ADMM based Fourier phase retrieval with untrained generative prior Liyuan MaHongxia WangNingyi LengZiyang Yuan

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ADMM based Fourier phase retrieval with untrained generative

prior

Liyuan Ma ∗Hongxia Wang †Ningyi Leng ‡Ziyang Yuan §

Abstract

Fourier phase retrieval (FPR) is an inverse problem that recovers the signal from its Fourier

magnitude measurement, it’s ill-posed especially when the sampling rates are low. In this paper,

an untrained generative prior is introduced to attack the ill-posedness. Based on the alternating

direction method of multipliers (ADMM), an algorithm utilizing the untrained generative network

called Net-ADM is proposed to solve the FPR problem. Firstly, the objective function is smoothed

and the dimension of the variable is raised to facilitate calculation. Then an untrained generative

network is embedded in the iterative process of ADMM to project an estimated signal into the

generative space, and the projected signal is applied to next iteration of ADMM. We theoretically

analyzed the two projections included in the algorithm, one makes the objective function descent,

and the other gets the estimation closer to the optimal solution. Numerical experiments show that

the reconstruction performance and robustness of the proposed algorithm are superior to prior

works, especially when the sampling rates are low.

Keywords: Fourier phase retrieval, untrained generative prior, alternating direction method

of multipliers

1 Introduction

Fourier phase retrieval (FPR) seeks to recover an unknown signal x∈Rnfrom its Fourier magnitude

measurement b=|Fx|+η∈Rm(m>n), where ηdenotes additive noise. It arises in various applica-

tions, such as X-ray crystallography [1;2], ptychography [3], diﬀraction imaging [4] and astronomical

imaging [5]. It is in general ill-posed since there are many diﬀerent signals which have the same Fourier

magnitude [6]. For x∈Rn, a theoretical work [7] indicates that the uniqueness of the solution can

be guaranteed when m≥2n−1 and every subset of nmeasurement vectors spans Rn. Here we use

sampling rate r=m/n to describe the ratio of measurement length to signal length in each dimension.

The signal is diﬃcult to be reconstructed when the sampling rate is low generally.

The models of addressing phase retrieval (PR) problems can be divided into convex models and

non-convex models. Convex models relax the non-convex problem into convex one, such as PhaseLift

[8], PhaseMax [9], etc. We consider the non-convex quadratic model min

2mkb− |Fx|k2because

that extensive numerical and experimental validation conﬁrms that the magnitude-based cost function

performs signiﬁcantly better than the intensity-based one [10;11]. Based on the model, it’s common

to add some prior information about the signal xto relieve the ill-posedness, i.e.

min

x∈S

2mkb− |Fx|k2,(1)

where Sis a set possibly constrained by priors. The common prior information is support constraint

[12], which considers that the signal outside boundaries is zero. Sparse prior is also used for FPR to

recover the desired signal using a minimal number of measurements [13], the signal has few non-zero

coeﬃcients in some basis under this assumption.

∗College of Science, National University of Defense Technology, Changsha, Hunan, 410073, P.R.China. Email:

maliyuan21@nudt.edu.cn

†College of Science, National University of Defense Technology, Changsha, Hunan, 410073, P.R.China. Corresponding

author. Email: wanghongxia@nudt.edu.cn

‡College of Science, National University of Defense Technology, Changsha, Hunan, 410073, P.R.China. Email:

lengningyi14@nudt.edu.cn

§Academy of Military Science of People’s Liberation Army, Beijing, P.R.China. Email: yuanziyang11@nudt.edu.cn

arXiv:2210.12646v1 [eess.SP] 23 Oct 2022

Recently, deep generative priors [14–18] which assume the signal is within the range of a generative

network are proposed to replace hand-crafted priors. Among them, trained generative networks [14–16]

build priors by learning the distribution information of signal from massive amounts of training data,

which achieve superior results. However, in many scenarios we can’t obtain suﬃcient training data, and

there may even be cases where the reconstruction performance of network is poor due to inconsistent

data distribution, so it’s necessary to introduce untrained generative priors. The untrained generative

networks such as Deep Image Prior (DIP) [19] and its variant Deep Decoder [20] are proposed, they

don’t rely on any external data to learn the distribution information of the signal, because their

structure can capture the low-level statistical priors of images implicitly [19].

