FASPR：利用上界概念的快速稀疏相位恢复算法
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本文是一篇发表在《数字信号处理》(Digital Signal Processing)2018年第80期的研究论文，标题为"FASPR: 一种基于上界概念的快速稀疏相位恢复算法"。作者Baoshun Shi、Shuzhen Chen、Ye Tian、Xiaoyu Fan和Qiusheng Li等人来自燕山大学信息科学与工程学院以及河北省信息传输与信号处理重点实验室。
论文主要探讨了在数字信号处理领域中的一个关键问题：相位恢复(Phase Retrieval, PR)，即通过测量一组线性变换后的幅度或强度来重构感兴趣图像。这一过程由于涉及非线性采样操作，使得恢复任务极具挑战性。为了克服这个难题，研究者们近年来开始利用稀疏性先验信息，将数据拟合度项与精细的稀疏诱导正则化相结合。
论文的核心贡献是提出了FASPR（Fast Sparse Phase Retrieval）算法，它采用了一种快速的、基于上界概念的方法。上界概念在这里是指利用数学上的epigraph set理论，这是一种优化工具，可以帮助约束和简化问题，特别是在处理非凸优化时。FASPR算法结合了总变分(Total Variation, TV)作为稀疏性表示，TV有助于保留图像的边缘和结构信息，这对于保持图像细节至关重要。
FASPR算法借鉴了快速迭代方法，如Fast Iterative Shrinkage-Thresholding Algorithm (FISTA)，该算法能够在保证收敛速度的同时，有效地处理大规模的数据集。通过这种方式，FASPR能够高效地在保证图像质量的同时，显著降低计算复杂度，这对于实时应用和大数据场景具有重要意义。
文章还提到了一些关键词，如"编码衍射图案"(Coded Diffraction Pattern)和"非凸优化"，这些概念都与实际实验和理论分析紧密相关。这篇论文在稀疏相位恢复领域提供了一个新颖且高效的解决方案，对于提升相位恢复算法的性能，特别是在高维度和大规模数据处理方面，具有很高的实用价值。
14 B. Shi et al. / Digital Signal Processing 80 (2018) 12–26
more difficult to recover compared to real-valued images, are con-
sidered
in this paper. Recovering the complex-valued image with
high quality from the undersampled and noise CDPs during a few
seconds is also one of our main contributions.
3. Problem formulation
We exploit the sparsity in the gradient domain for PR by using
the constraint based on TV, which has been widely utilized for
solving image inverse problems [29,30]. The TV-based constraint
of an image x can be defined as
TV(x) ≤ , (4)
where is a small positive value. The operator TV(x) for real-
valued
images is defined as TV
real
(x) =
i
(
∇
h
x
)
2
i
+
(
∇
v
x
)
2
i
(here,
∇
h
and ∇
v
represent the gradient operators in the horizontal and
vertical direction, respectively). For complex-valued images, the
magnitude image |x| and the phase-angle image
x (
x rep-
resents
the function that extracts the phase-angle image of a
complex-valued image x) are modeled by the sparse model of real-
valued
images separately. More discussions about this modeling
method of complex-valued images can be found in Ref. [31]. In
this paper, we define the TV model for complex-valued images as
λTV
real
(|x|) +TV
real
(
x); here, λ is a weighting factor that controls
the weight of the magnitude TV model in the two sparse models.
The weighting factor λ is related to the intensity ranges of both
the magnitude image and the phase-angle image. In this work, the
magnitude image is scaled to the interval [0.1, 1], and the phase-
angle
image is scaled to the interval [0, π ]. To balance the weight
of the two TV models, we set the weighting factor as λ = π . In
realistic applications, if the numerical ranges of the magnitude im-
age
and the phase-angle image are the same, the parameter λ can
be set as 1.
