  Segmentation remains an important problem in image processing. For
homogeneous (piecewise smooth) images, a number of important models have been
developed and refined over the past several decades. However, these models
often fail when applied to the substantially larger class of natural images
that simultaneously contain regions of both texture and homogeneity. This work
introduces a bi-level constrained minimization model for simultaneous
multiphase segmentation of images containing both homogeneous and textural
regions. We develop novel norms defined in different functional Banach spaces
for the segmentation which results in a non-convex minimization. Finally, we
develop a generalized notion of segmentation delving into approximation theory
and demonstrating that a more refined decomposition of these images results in
multiple meaningful components. Both theoretical results and demonstrations on
natural images are provided.
