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Sparse Dictionary Learning for Image Recovery by Iterative Shrinkage

Shima Shabani, Mohammadsadegh Khoshghiaferezaee, Michael Breuß

TL;DR

This work studies image recovery via online sparse dictionary learning (SDL) by comparing several shrinkage-based sparse solvers within a two-stage online SDL framework: sparse coding and dictionary updating. It demonstrates how the learned dictionary grows with more training data and examines reconstruction quality and computational efficiency across solvers, including $FISTA$, $FPC{-}BB$, $TwIST$, $SpaRSA$, $GSCG$, and $ISGA$, while enforcing an overcomplete dictionary $D\in\mathbb{R}^{m\times n}$ with $m\gg n$ and sparse codes $x$ under $\|x\|_0$ or $\|x\|_1$ penalties. The experiments on patch-based, grayscale images show that solver choice interacts with data availability: ISGA is typically fastest and competitive in accuracy, while TwIST can achieve rapid error reduction but may not benefit from larger training sets; FISTA and FPC-BB offer comparable accuracy to ISGA in some cases. These findings provide practical guidance for selecting sparse-coding solvers in online SDL for image recovery and highlight the impact of training data size on reconstruction performance.

Abstract

In this paper we study the sparse coding problem in the context of sparse dictionary learning for image recovery. To this end, we consider and compare several state-of-the-art sparse optimization methods constructed using the shrinkage operation. As the mathematical setting of these methods, we consider an online approach as algorithmical basis together with the basis pursuit denoising problem that arises by the convex optimization approach to the dictionary learning problem. By a dedicated construction of datasets and corresponding dictionaries, we study the effect of enlarging the underlying learning database on reconstruction quality making use of several error measures. Our study illuminates that the choice of the optimization method may be practically important in the context of availability of training data. In the context of different settings for training data as may be considered part of our study, we illuminate the computational efficiency of the assessed optimization methods.

Sparse Dictionary Learning for Image Recovery by Iterative Shrinkage

TL;DR

This work studies image recovery via online sparse dictionary learning (SDL) by comparing several shrinkage-based sparse solvers within a two-stage online SDL framework: sparse coding and dictionary updating. It demonstrates how the learned dictionary grows with more training data and examines reconstruction quality and computational efficiency across solvers, including , , , , , and , while enforcing an overcomplete dictionary with and sparse codes under or penalties. The experiments on patch-based, grayscale images show that solver choice interacts with data availability: ISGA is typically fastest and competitive in accuracy, while TwIST can achieve rapid error reduction but may not benefit from larger training sets; FISTA and FPC-BB offer comparable accuracy to ISGA in some cases. These findings provide practical guidance for selecting sparse-coding solvers in online SDL for image recovery and highlight the impact of training data size on reconstruction performance.

Abstract

In this paper we study the sparse coding problem in the context of sparse dictionary learning for image recovery. To this end, we consider and compare several state-of-the-art sparse optimization methods constructed using the shrinkage operation. As the mathematical setting of these methods, we consider an online approach as algorithmical basis together with the basis pursuit denoising problem that arises by the convex optimization approach to the dictionary learning problem. By a dedicated construction of datasets and corresponding dictionaries, we study the effect of enlarging the underlying learning database on reconstruction quality making use of several error measures. Our study illuminates that the choice of the optimization method may be practically important in the context of availability of training data. In the context of different settings for training data as may be considered part of our study, we illuminate the computational efficiency of the assessed optimization methods.

Paper Structure

This paper contains 23 sections, 2 theorems, 23 equations, 5 figures, 2 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Related Work
  3. Our Contribution
  4. Sparse Dictionary Learning
  5. Review of a Standard Sparse Recovery Method
  6. Review of ISTA Acceleration Methods
  7. FPC-BB:
  8. TwIST:
  9. FISTA:
  10. SpaRSA:
  11. GSCG:
  12. ISGA:
  13. Dictionary Learning Algorithm
  14. Sparse Coding Stage
  15. Dictionary Updating Stage
  16. ...and 8 more sections

Key Result

corollary thmcountercorollary

$x^*$ is a minimum for e:defprob1 if and only if

Figures (5)

  • Figure 1: Training and evaluation images of the Kaggle dataset.
  • Figure 2: Partial visualization of the final learned dictionaries with FISTA, FPC-BB, TwIST, GSCG, and ISGA.
  • Figure 3: Schematic grid to show patch averaging
  • Figure 4: Diagram of absolute reconstruction errors versus learned dictionaries (left column) and relative reconstruction errors versus learned dictionaries (right column) for FISTA, ISGA, and TwIST. Note that we decided to show here the natural (not unified) scales of results since we would like to emphasize the general behaviour of errors as they develop when expanding the dictionaries.
  • Figure 5: Diagram of absolute reconstruction errors versus learned dictionaries (left column) and relative reconstruction errors versus learned dictionaries (right column) for FPC-BB and GSCG. Note that we decided to show here the natural (not unified) scales of results since we would like to emphasize the general behaviour of errors as they develop when expanding the dictionaries.

Theorems & Definitions (4)

  • corollary thmcountercorollary
  • proof
  • proposition thmcounterproposition
  • proof