Alzheimer's disease (AD) is a no-cure disease that has been frustrating the scientists for many years. Analyzing the disease has become an important but challenging research topic. The shape analysis of the sub-cortical structure of AD patients has been commonly used to understand this disease. In this paper, we assess the feasibility of using shape information on the hippocampal (HP) surfaces to detect some sub-structural changes in AD patients. We propose a quasi-conformal statistical shape analysis model, which allows us to study local regional geometric changes in the HPs amongst normal control (NC) and AD groups. A shape index defined by the quasi-conformality and surface curvatures is used to characterize region-specific shape variations of the HP surfaces. Feature vectors can be extracted for each HPs, with which a classification model can be built using machine learning methods to classify HPs into NC and AD subjects. Experiments have been carried out on 99 normal controls and 41 patients with AD. Results demonstrate that the proposed quasi-conformal based model is effective for classifying HPs into NC and AD groups with high classification accuracy (with highest overall classification accuracy reaching 87.86% in a leave-one-out experiment using the whole dataset).

Shape registration has a wide range of applications in geometric modeling, medical imaging, and computer vision. This paper focuses on the registration of the genus-3 vestibular systems and studies the geometric differences between the normal and Adolescent Idiopathic Scoliosis (AIS) groups. The non-trivial topology of the VS poses great technical challenges to the geometric analysis. To tackle these challenges, we present an effective and practical solution to register the vestibular systems. We first extract six geodesic landmarks for the VS, which are stable, intrinsic, and insensitive to the VS’s resolution and tessellation. Moreover, they are highly consistent regardless of the AIS and normal groups. The detected geodesic landmarks partition the VS into three patches, a topological annulus and two topological disks. For each pair of patches of the AIS subject and the control, we compute a bijective map using the holomorphic 1-form and harmonic map techniques. With a carefully designed boundary condition, the three individual maps can be glued in a seamless manner so that the resulting registration is a homeomorphism with exact landmark matching. Our method is robust, automatic and efficient. It takes only a few seconds on a low-end PC, which significantly outperforms the non-rigid ICP algorithm. We conducted a student’s t-test on the test data. Computational results show that using the mean curvature measure E_H, our method can distinguish the AIS subjects and the normal subjects.

In this work, we introduce two sets of algorithms inspired by the ideas from modern geometry. One is computational conformal geometry method, including harmonic maps, holomorphic 1-forms and Ricci flow. The other one is optimization method using affine normals.

In this paper, we propose novel algorithms for inpainting and refinement of diffeomorphisms. We first represent a diffeomorphism by its Beltrami coefficient. Then it is possible to refine and inpaint the diffeomorphism by processing this Beltrami coefficient. With the inpainted/refined Beltrami coefficient, we construct a new diffeomorphism using the exact Beltrami holomorphic flow algorithm proposed in this paper. We apply our algorithms on several practical applications, which include the inpainting of a highly distorted diffeomorphism, the inpainting of image sequences of deforming shapes, the super-resolution of diffeomorphisms and the global parameterization of cortical surfaces by combining local parameterizations. Experiments show that our algorithm can solve these problems with natural and smooth results. We demonstrate how our proposed method can be widely applied in areas from texture mapping to video processing, and from computer graphics to medical imaging.

Surface conformal maps between genus-0 surfaces play important roles in applied mathematics and engineering, with applications in medical image analysis and computer graphics. Previous work (Gu and Yau in Commun Inf Syst 2(2):121–146, 2002) introduces a variational approach, where global conformal parameterization of genus-0 surfaces was addressed through minimizing the harmonic energy, with two weaknesses: its gradient descent iteration is slow, and its solutions contain undesired parameterization foldings when the underlying surface has long sharp features. In this paper, we propose an algorithm that significantly accelerates the harmonic energy minimization and a method that iteratively removes foldings by taking advantages of the weighted Laplace–Beltrami eigen-projection. Experimental results show that the proposed approaches compute genus-0 surface harmonic maps much faster than the existing algorithm in Gu and Yau (Commun Inf Syst 2(2):121–146, 2002) and the new results contain no foldings.