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.

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.

Computing centroidal Voronoi tessellations (CVT) has many applications in computer graphics. The existing methods, such as the Lloyd algorithm and the quasi-Newton solver, are efficient and easy to implement; however, they compute only the local optimal solutions due to the highly non-linear nature of the CVT energy. This paper presents a novel method, called manifold differential evolution (MDE), for computing globally optimal geodesic CVT energy on triangle meshes. Formulating the mutation operator using discrete geodesics, MDE naturally extends the powerful differential evolution framework from Euclidean spaces to manifold domains. Under mild assumptions, we show that MDE has a provable probabilistic convergence to the global optimum. Experiments on a wide range of 3D models show that MDE consistently outperforms the existing methods by producing results with lower energy. Thanks to its intrinsic and global nature, MDE is insensitive to initialization and mesh tessellation. Moreover, it is able to handle multiply-connected Voronoi cells, which are challenging to the existing geodesic CVT methods.

We address the registration problem of genus-one surfaces (such as vertebrae bones) with prescribed landmark constraints. The high-genus topology of the surfaces makes it challenging to obtain a unique and bijective surface mapping that matches landmarks consistently. This work proposes to tackle this registration problem using a special class of quasi-conformal maps called Teichmüller maps (T-Maps). A landmark constrained T-Map is the unique mapping between genus-1 surfaces that minimizes the maximal conformality distortion while matching the prescribed feature landmarks. Existence and uniqueness of the landmark constrained T-Map are theoretically guaranteed. This work presents an iterative algorithm to compute the T-Map. The main idea is to represent the set of diffeomorphism using the Beltrami coefficients (BC). The BC is iteratively adjusted to an optimal one, which corresponds to our desired T-Map that matches the prescribed landmarks and satisfies the periodic boundary condition on the universal covering space. Numerical experiments demonstrate the effectiveness of our proposed algorithm. The method has also been applied to register vertebrae bones with prescribed landmark points and curves, which gives accurate surface registrations.

Adolescent Idiopathic Scoliosis (AIS) characterized by the 3D spine deformity affects about 4% schoolchildren worldwide. One of the prominent theories of the etiopathogenesis of AIS was proposed to be the poor postural balance control due to the impaired vestibular function. Thus, the morphometry of the vestibular system (VS) is of great importance for studying AIS. The VS is a genus-3 structure situated in the inner ear and consists of three semicircular canals lying perpendicular to each other. The high-genus topology of the surface poses great challenge for shape analysis. In this work, we propose an effective method to analyze shapes of high-genus surfaces by considering their geodesic spectra. The key is to compute the canonical hyperbolic geodesic loops of the surface, using the Ricci flow method. The Fuchsian group generators are then computed which can be used to determine the geodesic spectra. The geodesic spectra effectively measure shape differences between high-genus surfaces up to the hyperbolic isometry. We applied the proposed algorithm to the VS of 12 normal and 15 AIS subjects. Experimental results show the effectiveness of our algorithm and reveal statistical shape difference in the VS between right-thoracic AIS and normal subjects.