A modified Chambolle-Pock primal-dual algorithm for Poisson noise removal
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A modified Chambolle‑Pock primal‑dual algorithm for Poisson noise removal Benxin Zhang1 · Zhibin Zhu2 · Zhijun Luo1 Received: 1 February 2020 / Revised: 4 June 2020 / Accepted: 20 July 2020 © Istituto di Informatica e Telematica (IIT) 2020
Abstract In this paper, we study the Poisson noise removal problem with total variation regularization term. Using the dual formulation of total variation and Lagrange dual, we formulate the problem as a new constrained minimax problem. Then, a modified Chambolle-Pock first-order primal-dual algorithm is developed to compute the saddle point of the minimax problem. The main idea of this paper is using different step size for different primal (dual) variables updating. Moreover, the convergence of the proposed method is also established under mild conditions. Numerical comparisons between new approach and several state-of-the-art algorithms are shown to demonstrate the effectiveness of the new algorithm. Keywords Poisson noise · Total variation · Minimax problem · Primal-dual method Mathematics Subject Classification 65K10 · 90C25
1 Introduction Restoring images corrupted by Poisson noise is an important task in various applications, such as fluorescence microscopy [1], astronomical imaging [2], X-ray computed tomography, positron emission tomography, and single particle This work is supported by the National Natural Science Foundation of China (11901137, 61967004, 11961010, 11961011), Guangxi Natural Science Foundation (2018GXNSFBA281023), Guangxi Key Laboratory of Automatic Detecting Technology and Instruments (YQ20113), and Guangxi Key Laboratory of Cryptography and Information Security (GCIS201927). * Zhibin Zhu [email protected] 1
School of Electronic Engineering and Automation, Guangxi Key Laboratory of Automatic Detecting Technology and Instruments, Guilin University of Electronic Technology, Guilin 541004, People’s Republic of China
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School of Mathematics and Computing Science, Guangxi Colleges and Universities Key Laboratory of Data Analysis and Computation, Guilin University of Electronic Technology, Guilin 541004, People’s Republic of China
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emission computed tomography [3]. Thus, it is a very active research area in image processing. In this paper, we focus on the following Poisson image reconstruction model:
g = Poisson(Hf + b𝟏),
(1)
where f ∈ Rn is a column vector concatenated from the original image with size m1 × m2 ( n = m1 × m2 ), g ∈ Rn is the observed image, 𝟏 ∈ Rn is a vector of all ones, b ≥ 0 is a fixed background, H ∈ Rn×n may model a convolution or some other linear operator, such as spatial-invariant blur matrix or emission tomography, and Poisson(⋅ ) denotes the degradation by Poisson noise. A classical approach to deal with the degrade image is the maximum a posterior (MAP) estimate. Assume that the observed data g is independent random variable and f is also a random vector. The MAP estimate f can be obtained by maximizing the conditional a posteriori probability p(f|g
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