A fresh framework for mesh optimization, the Filtered Hooke's Optimization, is proposed. With the notion of the elasticity theory, the Hooke's Optimization is developed by modifying the Hooke's law, in which an elastic force is simulated on the edges of a mesh so that adjacent vertices are either attracted to each other or repelled from each other, so as to regularize the mesh in terms of triangulation. A normal torque force is acted on vertices to guarantee smoothness of the surface. In addition, a filtering scheme, called the Newtonian Filtering, is proposed as a supplementary tool for the proposed Hooke's Optimization to preserve the geometry of the mesh. Numerical simulations on meshes with different geometry indicate an impressive performance of our proposed framework to significantly improve the mesh triangulation without noteworthy distortions of the mesh geometry.

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.

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.

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.

This paper proposes a novel approach for extracting two intrinsic feature curves on hippocampal (HC) surfaces. The hippocampus is a key target of study in medical imaging, as it degenerates in conditions such as epilepsy and Alzheimer's disease (AD), but its structure is complex. To facilitate HC morphometry, we generate two intrinsic feature curves that describe their global geometries. For example, the separation of them captures thickness changes in HC surfaces, which can be used to effectively measure HC atrophy found in patients with AD. They also separate HC surfaces into upper and lower surface patches where intrinsic shape analysis using conformal modules can be carried out. Based on these curves, we further propose a parameterization of HC surfaces called the eigen-harmonic parameterization (EHP). EHP maps each HC surface onto a parameter domain and imposes longitudinal and azimuthal coordinates on each surface, which follow the gradient and level sets of its first nontrivial Laplace-Beltrami eigenfunction, respectively. Each tubular domain is constructed according to the geometry of an individual HC surface. This gives a parameter domain with much less geometric distortion compared to spherical parameterization. With EHP, all HC surfaces are automatically registered with intrinsic feature curves preserved and geometric distortions minimized. This allows shape analysis on any number of HC surfaces to be performed consistently. We studied geometric changes over time in 138 HC surfaces of patients with AD and normal subjects scanned at two different times. We successfully located areas with significantly different shape changes over time between the two groups.