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Whiteness-based bilevel estimation of weighted TV parameter maps for image denoising

Monica Pragliola, Luca Calatroni, Alessandro Lanza

TL;DR

This work considers a bilevel optimisation strategy based on normalised residual whiteness loss for estimating the weighted total variation parameter maps for denoising images corrupted by additive white Gaussian noise that is fully unsupervised.

Abstract

We consider a bilevel optimisation strategy based on normalised residual whiteness loss for estimating the weighted total variation parameter maps for denoising images corrupted by additive white Gaussian noise. Compared to supervised and semi-supervised approaches relying on prior knowledge of (approximate) reference data and/or information on the noise magnitude, the proposal is fully unsupervised. To avoid noise overfitting an early stopping strategy is used, relying on simple statistics of optimal performances on a set of natural images. Numerical results comparing the supervised/unsupervised procedures for scalar/pixel-dependent \mbox{parameter maps are shown.

Whiteness-based bilevel estimation of weighted TV parameter maps for image denoising

TL;DR

This work considers a bilevel optimisation strategy based on normalised residual whiteness loss for estimating the weighted total variation parameter maps for denoising images corrupted by additive white Gaussian noise that is fully unsupervised.

Abstract

We consider a bilevel optimisation strategy based on normalised residual whiteness loss for estimating the weighted total variation parameter maps for denoising images corrupted by additive white Gaussian noise. Compared to supervised and semi-supervised approaches relying on prior knowledge of (approximate) reference data and/or information on the noise magnitude, the proposal is fully unsupervised. To avoid noise overfitting an early stopping strategy is used, relying on simple statistics of optimal performances on a set of natural images. Numerical results comparing the supervised/unsupervised procedures for scalar/pixel-dependent \mbox{parameter maps are shown.

Paper Structure

This paper contains 8 sections, 1 theorem, 13 equations, 7 figures, 2 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Bilevel estimation of WTV parameter maps
  3. Optimisation details
  4. Strongly-convex accelerated gradient descent lower-level solver
  5. Gradient-descent upper-level solver
  6. Early stopping
  7. Numerical examples
  8. Conclusions

Key Result

Proposition 1.1

The functional $F_{\epsilon}$ defined in eq:smooth_Feps is twice continuously differentiable and $1$-strongly convex on ${\mathbb R}^n$, hence it admits a unique minimiser. Its gradient $\boldsymbol{\nabla}_{\mathbf{x}} F_{\epsilon} \in {\mathbb R}^n$ and Hessian $\boldsymbol{\nabla}^2_{\mathbf{x}} where $\boldsymbol{\nabla} H_{\epsilon}$ and $\boldsymbol{\nabla}^2 H_{\epsilon}$ denote the vect

Figures (7)

  • Figure 1: Behavior of $\mathcal{Q}$, IPSNR and ISSIM values (top), and of the reconstructions and $\bm{\beta}$-maps (bottom) along the iterations of Alg. \ref{['alg:GD_bil']} when $\mathcal{Q}=\mathcal{Q}_{\text{white}}$.
  • Figure 2: Original test images of size $180\times 180$ from the BSD400 repository.
  • Figure 3: Output reconstructions for test image #1.
  • Figure 4: Output reconstructions for test image #2.
  • Figure 5: Output reconstructions for test image #3.
  • ...and 2 more figures

Theorems & Definitions (1)

  • Proposition 1.1