Title: Combining Contrast Invariant L1 Data Fidelities with Nonlinear Spectral Image Decomposition,
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.