# Combining Contrast Invariant L1 Data Fidelities with Nonlinear Spectral Image Decomposition

## TL;DR

This paper introduces a novel combination of contrast-invariant L1 data fidelities with nonlinear spectral decompositions, enhancing multi-scale image analysis and segmentation, especially in biomedical imaging.

## Contribution

It presents the first integration of L1-based multi-scale methods with nonlinear spectral decompositions for improved image segmentation.

## Key findings

- L1-TV promotes sparsity in spectral responses.
- Contrast invariance improves multi-scale decomposition.
- Enhanced segmentation results on biomedical images.

## Abstract

This paper focuses on multi-scale approaches for variational methods and corresponding gradient flows. Recently, for convex regularization functionals such as total variation, new theory and algorithms for nonlinear eigenvalue problems via nonlinear spectral decompositions have been developed. Those methods open new directions for advanced image filtering. However, for an effective use in image segmentation and shape decomposition, a clear interpretation of the spectral response regarding size and intensity scales is needed but lacking in current approaches. In this context, $L^1$ data fidelities are particularly helpful due to their interesting multi-scale properties such as contrast invariance. Hence, the novelty of this work is the combination of $L^1$-based multi-scale methods with nonlinear spectral decompositions. We compare $L^1$ with $L^2$ scale-space methods in view of spectral image representation and decomposition. We show that the contrast invariant multi-scale behavior of $L^1-TV$ promotes sparsity in the spectral response providing more informative decompositions. We provide a numerical method and analyze synthetic and biomedical images at which decomposition leads to improved segmentation.