated behaviour. For example, <cit.> suggest that a quasi-central spark can account for the non-drifting core components in profiles. The well-known phenomenon of bi-drifting may be explained in terms of the presence of an inner annular gap <cit.> or an inner acceleration gap <cit.>, or non-circular spark motions <cit.>. Similarly, the phenomenon of drift rate reversal, shown by some pulsars, can be explained by the modified carousel model, where sparks rotate around the location of the electric potential extremum of the polar cap instead of the magnetic axis <cit.>. These extensions/modifications are generally developed to explain specific drifting behaviours observed in a relatively small subset of subpulse drifting pulsars. However, there is still no comprehensive theory that can describe all the observed drifting behaviours.
Several theories have also been suggested to interpret the drifting subpulses geometrically. <cit.> and <cit.> suggested that drifting subpulses result from modulation in the emission region caused by drift waves in some form of magnetospheric oscillations. In their model, the subpulses result from periodic variations in the magnetospheric plasma, which may cause the emission region to move across the observer's line of sight. <cit.> suggest that non-radial oscillations in the emission region could be responsible for subpulse drifting without invoking circulations in the magnetosphere. They propose that the drifting subpulses could be due to non-radial oscillations in the magnetosphere. Although these models are able to explain phenomena such as mode changing, other phenomena, such as bi-drifting, memory across nulls, etc., cannot be convincingly accounted for.
Another phenomenon often seen in conjunction with subpulse drifting is `nulling', where the emission from a pulsar ceases abruptly for a few to hundreds of pulse periods <cit.> before it is restored. To date, pulse nulling has been reported in more than 200 pulsars <cit.>, which is less than 10% of the known pulsar population. Nulls lasting for one or two pulses are generally attributed to the stochastic processes within the pulsar magnetosphere <cit.>. However, in subpulse drifting pulsars, short nulls can be attributed to a slight variation of the spark distribution, where nulls are caused by an empty line-of-sight <cit.>. Long nulls, on the other hand, are thought to be related to changes in the plasma processes within the pulsar magnetosphere <cit.>, or even the spin-down energy loss in the most extreme cases <cit.>. If nulls and changes in the drift modes are, in fact, caused by intrinsic changes in the pulsar magnetosphere, their interactions could be crucial in understanding the mechanisms behind changes between different magnetospheric states. Nulling itself may be an extreme form of mode-changing, where a pulsar switches between different magnetospheric states, as suggested by the broadband behaviour of three nulling pulsars reported by <cit.>. Hence, the study of pulsars exhibiting both nulling and drifting phenomena is crucial for comprehending the true origin and nature of the nulling phenomenon.
Pulsars which exhibit complicated drifting behaviour such as mode changing and nulling, and are also bright enough for single pulse analysis, are relatively rare. However, this combination is essential in shaping ideas concerning the pulsar radio emission process. Recently, the Murchison Widefield Array (MWA) independently discovered PSR J0026–1955 <cit.> in the shallow pass of their Southern-Sky MWA Rapid Two-metre (SMART) pulsar survey <cit.>. The pulsar was originally detected in 2018 in the Green Bank Northern Celestial Cap (GBNCC) pulsar survey <cit.>. However, the discovery was not followed up until the recent re-discovery by the MWA. J0026–1955 is a bright pulsar which has a period of 1.306150 s and a dispersion measure (DM) of 20.869 pc cm^-3. The pulsar exhibits complex subpulse drifting behaviour and mode switching in addition to a large nulling fraction (∼77% at 155 MHz). <cit.> found two distinct subpulse drifting modes A and B, with slow and fast drift rates, respectively, which were further categorised (A1/A2 and B1/B2) depending upon the qualitative properties of modal appearances and context. The pulsar was sometimes seen to abruptly change its drift rate, while at other times, it exhibits a consistent evolution of the drift rate within its drifting modes.
The most distinctive feature of PSR J0026–1955 is its slow drift rate evolution, which has been found in only a handful of other pulsars like B0031-07 <cit.> and B0818-13 <cit.>. Furthermore, with its variable drift rates, the pulsar also poses an essential question to the stability of the carousel, as the basic models assume a stable configuration, leading to a non-variable drift rate throughout a drift mode. For J0026–1955, <cit.> attempt to model the observed drifting behaviour with an exponentially decaying drift rate, similar to what is seen in PSR B0818-13 and PSR B0809+74 <cit.>. However, they were unable to fully characterise the observed drifting behaviour owing to their limited data sets and, consequently, an insufficient number of drift sequences. They also suggest a possibility of subpulse phase memory across short null sequences, which would benefit from more observations. Using longer multi-frequency observations, the modal taxonomy presented by <cit.> can be tested for viability and to examine whether there is a need for something more sophisticated than an exponential model of drift rates. The complex drifting and nulling behaviour of this new pulsar thus warrants deeper investigations of its properties and their nature at different frequencies.
The unusual behaviour of J0026–1955 reported in <cit.> will also benefit from a detailed study at higher frequencies, allowing us to test the frequency dependence of such characteristics. Furthermore, given the slow period and large nulling fraction of the pulsar, long-duration observations with a higher signal-to-noise ratio (S/N) are necessary to collect a sufficiently large number of complete burst sequences in order to undertake a more robust statistical analysis.
In this study, we present a detailed investigation of subpulse drifting and nulling exhibited by J0026–1955, with new observations obtained using the upgraded Giant Metrewave Radio Telescope (uGMRT) at 300-500 MHz. This paper is organised as follows. In section <ref>, we briefly describe the observation details and data-reduction procedures; the subpulse drifting and nulling analysis are presented in section <ref>; our findings are discussed in section <ref>; and a summary of the paper is given in section <ref>.
