An analysis of variational alignment of curves in images

Abstract

In this paper a common variational formulation for alignment of curves to vector fields is analyzed. This variational approach is often used to solve the problem of aligning curves to edges in images by choosing the vector field to be the image gradient. The main contribution of this paper is an analysis of the Gâteaux derivative and the descent motion of the corresponding alignment functional, improving on earlier research in this area. Several intermediate results are proved and finally a theorem concerning necessary conditions for extremals of the alignment functional is derived. The analysis of the evolution is performed using a level set formulation and results from distribution theory. © Springer-Verlag Berlin Heidelberg 2005.