  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.
