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Wavelets on the 2sphere: A grouptheoretical approach
 APPLIED AND COMPUTATIONAL HARMONIC ANALYSIS
, 1999
"... We present a purely grouptheoretical derivation of the continuous wavelet transform (CWT) on the 2sphere S 2, based on the construction of general coherent states associated to square integrable group representations. The parameter space X of our CWT is the product of SO(3) for motions and R + ∗ f ..."
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Cited by 46 (10 self)
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We present a purely grouptheoretical derivation of the continuous wavelet transform (CWT) on the 2sphere S 2, based on the construction of general coherent states associated to square integrable group representations. The parameter space X of our CWT is the product of SO(3) for motions and R + ∗ for dilations on S 2,which are embedded into the Lorentz group SO0(3, 1) via the Iwasawa decomposition, so that X � SO 0(3, 1)/N, where N � C. We select an appropriate unitary representation of SO0(3, 1) acting in the space L 2 (S 2,dµ) of finite energy signals on S 2. This representation is square integrable over X; thus it yields immediately the wavelets on S 2 and the associated CWT. We find a necessary condition for the admissibility of a wavelet, in the form of a zero mean condition. Finally, the Euclidean limit of this CWT on S 2 is obtained by redoing the construction on a sphere of radius R and performing a group contraction for R →∞. Then the parameter space goes into the similitude group of R 2 and one recovers exactly the CWT on the plane, including the usual zero mean necessary condition for admissibility.
Allfrequency precomputed radiance transfer using spherical radial basis functions and clustered tensor approximation
 ACM Trans. Graph
, 2006
"... This paper introduces a new data representation and compression technique for precomputed radiance transfer (PRT). The light transfer functions and light sources are modeled with spherical radial basis functions (SRBFs). A SRBF is a rotationinvariant function that depends on the geodesic distance b ..."
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Cited by 37 (0 self)
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This paper introduces a new data representation and compression technique for precomputed radiance transfer (PRT). The light transfer functions and light sources are modeled with spherical radial basis functions (SRBFs). A SRBF is a rotationinvariant function that depends on the geodesic distance between two points on the unit sphere. Rotating functions in SRBF representation is as straightforward as rotating the centers of SRBFs. Moreover, highfrequency signals are handled by adjusting the bandwidth parameters of SRBFs. To exploit intervertex coherence, the light transfer functions are further classified iteratively into disjoint clusters, and tensor approximation is applied within each cluster. Compared with previous methods, the proposed approach enables realtime rendering with comparable quality under highfrequency lighting environments. The data storage is also more compact than previous allfrequency PRT algorithms. CR Categories: I.3.7 [Computer Graphics]: ThreeDimensional Graphics and Realism—Color, shading, shadowing, and texture; G.1.2 [Numerical Analysis]: Approximation—Special function approximations; E.4 [Coding and Information Theory]: Data compaction and compression
Error estimates for scattered data interpolation on spheres
 MATH. COMP
, 1999
"... We study Sobolev type estimates for the approximation order resulting from using strictly positive definite kernels to do interpolation on the nsphere. The interpolation knots are scattered. Our approach partly follows the general theory of Golomb and Weinberger and related estimates. These error e ..."
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Cited by 34 (4 self)
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We study Sobolev type estimates for the approximation order resulting from using strictly positive definite kernels to do interpolation on the nsphere. The interpolation knots are scattered. Our approach partly follows the general theory of Golomb and Weinberger and related estimates. These error estimates are then based on series expansions of smooth functions in terms of spherical harmonics. The Markov inequality for spherical harmonics is essential to our analysis and is used in order to find lower bounds for certain sampling operators on spaces of spherical harmonics.
Scattered Data Fitting on the Sphere
 in Mathematical Methods for Curves and Surfaces II
, 1998
"... . We discuss several approaches to the problem of interpolating or approximating data given at scattered points lying on the surface of the sphere. These include methods based on spherical harmonics, tensorproduct spaces on a rectangular map of the sphere, functions defined over spherical triangulat ..."
