Ka Chun Lam

Teichmuller Mapping (T-Map) and Its Applications to Landmark Matching Registration

Registration, which aims to find an optimal 1-1 correspondence between shapes, is an important process in different research areas. Landmark-based surface registration has been widely studied to obtain a mapping between shapes that matches important features. Obtaining a unique and bijective surface registration that matches features consistently is generally challenging, especially when a large number of landmark constraints are enforced. This motivates us to search for a unique landmark matching surface diffeomorphism, which minimizes the local geometric distortion. For this purpose, we propose a special class of diffeomorphisms called the Teichmuller mappings (T-Maps). Under suitable conditions on the landmark constraints, a unique T-Map between two surfaces can be obtained, which minimizes the maximal conformality distortion. The conformality distortion measures how far the mapping deviates from a conformal mapping, and hence it measures the local geometric distortion. In this paper, we propose an eff...

Texture Map and Video Compression Using Beltrami Representation

Surface parameterizations and registrations are important in computer graphics and imaging, where 1-1 correspondences between meshes are computed. In practice, surface maps are usually represented and stored as three-dimensional coordinates each vertex is mapped to, which often requires lots of memory. This causes inconvenience in data transmission and data storage. To tackle this problem, we propose an effective algorithm for compressing surface homeomorphisms using Fourier approximation of the Beltrami representation. The Beltrami representation is a complex-valued function defined on triangular faces of the surface mesh with supreme norm strictly less than 1. Under suitable normalization, there is a 1-1 correspondence between the set of surface homeomorphisms and the set of Beltrami representations. Hence, every bijective surface map is associated with a unique Beltrami representation. Conversely, given a Beltrami representation, the corresponding bijective surface map can be exactly reconstructed usin...

FLASH: Fast Landmark Aligned Spherical Harmonic Parameterization for Genus-0 Closed Brain Surfaces

Surface registration between cortical surfaces is crucial in medical imaging for performing systematic comparisons between brains. Landmark-matching registration that matches anatomical features, called the sulcal landmarks, is often required to obtain a meaningful 1-1 correspondence between brain surfaces. This is commonly done by parameterizing the surface onto a simple parameter domain, such as the unit sphere, in which the sulcal landmarks are consistently aligned. Landmark-matching surface registration can then be obtained from the landmark aligned parameterizations. For genus-0 closed brain surfaces, the optimized spherical harmonic parameterization, which aligns landmarks to consistent locations on the sphere, has been widely used. This approach is limited by the loss of bijectivity under large deformations and the slow computation. In this paper, we propose FLASH, a fast algorithm to compute the optimized spherical harmonic parameterization with consistent landmark alignment. This is achieved by f...

Landmark- and Intensity-Based Registration with Large Deformations via Quasi-conformal Maps

Registration, which aims to find an optimal one-to-one correspondence between different data, is an important problem in various fields. This problem is especially challenging when large deformations occur. In this paper, we present a novel algorithm to obtain diffeomorphic image or surface registrations with large deformations via quasi-conformal maps. The basic idea is to minimize an energy functional involving a Beltrami coefficient term, which measures the distortion of the quasi-conformal map. The Beltrami coefficient effectively controls the bijectivity and smoothness of the registration, even with very large deformations. Using the proposed algorithm, landmark-based registration between images or surfaces can be effectively computed. The obtained registration is guaranteed to be diffeomorphic (1-1 and onto), even with a large deformation or large number of landmark constraints. The proposed algorithm can also be combined with matching intensity (such as image intensity or surface curvature) to improve the accuracy of the registration. Experiments have been carried out on both synthetic and real data. Results demonstrate the efficacy of the proposed algorithm to obtain diffeomorphic registration between images or surfaces.

Genus-One Surface Registration via Teichmüller Extremal Mapping

This paper presents a novel algorithm to obtain landmark-based genus-1 surface registration via a special class of quasi-conformal maps called the Teichmüller maps. Registering shapes with important features is an important process in medical imaging. However, it is challenging to obtain a unique and bijective genus-1surface matching that satisfies the prescribed landmark constraints. In addition, as suggested by [11], conformal transformation provides the most natural way to describe the deformation or growth of anatomical structures. This motivates us to look for the unique mapping between genus-1 surfaces that matches the features while minimizing the maximal conformality distortion. Existence and uniqueness of such optimal diffeomorphism is theoretically guaranteed and is called the Teichmüller extremal mapping. In this work, we propose an iterative algorithm, called the Quasi-conformal (QC) iteration, to find the Teichmüller extremal mapping between the covering spaces of genus-1 surfaces. By representing the set of diffeomorphisms using Beltrami coefficients (BCs), we look for an optimal BC which corresponds to our desired diffeomorphism that matches prescribed features and satisfies the periodic boundary condition on the covering space. Numerical experiments show that our proposed algorithm is efficient and stable for registering genus-1 surfaces even with large amount of landmarks. We have also applied the algorithm on registering vertebral bones with prescribed feature curves, which demonstrates the usefulness of the proposed algorithm.

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

Landmark constrained genus-one surface Teichmüller map applied to surface registration in medical imaging.

(n \ge 3)

Quasi-Conformal Hybrid Multi-modality Image Registration and its Application to Medical Image Fusion

(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.