We develop a new algorithm to automatically register hippocampal (HP) surfaces with complete geometric matching, avoiding the need to manually label landmark features. A good registration depends on a reasonable choice of shape energy that measures the dissimilarity between surfaces. In our work, we first propose a complete shape index using the Beltrami coefficient and curvatures, which measures subtle local differences. The proposed shape energy is zero if and only if two shapes are identical up to a rigid motion. We then seek the best surface registration by minimizing the shape energy. We propose a simple representation of surface diffeomorphisms using Beltrami coefficients, which simplifies the optimization process. We then iteratively minimize the shape energy using the proposed Beltrami Holomorphic flow (BHF) method. Experimental results on 212 HP of normal and diseased (Alzheimer's disease) subjects show our proposed algorithm is effective in registering HP surfaces with complete geometric matching. The proposed shape energy can also capture local shape differences between HP for disease analysis.

We propose a novel framework for reconstructing lightweight polygonal surfaces from point clouds. Unlike traditional methods that focus on either extracting good geometric primitives or obtaining proper arrangements of primitives, the emphasis of this work lies in intersecting the primitives (planes only) and seeking for an appropriate combination of them to obtain a manifold polygonal surface model without boundary. We show that reconstruction from point clouds can be cast as a binary labeling problem. Our method is based on a hypothesizing and selection strategy. We first generate a reasonably large set of face candidates by intersecting the extracted planar primitives. Then an optimal subset of the candidate faces is selected through optimization. Our optimization is based on a binary linear programming formulation under hard constraints that enforce the final polygonal surface model to be manifold and watertight. Experiments on point clouds from various sources demonstrate that our method can generate lightweight polygonal surface models of arbitrary piecewise planar objects. Besides, our method is capable of recovering sharp features and is robust to noise, outliers, and missing data.

Geodesic based Voronoi diagrams play an important role in many applications of computer graphics. Constructing such Voronoi diagrams usually resorts to exact geodesics. However, exact geodesic computation always consumes lots of time and memory, which has become the bottleneck of constructing geodesic based Voronoi diagrams. In this paper, we propose the window-VTP algorithm, which can effectively reduce redundant computation and save memory. As a result, constructing Voronoi diagrams using the proposed window-VTP algorithm runs 3-8 times faster than Liu et al.’s method [LCT11], 1.2 times faster than its FWP-MMP variant and more importantly uses 10-70 times less memory than both of them.

The study of 2D shapes is a central problem in the field of computer vision. In 2D shape analysis, classification and recognition of objects from their observed silhouettes are extremely crucial and yet difficult. It usually involves an efficient representation of 2D shape space with natural metric, so that its mathematical structure can be used for further analysis. Although significant progress has been made for the study of 2D simply-connected shapes, very few works have been done on the study of 2D objects with arbitrary topologies. In this work, we propose a representation of general 2D domains with arbitrary topologies using conformal geometry. A natural metric can be defined on the proposed representation space, which gives a metric to measure dissimilarities between objects. The main idea is to map the exterior and interior of the domain conformally to unit disks and circle domains, using holomorphic 1-forms. A set of diffeomorphisms from the unit circle S^1 to itself can be obtained, which together with the conformal modules are used to define the shape signature. We prove mathematically that our proposed signature uniquely represents shapes with arbitrary topologies. We also introduce a reconstruction algorithm to obtain shapes from their signatures. This completes our framework and allows us to move back and forth between shapes and signatures. Experiments show the efficacy of our proposed algorithm as a stable shape representation scheme.

Computing geodesic distances on triangle meshes is a fundamental problem in computational geometry and computer graphics. To date, two notable classes of algorithms, the Mitchell-Mount-Papadimitriou (MMP) algorithm and the Chen-Han (CH) algorithm, have been proposed. Although these algorithms can compute exact geodesic distances if numerical computation is exact, they are computationally expensive, which diminishes their usefulness for large-scale models and/or time-critical applications. In this paper, we propose the fast wavefront propagation (FWP) framework for improving the performance of both the MMP and CH algorithms. Unlike the original algorithms that propagate only a single window (a data structure locally encodes geodesic information) at each iteration, our method organizes windows with a bucket data structure so that it can process a large number of windows simultaneously without compromising wavefront quality. Thanks to its macro nature, the FWP method is less sensitive to mesh triangulation than the MMP and CH algorithms. We evaluate our FWP-based MMP and CH algorithms on a wide range of large-scale real-world models. Computational results show that our method can improve the speed by a factor of 3-10.