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Cited by 34 (5 self)
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. We discuss several approaches to the problem of interpolating or approximating data given at scattered points lying on the surface of the sphere. These include methods based on spherical harmonics, tensorproduct spaces on a rectangular map of the sphere, functions defined over spherical triangulations, spherical splines, spherical radial basis functions, and some associated multiresolution methods. In addition, we briefly discuss spherelike surfaces, visualization, and methods for more general surfaces. The paper includes a total of 206 references. x1. Introduction Let S be the unit sphere in IR 3 , and suppose that fv i g n i=1 is a set of scattered points lying on S. In this paper we are interested in the following problem: Problem 1. Given real numbers fr i g n i=1 , find a (smooth) function s defined on S which interpolates the data in the sense that s(v i ) = r i ; i = 1; : : : ; n; (1) or approximates it in the sense that s(v i ) ß r i ; i = 1; : : : ; n: (2) Data f...
Optimal Functions for a Periodic Uncertainty Principle and Multiresolution Analysis
, 1995
"... . In this paper, it is shown that certain Theta functions are asymptotically optimal for the periodic time frequency uncertainty principle described by Breitenberger in [3]. These extremal functions give rise to a periodic multiresolution analysis where the corresponding wavelets also show similar l ..."
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Cited by 17 (3 self)
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. In this paper, it is shown that certain Theta functions are asymptotically optimal for the periodic time frequency uncertainty principle described by Breitenberger in [3]. These extremal functions give rise to a periodic multiresolution analysis where the corresponding wavelets also show similar localization properties. Subject Classification. Primary 42A16, Secondary 26D05, 26D10, 26D15. x1. Introduction A fundamental result on time and frequency localization of squareintegrable functions on the real line is given by the Heisenberg uncertainty principle, and it is well known that the Gaussian functions serve as extremal functions for this inequality. On the other hand, uncertainty relations for periodic functions have not been studied as thoroughly as the original inequality on the real axis. In this paper, we want to focus on a concept discussed by Breitenberger in [3], where uncertainty of a periodic squareintegrable function is described in terms of the product of frequency and...
An Uncertainty Principle For Ultraspherical Expansions
, 1996
"... Motivated by HeisenbergWeyl type uncertainty principles for the torus T and the sphere S 2 due to Breitenberger, Narcowich, Ward and others, we derive an uncertainty relation for radial functions on the spheres S n ae IR n+1 and, more generally, for ultraspherical expansions on [0; ß]: In thi ..."
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Cited by 13 (1 self)
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Motivated by HeisenbergWeyl type uncertainty principles for the torus T and the sphere S 2 due to Breitenberger, Narcowich, Ward and others, we derive an uncertainty relation for radial functions on the spheres S n ae IR n+1 and, more generally, for ultraspherical expansions on [0; ß]: In this setting, the "frequency variance" of a L 2 function on [0; ß] is defined by means of the ultraspherical differential operator, which plays the role of a Laplacian. Our proof is based on a certain firstorder differentialdifference operator on the doubled interval [\Gammaß; ß] . Moreover, using the densities f t of "Gaussian measures" on [0; ß] with the time t tending to 0 , we show that the bound of our uncertainty principle is optimal. 1. An uncertainty principle for n spheres There exist different versions of the HeisenbergWeyl inequality for the torus T := fz 2 C : jzj = 1g . One possible version is the localizationfrequency uncertainty principle discussed by Breitenberger [1...
Wavelets on the NSphere and Related Manifolds
 J. Math. Phys
, 1997
"... We present a purely grouptheoretical derivation of the continuous wavelet transform (CWT) on the (n \Gamma 1)sphere S n\Gamma1 , based on the construction of general coherent states associated to square integrable group representations. The parameter space of the CWT, Y ¸ SO(n) \Theta R + , ..."
