Parametric methods for image processing using active contours with topology changes

Abstract

In this thesis we consider parametric methods for image processing based on active contours. We introduce an efficient scheme for image segmentation by evolving parametric hypersurfaces. More precisely, we present methods for segmentation of 1) two-dimensional, planar images, of 2) images on curved surfaces and of 3) three-dimensional images. The developed methods can handle complex curve networks with possible triple junctions and intersections of the curves with the image boundary. Also curves with free endpoints are supported. The methods can be used to segment a given image in regions of arbitrary number, separated by hypersurfaces. Numerically, the evolving curves and surfaces are discretized and the resulting schemes are solved by finite differences and finite elements. We show that the parametric approach for curve evolution in the plane and on surfaces has good properties concerning the equidistribution of mesh points along the discretized curves. For evolving surfaces, we observe problems with the quality of the triangulated meshes in rare cases only. We propose a method for an efficient mesh regularization which is incorporated into the evolution scheme for surfaces. Standard parametric approaches cannot automatically handle topology changes like splitting and merging of curves and surfaces, creating and deleting triple junctions and boundary intersection points of curves as well as changing the genus of a surface. Therefore, we introduce an efficient method to detect and execute such topology changes. Using our approach, the computational effort to detect a topology change depends only linearly on the number of mesh points. In addition to image segmentation, we propose a method for edge-preserving image smoothing. The denoising of the image is executed as a postprocessing step, subsequently to the segmentation. Thereby, diffusion equations with Neumann boundary conditions are solved in the already segmented regions. In the case of images defined on surfaces, this results in partial differential equations on manifolds. Finally, we demonstrate the developed methods on various artificial and real images and show the efficiency of the methods and their application to real, practical image processing tasks arising in medicine, navigation, Earth observation and in many other areas.