Wavelets invariant under finite reflection groups

Bernardes, Gil; Bernstein, Swanhild; Cerejeiras, Paula; Kähler, Uwe

doi:10.1002/mma.1220

Cited by 3 publications

(3 citation statements)

References 20 publications

(19 reference statements)

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“…Another approach is the use of the Dunkl setting to achieve invariance under finite reflection groups (cf. [6] and the references therein) which finds applications in crystallography.…”

Section: Introductionmentioning

confidence: 99%

Spherical fast multiscale approximation by locally compact orthogonal wavelets

Bauer¹,

Gutting

2011

Int J Geomath

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Using a stereographical projection to the plane we construct an O(N log(N )) algorithm to approximate scattered data in N points by orthogonal, compactly supported wavelets on the surface of a 2-sphere or a local subset of it. In fact, the sphere is not treated all at once, but is split into subdomains whose results are combined afterwards. After choosing the center of the area of interest the scattered data points are mapped from the sphere to the tangential plane through that point. By combining a k-nearest neighbor search algorithm and the two dimensional fast wavelet transform a fast approximation of the data is computed and mapped back to the sphere. The algorithm is tested with nearly 1 million data points and yields an approximation with 0.35% relative errors in roughly 2 minutes on a standard computer using our MATLAB R ⃝implementation. The method is very flexible and allows the application of the full range of two dimensional wavelets.

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“…Another approach is the use of the Dunkl setting to achieve invariance under finite reflection groups (cf. [6] and the references therein) which finds applications in crystallography.…”

Section: Introductionmentioning

confidence: 99%

Spherical fast multiscale approximation by locally compact orthogonal wavelets

Bauer¹,

Gutting

2011

Int J Geomath

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“…To incorporate crystal symmetry, i.e. invariance under point groups (as finite reflection groups are called in crystallography) spherical wavelets which are invariant under finite reflection groups had been constructed with the help of the theory of Dunkl operators 9. In this paper we will use the diffusion semi‐group to construct wavelets on conformally flat tori.…”

Section: Introductionmentioning

confidence: 99%

On the diffusion equation and diffusion wavelets on flat cylinders and the n-torus

Bernstein

Ebert

Kraußhar

2010

Math. Meth. Appl. Sci.

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In this paper we study the solutions to the diffusion equation on some conformally flat cylinders and on the n-torus. Using the Clifford algebra calculus with an appropriate Witt basis, the solutions can be expressed as multiperiodic eigensolutions to the parabolic Dirac operator. We study their fundamental properties, give representation formulas of all these solutions and develop some integral representation formulas. In particular we set up a Green type formula for the solutions to the homogeneous diffusion equation on cylinders and tori. Then we also treat the inhomogeneous diffusion equation diffusion with prescribed boundary conditions in Lipschitz domains on these manifolds. As main application, we construct well localized diffusion wavelets on this class of cylinders and tori by means of multiperiodic eigensolutions to the parabolic Dirac operator. We round off with presenting some concrete numerical simulations for the three dimensional case.

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“…The combination of this idea with the approach by Freeden makes it possible to construct spherical Dunkl wavelets. The praciticallity of this approach has been shown for Abel-Poisson wavelets in [6].…”

Section: Introductionmentioning

confidence: 99%