Multiscale Segmentation via Bregman Distances and Nonlinear Spectral Analysis

In biomedical imaging reliable segmentation of objects (e. g., from small cells up to large organs) is of fundamental importance for automated medical diagnosis. New approaches for multiscale segmentation can considerably improve performance in case of natural variations in intensity, size, and shape. This paper aims at segmenting objects of interest based on shape contours and automatically finding multiple objects with di ff erent scales. The overall strategy of this work is to combine nonlinear segmentation with scales spaces and spectral decompositions. We generalize a variational segmentation model based on total variation using Bregman distances to construct an inverse scale space. This o ff ers the new model to be accomplished by a scale analysis approach based on a spectral decomposition of the total variation. As a result we obtain a very e ffi cient, (nearly) parameter-free multiscale segmentation method that comes with an adaptive regularization parameter choice. To address the variety of shapes and scales present in biomedical imaging we analyze synthetic cases clarifying the role of scale and the relationship of Wulff shapes and eigenfunctions. To underline the potential of our approach and to show its wide applicability we address three different experimental biomedical applications. In particular, we demonstrate the added benefit for identifying and classifying circulating tumor cells and present interesting results for network analysis in retina imaging. Due to the nature of underlying nonlinear di ff usion, the mathematical concepts in this work offer promising extensions to nonlocal classi fi cation problems.