In this chapter we show how a number of problems in image processing and computer vision can be formulated as the computation of paths or surfaces of minimal energy.We start with the basic formulation, connecting classical work on segmentation with the computation of geodesic curves in two dimensions.We then extend this work to three dimensions and show the application of this framework to object tracking and stereo. The geodesic or minimal surface is computed by means of geometric PDEs, obtained from gradient descent flows. These flows are driven by intrinsic curvatures as well as forces that are derived from the image (data). From this point of view, with this chapter we move one step forward in the theory of curve evolution and PDEs: from equations that included only intrinsic velocities to equations that combine intrinsic with external velocities.

Basic Two-Dimensional Derivation

Since the original pioneering work by Kass et al. [198], extensive research has been done on “snakes” or active-contour models for boundary detection. The classical approach is based on deforming an initial contour C0 toward the boundary of the object to be detected. We obtain the deformation by trying to minimize a functional designed so that its (local) minimum is obtained at the boundary of the object. These active contours are examples of the general technique of matching deformable models to image data by means of energy minimization [38, 387]. The energy functional is basically composed of two components; one controls the smoothness of the curve and the other attracts the curve toward the boundary. This energy model is not capable of handling changes in the topology of the evolving contour when direct implementations are performed.