§ OBSERVATIONS AND DATA REDUCTION
The Giant Metrewave Radio Telescope (GMRT) is a radio interferometric array consisting of 30 antennas, each with a 45-meter diameter, and spread over an area of 28 square kilometres in a Y-shape <cit.>. The GMRT recently underwent an upgrade, which included the addition of wide-band receivers and digital instrumentation, allowing for near-seamless coverage in frequency from 120 MHz to 1600 MHz <cit.>. For our observations of PSR J0026–1955, we used the upgraded GMRT in the phased array mode, where signals from each antenna are coherently added for maximum sensitivity. J0026–1955 was observed with the uGMRT at Band 3 (300-500 MHz) and Band 4 (550-750 MHz), over two epochs at each frequency band. However, due to the presence of higher levels of radio frequency interference (RFI), Band 4 data quality was not adequate for meaningful single-pulse analysis. Thus for the work presented in this paper, we limit our analysis to Band 3 (300-500 MHz) data. Details of observations, including the number of pulses which had pulsar emission (burst) and lacked any emission (null), are summarised in Table <ref>.
The data were recorded at 655.56μs time resolution, spread across 2048 channels, thus providing a frequency resolution of 97.65 kHz, and were converted to single-pulse archives using the package <cit.>. The single pulse files were then frequency scrunched and combined using the routines from <cit.>. Finally, each frequency-scrunched single-pulse sequence file was manually searched for RFI using the interactive RFI zapping subroutine of . The RFI-excised file was then converted into an ASCII format that contained the pulse time series and was used for all subsequent analyses.
Following the methodology detailed above, the pulsar time series data obtained in the last step were used to generate the pulse stacks (pulse phase vs pulse number) as shown in Fig. <ref>. The bright yellow diagonally arranged pixels between pulse phase -20^∘ and +20^∘ are the subpulse driftbands, which can be clearly seen to march from a positive to a negative phase, with increasing pulse number. Despite multiple rounds of RFI excision using the subroutine, it is evident from Fig. <ref> that there is some residual RFI. For example, several seconds of RFI can be seen right before pulse number 850 in panel <ref>. However, cases where the subpulses are bright enough to be visually recognised (despite the RFI), were retained. With such a tradeoff, we were able to salvage data that could still be used for exploration without affecting the subpulse drifting analysis. Further, panels <ref> and <ref>, provide clear examples of short and long nulls, where any emission from the pulsar is absent for a certain duration ranging from a few to a few hundred pulses. Table <ref> presents a comprehensive summary of nulls and bursts observed in different observations.
§ ANALYSIS
In subsequent analysis, we elaborate on different subpulse behaviours and attempt to characterise the drifting nature to learn the underlying mechanism in the context of the carousel model <cit.>.
In general, the preliminary analysis of any subpulse drifting pulsar aims at the classification of drift modes. However, considering that J0026–1955 does not always exhibit well-defined discrete modes but rather a drift rate evolution, such an analysis is complicated. In section <ref>, we discuss a drift rate evolution-based classification scheme for deciding the mode boundaries, using linear and exponential models for drift rate evolution. Section <ref> discusses the modes identified using this scheme. The observed nulling behaviour of the pulsar at 400 MHz is discussed in section <ref>. Further, the evolutionary drift rate behaviour of the pulsar is studied in detail in section <ref>. The exponential drift rate model used in section <ref> was employed by <cit.> to demonstrate memory across nulls for the first time. Following their lead, we have also examined J0026–1955 for possible instances of memory across nulls, detailed in section <ref>.
§.§ Drift Mode Boundaries
PSR J0026–1955 poses a unique challenge for identifying the drift modes. Generally, subpulse-drifting pulsars exhibit stable modes, which can be uniquely characterised by their P_3 values. However, the slowly evolving drift rates and P_3 of J0026–1955 complicate the mode identification. Thus, to identify the drift modes, we employed a different strategy. Given that the pulsar exhibits both evolutionary and non-evolutionary drift rates, we modelled the drift rate behaviours to make a drift rate-based modal classification, which also accounts for the evolution. The linear model is the most straightforward generalisation of a constant drift rate, essentially including the next term in the Taylor expansion – and valid in situations where the drift sequences are sufficiently short relative to the rate of evolution.. Further, to account for the drift rate evolution across individual driftbands (which can show significant curvature), we follow the lead of <cit.> to use an exponential model. Hence, we used the linear model for non-evolving drift rates and an exponential model for evolutionary drift rates. To account for both kinds of drift rate behaviour, we used the following two models:
* Linear model of drift rates <cit.>
This model assumes a linearly evolving drift rate (D) with respect to increasing pulse number. Thus, devising an equation which depends linearly on pulse number (p) since the onset of the drift sequence, we get
D = dϕ/dp = a_1 p + a_2
where a_1 and a_2 are constants, and ϕ is the pulse phase. We can integrate eqn. <ref> to get the dependence of pulse phase on pulse number which would be a quadratic relationship.
ϕ(p) = a_1 p^2 + a_2 p + C
Here C = P_2 d + ϕ_0 is the integration constant that can be associated with a physical parameter, P_2, which is the “horizontal” separation between driftbands and is assumed to be a constant for the model fit of each drift sequence. Thus, the model has four free parameters, a_1, a_2, a_3, and P_2, to be accounted for.
* Exponential model of drift rates <cit.>
For the cases where individual driftbands can show significant curvature, the exponential model can be a better fit. In this model, the driftbands are modelled with an exponential function which assumes an exponential decay rate for the drift rate, D
D = dϕ/dp = D_0 e^-p/τ_ r + D_ f
where D_ f is the asymptotic drift rate, D_0 is the difference between D_ f and the drift rate at the beginning of the drift sequence, p is the number of pulses since the onset of the drift sequence, and τ_ r is the drift rate relaxation time (in units of the rotation period). To get the relationship with ϕ and p, we integrate eqn. <ref>
ϕ = τ_ r D_0 ( 1 - e^-p/τ_ r) + D_ f p + ( ϕ_0 + P_2 d )
where ϕ is the phase of a sub-pulse; ϕ_0 is an initial reference phase; d is the (integer) driftband number; and P_2 is the longitudinal spacing between successive driftbands. Thus, the model has five free parameters, D_0, D_ f, τ_ r, ϕ_0, and P_2, of which the expression ϕ_0+P_2d defines the pulse phase at p = 0. The last term in eqn. <ref> is the same as the constant in eqn. <ref>.
Using eqn. <ref> and <ref> from the linear and exponential drift rate models, we employed either of the two on different mode sequences. The fitting procedure for either of the models was carried out following the method described in <cit.>.