Incorporating the TV constraint into PR, we formulate the fol-
lowing
problem
min
x,c
1
2
||c
√
y −x||
2
2
s.t. TV(x) ≤, |c|=1, (5)
where c is an auxiliary vector that is consistent with the missing
phase, and 1 is a vector whose elements are all 1. The data-fidelity
term indicates that the linear transform of the signal matches the
recorded magnitude. Compared with the optimization model (2),
the proposed model (5)is more flexible. The proposed formulation
is not only suitable for recovering the image from the recorded in-
tensity,
but also suitable for the recorded magnitude case. One may
argue that the parameter must be determined; however, in this
paper this parameter is automatically determined by the orthogo-
nal
projection of the estimated signal. Problem (5)is a non-convex
optimization problem. In the next section, we present the numeri-
cal
algorithm of its solution in detail.
4. Proposed algorithm
We exploit the alternating optimization approach to solve prob-
lem
(5) that contains two optimization variables. At the tth itera-
tion,
we solve problem (5)via the following steps:
(1)
Phase-recovering step
Fixing
the estimated image x
(t−1)
, the sub-problem of recover-
ing
the phase as well as the auxiliary vector c
(t)
is
min
c
1
2
||c
√
y −x
(t −1)
||
2
2
s.t. |c|=1. (6)
The above problem can be regarded as a projection problem, which
has the closed-form solution:
c
(t )
=
x
(t −1)
|x
(t −1)
|
,
(7)
where
x
|x|
represents the element-wise division.
(2) Image-updating step
Fixing
the auxiliary vector c
(t)
, the sub-problem of updating the
image is
min
x
1
2
||c
(t )
√
y −x||
2
2
s.t. TV(x) ≤. (8)
The above problem can be solved using the projected gradient de-
scent
method. Concretely, to reduce the computational complexity,
we exploit the one-step FISTA method to solve problem (8). The
FISTA method used to solve this problem includes the following
updating scheme [28]:
x
(t )
= p[z
(t )
], (9)
h
(t +1)
=
1 +
1 +4[h
(t )
]
2
2
, and (10)
z
(t +1)
=x
(t )
+
h
(t )
−1
h
(t +1)
[x
(t )
−x
(t −1)
]. (11)
Here, p[z
(t)
] is the operator defined as
p[z
(t )
]=arg min
x
||x −[z
(t )
−τ ∇ f (z
(t )
)]||
2
2
s.t. TV(x) ≤ , (12)
where f (x) = 1/2||c
(t)
√
y − x||
2
2
, and ∇ f (z
(t)
) =
H
(z
(t)
−
c
(t)
√
y) represents its gradient at the point z
(t)
. τ is a step size,
which is often tuned manually. Different from the hand-tuning
manner for this parameter, we select it adaptively by solving the
following problem:
ˆ
τ =arg min
τ
{f [x −τ ∇ f (x)]}. (13)
Problem (13)can be recast as
ˆ
τ =arg min
τ
{
1
2
||c
(t )
√
y −[x − τ ∇ f (x)]||
2
2
}. (14)
Denoting the cost function in (14)as F (x) =
1
2
||c
(t)
√
y − [x −
τ ∇ f (x)]||
2
2
, and ignoring the constant in the optimization process,
the cost function F (x) can then be recast as
F (x) =
1
2
||∇ f (x)||
2
2
τ
2
−τ x −c
(t )
√
y, ∇ f (x), (15)
where x −c
(t)
√
y, ∇ f (x) =[∇ f (x)]
H
(x − c
(t)
√
y).
Therefore, the closed-form solution to (14)is
ˆ
τ =
x −c
(t )
√
y, ∇ f (x)
||∇ f (x)||
2
2
. (16)
At the tth iteration, we can calculate the step size τ
(t)
by setting
x = x
(t−1)
. In particular, if the matrix admits
H
= I, the step
size is a constant τ =1.
Letting
v
(t)
=z
(t)
−τ ∇ f (z
(t)
), problem (12)can be rewritten as
proj
TV
(v
(t )
) = arg min
x
{||x −v
(t )
||
2
2
} s.t. TV(x) ≤ (17)
where proj
TV
(v
(t)
) represents a projection operator. This projection
operator indicates the projection of the signal v
(t)
onto the TV-
based
constraint set. In general, the parameter is a fixed upper
bound on the TV function value of the signal, and it has to be
finely tuned. To avoid this tedious tuning process, we exploit the
epigraph concept of the TV function for solving problem (17). In
other words, the aim of using the epigraph concept is to avoid
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