A Comparison Study on Low Rank Matrix Completion Algorithms in Image and Painting Restoration Público

Abstract

Low rank matrix completion solves prevalent real life problems including image restoration, movie recommendation and photo depth enhancement. However, there is not a systematic comparison and evaluation for different low rank matrix completion algorithms. This thesis aims to focus on painting restoration and analyze the run- ning time and relative Frobenius norm for three different algorithm: Singular Value Thresholding (SVT), Iteratively Reweighted Least Squares (IRLS) and Low-rank Ma- trix Fit (LMaFit). Using custom testing images generated from a publicly available dataset of contemporary abstract paintings, it is observed that the LMaFit algorithm has the fastest running time but the worst accuracy. SVT has similar accuracy as IRLS but has faster running time. However, IRLS has more options for parameter tuning than SVT, especially the rank of desired matrix. So each algorithm has its own benefits and drawbacks in terms of image restoration or denoising. 

Table of Contents

1 Introduction..........................1

1.1 Introduction................................ 1 1.2 Problem Setup .............................. 3 1.3 Dataset .................................. 5

2 Singular Value Thresholding..........................8

2.1 Choosing the Threshold ......................... 9 2.2 Choosing the Step Size and Stopping Criteria . . . . . . . . . . . . . 10 2.3 Algorithm................................. 11 2.4 Numerical Experiments.......................... 12

3 Iteratively Reweighted Least Squares..........................15

3.1 Least Squares ............................... 17 3.2 Choosing Threshold............................ 18 3.3 Algorithm................................. 19 3.4 Numerical Experiments.......................... 21

4 Low Rank Matrix Fitting..........................23

4.1 Iterative Steps............................... 24 4.2 Algorithm................................. 25 4.3 Numerical Experiments.......................... 25

5 Comparison Across Algorithms..........................29

5.1 Problem of Negative Numbers............................... 31 5.2 Algorithm................................. 34 5.3 Modified LMaFit Result.......................... 34

6 Conclusions..........................37

Bibliography..........................40

About this Honors Thesis

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School	Emory College
Department	Mathematics
Degree	B.S.
Submission	Honors Thesis
Language	English
Research Field	Applied Mathematics
Palavra-chave	Image Denoising Matrix Completion Low Rank Matrix Completion Singular Value Thresholding Low-rank Matrix Fit Iteratively Reweighted Least Squared
Committee Chair / Thesis Advisor	James Nagy, Emory University
Committee Members	Joyce Ho, Emory University Li Xiong, Emory University

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