Although a variety of excellent PR models have been proposed, designing eﬃcient algorithms to

solve these models is still challenging. Algorithms to solve PR problem can be divided into classical

algorithms and learning-based algorithms. The earliest classical methods are based on alternating

projections, such as Gerchberg-Saxton (GS) [21] and its variant, hybrid-input-output (HIO) [22]. The

algorithms based on gradient descent are proposed later, such as Wirtinger Flow (WF) [23], Truncated

Wirtinger Flow (TWF) [24] and related variants [25]. In addition, there are also algorithms based on

second-order derivatives [26], which are faster than WF. Most of the classical algorithms are based on

the above algorithms or their variants.

With the development of neural networks, learning-based algorithms gain a lot of momentum.

There are some algorithms which take deep neural networks as denoisers in the iterative process of

PR [27;28], and other algorithms utilize deep neural networks to learn a mapping that reconstructs

the images from measurements [29;30]. Deep generative networks as image priors have been recently

introduced into PR problem. There are some algorithms [14–16] that use trained or pre-trained

generative networks to solve the Fourier or Gaussian PR problems (which replaces Fourier matrix

Fwith a Gaussian random matrix). However, it’s inevitable that their reconstruction performance

depends on massive amounts of training data.

The algorithms [17;31] based on untrained generative priors named Net-GD and Net-PGD have

been proposed recently and applied to Gaussian PR, they adopt gradient descent and projected gradient

descent respectively. Even though there have been many researches about Gaussian PR, FPR is more

challenging [32] and applicable in a wide range of scenarios. When Net-GD and Net-PGD are applied to

FPR problem, their reconstruction performance and robustness are not ideal, even when the sampling

rates are high. In summary, the application of untrained generative networks to FPR problems remains

to be further explored, including theoretical guarantees and algorithms that can be adopted at low

sampling rates.

This paper proposes an algorithm named Net-ADM that combines the alternating direction method

of multipliers (ADMM) with untrained generative network to tackle FPR problem. Speciﬁcally, we

smooth the amplitude-based loss function (1) and improve the dimension of the variable, so that

the objective function is Lipschitz diﬀerential and the Fourier transform can be calculated by fast

Fourier transform (FFT) [33]. Then the FPR problem is modeled as an constrained optimization

problem and ADMM is adopted to solve the optimization problem alternately. Additionally, the

untrained generative network is embedded into the iterative process to project an estimated signal of

the ADMM into the generative space that captures some image characteristics, the projected signal can

be applied to next iteration of ADMM. We theoretically analyze the two projections included in Net-

ADM algorithm, one makes the objective function descent, and the other makes the estimation closer

to the optimal solution, these good properties may be the reason for the gain compared to ADMM.

Numerical experiments show that the reconstruction performance and robustness of Net-ADM are

superior as compared to prior works, especially when the sampling rates are low.

The remainder of the paper is organized as follows. In section 2, we establish the FPR model based

on untrained generative prior, describe the Net-ADM algorithm and give relevant theoretical analysis.

Section 3describes the experimental settings, conducts numerical experiments under diﬀerent sampling

rates and noise levels, the eﬀect of parameters in the algorithm is discussed ﬁnally. In section 4, some

conclusions are drawn and new ideas are proposed for future work.

In this paper, the bold lowercase letters or numbers represent vectors, calligraphic uppercase letters

denote operators and the uppercase letters on the blackboard bold represent spaces or sets. √·,| · |

and ÷are element-wised operators. k·kis the Euclidean norm of a vector or the spectral norm of

a matrix. Hadamard product is represented by . Vectorization of a matrix is written as vec(·).

