利用结构假设加速相位恢复：成像问题的 convex 优化方法
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收藏 428KB PDF 举报 "本文主要探讨了成像问题中的相位恢复（Phase Retrieval）算法。由Fajwel Fogel, IRENEWAL DSPurger, 和Alexandre d'Aspremont共同研究，该领域关注的是如何在只有线性测量的幅度信息，即求解复数向量x，使得|Ax| = b（其中A是实数矩阵，b是实数向量）的情况下重构信号。这一问题在实际应用中尤为重要，尤其是在X射线晶体学、衍射成像、傅里叶光学以及显微镜等领域，由于物理限制，往往只能获取观测值的强度，而无法得到其相位。
相位恢复问题的关键在于利用信号和观测特征来优化算法性能。论文中强调了结构假设的作用，例如稀疏性、平滑性或正性，这些假设能够加速收敛速度并提高恢复效果。实验部分特别关注了分子成像问题，通过模拟PDB数据进行测试，展示了算法在处理这类实际场景时的有效性。
具体来说，文章介绍了在衍射成像中，A通常是一个包含一或多个掩码的傅里叶变换，这种技术有时被称为“缺失相位恢复”（Phase Retrieval from Missing Data）。作者详细地分析了如何将这些先验知识融入到凸优化算法中，以提升图像重建的精度和效率。
本研究不仅提供了理论框架，还展示了实证结果，为解决实际成像问题中的相位恢复挑战提供了有价值的策略。对于那些依赖于相位信息恢复的科学家和工程师来说，这篇论文具有重要的参考价值，它展示了如何结合特定领域的知识和技术改进相位恢复算法的性能。"
A classical “input-output” variant, detailed here as algorithm Fienup, introduced by [Fienup, 1982] adds
an extra penalization step which usually speeds up convergence and improves recovery performance when
additional information is available on the support of the signal. Of course, in all these cases, convergence to
a global optimum cannot be guaranteed but empirical performance is often quite good.
Algorithm 2 Fienup
Input: An initial y
1
∈ F, i.e. such that |y
1
| = b, a parameter β > 0.
1: for k = 1, . . . , N − 1 do
2: Set
w
i
=
(AA
†
y
k
)
i
|(AA
†
y
k
)
i
|
, i = 1, . . . , n.
3: Set
y
k+1
i
= y
k
i
− β(y
k
i
− b
i
w
i
) (Fienup)
4: end for
Output: y
N
∈ F.
2.2. Semidefinite programming. In [Chai et al., 2011; Candes et al., 2011b], the phase recovery problem
is then written as
minimize Rank(X)
subject to Tr(a
i
a
∗
i
X) = b
2
i
, i = 1, . . . , n
X 0
in the variable X ∈ H
p
, where X = xx
∗
when exact recovery occurs. This last problem can be relaxed as
minimize Tr(X)
subject to Tr(a
i
a
∗
i
X) = b
2
i
, i = 1, . . . , n
X 0
(PhaseLift)
which is a semidefinite program (called PhaseLift by Candes et al. [2011b]) in the variable X ∈ H
p
. This
problem is solved in [Candes et al., 2011b] using first order methods.
In [Waldspurger et al., 2012], it was shown that the phase problem retrieval problem (1) could be split
between phase and signal reconstruction, exploiting the fact that given the phase, signal reconstruction is
a simple least squares problem. Problem (2) is non-convex in the phase variable u. However, given u we
obtain x as
x = A
†
diag(b)u (4)
where A
†
is the pseudo inverse of A. Replacing x by its value in (2), the phase recovery problem becomes
minimize u
∗
Mu
subject to |u
i
| = 1, i = 1, . . . n,
(5)
in the variable u ∈ C
n
, where the Hermitian matrix
M = diag(b)(I − AA
†
) diag(b)
is positive semidefinite. [Waldspurger et al., 2012] detailed greedy algorithm Greedy to optimize (5) in the
phase variable.
A convex relaxation to (1) was also derived in Waldspurger et al. [2012] using the classical lifting argument
for nonconvex quadratic programs developed in [Shor, 1987; Lov
´
asz and Schrijver, 1991]. This relaxation
is written
min. Tr(UM )
subject to diag(U ) = 1, U 0,
(PhaseCut)
3
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