The pulsar exhibits both stable subpulse drifting (no evolution) and evolutionary drifting. Therefore, firstly we determined the mode boundaries (the beginning and end of a drift sequence) by visually inspecting the drift rate evolution in the pulse stack. Then, the evolutionary and non-evolutionary drift sequences were separated, with the caution that not too many mode boundaries are made. Sub-pulses were first smoothed using a Gaussian kernel of width ∼3.6 ms (i.e. 1^∘ of pulsar rotation, the approximate width of a subpulse). Then, the subpulses in each drifting sequence were identified by determining the peaks above a certain flux density threshold. This threshold was chosen such that for a minimum pixel value, no sub-pulse is identified in the off-pulse region. Each sub-pulse in the drift sequence is then assigned a driftband number. Finally, depending upon the drift rate model of choice, the driftbands are fitted (using SciPy’s method) with the functional form of the sub-pulse phases as mentioned in eqn. <ref> and <ref>. Examples of drift rate fitting are shown in Fig. <ref>, where the bright diagonally arranged patterns are the driftbands and the white overlayed lines are the driftband fits. The drift rate evolution across an entire observation (scan 2 of observation made on MJD 59529) using the above methodology is shown in Fig. <ref>. Here the x-axis shows the pulse number and the y-axis shows the drift rate in ^∘/P_1 units. The different curves correspond to either a linear or an exponentially varying drift rate across a drift sequence.
The models described above do not take into account the geometric curvature that must be present (to some degree), as discussed earlier. However, we argue that the geometric curvature must be negligible in J0026–1955's pulse window. As seen in Fig. <ref> (and Fig. <ref>), the pulsar exhibits a variety of drifting modes with inconsistent drift rates. If the geometric curvature was significant across the pulse window, it should be visible in all the modes, despite their evolutionary and non-evolutionary features. The fact that the characteristic `S'-shape of geometric curvature is not visible throughout leads us to conclude that it must be negligible across the pulse window for this pulsar. We, therefore, do not attempt to include geometric curvature in our models. In the next subsection, we describe the drifting modes and their various sub-classes, among which is a non-evolutionary mode (A0), in which the driftbands appear straight (see the left panel of Fig. <ref>). This mode demonstrates the lack of a significant presence of geometric curvature.
§.§ Drift Mode Classification
<cit.> categorised the drifting behaviour into two different classes: A and B. They further made sub-categories of modes A and B depending on the qualitative properties of drift sequences, their appearance and context. In this work, the broad classification into modes A and B follows <cit.>, but our subcategories of these modes are completely different, and are based on the drift rate modulation instead of organisation in drifting patterns. In our analysis, we first modelled the drift rates exhibited by the pulsar and then categorised them. The drift rate modelling provided insight into the drifting behaviour, which was the basis for our mode classification, as described below.
* Mode A: Mode A is classified as an umbrella mode category which encompasses the slower drift rates. The pulsar in mode A exhibits organised as well as unorganised driftbands. All of the evolutionary subpulse drifting behaviour exhibited by the pulsar can also be sub-categorised under mode A.
* Mode A0: This is the non-evolutionary sub-category of mode A. These are mode sequences which possess an almost constant drift rate and do not exhibit any evolutionary behaviour, as shown in Fig. <ref>. Along with a stable drifting rate (∼ -0.6^∘/P_1), mode A0 also has the largest mode length. The sequences, at times, do show frequent interruptions and rapid but temporary deviations in the drift rate. However, the overall drift rate still hovers around a constant number. The occurrence fraction of mode A0 in the complete set of observations was about 12%.
* Mode A1: According to our drift rate classification, sequences which demonstrate a slow evolution from fast to slow drift rates, as shown in Fig. <ref>, are labelled as mode A1. In an extreme case of drift rate evolution in mode A1, the sequence begins with a small P_3 value of about 16P_1 and ends after 110 pulses with a much different P_3 of about 60P_1. Mode A1 also had the largest occurrence fraction of ∼17% among all the subpulse drifting modes.
* Mode A2: In addition to the evolution from faster to slower drift rates, the pulsar also exhibits the opposite evolutionary behaviour. Mode A2 sequences begin with a slow drift rate, where the driftbands are far apart and evolve towards a faster drift rate. In our data, mode A2 had a total occurrence fraction of ∼7%. The sequences in mode A2 are generally short-lived and consist of 3-4 driftbands before the sequence ends with a faster drift rate (see fig. <ref>). We also note that most occurrences of mode A2 are followed by a null. This possible correlation is discussed in detail in section <ref>. <cit.> assumed this mode as a combination of mode A and the faster drifting mode (mode B). However, we believe that this is yet another evolutionary mode of J0026–1955, as the driftbands are fully connected throughout the drift sequence and exhibit a slow evolution rather than a sudden change in drift rate.
As expected, from Fig. <ref> it can be seen that the modal profiles of all the sub-categories of mode A show similar features despite the dissimilar drift rate behaviour, lending credibility to our classification scheme.
* Mode B: The pulsar also exhibits a faster drift rate on its own without being a part of any evolutionary behaviour. This was present in only ∼4% of the total observation. Mode B sequences are short-lived, are generally isolated occurrences, and do not show a drift rate evolution, as shown in Fig. <ref>. Mode B sequences can be found anywhere in the pulse stack, even in the midst of long, otherwise uninterrupted null sequences. The average drift rate for mode B is ∼ -1.6^∘/P_1. The average modal profile of mode B is shown in Fig. <ref>, which shows slightly different features with a skewed average profile, as compared to mode A profiles.
An additional feature was sometimes noted in the drift sequences of J0026–1955, where an extra driftband seems to appear towards the leading edge. An example can be seen at around pulse number 2125 in panel (c) of Fig. <ref>, in mode A2. There is a sudden break in the middle of the driftband, and both pieces look disassociated. It appears that towards the end of the first driftband, the drift rate fastens, and for the second driftband, the drift rate begins at a faster rate and then slows down. Overall, if the sudden drift rate change and the break are ignored, they seem to form a full driftband. During our analysis, we have not accounted for the break and considered the driftband in full wherever such a deviation was noted.
