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Whiteness-based bilevel learning of regularization parameters in imaging

Carlo Santambrogio, Monica Pragliola, Alessandro Lanza, Marco Donatelli, Luca Calatroni

Abstract

We consider an unsupervised bilevel optimization strategy for learning regularization parameters in the context of imaging inverse problems in the presence of additive white Gaussian noise. Compared to supervised and semi-supervised metrics relying either on the prior knowledge of reference data and/or on some (partial) knowledge on the noise statistics, the proposed approach optimizes the whiteness of the residual between the observed data and the observation model with no need of ground-truth data.We validate the approach on standard Total Variation-regularized image deconvolution problems which show that the proposed quality metric provides estimates close to the mean-square error oracle and to discrepancy-based principles.

Whiteness-based bilevel learning of regularization parameters in imaging

Abstract

We consider an unsupervised bilevel optimization strategy for learning regularization parameters in the context of imaging inverse problems in the presence of additive white Gaussian noise. Compared to supervised and semi-supervised metrics relying either on the prior knowledge of reference data and/or on some (partial) knowledge on the noise statistics, the proposed approach optimizes the whiteness of the residual between the observed data and the observation model with no need of ground-truth data.We validate the approach on standard Total Variation-regularized image deconvolution problems which show that the proposed quality metric provides estimates close to the mean-square error oracle and to discrepancy-based principles.
Paper Structure (9 sections, 1 theorem, 23 equations, 3 figures, 1 table, 2 algorithms)

This paper contains 9 sections, 1 theorem, 23 equations, 3 figures, 1 table, 2 algorithms.

Table of Contents

  1. Introduction
  2. Methods
  3. Hyper-parameter bilevel learning
  4. Residual whiteness principle
  5. Optimization algorithms
  6. Accelerated gradient-descent lower-level solver
  7. Gauss-Newton upper-level solver
  8. Experimental results
  9. Conclusions

Key Result

Proposition 3.1

The functional $F_{\epsilon}$ defined in (eq:TV_smoothed)-(eq:smooth_Feps) is twice continuously differentiable and convex on $\mathbb{R}^n$. Moreover, if ${\rm ker}(\mathbf{A})\cap {\ker}(\mathbf{D}) = \{\bm{0}_n\}$, $F_{\varepsilon}$ is also coercive and, hence, admits a compact convex set of glob where $\nabla H_{\varepsilon}$ and $\nabla^2 H_{\varepsilon}$ denote the vector (respectively, ma

Figures (3)

  • Figure 1: Average PSNR and SSIM and dispersion bands for MSE (S), Gaussianity (SS) and Whiteness (U) loss computed over 30 test images corrupted by Gaussian blur and AWGN of different levels.
  • Figure 2: Average PSNR and SSIM and dispersion bands for MSE (S), Gaussianity (SS) and Whiteness (U) loss computed over 30 test images corrupted by motion blur and AWGN of different levels.
  • Figure 3: From top to bottom, row-wise: ground-truth images, data $\mathbf{y}$ corrupted by motion blur and AWGN with BSNR=10, optimal reconstructions achieved by bilevel optimization of MSE (S), Gaussianity (SS) and Whiteness (U) loss.

Theorems & Definitions (3)

  • Proposition 3.1
  • proof
  • Remark