This paper presents a method to compute the quasi-conformal parameterization (QCMC) for a multiply-connected 2D domain or surface. QCMC computes a quasi-conformal map from a multiply-connected domain S onto a punctured disk D_S associated with a given Beltrami differential. The Beltrami differential, which measures the conformality distortion, is a complex-valued function μ: S→ℂ with supremum norm strictly less than 1. Every Beltrami differential gives a conformal structure of S. Hence, the conformal module of D_S, which are the radii and centers of the inner circles, can be fully determined by μ, up to a Möbius transformation. In this paper, we propose an iterative algorithm to simultaneously search for the conformal module and the optimal quasi-conformal parameterization. The key idea is to minimize the Beltrami energy subject to the boundary constraints. The optimal solution is our desired quasi-conformal parameterization onto a punctured disk. The parameterization of the multiply-connected domain simplifies numerical computations and has important applications in various fields, such as in computer graphics and vision. Experiments have been carried out on synthetic data together with real multiply-connected Riemann surfaces. Results show that our proposed method can efficiently compute quasi-conformal parameterizations of multiply-connected domains and outperforms other state-of-the-art algorithms. Applications of the proposed parameterization technique have also been explored.

The medial axis is an important shape representation that finds a wide range of applications in shape analysis. For large-scale shapes of high resolution, a progressive medial axis representation that starts with the lowest resolution and gradually adds more details is desired. In this paper, we propose a fast and robust geometric algorithm that computes progressive medial axes of a large-scale planar shape. The key ingredient of our method is a novel structural analysis of merging medial axes of two planar shapes along a shared boundary. Our method is robust by separating the analysis of topological structure from numerical computation. Our method is also fast and we show that the time complexity of merging two medial axes is O(n log nv), where n is the number of total boundary generators, nv is strictly smaller than n and behaves as a small constant in all our experiments. Experiments on large-scale polygonal data and comparison with state-of-the-art methods show the efficiency and effectiveness of the proposed method.

Finding a meaningful 1–1 correspondence between different data, such as images or surface data, has important applications in various fields. It involves the optimization of certain energy functionals over the space of all diffeomorphisms. This type of optimization problems (called the diffeomorphism optimization problems, DOPs) is especially challenging, since the bijectivity of the mapping has to be ensured. Recently, a method, called the Beltrami holomorphic flow (BHF), has been proposed to solve the DOP using quasi-conformal theories (Lui et al. in J Sci Comput 50(3):557–585, 2012). The optimization problem is formulated over the space of Beltrami coefficients (BCs), instead of the space of all diffeomorphisms. BHF iteratively finds a sequence of BCs associated with a sequence of diffeomorphisms, using the gradient descent method, to minimize the energy functional. The use of BCs effectively controls the smoothness and bijectivity of the mapping, and hence makes it easier to handle the constrained optimization problem. However, the algorithm is computationally expensive. In this paper, we propose an efficient splitting algorithm, based on the classical alternating direction method of multiplier (ADMM), to solve the DOP. The basic idea is to split the energy functional into two energy terms: one involves the BC whereas the other involves the quasi-conformal map. Alternating minimization scheme can then be applied to minimize the energy functional. The proposed method significantly speeds up the previous BHF approach. It also extends the previous BHF algorithm to Riemann surfaces of arbitrary topologies, such as multiply-connected shapes. Experiments have been carried out on synthetic together with real surface data, which demonstrate the efficiency and efficacy of the proposed algorithm to solve the DOP.

There is an unmet medical need to identify neuroimaging biomarkers that allow us to accurately diagnose and monitor Alzheimer's disease (AD) at its very early stages and to assess the response to AD-modifying therapies. To a certain extent, volumetric and functional magnetic resonance imaging (fMRI) studies can detect changes in structure, cerebral blood flow, and blood oxygenation that distinguish AD and mild cognitive impairment (MCI) subjects from healthy control (HC) subjects. However, it has been challenging to use fully automated MRI analytic methods to identify potential AD neuroimaging biomarkers. We have thus proposed a method based on independent component analysis (ICA) for studying potential AD-related MR image features that can be coupled with the use of support vector machine (SVM) for classifying scans into categories of AD, MCI, and HC subjects. The MRI data were selected from

Here we propose a novel method to compute Teichmller shape space based shape index to study brain morphometry. Such a shape index is intrinsic, and invariant under conformal transformations, rigid motions and scaling. We conformally map a genus-zero open boundary surface to the Poincar disk with the Yamabe flow method. The shape indices that we compute are the lengths of a special set of geodesics under hyperbolic metric. Tests on longitudinal brain imaging data were used to demonstrate the stability of the derived feature vectors. In leave-one-out validation tests, we achieved 100% accurate classification (versus only 68% accuracy for volume measures) in distinguishing 11 HIV/AIDS individuals from 8 healthy control subjects, based on Teichmller coordinates for lateral ventricular surfaces extracted from their 3D MRI scans.