Landmark-Matching Transformation with Large Deformation Via n-dimensional Quasi-conformal Maps

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 ≥ 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 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

Multiple Shape Registration using Constrained Optimal Control

Lagrangian particle formulations of the large deformation diffeomorphic metric mapping algorithm (LDDMM) only allow for the study of a single shape. In this paper, we introduce and discuss both a theoretical and practical setting for the simultaneous study of multiple shapes that are either stitched to one another or slide along a submanifold. The method is described within the optimal control formalism, and optimality conditions are given, together with the equations that are needed to implement augmented Lagrangian methods. Experimental results are provided for stitched and sliding surfaces

Shape analysis in shape space

This study aims to classify different deformations based on the shape space concept. A shape space is a quotient space in which each point corresponds to a class of shapes. The shapes of each class are transformed to each other by a transformation group preserving a geometrical property in which we are interested. Therefore, each deformation is a curve on the high dimensional shape space manifold, and one can classify the deformations by comparison of their corresponding deformation curves in shape space. Towards this end, two classification methods are proposed. In the first method, a quasi conformal shape space is constructed based on a novel quasi-conformal metric, which preserves the curvature changes at each vertex during the deformation. Besides, a classification framework is introduced for deformation classification. The results on synthetic and real datasets show the effectiveness of the metric to estimate the intrinsic geometry of the shape space manifold, and its ability to classify and interpolate different deformations. In the second method, we introduce the medial surface shape space which classifies the deformations based on the medial surface and thickness of the shape. This shape space is based on the log map and uses two novel measures, average of the normal vectors and mean of the positions, to determine the distance between each pair of shapes on shape space. We applied these methods to classify the left ventricle deformations. The experimental results shows that the first method can remarkably classify the normal and abnormal subjects but this method cannot spot the location of the abnormality. In contrast, the second method can discriminate healthy subjects from patients with cardiomyopathy, and also can spot the abnormality on the left ventricle, which makes it a valuable assistant tool for diagnostic purposes