Nonlinear scale space analysis in image processing

Abstract

The objective of this work is to develop and analyze robust and fast image segmentation algorithms. They must be robust to pervasive, large-amplitude noise, which cannot be well characterized in terms of probabilistic distributions. This is because the applications of interest include synthetic aperture radar (SAR) segmentation in which speckle noise is a well-known problem that has defeated many algorithms. The methods must also be robust to blur, because many imaging techniques result in smoothed images. For example, SAR image formation has a natural blur associated with it, due to the finite aperture used in forming the image. We introduce a family of first-order multi-dimensional ordinary differential equations with discontinuous right-hand sides and demonstrate their applicability to segmenting both scalar-valued and vector-valued images, as well as images taking values on a circle. An equation belonging to this family is an inverse diffusion everywhere except at local extrema, where some stabilization is introduced. For this reason, we call these equations "stabilized inverse diffusion equations" ( "SIDEs" ). Existence and uniqueness of solutions, as well as stability, are proven for SIDEs. A SIDE in one spatial dimension may be interpreted as a limiting case of a semi-discretized Perona-Malik equation [49,50], which, in turn, was proposed in order to overcome certain shortcomings of Gaussian scale spaces [72]. These existing techniques are reviewed in a background chapter. SIDEs are then described and experimentally shown to suppress noise while sharpening edges present in the input image. Their application to the detection of abrupt changes in 1-D signals is also demonstrated. It is shown that a version of the SIDEs optimally solves certain change detection problems. Its relations to the Mumford-Shah functional [44] and to linear programming are discussed. Theoretical performance analysis is carried out, and a fast implementation of the algorithm is described.