From Landmark Matching to Shape and Open Curve Matching: A Level Set Approach

In this paper, we present a new framework for warping shapes and open curves between two images. This method could also handle multiple pairs of shapes or curves. When implemented in 3-D, the same framework could be used to warp 3-D objects with minimal modification. Our approach is to use the level set formulation to represent the shapes or curves to be matched. Using this representation, the problem becomes an energy minimization problem. Cost functions for warping overlapping, non-overlapping, and open curves are proposed. For overlapping shapes, a cost function based on minimizing the symmetric difference of the two shapes is used. This cost function is then generalized to deal with non-overlapping shapes as well. For warping open curves, we are inspired by the idea of geodesic active contours. A similar approach, which we will refer to as geodesic curve matching, is proposed. Euler-Lagrange equations are applied and gradient descent is used to solve the corresponding partial differential equations.