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Cited by 12 (0 self)
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We present a purely grouptheoretical derivation of the continuous wavelet transform (CWT) on the (n \Gamma 1)sphere S n\Gamma1 , based on the construction of general coherent states associated to square integrable group representations. The parameter space of the CWT, Y ¸ SO(n) \Theta R + , is embedded into the generalized Lorentz group SO o (n; 1) via the Iwasawa decomposition, so that X ' SO o (n; 1)=N , where N ' R n\Gamma1 . Then the CWT on S n\Gamma1 is derived from a suitable unitary representation of SO o (n; 1) acting in the space L 2 (S n\Gamma1 ; d¯) of finite energy signals on S n\Gamma1 , which turns out to be square integrable over X . We find a necessary condition for the admissibility of a wavelet, in the form of a zero mean condition, which entails all the usual filtering properties of the CWT. Next the Euclidean limit of this CWT on S n\Gamma1 is obtained by redoing the construction on a sphere of radius R and performing a group contraction for R !...
Kernels of Spherical Harmonics and Spherical Frames
 in Advanced Topics in Multivariate Approximation
, 1996
"... . Our concern is with the construction of a frame in L 2 (S) consisting of smooth functions based on kernels of spherical harmonics. The corresponding decomposition and reconstruction algorithms utilize discrete spherical Fourier transforms. Numerical examples confirm the theoretical expectations. ..."
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Cited by 11 (0 self)
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. Our concern is with the construction of a frame in L 2 (S) consisting of smooth functions based on kernels of spherical harmonics. The corresponding decomposition and reconstruction algorithms utilize discrete spherical Fourier transforms. Numerical examples confirm the theoretical expectations. x1. Introduction Traditionally, wavelets were tailored to problems on the Euclidean space IR d . However, in most applications one has to analyze functions defined on compact domains. In particular, in geophysics wavelets on the unit sphere S of IR 3 are of interest. There exist different approaches to the constructions of spherical wavelets. Having spherical coordinates in mind, the idea of using tensorproducts of periodic wavelets and wavelets on the interval was suggested in [7]. Applying tensorproducts of periodic exponential splinewavelets and splinewavelets on the interval, wavelets on S were constructed in [3]. Unfortunately, tensorproduct wavelets can possess singular...
Orthogonal and Nonorthogonal Multiresolution Analysis, Scale Discrete and Exact Fully Discrete Wavelet Transform on the Sphere
, 1998
"... Based on a new definition of dilation a scale discrete version of spherical multiresolution is described, starting from a scale discrete wavelet transform on the sphere. Depending on the type of application, different families of wavelets are chosen. In particular, spherical Shannon wavelets are con ..."
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Cited by 11 (3 self)
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Based on a new definition of dilation a scale discrete version of spherical multiresolution is described, starting from a scale discrete wavelet transform on the sphere. Depending on the type of application, different families of wavelets are chosen. In particular, spherical Shannon wavelets are constructed that form an orthogonal multiresolution analysis. Finally fully discrete wavelet approximation is discussed in case of bandlimited wavelets. Abbreviated title: Scale Discrete Wavelets AMS Subject classification: 41A58, 42C15, 44A35, 45E99 Key words: Spherical multiresolution analysis, scaling function, scale discrete wavelets, rational wavelets, exponential wavelets, orthogonal (Shannon) wavelets, de la Vall'e Poussin wavelets, exact fully discrete wavelet transform 1 This research was supported by "Stiftung RheinlandPfalz fur Innovation" i Contents 1 Introduction 1 2 Preliminaries 2 2.1 Spherical Harmonics : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : ...
A Framework for Interpolation and Approximation on Riemannian Manifolds
"... In this paper we provide a framework for studying the approximation order resulting from using strictly positive definite kernels to do generalized Hermite interpolation and approximation on a compact Riemannian manifold. We apply this framework to obtain explicit estimates in cases of the circle an ..."
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Cited by 7 (1 self)
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In this paper we provide a framework for studying the approximation order resulting from using strictly positive definite kernels to do generalized Hermite interpolation and approximation on a compact Riemannian manifold. We apply this framework to obtain explicit estimates in cases of the circle and 2sphere. In addition, we provide a technique for constructing strictly positive definite spherical functions out of radial basis functions, and we use it to make a spherical function that is locally supported.