§.§ Nulling
Nulling is the temporary disappearance of emission from a pulsar for brief periods of time. After deciding the mode boundaries using the method described in <ref>, sequences with no subpulse detection were counted as nulls. In our observations, PSR J0026–1955 showed evidence of both long bursts of pulses and long nulls. The long nulls are sometimes interrupted with short mode B sequences. Fig. <ref> shows the entire length of null and burst sequences in all our observation scans. The longest null sequence goes on for 1117 pulses (∼25 minutes). In contrast, the most prolonged burst in our observations lasts for 867 pulses (∼19 minutes).
Fig. <ref> shows intensity as a function of time for all the observations. The shaded grey regions show detected subpulses, and the rest are nulls. The degree of nulling in a pulsar can be quantified in terms of the nulling fraction (NF), which is the fraction of pulses with no detectable emission. Overall, J0026–1955 was in a null state for more than half of our observations, with an estimated total nulling fraction of ∼58%. This differs from the nulling fraction of ∼77% obtained at 155 MHz using the MWA <cit.>. This discrepancy and the nulling behaviour of J0026–1955 are discussed in detail in section <ref>.
§.§ Drift Rate Evolution
PSR J0026–1955 exhibits multiple drift rates and evolutionary features throughout the drift sequences and individual driftbands. Initially, we used the linear and exponential models to understand the drift rate behaviour within a sequence, as described in section <ref>. However, driftbands and sequences in J0026–1955 exhibit more complicated evolutionary features, as seen in the mode A1 and A2 occurrences in Fig. <ref>, where even an exponential model fails to accurately describe the evolutionary drift rates correctly.
We further explored the drift rate behaviour of J0026–1955 by studying the variation in drift rate with each pulse in a driftband. To calculate the evolution of the drift rate with each pulse, we followed the methodology described in <cit.>. We first employed a cubic smoothing spline estimate using the SmoothingSplines[<https://github.com/nignatiadis/SmoothingSplines.jl>] software package on each of the driftbands, irrespective of their mode identity or drift mode boundaries. Then, we obtained the drift rate (phase/pulse number) by calculating the gradient of the fitted spline function at every pulse number for each driftband. As seen in J0026–1955 pulse stacks, there can be two driftbands at a given pulse number. Fig. <ref> shows examples of some drift sequences, where the top panel shows part of the pulse stack; where the red dots indicate the location of subpulses. Here, the green line is the cubic spline fit, which was obtained with the smoothing parameter λ = 100. In the bottom panel of Fig. <ref>, the black lines show the gradient calculated at each pulse number for every driftband. As some pulses contain two subpulses, yielding two measurements of the drift rate for that pulse number, we estimate multiple drift rates at some of the pulse numbers. The solid grey envelope shows the mean drift rate at every pulse number where the contribution from multiple driftbands at any given pulse is averaged.
A subset of pulsars that exhibit multiple subpulse drifting modes shows a harmonic relationship between P_3, as reported in previous studies <cit.>. In the case of J0026–1955, a similar analysis with drift rates leads to interesting implications. A cumulative and modal histogram of individual drift rates obtained by taking a gradient of the smoothing spline fit of driftbands at each pulse number can be seen in Fig. <ref>. The top panel shows a distribution of all drift rates, where each colour corresponds to the different modes classified for J0026–1955. The lower panels of Fig. <ref> display the distribution of drift rates for all observed subpulse drifting modes of J0026–1955. Apart from two distinct drift rate peaks at approximately -0.5^∘/P_1 and -1.6^∘/P_1, a third peak is visible at around -2.7^∘/P_1. The peak values of the drift rates form an arithmetic sequence with a common difference of approximately -1.1^∘/P_1. Further, considering eqn. 1 and 2 in <cit.>, it can be implied that if an arithmetic spacing exists between drift rates, then the corresponding number of sparks will also have an arithmetic relationship, assuming a constant carousel rotation rate (P_4).
§.§.§ A fourth-order polynomial fit of drift rates
On a closer examination of Fig. <ref>, it is evident that the drift rate is irregular. Furthermore, the drift rate does not simply evolve towards a higher or lower rate but shows variability, even within a particular mode. A linear or exponential drift rate could not accurately comprehend the complexity of this drift rate evolution. Hence, we tried to fit the average drift rate with a polynomial. Employing the polynomial regression method using , we successfully fit a fourth-order (quartic) polynomial function to the evolving drift rates. A higher-order polynomial could also describe the evolutionary drift rate. However, such a model might be counter-intuitive and would only provide a customised fitting rather than a general model. Fig. <ref> shows the fourth-order polynomial fit of the drift rates (blue dots) for scan 2 of the observation made on MJD 59529 (November 11, 2022). The black line is the average drift rate at each pulse number, same as the grey envelope in Fig. <ref>. Different colours of the fourth-order polynomial fit correspond to different modes, following the colour scheme of Fig. <ref>. A direct comparison between the drift rate models in Fig. <ref> and Fig. <ref> can be made, where the latter shows the evolution of drift rate within a drift mode sequence. The fourth-order polynomial model can be seen to better describe the drift rate modulation as compared to the more simplistic, linear and exponential models, which lacked the detail.
To understand the overall drift rate modulation in various modes, we overlayed the drift rate fits for each mode, as shown in Fig. <ref>, where the x-axis shows mode length and y-axis the drift rate. The black line in each subplot shows the mean of drift rates (from the polynomial fits) with the pulse number. The grey envelope corresponds to the 1σ deviation from the mean drift rate. As expected, mode A0, which does not show any noticeable evolutionary behaviour had an almost constant average drift rate across all instances, around -0.6. In contrast, the drift rate evolution of modes A1 and A2 is observable in their respective subplots. Drift rates in mode A1 can be seen to evolve towards a slower drift rate as compared to the commencing rate, whereas mode A2 shows an overall evolution towards a faster drift rate as it reaches the end of a drift sequence.
§.§ Memory across nulls
We also investigated the presence of memory across nulls, as reported by <cit.>. Our analysis shows that the short-lived nulls of PSR J0026–1955 indicate evidence of subpulse memory across nulls. This could be a subpulse phase memory or a drift rate memory. In the first scenario, the phase of the last subpulse before the null and the first subpulse after the null are similar. On the other hand, a drift rate memory could be a case where the drift rate across the null stays the same, and the driftbands could be extrapolated.
