Nested Bregman Iterations for Decomposition Problems

TL;DR

The authors address image decomposition into structured components by leveraging infimal convolution regularizers and the Bregman iteration framework. They introduce Nested Bregman iterations to transform parameter tuning into a principled stopping rule based on cross-correlation, enabling robust decomposition under noise. The paper proves well-definedness and convergence aspects for selected regularizers (including $L^1$–$H^1$, TV–H^1, and $TGV$–$TGV^{osci}$) and demonstrates the method through numerical experiments that achieve competitive PSNR and improved component separation compared to bilevel or grid-search approaches. This approach reduces manual tuning, offers a flexible alternative to bilevel optimization, and broadens applicability to complex regularizers in denoising and deblurring tasks.

Abstract

We consider the task of image reconstruction while simultaneously decomposing the reconstructed image into components with different features. A commonly used tool for this is a variational approach with an infimal convolution of appropriate functions as a regularizer. Especially for noise corrupted observations, incorporating these functionals into the classical method of Bregman iterations provides a robust method for obtaining an overall good approximation of the true image, by stopping early the iteration according to a discrepancy principle. However, crucially, the quality of the separate components depends further on the proper choice of the regularization weights associated to the infimally convoluted functionals. Here, we propose the method of Nested Bregman iterations to improve a decomposition in a structured way. This allows to transform the task of choosing the weights into the problem of stopping the iteration according to a meaningful criterion based on normalized cross-correlation. We discuss the well-definedness and the convergence behavior of the proposed method, and illustrate its strength numerically with various image decomposition tasks employing infimal convolution functionals.

Figures (14)

• Figure 1: Components $u^\dagger$ and $v^\dagger$, true image $x^\dagger = u^\dagger+v^\dagger$, as well as the blurred and noisy observation $f^\delta$.
• Figure 2: Different decompositions for the deblurring problem, obtained by Tikhonov regularization with $J(u,v) = \frac{\alpha}{2}\int\limits_\Omega \left| \nabla u \right|^2 + \beta \int\limits_\Omega \left| \nabla v \right|$ for different ratios $\frac{\alpha}{\beta}$. First line: true decomposition and observation. Lines $2-5$ (top to bottom): Reconstructions with ration $\frac{\alpha}{\beta} = 1000,47,10,2$
• Figure 3: Different decompositions of the deblurring problem, obtained by Bregman iterations with $J(u,v) = \frac{\alpha}{2}\left\Vert \nabla u \right\Vert _{L^2}^2 + \beta \left\Vert D v \right\Vert _\mathcal{M}$ stopped via discrepancy principle $(\tau =1.001)$ First line: true decomposition and observation. Lines $2-5$ (top to bottom): Reconstructions with ration $\frac{\alpha}{\beta} = 1000,47,10,2$
• Figure 4: PSNR values for Morozov regularization for different values of $\alpha$, Nested Bregman iterations with weight $\alpha = 1000$ at the stopping index and maximal PSNR from Nested Bregman iterations. From top left to bottom right: $u-$component, $v-$component, full reconstruction, sum of $u-$component and $v-$component PSNR values.
• Figure 5: Sum of the componentwise PSNR values obtained by Algorithm \ref{['algo:NestedBregman_Noisy_Morozov']} (left) and Algorithm \ref{['algo:NestedBregman_Noisy']} (right). The circle and the asterisk mark the maximum of the PSNR and the first local minimum of the scalar normalized cross-correlation, respectively.

Theorems & Definitions (39)

• Definition 2.1
• Proposition 2.1
• proof
• Definition 2.2
• Proposition 2.2
• proof
• Example 3.1
• Remark 3.1
• Theorem 3.1
• Theorem 3.2