(·)[n+1:m]denotes a sub-vector constructed by the last m−nelements.

2 Net-ADM

2.1 FPR Model based on untrained generative prior

The untrained generative network incorporates implicit priors about the signals they generate, and

these prior assumptions are built into the network structure. Thus we expand the components of the

network in detail and analyze the prior information that may be carried by its structure.

Under the untrained generative prior, Sin (1) represents the range of a network which is denoted by

G(w;z). The input z∈Rd(dn) of network represents the low-dimensional latent code, its elements

are ﬁxed and generated from uniform random distribution; wrepresents the weights that need to be

learned but not pre-trained. This paper draws on the network structure in [31], which exchanges the

order of upsampling with a composition of activation function and channel normalization compared to

deep decoder [20].

For a J-layer network, we denote the input of j+ 1-th layer as Zj∈Rcj×dj(j= 0,1,··· , J −1)

and ZJ∈RcJ×dJrepresents the output of J-th layer, in addition, z=vec(Z0) and d=c0×d0.

The network consists of 1 ×1 convolutions’ weights Wj∈Rcj+1×cj, ReLU activation function relu(·),

channel normalization operator cn(·) and bi-linear upsampling operators Uj. The operators from the

last layer to the output includes the 1 ×1 convolutions’ weight WJ∈Rcout ×cJand sigmoid activation

function sig(·), where cout = 1 for a grayscale image and cout = 3 for an RGB image. Thus, the

network at layer j+ 1 can be expressed as

Zj+1 =Ujcn (relu (WjZj)) , j = 0,1,··· , J −1.(2)

The output of network is G(w;z) = sig (WJZJ), where w={W0, W1,··· , WJ}are weights to learn.

It’s apparent that the manually adjustable parameters of the network are network depth and number

of weight channels in each layer. We can use {c0, c1,··· , cJ}to represent the designed structure for a

J-layer network. Additionally, simplifying (2) gives

G(w;z)=(σJ◦ ZJ◦σJ−1◦ ZJ−1◦ ··· ◦ σ0◦ Z0) (w),(3)

where Zj(j= 0,1,··· , J) denote the linear operators in the network because the 1 ×1 convolution

which is the operation between Zjand wcan be viewed as a linear combination of their channels,

σj=Uj◦cn ◦relu (j= 0,1,··· , J −1) denote the composite operators of ReLU activation function,

channel normalization operator and bi-linear upsampling, σJ=sig denotes the sigmoid activation

function. These operators are ﬁxed, thus the network can be regarded as a function of w.

Remark 1. Since the input Z0∈Rc0×d0is randomly generated from the [0, 0.1] uniform distribution,

it spans Rc0×d0with probability 1. The structure of the network restricts the representation space of

G(w;z)to a certain range S, which captures some characteristics of images. For example, the bi-linear

upsampling operators Uj(j= 0,1,··· , J −1) reﬂect the correlation of adjacent pixels in an image.

Speciﬁcally, which image in the space Sis represented by the network is determined by w.

Our goal is to ﬁnd optimal weights w∗to represent the optimal solution x∗, which is equivalent to

substituting the surjective mapping [31]G:w→xand minimizing the loss function (4) over w,

min

w∈Wψ(G(w;z)), where ψ(x) = 1

2mkb− |Fx|k2.(4)

where Wdenotes the set of weights. Inspired by [34], we can obtain the existence of the solution of

model (4).

Theorem 1. For the network (3), if the set of weights Wis a bounded weakly closed subset in reﬂexive

Banach space, there at least exists a solution w∗of model (4).