We followed the model fitting technique described in <ref> to explore the latter. To check if there is indeed a drift rate connection between the sequence before the null and the sequence after the null, we fit the previous drift sequence (before null) using the model of choice (following the method in <ref>). We then extrapolate the model to a subpulse right after the null sequence ends, allocating the subpulse a reasonable driftband number. We can consider a drift rate memory if the projected phase of the subpulse after the null matches the real subpulse within an error range. The error on phase prediction is calculated from the covariance matrix of the model fit (using standard uncertainty propagation). If the phase of the real pulse falls outside the subpulse phase range projected by the model, then we consider that there is no memory across the null. On the other hand, if the projected phase and the phase of the real pulse are within the error bars and smaller than P_2 (phase distance between two subpulses within a pulse), then we classify that null as being consistent with being phase-connected subpulse across the null. An example of drift rate memory across nulls is shown in the top panel of Fig. <ref>, where the white lines depict the drift rate fits. The top panel in Fig. <ref> shows two drift sequences on either end and a null. By fitting the sequence before the null and projecting the drift rate behaviour to the latter sequence, one can note that the drift sequences before and after the null are consistent.
We, however, do not find many instances of subpulse phase memory across nulls in our data. A handful of instances were initially recognised by visual inspection, as they did not show a drift rate memory across nulls. They were further investigated by calculating the phase of the subpulse before and after null. One such example is shown in the bottom panel of Fig. <ref>, where the subpulse phase before and after the null are almost the same. In contrast, since the sequences have different drift rates, a drift rate memory might not be present. A more careful analysis of longer observations is required to verify this fully.
§ DISCUSSION
PSR J0026–1955 exhibits a multitude of pulse-to-pulse modulation phenomena, viz. subpulse drifting, mode switching, and nulling, as shown in Fig. <ref>. For each of the driftbands, subpulses arrive at earlier phases with pulsar rotation, thus conforming to the `positive drifting' class of subpulse drifters <cit.>. However, the vertical separation between driftbands (i.e., P_3) can be seen to vary between different sequences, as well as within a drift sequence. There are instances where J0026–1955 can be seen to abruptly switch from one subpulse drifting mode to another, a behaviour exhibited by many other pulsars <cit.>. However, in addition to the abrupt mode change, the drift rate can also sometimes be seen to evolve gradually, which is a relatively rare phenomenon, observed in only a small subset of subpulse drifting pulsars, e.g., PSR B0943+10 <cit.>, PSR B0809+74 and PSR B0818–13 <cit.>. For PSR J0026–1955, a quick glance at mode A1 and A2 sequences (evolutionary drift modes) in Fig. <ref> shows this variable drift rate. The pulsar also exhibits long and short-duration nulls, where an association between drifting and nulling can be drawn. Below, we discuss the variety of phenomena exhibited by J0026–1955 in light of the analysis presented in section <ref>.
§.§ Nulling Behaviour of J0026–1955
The analysis presented in section <ref> highlights the unique nulling behaviour of J0026–1955, which exhibits complex emission properties. The nulling fraction at 400 MHz is approximately 58%, an estimate reached from 330 minutes of observation, whereas, at 155 MHz, it was estimated to be 77% from 192 minutes of MWA observation <cit.>. This discrepancy suggests a possibility that the nulling behaviour of J0026–1955 may not be broadband and that there may be frequency-dependent mechanisms at play. Frequency-dependent nulling has been observed in some pulsars, where the nulling behaviour varies depending on the observed frequency <cit.>. However, given the modest separation in the frequency bands (with band edges separated by ∼130 MHz and the centre frequencies differing by a factor of ∼2.5), it is unclear if the observed discrepancy is entirely attributable to frequency-dependent nulling. Alternatively, the observed inconsistency could simply be an unintended observational bias, where the MWA observations were incidentally made around the long nulls. The perceived discrepancy could also be because of the presence of an emission component with a shallow spectral index leading to a null at 155 MHz. Assuming that J0026–1955 exhibits broadband nulling, a combination of the number of nulls out of the total number of pulses observed at 155 MHz and 400 MHz will imply a nulling fraction of ∼65%.
It is also possible that the nulling behaviour of J0026–1955 is complex and multi-faceted, involving both broadband and frequency-dependent mechanisms. Different models attribute pulsar nulling to intrinsic changes in the magnetosphere, such as temperature fluctuations altering coherence conditions <cit.>, switching between gap discharge mechanisms <cit.>, variations in magnetospheric currents <cit.>, change in pulsar beam geometry <cit.>, disruption of the entire particle flow in the magnetosphere <cit.>, etc. Though such models may be able to describe the long-period nulls seen for J0026–1955, the presence of subpulse (phase or drift rate) memory across nulls challenges these theories for at least the case of short nulls where such memory exists. Further investigation is needed to fully understand the nature of the observed disparity and the complex emission properties of J0026–1955.
§.§ Drift Rate - Nulling Correlation
There have been only a limited number of investigations that explored a correlation between nulling and subpulse drifting. For example, PSR B0818–41 presents a case where a decrease in the pulsar drift rate is accompanied by a gradual decrease in intensity before the onset of a null <cit.>. PSR B0809+74 also shows an association between nulls and subpulse drifting, where the drift rate deviates from normal after the nulls <cit.>. Using the partially screened gap model, the authors suggest that some kind of “reset” of the pulsar’s radio emission engine occurs during the nulls, which is responsible for the conditions of the magnetosphere. For PSR B0809+74, <cit.> suggest that nulling and subpulse drifting may be related, with changes in emission beam geometry potentially causing both the nulling and changes in the drift behaviour. <cit.> suggested that emission can cease or commence suddenly when the charge or magnetic configuration in the magnetosphere reaches the so-called “tipping point”; however, the triggering mechanism for such stimulus is unknown. In the case of PSR J1822–2256, <cit.> also showcase a relationship between nulling and mode changing, where a null preceded most occurrences of their mode D. Such correlations suggest that emission mechanisms and magnetosphere dynamics between nulls and subpulse drifting may be strongly related.
