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Proximal Comixture Minimization Models for Image Recovery and Data Analysis

Patrick L. Combettes, Diego J. Cornejo

TL;DR

This work proposes an alternative minimization model based on proximal comixtures, an operation which combines functions and linear operators in such a way that the proximity operator of the resulting function is computable explicitly in terms of the individual proximity and linear operators.

Abstract

In minimization models for image recovery and data analysis problems, loss functions and linear operators are typically aggregated as an average of composite terms. Each term in the aggregate models a desired property of the ideal solution arising from the \emph{a priori} knowledge and the observed data. We propose an alternative minimization model based on proximal comixtures, an operation which combines functions and linear operators in such a way that the proximity operator of the resulting function is computable explicitly in terms of the individual proximity and linear operators. The mathematical properties of this operation are analyzed and comparisons between proximal comixtures and standard composite averages are made. Numerical illustrations of the benefits of minimization models based on proximal comixtures are provided in the context of image recovery and machine learning applications.

Proximal Comixture Minimization Models for Image Recovery and Data Analysis

TL;DR

This work proposes an alternative minimization model based on proximal comixtures, an operation which combines functions and linear operators in such a way that the proximity operator of the resulting function is computable explicitly in terms of the individual proximity and linear operators.

Abstract

In minimization models for image recovery and data analysis problems, loss functions and linear operators are typically aggregated as an average of composite terms. Each term in the aggregate models a desired property of the ideal solution arising from the \emph{a priori} knowledge and the observed data. We propose an alternative minimization model based on proximal comixtures, an operation which combines functions and linear operators in such a way that the proximity operator of the resulting function is computable explicitly in terms of the individual proximity and linear operators. The mathematical properties of this operation are analyzed and comparisons between proximal comixtures and standard composite averages are made. Numerical illustrations of the benefits of minimization models based on proximal comixtures are provided in the context of image recovery and machine learning applications.
Paper Structure (9 sections, 10 theorems, 79 equations, 9 figures)

This paper contains 9 sections, 10 theorems, 79 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Notation and background
  3. Properties of proximal comixtures
  4. Algorithms
  5. Numerical experiments
  6. Experiment 1: Proximal total variation denoising
  7. Experiment 2: Multiview image reconstruction
  8. Experiment 3: Image reconstruction from phase
  9. Experiment 4: Linear regression

Key Result

Lemma 2.1

Let $\varphi\in\Gamma_0(\mathcal{H})$ and let $\gamma\in\intv[o]0{0}{{+}\infty}$. Then the following hold:

Figures (9)

  • Figure 1: (left): Standard composite average \ref{['e:cav2']}. (right): Proximal comixture \ref{['e:pcm2']} in terms of the Moreau envelopes of \ref{['e:m1']} and \ref{['e:m2']} using Proposition \ref{['p:0']}\ref{['p:0i']}.
  • Figure 2: (a) Original signal $\overline{x}$. (b) Noisy observation $z$. (c) Solution to Problem \ref{['p:ex4b']} for $\gamma=10^{-3}$ (the solution to Problem \ref{['p:ex4a']} is essentially identical).
  • Figure 3: (a) Original image $\overline{x}$. (b) Degraded image $z_1$. (c) Degraded image $z_2$.
  • Figure 4: Images restored by Problems \ref{['p:ex3a']} and \ref{['p:ex3c']}.
  • Figure 5: Normalized error $20\log_{10}(\|x_n-x_\infty\|/\|x_0-x_\infty\|)$ (dB) versus time (s).
  • ...and 4 more figures

Theorems & Definitions (23)

  • Lemma 2.1
  • proof
  • Lemma 2.2: Bern10
  • Lemma 2.3
  • proof
  • Remark 2.4
  • Definition 3.1: Svva23
  • Proposition 3.2
  • proof
  • Proposition 3.3
  • ...and 13 more