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Adjoint Method in PDE-based Image Compression

Zakaria Belhachmi, Thomas Jacumin

TL;DR

A shape optimization based method for finding the best interpolation data in the compression of images with noise and a numerical approach to determine the optimal set is proposed and numerical experiments showing the efficiency of this method are presented.

Abstract

We consider a shape optimization based method for finding the best interpolation data in the compression of images with noise. The aim is to reconstruct missing regions by means of minimizing a data fitting term in an $L^p$-norm between original images and their reconstructed counterparts using linear diffusion PDE-based inpainting. Reformulating the problem as a constrained optimization over sets (shapes), we derive the topological asymptotic expansion of the considered shape functionals with respect to the insertion of small ball (a single pixel) using the adjoint method. Based on the achieved distributed topological shape derivatives, we propose a numerical approach to determine the optimal set and present numerical experiments showing, the efficiency of our method. Numerical computations are presented that confirm the usefulness of our theoretical findings for PDE-based image compression.

Adjoint Method in PDE-based Image Compression

TL;DR

A shape optimization based method for finding the best interpolation data in the compression of images with noise and a numerical approach to determine the optimal set is proposed and numerical experiments showing the efficiency of this method are presented.

Abstract

We consider a shape optimization based method for finding the best interpolation data in the compression of images with noise. The aim is to reconstruct missing regions by means of minimizing a data fitting term in an -norm between original images and their reconstructed counterparts using linear diffusion PDE-based inpainting. Reformulating the problem as a constrained optimization over sets (shapes), we derive the topological asymptotic expansion of the considered shape functionals with respect to the insertion of small ball (a single pixel) using the adjoint method. Based on the achieved distributed topological shape derivatives, we propose a numerical approach to determine the optimal set and present numerical experiments showing, the efficiency of our method. Numerical computations are presented that confirm the usefulness of our theoretical findings for PDE-based image compression.
Paper Structure (9 sections, 12 theorems, 97 equations, 17 figures, 6 tables)

This paper contains 9 sections, 12 theorems, 97 equations, 17 figures, 6 tables.

Table of Contents

  1. Problem Formulation
  2. Adjoint Method and variations of the cost functional
  3. The Adjoint Problem and Related Estimates
  4. Variations of the Bilinear Form
  5. Variations of the Linear Form
  6. Computation of the topological derivative
  7. Numerical Results
  8. Salt and Pepper Noise
  9. Gaussian Noise

Key Result

Theorem 2.1

Let $v_\varepsilon \in V$ be the solution of the following problem : find $v\in V$ such that, Let $w_0$ be the solution of the so-called adjoint problem : find $w\in V$ such that Then,

Figures (17)

  • Figure 1: Illustration of the splitting.
  • Figure 2: Input images.
  • Figure 3: Masks and reconstructions from image without noise and with $10\%$ of total pixels saved.
  • Figure 4: Masks and reconstructions from image with $2\%$ of salt noise and with $10\%$ of total pixels saved.
  • Figure 5: Masks and reconstructions from image with $2\%$ of pepper noise and with $10\%$ of total pixels saved.
  • ...and 12 more figures

Theorems & Definitions (21)

  • Theorem 2.1
  • Proposition 2.1
  • proof
  • Proposition 2.2
  • proof
  • Remark
  • Proposition 2.3
  • proof
  • Proposition 2.4
  • Proposition 2.5
  • ...and 11 more