Numerical Construction of Diffeomorphism and the Applications to Grid Generation and Image Registration

Abstract

Diffeomorphism is an active research topic in differential geometry. In this area, the existence and construction of diffeomorphism under certain constraints is an interesting and meaningful task. J. Moser first proved the existence of diffeomorphism under a Jacobian determinant constraint. Later, Dr. Liao along with his co-authors, proposed the deformation method to construct diffeomorphisms. A div-curl system is created in the construction of diffeomorphisms. Since the Jacobian determinant has a direct physical meaning in grid generation, i.e. the grid cell size, the deformation method was applied successfully to grid generation and adaptation problems.

In this dissertation, we review the deformation method, focus again on the construction of diffeomorphisms, address clearly a new formation of the deformation problem especially for moving domains. In theory, the deformation method provides one diffeomorphic solution to a nonlinear differential equation.

Inspired by the div-curl system in the deformation method, we developed a new method to construct diffeomorphisms, through a completely different approach. The idea is to control directly the Jacobian determinant and the curl vector of a transformation. Based on calculus of variation and optimization, we proposed a new variational method with prescribed Jacobian determinant and curl vector.

In the study of the two methods of diffeomorphisms construction, we observed the important role of the Jacobian determinant and the curl vector in determining a diffeomorphism. Hence, the corresponding uniqueness problem deserves an investigation. In this dissertation, we discuss this problem by both numerical experiments and theoretical analysis.

Last, we turn to non-rigid image registration, which shares the basic idea of finding transformations. The same equations for divergence and curl vectors are used as constraints to minimize a similarity measure. We designed graphical user interfaces for grid generation and image registration to demonstrate all the methods discussed in this dissertation.