Continuous wavelet transform on a special homogeneous space

Fashandi, Massoumeh; Gol, Rajab Ali Kamyabi; Niknam, Assadollah; Pourabdollah, M. A.

doi:10.1063/1.1591055

Cited by 9 publications

(6 citation statements)

References 9 publications

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“…The case where the underlying group is R n has been studied e.g. in [2,9,15], the generalization to arbitrary LCA groups was considered in [4].…”

Section: Wavelet Transforms From Semidirect Productsmentioning

confidence: 99%

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Generalized Calderón conditions and regular orbit spaces

Führ¹

2010

Colloq. Math.

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The construction of generalized continuous wavelet transforms on locally compact abelian groups A from quasi-regular representations of a semidirect product group G = A ⋊ H acting on L 2 (A) requires the existence of a square-integrable function whose Plancherel transform satisfies Calderón-type resolution of the identity. The question then arises under what conditions such square-integrable functions exist.The existing literature on this subject leaves a gap between sufficient and necessary criteria. In this paper, we give a characterization in terms of the natural action of the dilation group H on the character group of A. We first prove that a Calderón-type resolution of the identity gives rise to a decomposition of Plancherel measure of A into measures on the dual orbits, and then show that the latter property is equivalent to regularity conditions on the orbit space of the dual action.Thus we obtain, for the first time, sharp necessary and sufficient criteria for the existence of a wavelet inversion formula. As a byproduct and special case of our results we obtain that discrete series subrepresentations of the quasiregular representation correspond precisely to dual orbits with positive Plancherel measure and associated compact stabilizers. Only sufficiency of the conditions was previously known.

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“…The case where the underlying group is R n has been studied e.g. in [2,9,15], the generalization to arbitrary LCA groups was considered in [4].…”

Section: Wavelet Transforms From Semidirect Productsmentioning

confidence: 99%

“…[2,8,9,15]. A further extension, replacing R d by a general locally compact group A and H by a group of topological automorphisms, was considered in [4].…”

Section: Introductionmentioning

confidence: 99%

Generalized Calderón conditions and regular orbit spaces

Führ¹

2010

Colloq. Math.

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“…This theorem and other generalizations of this theorem, such as [29], [17] are not applicable to the reducible left regular action of the Euclidean motion group onto ‫ތ‬ 2 ‫ޒ(‬ d ) given by (2.6), which is needed for our orientation score application in image analysis. Therefore, we give a true generalization to the wavelet theorem, where irreducibility is neither a requirement nor replaced by another requirement.…”

Section: × τ ‫ޔ‬mentioning

confidence: 99%

“…Let ψ be a proper wavelet; then there exists a 1-to-1 correspondence between bounded operators Φ ∈ Ꮾ( ) on orientation scores and bounded operators ϒ ∈ Ꮾ(‫ތ‬ 2 ‫ޒ(‬ d )) on band-limited images, (8.66) which allows us to relate operations on orientation scores to operations on images in a robust manner 17 .…”

Section: Image Enhancement By Means Of Left-invariant Operations On Omentioning

confidence: 99%

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Invertible orientation scores as an application of generalized wavelet theory

Duits¹,

Duits²,

Almsick³

et al. 2007

Pattern Recognit. Image Anal.

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-Inspired by the visual system of many mammals, we consider the construction of-and reconstruction from-an orientation score of an image, via a wavelet transform corresponding to the left-regular representation of the Euclidean motion group in ‫ތ‬ 2 ( ‫ޒ‬ 2 ) and oriented wavelet ψ ∈ ‫ތ‬ 2 ( ‫ޒ‬ 2 ). Because this representation is reducible, the general wavelet reconstruction theorem does not apply. By means of reproducing kernel theory, we formulate a new and more general wavelet theory, which is applied to our specific case. As a result we can quantify the well-posedness of the reconstruction given the wavelet ψ and deal with the question of which oriented wavelet ψ is practically desirable in the sense that it both allows a stable reconstruction and a proper detection of local elongated structures. This enables image enhancement by means of left-invariant operators on orientation scores.

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Wavelet Transforms for Semidirect Product Groups with Not Necessarily Commutative Normal Subgroups

Ishi

2006

J Fourier Anal Appl

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Let G be the semidirect product group of a separable locally compact unimodular group N of type I with a closed subgroup H of Aut(N ). The group N is not necessarily commutative. We consider irreducible subrepresentations of the unitary representation of G realized naturally on L 2 (N ), and investigate the wavelet transforms associated to them. Furthermore, the irreducible subspaces are characterized by certain singular integrals on N analogous to the Cauchy-Szegö integral.Math Subject Classifications. 42C40, 22D10, 43A32.

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Continuous wavelet transform on a special homogeneous space

Cited by 9 publications

References 9 publications

Generalized Calderón conditions and regular orbit spaces

Generalized Calderón conditions and regular orbit spaces

Invertible orientation scores as an application of generalized wavelet theory

Wavelet Transforms for Semidirect Product Groups with Not Necessarily Commutative Normal Subgroups

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