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 formulating the optimization problem to $\overline{\mathbb{C}}$ and thereby linearizing the problem. Errors introduced near the pole are corrected using quasi-conformal theories. Also, by adjusting the Beltrami differential of the mapping, a diffeomorphic (1-1, onto) spherical parameterization can be effectively obtained. The proposed algorithm has been tested on 38 human brain surfaces. Experimental results demonstrate that the computation of the landmark aligned spherical harmonic parameterization is significantly accelerated using the proposed algorithm.

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 using the linear Beltrami solver introduced in this paper. Using the Beltrami representation, the surface homeomorphism can be easily compressed by Fourier approximation, without distorting the bijectivity of the map. The storage requirement can be effectively reduced, which is useful for many practical problems in computer graphics and imaging. In this paper, we propose applying the algorithm to texture map compression and video compression. With our proposed algorithm, the storage requirement for the texture properties of a textured surface can be significantly reduced. Our algorithm can further be applied to compressing motion vector fields for video compression, which effectively improves the compression ratio.

We present a new approach to obtain diffeomorphic registrations with large deformations using landmark and intensity information 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. In this paper, we first propose the quasi-conformal landmark registration (QCLR) algorithm to obtain diffeomorphic (1-1 and onto) registrations between images or surfaces. Using QCLR, landmark-aligned diffeomorphisms between images or surfaces can be obtained, even with a large geometric difference or a large number of landmark constraints. This algorithm is then extended to the quasi-conformal hybrid registration (QCHR) algorithm, which combines landmark and intensity (such as image intensity or surface curvature) information to achieve a more accurate registration result. Experiments have been carried out on both synthetic and real data. Results demonstrate the stability and efficacy of the proposed algorithm to obtain diffeomorphic registrations between images or surfaces.

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 Teichmüller 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 efficient iterative algorithm, called the quasi-conformal (QC) iteration, to compute the T-Map. The basic idea is to represent the set of diffeomorphisms using Beltrami coefficients (BCs) and look for an optimal BC associated to the desired T-Map. The associated diffeomorphism can be efficiently reconstructed from the optimal BC using the linear Beltrami solver (LBS). Using BCs to represent diffeomorphisms guarantees the diffeomorphic property of the registration, even with very large deformation. Using our proposed method, the T-Map can be accurately and efficiently computed. The obtained registration is guaranteed to be bijective. The proposed algorithm can also be extended to compute T-Map with soft landmark constraints. We applied the proposed algorithm to real applications, such as brain landmark matching registration, constrained texture mapping, and human face registration. Experimental results shows that our method is both effective and efficient in computing a nonoverlap landmark matching registration with the least amount of conformality distortion.

In this paper, we propose an adaptive fast solver for a general class of symmetric positive definite (SPD) matrices which include the well-known graph Laplacian. We achieve this by developing an ad...

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 \((n \ge 3)\). 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.