DOLPH Diffusion Models for Phase Retrieval Shirin Shoushtariy Jiaming Liuy and Ulugbek S. Kamilov

2025-08-18 2 0 3.56MB 9 页 10玖币

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DOLPH: Diffusion Models for Phase Retrieval

Shirin Shoushtari†, Jiaming Liu†, and Ulugbek S. Kamilov*

Computational Imaging Group (CIG), Washington University in St. Louis, MO, USA

†These authors contributed equally.

Abstract

Phase retrieval refers to the problem of recovering an image from the magnitudes of its

complex-valued linear measurements. Since the problem is ill-posed, the recovery requires

prior knowledge on the unknown image. We present DOLPH as a new deep model-based

architecture for phase retrieval that integrates an image prior speciﬁed using a diffusion model

with a nonconvex data-ﬁdelity term for phase retrieval. Diffusion models are a recent class of

deep generative models that are relatively easy to train due to their implementation as image

denoisers. DOLPH reconstructs high-quality solutions by alternating data-consistency updates

with the sampling step of a diffusion model. Our numerical results show the robustness of DOLPH

to noise and its ability to generate several candidate solutions given a set of measurements.

1 Introduction

Phase retrieval (PR) refers to the problem of recovering phase information from noisy amplitude-only

measurements. In the context of computational imaging, it is often formulated as the recovery of an

unknown image xfrom the measurements

y=|Ax|+e,(1)

where

A∈Cm×n

is the measurement matrix,

|·|

is the element-wise absolute value, and

is the noise.

The popularity of PR stems from its broad applicability in microscopy [1], optics [2], astronomical

imaging [3], and inverse scattering [4]. PR is known to be challenging due to the nonlinearity of

the measurements and ill-posedness of the corresponding inverse problem, necessitating algorithms

that can efﬁciently integrate image priors. The literature on PR is vast (see the review [5]) with a

large number of existing methods [6

–

12]. Nonetheless, there is a strong interest in the development

of PR methods that can use modern deep learning (DL) priors.

The focus of this paper is to design and validate a PR algorithm that can leverage an image prior

speciﬁed by a diffusion model. Diffusion models are a recent class of DL methods for generating

high-quality images using pre-trained image denoisers [13, 14]. The image denoiser in diffusion

models can be interpreted as the gradient of the log of the image probability density function,

leading to its view as a compact representation of an image distribution. This interpretation has

led to the use of diffusion models as image priors in various imaging inverse problems [15

–

18]. It

*This material is based upon work supported by the NSF CAREER award under grant CCF-2043134.

arXiv:2211.00529v2 [eess.IV] 2 Nov 2022

Sampling from DDPM Gradient Correction Step

(a)

(b)

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Figure 1: (a) Illustration of the denoising diffusion probabilistic model (DDPM) Markov chain that

relates the distribution of high-quality images with that of the Gaussian noise. (b) DOLPH uses a

pre-trained DDPM for recovering images from amplitude measurements. DOLPH can be viewed as a

modiﬁed reverse diffusion process that has a gradient correction step to ensure consistency with the

measured amplitudes.

is also worth noting the connection between use of diffusion models in inverse problems and the

plug-and-play priors (PnP) framework, which also uses image denoisers as image priors [19–21].

In this work, we present

iffusion m

s for

ase retrieval (DOLPH) as a new method for using

diffusion models as image priors for solving PR problems of form

(1)

. DOLPH combines the

sampling procedure in diffusion models with the data-consistency updates ensuring that the predicted

amplitudes match the measured ones in

. Our numerical results on phase retrieval from coded

diffraction patters (CDP) [22] show that DOLPH can generate realistic looking images from severely

noisy measurements, where the traditional approaches lead to overly smooth images.

2 Proposed Method

DOLPH builds on the sampling process of diffusion models to make it applicable to the problem

in eq.

(1)

. The sampling process uses a denoising convolutional neural network (CNN) pre-

trained using the denoising diffusion probabilistic models (DDPM) framework [13, 15]. DOLPH then

uses the pre-trained CNN denoiser to generate samples consistent with a given set of amplitude

measurements.

2.1 Denoising diffusion probabilistic models

As illustrated in Figure 1(a), DDPM consists of two Markov processes: the ﬁxed forward process and

the learning-based reverse process. The forward process consists of

T≥1

steps, where each step

adds a Gaussian noise with a pre-designated variance, so that the at step

the statistical distribution

of data corresponds to the standard Gaussian distribution. Each step of the forward process can be

expressed as

q(xt|xt−1):=N(xt;p1−βtxt−1, βtI),(2)

where

N(x;µ,Σ)

denotes a Gaussian probability density function with the mean vector

and the

covariance matrix

. The vectors

x1,··· ,xT

are the latent variables that have the same dimensions

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DOLPH:DiffusionModelsforPhaseRetrievalShirinShoushtariy,JiamingLiuy,andUlugbekS.Kamilov*ComputationalImagingGroup(CIG),WashingtonUniversityinSt.Louis,MO,USAyTheseauthorscontributedequally.AbstractPhaseretrievalreferstotheproblemofrecoveringanimagefromthemagnitudesofitscomplex-valuedlinearmeasurement...

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DOLPH Diffusion Models for Phase Retrieval Shirin Shoushtariy Jiaming Liuy and Ulugbek S. Kamilov

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