Landmark-matching Transformation with Large Deformation via n-dimensional Quasi-conformal Maps

Abstract

We propose a new method to obtain landmark-matching transformations between n-dimensional Euclidean spaces with large deformations. Given a set of feature correspondences, our algorithm searches for an optimal folding-free mapping that satisfies the prescribed landmark constraints. The standard conformality distortion defined for mappings between 2-dimensional spaces is first generalized to the $n$-dimensional conformality distortion $K(f)$ for a mapping $f$ between $n$-dimensional Euclidean spaces $(n \geq 3)$. We then propose a variational model involving $K(f)$ to tackle the landmark-matching problem in higher dimensional spaces. The generalized conformality term $K(f)$ enforces the bijectivity of the optimized mapping and minimizes its local geometric distortions even with large deformations. Another challenge is the high computational cost of the proposed model. To tackle this, we have also proposed a numerical method to solve the optimization problem more efficiently. Alternating direction method with multiplier (ADMM) is applied to split the optimization problem into two subproblems. Preconditioned conjugate gradient method with multi-grid preconditioner is applied to solve one of the sub-problems, while a fixed-point iteration is proposed to solve another subproblem. Experiments have been carried out on both synthetic examples and lung CT images to compute the diffeomorphic landmark-matching transformation with different landmark constraints. Results show the efficacy of our proposed model to obtain a folding-free landmark-matching transformation between $n$-dimensional spaces with large deformations.