Linear and non-linear geometric object matching with implicit representation

This paper deals with the matching of geometric objects including points, curves, surfaces, and subvolumes using implicit object representations in both linear and non-linear settings. This framework can be applied to feature-based non-linear image warping in biomedical imaging with the deformation constrained to be one-to-one, onto, and diffeomorphic. Moreover, a theoretical connection is established between the well known Hausdorff metric and the framework proposed in this paper. A general strategy for matching geometric objects in both 2D and 3D is discussed. The corresponding Euler-Lagrange equations are presented and gradient descent method is employed to solve the time dependent partial differential equations.