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All orientable metric surfaces are Riemann surfaces and admit global conformal parameterizations. Riemann surface structure is a fundamental structure and governs many natural physical phenomena, such as heat diffusion, electric-magnetic fields on the surface. Good parameterization is crucial for simulation and visualization. This paper gives an explicit method for finding optimal global conformal parameterizations of arbitrary surfaces. It relies on certain holomorphic differential forms and conformal mappings from differential geometry and Riemann surface theories. Algorithms are developed to modify topology, locate zero points, and determine cohomology types of differential forms. The implementation is based on finite dimensional optimization method. The optimal parameterization is intrinsic to the geometry, preserving angular structure, and can play an important role in various applications including texture mapping, remeshing, morphing and simulation. The method is demonstrated by visualizing the Riemann surface structure of real surfaces represented as triangle meshes.
The throughput characteristics of contention-based random access channels which use Q-ary splitting algorithms (where Q is the number of groups into which colliding users are split) are analyzed. The algorithms considered are of the Capetanakis-Tsybakov- Mikhailov-Vvedenskaya (CTMV) type and are studied for infinite populations of identical users generating packets according to a discrete time batch Markovian arrival process (D-BMAP). D-BMAPs are a class of tractable Markovian arrival processes, which, in general, are non-renewal. Free channel-access is assumed in combination with Q-ary collision resolution algorithms that exploit either binary or ternary feedback. For the resulting schemes, tree structured Quasi-Birth-Death (QBD) Markov chains are constructed and their stability is determined. The maximum achievable throughput is determined for a variety of arrival processes and splitting factors Q. It is concluded that binary (Q=2) and ternary (Q=3) algorithms should be preferred above other splitting factors Q as the throughput for Q > 3 quickly degrades when subject to bursty arrival streams. If packets arrivals are correlated and bursty, higher throughput rates can be achieved by making use of biased coins.
Surface segmentation is a fundamental problem in computer graphics. It has various applications such as metamorphosis, surface matching, surface compression, 3D shape retrieval, texture mapping, etc. All orientable surfaces are Riemann surfaces, and admit conformal structures. This paper introduces a novel surface segmentation algorithm based on its conformal structure. Each segment can be conformally mapped to a planar rectangle, and the transition maps are planar translations. The segmentation is intrinsic to the surface, independent of the embedding, and consistent for surfaces with similar geometries. By using segmentation based on conformal structure, the mapping between surfaces with arbitrary topologies can be constructed explicitly. The method is rigorous, efficient and automatic. The segmentation can be applied to surface morphing, construct conformal geometry image, convert mesh to Spline surface, solve Partial Differential Equations on arbitrary surfaces, etc.
An image stored in image database systems is assumed to be associated with some content-based meta-data about that image, that is, information about objects in the image and absolute/relative spatial relationships among them. An image query for such an image database system can generally be handled in two ways: exact picture matching and approximate picture matching. In this paper we show the intractability of matching of spatial relationships between a query image and an image stored in the database. In particular, our results suggest that one would not expect to have polynomial-time algorithms for finding the exact picture-matching and computing the maximal similarity between a query picture and a database picture, unless P = NP.