PSR J0026–1955 also provides some compelling evidence of a possible correlation between subpulse drifting and nulling. Our observations suggest a complex and dynamic mechanism underlying the pulsar radio emission, with important implications for understanding astrophysical processes in extreme environments. In particular, we have found that the pulsar J0026–1955 likely switches to a null state after mode A2. Mode A2 is an evolutionary mode, where the drifting begins at a slower drift rate and evolves towards a faster drift rate before the mode eventually ends. In 27 out of 31 instances, mode A2 is followed by a null, either short or long. In such cases, the ramping up towards the faster drift rate begins remarkably consistently about 20 pulsar rotations prior to the null (see Fig. <ref>), which suggests that the null itself might be causally related to the preceding drifting behaviour. In the rest of the four sequences, mode A2 was once at the end of our observation and was followed by either mode A0 or A1 in three instances. These occurrences can also be seen as a drift reset, where the pulsar temporarily evolves to a faster drift rate and then returns to a slower one. This reset could also be due to a change in the magnetospheric conditions before a “tipping point” was reached. It was also noted that not every null sequence followed mode A2, but a null followed most occurrences of mode A2. We did not find any strong correlation between the intensity of subpulses towards the onset or end of a null. The variation in pulse intensity of transitions from burst to null was sometimes abrupt and smooth at other times and lacked any compelling evidence of intensity dependence. It is worth noting that this kind of phenomenon, where nulls affect the drifting behaviour leading up to them, is relatively rare in contrast with the more commonly studied effect of nulls on burst sequences that come after them <cit.>. A deeper understanding of these rare cases can provide useful insights into the complex dynamics relating nulls and subpulse drifting.
§.§ Memory Across Nulls
Only a limited number of pulsars retain the information about previous subpulses during nulls, as demonstrated by, e.g., <cit.> and <cit.>. This memory retention can provide valuable insights into the true nature of the nulling phenomenon and any correlation with subpulse drifting it may have. If intrinsic changes in the pulsar magnetosphere lead to both nulls and drift-rate modulations, studying the interactions between these phenomena could offer valuable insights into the mechanisms that trigger and facilitate transitions between the different states. Subpulse memory across nulling in drifting pulsars has been generally attributed to either of the two reasons: (1) the polar cap continues to discharge, but the emission is not observed <cit.>, or (2) the subpulse ceases drifting for the duration of the null <cit.>. In the first case, the absence of radio emission is attributed to the lack of coherent structure, and the drift rate remains the same before and after the null. Thus, a part of the drift sequence would be missing, though one could still map the driftbands with similar drift rates pre- and post-null, showing `drift rate' memory across nulls. Whereas in the second scenario, the subpulse drifting is thought to resume at the phase where it left off, demonstrating a subpulse `phase memory' across nulls. As described in section <ref> have encountered possibilities of both `drift rate' memory and `subpulse phase' memory across nulls in the case of J0026–1955.
§.§.§ Drift rate memory
J0026–1955 exhibits a clear case of subpulse drift memory (top panel in Fig. <ref>), where the pulsar seems to remember the drift rate even after a short (∼20 pulses) null. The study described in <cit.> suggests that nulling in pulsars results from an uninterrupted and stable discharge in the polar gap rather than a complete cessation of sparks.
This observation is similar to the case of J1840–0840, where the drift rate stays the same across the null, and an entire driftband is missing from the sequence <cit.>. PSR J1840–0840 also shows `subpulse phase' memory in conjunction with the `drift rate' memory.
The cause of undetectable radiation during nulling can be attributed to the absence of the dominant coherence mechanism present during regular pulsar operation rather than the lack of particle flux from the polar cap <cit.>. Therefore, during the null states, the subpulses on the polar cap may continue to drift during the null state, either at similar or different rotation speeds. In this case, the phase of the subpulse after the null sequence can be anticipated based on the duration of nulls. In the case of J0026–1955, when the pulsar switches its emission back on after a null, the phase of the subpulse can be extrapolated from the drift rate model before the null. The predicted phase was well within the error range for drift sequences around short nulls. This finding supports the idea that drifting and sparking may still be operational during nulls, providing further evidence for the two scenarios previously observed in drifting pulsars.
§.§.§ Subpulse phase memory
As shown in the lower panel of Fig. <ref>, J0026–1955 also presents evidence of a `phase' memory across nulls, where the drift rate does not necessarily stay the same across the nulls. Still, the phase of the last subpulse before the null is almost identical to that of the first subpulse after the null. Similar behaviour is observed in a few other pulsars like B0809+74 and B0818–13, where the drift rate changes after null sequences <cit.>. During some of these interactions, the pulsars also seem to indicate some kind of phase memory, such that information regarding the phase of the last subpulse was retained during the null state. <cit.> propose that during the null, the drifting stops and the position of the sparks are remembered by the presence of a hotspot on the pulsar surface. This would imply that once the drifting resumes, the sparks will reform at their previous position. PSR B0031–07 was shown to retain the memory of its pre-null burst phase across short nulls <cit.>. In their study by <cit.>, PSR J1840–0840 presents a unique example of both `subpulse phase' and `drift rate' memory across null. In our data, we only had a few occurrences of memory across nulls of both kinds. A more detailed analysis of drift rate and subpulse phase memory around nulls needs to be conducted for J0026–1955 using longer observations, thus increasing the sample size of such events.
§.§ Subpulse Drifting Model for J0026–1955
J0026–1955 presents an exciting and rare phenomenon of drift rate evolution (modes A1 and A2), in addition to regular subpulse drifting with a constant drift rate (modes A0 and B). The pulsar exhibits changes in drift rate over sequences, with mode A1 showing a trend towards a slower drift rate and mode A2 showing a trend towards a faster drift rate with increasing pulse number. Furthermore, irrespective of the modal transitions, we have also observed an inter-modal driftband connectivity in most mode-switching cases. We utilised the techniques described in section <ref> to analyse the evolution of drift rates across the driftbands and sequences. Measuring the slopes of individual driftbands was essential to scrutinise any changes in the drifting pattern.
