Computing smooth and optimal one-to-one maps between surfaces of same topology is a fundamental problem in computer graphics and such a method provides us a ubiquitous tool for geometric modeling and data visualization. Its vast variety of applications includes shape registration/matching, shape blending, material/data transfer, data fusion, information reuse, etc. The mapping quality is typically measured in terms of angular distortions among different shapes. This paper proposes and develops a novel quasi-conformal surface mapping framework to globally minimize the stretching energy inevitably introduced between two different shapes. The existing state-of-the-art inter-surface mapping techniques only afford local optimization either on surface patches via boundary cutting or on the simplified base domain, lacking rigorous mathematical foundation and analysis. We design and articulate an automatic variational algorithm that can reach the global distortion minimum for surface mapping between shapes of arbitrary topology, and our algorithm is sorely founded upon the intrinsic geometry structure of surfaces. To our best knowledge, this is the first attempt towards numerically computing globally optimal maps. Consequently, our mapping framework offers a powerful computational tool for graphics and visualization tasks such as data and texture transfer, shape morphing, and shape matching.

This paper presents a new approach to the physically-based thin-shell simulation of point-sampled geometry via explicit, global conformal point-surface parameterization and meshless dynamics. The point-based global parameterization is founded upon the rigorous mathematics of Riemann surface theory and Hodge theory. The parameterization is globally conformal everywhere except for a minimum number of zero points. Within our parameterization framework, any well-sampled point surface is functionally equivalent to a manifold, enabling popular and powerful surface-based modeling and physically-based simulation tools to be readily adapted for point geometry processing and animation. In addition, we propose a meshless surface computational paradigm in which the partial differential equations (for dynamic physical simulation) can be applied and solved directly over point samples via Moving Least Squares (MLS) shape functions defined on the global parametric domain without explicit connectivity information. The global conformal parameterization provides a common domain to facilitate accurate meshless simulation and efficient discontinuity modeling for complex branching cracks. Through our experiments on thin-shell elastic deformation and fracture simulation, we demonstrate that our integrative method is very natural, and that it has great potential to further broaden the application scope of point-sampled geometry in graphics and relevant fields.

People in different places talk about different things. This interest distribution is reflected by the newspaper articles circulated in a particular area. We use data from our large-scale newspaper analysis system (Lydia) to make entity datamaps, a spatial visualization of the interest in a given named entity. Our goal is to identify entities which display regional biases. We develop a model of estimating the frequency of reference of an entity in any given city from the reference frequency centered in surrounding cities, and techniques for evaluating the spatial significance of this distribution.

Abstract This paper presents an innovative method for naturally and smoothly morphing point‐sampled surfaces via dynamic meshless simulation on point‐sampled surfaces. While most existing literature on shape morphing emphasizes the issue of finding a good correspondence map between two object representations, this research primarily investigates the challenging problem of how to find a smooth, physically‐meaningful transition path between two homeomorphic point‐set surfaces. We analyze the deformation of surface involved in the morphing process using concepts in differential geometry and continuum mechanics. The morphing paths can be determined by optimizing an energy functional, which characterizes the intrinsic deformation of the surface away from its rest shape. As demonstrated in the examples, our method automatically produces a series of natural and physically‐plausible in‐between shapes, which greatly alleviates the shrinking, stretching, and self‐intersection problems that often occur when linear interpolation is employed for the morphing of two objects. We envision that our new technique will continue to broaden the application scope of point‐set surfaces and their dynamic animation. Copyright © 2005 John Wiley & Sons, Ltd.