Proof 1. Since the form of (4) and (3) is similar to (1.2) and (2.1) of [34] respectively, they satisfy the

conditions similar to (A1) of Condition 2.2, which guarantees the weakly lower semicontinuity of the

objective function F(w) = ψ(G(w;z)). Speciﬁcally, since the elements in zbelong to [0,0.1] generally,

Zjare bounded linear when Wis bounded. In addition, σjare weakly continuous, and the function

ψ(·) = 1

2mkb− |F(·)|k2is weakly lower semi-continuous. Thus, the objective function F(w)is weakly

lower semi-continuous. For w∈W, when Wis a bounded weakly closed subset in reﬂexive Banach

space, F(w)is able to reach inﬁmum on W, thus there at least exits a solution w∗of model (4).

Note that the conclusion in [34] assumes all free parameters in F(w) are trained before minimization

of (4), it is also satisﬁed here since the untrained generative network (2) does not require to be pre-

trained and all operators are ﬁxed. Furthermore, theorem 1 requires that the set of weights Wis a

closed set. We can use strategies such as weight decay and so on in the learning process of network to

ensure that the condition is satisﬁed.

For the convenience of algorithm implementation, we rewrite the model (4) as the following opti-

mization model by using the intermediate variable x.

min

x,wψ(x),

s.t. x−G(w;z)=0.

(5)

2.2 Algorithm design

In this section, we will design an algorithm to solve (5), which combines the ADMM with untrained

generative network, thus it’s called Net-ADM method in this paper. Firstly, in order to execute Fourier

transform by FFT algorithm, we introduce an auxiliary variable u∈Rm(m>n) whose last m−n

elements are zeros u[n+1:m]= 0 such that Frepresents FFT operator. Secondly, in order to make

the objective function ψ(·) Lipschitz diﬀerential, inspired by Chang [35], we add a penalty ε1on both

sides of b=|Fu|, then use radical smoothing, the loss function ψ(x) can be rewritten as follows:

f(u) = 1

2mkpb2+ε1−q|Fu|2+ε1k2,(6)

where ε > 0 represents penalization parameter, 1∈Rmrepresents a vector whose elements are all

ones. The gradient of f(u) is shown in (7), its continuity is proved in Lemma 1.

∇f(u) = u− F−1 √b2+ε1

p|F(u)|2+ε1 F(u)!.(7)

Lemma 1. The function f(u)is gradient Lipschitz continuous,

k∇f(u2)− ∇f(u1)k ≤ Lku2−u1k,∀u1,u2∈Rm(8)

where L= 1 + 2

√εk√b2+ε1k∞.

Proof 2.

k∇f(u2)− ∇f(u1)k

=ku2− F−1 √b2+ε1

p|Fu2|2+ε1 Fu2!−u1+F−1 √b2+ε1

p|Fu1|2+ε1 Fu1!k

≤ ku2−u1k+kF−1 √b2+ε1

p|Fu2|2+ε1 Fu2−√b2+ε1

p|Fu1|2+ε1 Fu1!k

=ku2−u1k+1

√mk√b2+ε1

p|Fu2|2+ε1 Fu2−√b2+ε1

p|Fu1|2+ε1 Fu1k(by Parseval’s theorem)

≤ ku2−u1k+1

√mk√b2+ε1

p|Fu2|2+ε1(Fu2− Fu1)k

√mk √b2+ε1

p|Fu2|2+ε1−√b2+ε1

p|Fu1|2+ε1! Fu1k

≤ ku2−u1k+2

√εkpb2+ε1k∞ku2−u1k

(9)

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ADMMbasedFourierphaseretrievalwithuntrainedgenerativepriorLiyuanMa*HongxiaWangNingyiLengZiyangYuan§AbstractFourierphaseretrieval(FPR)isaninverseproblemthatrecoversthesignalfromitsFouriermagnitudemeasurement,it'sill-posedespeciallywhenthesamplingratesarelow.Inthispaper,anuntrainedgenerativepriorisi...

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ADMM based Fourier phase retrieval with untrained generative prior Liyuan MaHongxia WangNingyi LengZiyang Yuan

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