Furthermore, we modelled the drift rate behaviour across the drift sequences using a quartic polynomial. The quartic model fits were then used to understand the global drift rate variation for the different modes, as shown in Fig. <ref>. The figure shows that the modes A1 and A2 for J0026–1955 display a gradual evolution from their initial drift rates. It is noteworthy that the change of drift rate for mode A2 (∼ -0.5^∘/P_1 to ∼ -2.0^∘/P_1) is almost twice as much as the change of drift rate in mode A1 (∼ -1.3^∘/P_1 to ∼ -0.5^∘/P_1). Hereafter, we discuss the possible modifications/additions to the existing carousel model that can explain the observed unique behaviour of J0026–1955.
§.§.§ Variable Spark Configuration in Carousel Model
As discussed in <cit.>, <cit.>, and <cit.>, different modes and drift rates of a pulsar can be attributed to a carousel with varying numbers of sparks for each subpulse drifting mode. In such cases, the drift rate is observed to change abruptly. This sudden change cannot be ascribed to the carousel rotation rate, as it would imply significant magnetosphere reconfiguration over a short time scale. However, the drift rate change within a single pulsar rotation can be due to the reconfiguration of spark distribution. In the case of the evolutionary drifting modes of J0026–1955, the idea of a carousel with a constantly changing number of sparks to describe the changing drift rate may be counter-intuitive. Nevertheless, the non-evolutionary modes (A0 and B) may still have a fixed spark configuration.
An alternative hypothesis which could involve a changing carousel rotation rate, would need a slow change in the spark carousel itself, where the gradual evolution is a signature of the spark configuration relaxing into a new arrangement after a spark suddenly appears or disappears from the carousel <cit.>. Such “relaxation” of the drift rate is reminiscent of the behaviour observed in PSR B0809+74 <cit.>, where, after a null, the pulsar would temporarily attain a faster drift rate before relaxing into a steady drift rate. They further suggest that a perturbation might alter the drift rate (and emission, thus causing a null), after which the drift rate recovers exponentially to its normal value.
If the number of sparks in the carousel is changing, then the carousel will take some finite amount of time for the sparks to rearrange themselves, for example, into the new configuration in which the sparks are equidistant from each other. During this relaxation time, the angular speed of individual sparks may differ from the angular speed of the whole carousel. In that case, the observed drift rate at any one moment will only depend on the spark that is “under” the line of sight during any given rotation period. In this view, the observed drift rate can appear to change slowly without requiring the average carousel rotation speed to change, as long as the time scale for the spark reconfiguration is relatively long.
If, as the above suggests, the sparks reconfigure themselves only slowly after one of the sparks either appears or disappears, one observational consequence of this is that the driftbands should always look connected, which appears to be the case for J0026–1955. This still remains true in the presence of aliasing, although the observed drift rate changes may be magnified. In comparison, <cit.> point out that connectivity of driftbands is impossible if the carousel rotation rate transitions from non-aliased to aliased regimes.
§.§.§ Carousel Model in Partially Screened Gap
Alternatively, a steady change in the carousel rotation rate (P_4) with a constant spark configuration is also a plausible explanation for the evolutionary drift modes of J0026–1955. In such a case, some conditions might change at the pulsar surface, making the spark carousel change its rotation speed. As a result, the carousel rotation rate varies smoothly, resulting in a gradual drift rate evolution. Below we provide a hypothesis, using the partially screened gap model <cit.>, for such a phenomenon that can gradually alter the carousel rotation rate.
According to the PSG, a reverse flow of electrons towards the polar cap lead to an ion discharge in the gap region, which ultimately acts as a screen. Due to this screen, the electric field in the polar cap reduces, which reduces the spark velocity <cit.>. <cit.> also suggests that the dependence of the drift rate on the electric and magnetic fields boils down to a dependence on the variation of the accelerating potential across the polar cap. Hence, a variation in the polar gap results in an increase/decrease in the spark velocity, which in turn may also increase/decrease the carousel rotation rate (regardless of whether the drifting is aliased). Consequently, the carousel rotation rate will also affect P_3. In the case of no aliasing, the drift rate will be more negative for a faster carousel rotation rate, and the P_3 value will be smaller. Further, as discussed in <cit.>, a change in drift rate is reflected in the emission heights, where faster drift rate (lower P_3 value) emission is thought to originate from higher emission heights; and at lower emission heights, emission from a slower drift rate (higher P_3 value) is observed. For curvature radiation, the only way to observe a change in the emission height at one observation frequency is if there is a change in the magnetic field lines. For a changing emission height, the foot points of the magnetic field lines will be closer to or further from the (dipolar) magnetic pole, implying a changing size of the observed spark carousel. For the evolutionary drift modes of J0026–1955, where the drift rate is seen to vary drastically within a drift sequence, a change in emission height (at a fixed observational frequency) would automatically suggest a variable carousel rotation rate for a fixed carousel configuration.
Extrapolating the PSG model, if the screening increases due to the ion discharge in the polar cap region, the electric field in the polar gap region will decrease, lowering the spark velocity and, consequently, lowering the carousel rotation rate. Given the proposed relation between P_4 and emission heights, the emission from a carousel with lower P_4 will come from a lower altitude in the pulsar magnetosphere. We observe a slow drift rate evolution in mode A1, which could be where the screening steadily increases, causing an evolution towards a slower carousel rotation rate. Similarly, mode A2 can be the case where the screening lowers, causing the drift rate to increase and the emission to come from higher altitudes. Since the typical mode length of observed mode A2 occurrences is not as long as mode A0 or A1, we can only conjecture that the screening cannot lower beyond a certain extent.
For modes A1 and A2, a gradual change in the carousel rotation rate might be an acceptable model. This is also consistent with the general understanding that the rotation rate of the carousel cannot change its magnitude or direction abruptly during a single pulsar rotation (as it implies a rapid change in the pulsar magnetosphere), although it may exhibit a slow evolution.
§.§.§ Revisiting Mode A2 - Null Correlation
Our findings suggest a strong correlation between mode A2 and nulling across multiple observation epochs, as discussed in section <ref>. Such an association indicates the intrinsic nature of these changes and their close relationship with the nulling process.
