Viscosity solutions for geodesic active contour under geometrical conditions

Abstract

This paper addresses the issue of introducing geometrical constraints into segmentation processes in image analysis. This question has emerged from the analysis of classical tools: indeed, usual methods such as deformable models, fast marching prove to be fruitless when image data are missing or of poor quality or when the image owns homogeneous-textured regions. To cope with these hindrances, the idea developed hereafter consists of integrating geometrical constraints in the modelling to make the segmentation process easier to handle. We have devised a geodesic active contour-based model, in which we are trying to determine a curve that best approaches the given points (geometrical data) while detecting the object under consideration (see also Apparato et al., Segmentation of medical image sequence under constraints: application to non-invasive assessment of pulmonary arterial hypertension, Int. J. Comput. Math. 5(2004), pp. 527–536 and Gout and Vieira-Testé C. Gout and S. Vieira-Testé, An algorithm for segmentation under interpolation conditions using deformable models, Int. J. Comput. Math. 80(1) (2003), pp. 47–54 where other approaches are given for this kind of problems). In this paper the main results concern the study of the existence and uniqueness of the viscosity solution of this problem (following three different approaches). We also give numerical examples on real data sets.