Nested Bregman Iterations for Decomposition Problems

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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 the iteration early according to a discrepancy principle. However, crucially, the quality of the separate components depends further on the proper choice of the regularization weights associated with the infimally convoluted functionals. Here, we propose the method of Nested Bregman iterations to improve a decomposition in a structured way. This allows for the transformation of 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.
Funding Information
  • Austrian Science Fund (10.55776/DOC78)

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