Efficient Spherical Harmonics Representation of 3D Objects
ABSTRACT In this paper, we present a new and efficient spherical harmonics decomposition for spherical functions defining 3D triangulated objects. Such spherical functions are intrinsically associated to star-shaped objects. However, our results can be extended to any triangular object after segmentation into star-shaped surface patches and recomposition of the results in the implicit framework. There is thus no restriction about the genus number of the object. We demonstrate that the evaluation of the spherical harmonics coefficients can be performed by a Monte Carlo integration over the edges, which makes the computation more accurate and faster than previous techniques, and provides a better control over the precision error in contrast to the voxel-based methods. We present several applications of our research, including fast spectral surface reconstruction from point clouds, local surface smoothing and interactive geometric texture transfer.
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ABSTRACT: Spherical Harmonics arise on the sphere S 2 in the same way that the (Fourier) exponential functions fe ik` g k2Z arise on the circle. Spherical Harmonic series have many of the same wonderful properties as Fourier series, but have lacked one important thing: a numerically stable fast transform analogous to the Fast Fourier Transform. Without a fast transform, evaluating (or expanding in) Spherical Harmonic series on the computer is slow---for large computations prohibitively slow. This paper provides a fast transform. For a grid of O(N 2 ) points on the sphere, a direct calculation has computational complexity O(N 4 ), but a simple separation of variables and Fast Fourier Transform reduce it to O(N 3 ) time. Here we present algorithms with times O(N 5=2 log N) and O(N 2 (log N) 2 ). The problem quickly reduces to the fast application of matrices of Associated Legendre Functions of certain orders. The essential insight is that although these matrices are dense and osc...Journal of Fourier Analysis and Applications 07/1999; · 1.08 Impact Factor
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ABSTRACT: In this paper, we present the Symmetry Descriptors of a 3D model. This is a collection of spherical functions that describes the measure of a model's rotational and reflective symmetry with respect to every axis passing through the center of mass. We show that Symmetry Descriptors can be computed efficiently using fast signal processing techniques, and demonstrate the empirical value of Symmetry Descriptors by showing that they improve matching performance in a variety of shape retrieval experiments.06/2004;
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ABSTRACT: We show how spectral methods may be applied to 3D mesh data to obtain compact representations. This is achieved by projecting the mesh geometry onto an orthonormal basis derived from the mesh topology. To reduce complexity, the mesh is partitioned into a number of balanced submeshes with minimal interaction, each of which are compressed independently. Our methods may be used for compression and progressive transmission of 3D content, and are shown to be vastly superior to existing methods using spatial techniques, if slight loss can be tolerated.08/2001;