Partial convolution for total variation deblurring and denoising by new linearized alternating direction method of multipliers with extension step

Yuan Shen; Lei Ji

doi:10.3934/jimo.2018037

Article Contents

2019, Volume 15, Issue 1: 159-175. Doi: 10.3934/jimo.2018037

This issue Previous Article Mechanism design in project procurement auctions with cost uncertainty and failure risk Next Article Improved particle swarm optimization and neighborhood field optimization by introducing the re-sampling step of particle filter

Partial convolution for total variation deblurring and denoising by new linearized alternating direction method of multipliers with extension step

Yuan Shen^, and
Lei Ji

School of Applied Mathematics, Nanjing University of Finance & Economics, Nanjing 210023, China

_* Corresponding author: Yuan Shen
_* Corresponding author: Yuan Shen

Received: May 2017

Revised: September 2017

Early access: April 2018

Published: January 2019

Research supported by National Natural Science Foundation of China under grant 11401295 and by National Natural Science Foundation Tianyuan Project under grant 11726618 and by Jiangsu Provincial Natural Science Foundation under grant BK20141007 and by Major Program of the National Social Science Foundation of China under Grant 12&ZD114 and by National Social Science Foundation of China under Grant 15BGL58 and by Jiangsu Provincial Social Science Foundation under Grant 14EUA001 and by Qinglan Project of Jiangsu Province

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Abstract

In this paper, we propose a partial convolution model for image deblurring and denoising. We also devise a new linearized alternating direction method of multipliers (ADMM) with an extension step. As the computation of its subproblem is simple enough to have closed-form solutions, its per-iteration cost is low; however, the relaxed parameter condition together with the extra extension step inspired by Ye and Yuan's ADMM enables faster convergence than the original linearized ADMM. Preliminary experimental results show that our algorithm can produce better quality results than some existing efficient algorithms within a similar computation time. The performance advantage of our algorithm is particularly evident at high noise ratios.

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Mathematics Subject Classification: Primary: 90C30, 65K05; Secondary: 94A08.

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Figure 1. From left to right are the results of FTVD, New algorithm, TVAL3, and SNR history plot. First row: $NR$ = 60%, $airplane$; second row: $NR$ = 80%, $lena$

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Figure 2. Performance v.s. noise ratio. Left: time; right: SNR. First row: $RC = 0.1$, $coins$; second row: $RC = 0$, $westconcordorthophoto$

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Figure 3. Performance v.s. indices matrix quality. Left: time, right: SNR. First row: $moon$, $NR = 60%$; second row: $tire$, $NR = 80%$

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Figure 4. From left to right to below are the results of FTVD, New algorithm, and TVAL3, and SNR history plot. First row: "average" kernel with dirichlet boundary condition; second row: "Gaussian" kernel; third row: "Disk" kernel; fourth row: "motion" kernel

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Table 1. Numerical results of basic test

	FTVD		New LADMM		TVAL3
Noise Ratio	time	SNR	time	SNR	time	SNR
60%	3.806	20.894	20.155	23.594	14.212	11.806
80%	8.174	8.751	19.687	15.226	26.442	7.106

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