The gradual transition from slow to fast drift rate, via addition/reduction of sparks as discussed in section <ref>, might trigger a “reset” of the pulsar's radio emission engine, causing the emission to cease for several pulses. J0026–1955 provides compelling evidence for a scenario in which the electromagnetic conditions in the magnetosphere region responsible for radio emission attain a null state after the considerable drift rate evolution seen in mode A2. Additionally, the stability of sparks for fast drift rates could be a reason for short sequences. Implying that for mode A2, the spark configuration becomes unstable once the drift rate is sufficiently fast due to an increase in the number of sparks, causing a null. The instability argument can also explain why mode B sequences only last for a short time compared to all the other modes.
Alternatively, following the subpulse drifting model suggested in section <ref>, mode A2 might indicate the scenario where screening decreases, causing the electric field in the polar gap region to increase. This impacts the spark velocity and, thus, the carousel rotation rate. As the carousel rotation rate increases (due to a decrease in screening), P_3 decreases and the driftbands appear closer in the pulse stack. Using the direct dependence between P_4 and emission heights (derived from the inverse relation between P_3 and emission heights), we can deduce that the emission comes from increasingly higher emission heights for mode A2. Alternatively, it is possible that due to our line of sight, we cannot observe the pulsar past a certain emission height. The drift sequence ends very soon after mode A2 achieves a significantly low drift rate. Similarly, due to the line-of-sight constraint, we might explain why modes with lower drift rates, like mode B, have comparatively shorter mode lengths.
§ SUMMARY
We have conducted a thorough analysis of subpulse drifting behaviour in PSR J0026–1955 at 300-500 MHz from uGMRT observations. Our results and conclusions from this study are summarised below.
* From our observations, we have found that the pulsar exhibits short- and long-duration nulls, with an estimated nulling fraction of ∼58% from our uGMRT observations. The nulling fraction is in stark difference from ∼77% that was observed at 155 MHz in MWA observations. This disparity could be due to differences in the lengths of observations, or a shallow spectral index component, or a frequency dependence of nulling.
* The pulsar exhibits unusual drifting behaviour, with both evolutionary and non-evolutionary drift rates. Further, we categorise the unusual subpulse drifting behaviour of this pulsar into two drifting modes: A and B, where mode B is a non-evolutionary mode with a faster drift rate. Mode A was further sub-categorised depending on its evolutionary behaviour. Mode A0 is a non-evolutionary mode with a drift rate 3-4 times slower than mode B. The lack of any curvature in A0 suggests that the viewing geometry must be such that the driftband curvature arising from purely geometric considerations (the “geometric curvature”) is negligible across the pulse window. Mode A1 is an evolutionary mode which shows a smooth evolution of drift rate from fast to slow. On the other hand, the drift rate in mode A2 evolves from a slower to a faster drift rate.
* The individual driftbands for J0026–1955 are not linear and have variable drift rates. We used a cubic smoothing spline estimate on individual driftbands and calculated the gradient (drift rate) at each pulse. To understand the overall drift rate modulation, we fit the drift rates using a quartic polynomial. The model helped in recognising the inter-band and inter-mode variability in drift rates. Though a simplistic higher-order polynomial can describe the global evolution empirically, understanding the local drift rate variability requires more complex modelling.
* The pulsar J0026–1955 shows an evolution in drift rate for modes A1 and A2. We advocate the following two models to explain their behaviour:
* Variable spark configuration - The gradual evolution of drift rate in modes A1 and A2 could be caused by a slow change in the spark configuration as the carousel reconfigures into an optimal arrangement after a spark appears or disappears. During this reconfiguration, the angular speed of individual sparks may differ from the average carousel rotation. In this case, the observed evolution in drift rate will be due to the motion of sparks “under” the line-of-sight as the entire carousel slowly recomposes.
* Variable carousel rotation rate - In this model, we propose that the evolution in drift modes can be explained by a smoothly varying carousel rotation rate with a direct correlation with emission height rather than changes in the number of sparks. The evolution in carousel rotation rate is thought to originate from an increase or decrease in screening in the polar gap region. Therefore, as the screening decreases/increases, the carousel rotates faster/slower (mode A2/A1), with the emission coming from higher/lower altitudes in the pulsar magnetosphere.
A combination of the two suggested models might also be plausible, a possibility that should be explored in future.
* J0026–1955 shows robust evidence of subpulse memory across nulls. In multiple instances, we have found the possibility of `drift rate' and `subpulse phase' memory across nulls. We believe that there could be an uninterrupted stable discharge in the polar gap during the null, which is not observed due to the absence of a dominant coherence mechanism or a partially screened gap making the generation of detectable radio emission difficult.
* J0026–1955 exhibits an almost consistent behaviour, where a null often follows mode A2. We propose two hypotheses for this behaviour:
* The transition from slow to fast drift rates due to the appearance/disappearance of a spark often triggers a “reset” of the pulsar’s radio emission engine, which often culminates in a null. This reset is most likely to take place after the occurrence of mode A2, where the pulsar transitions from a slow to a fast drift rate. Not every occurrence of a null sequence is preceded by mode A2. However, most occurrences of mode A2 are followed by a null state. A null, followed by mode A2, must relate to a defined pathway in the pulsar emission as it is consistently observed across all independent data sets (from two epochs of observations).
* Using the proposed carousel model, we advocate the idea that a decrease in screening increases the electric field in the polar gap region and impacts the spark velocity and carousel rotation rate. This results in a decrease in P_3 and closer driftbands in the pulse stack, suggesting that emission comes from higher emission heights for mode A2. The sequence ends soon after mode A2 achieves a low drift rate, possibly due to our line of sight not being able to observe the pulsar beyond a certain emission height.
§ ACKNOWLEDGEMENTS
We thank the anonymous referee for several useful comments that helped improve the paper. PJ acknowledges the Senior Research Fellowship awarded by the Council of Scientific & Industrial Research, India. We thank S. Kudale for their help with conducting these observations. The GMRT is run by the National Centre for Radio Astrophysics of the Tata Institute of Fundamental Research.
§ DATA AVAILABILITY
This paper includes data taken from the uGMRT in the 41st observing cycle.
mnras
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