&lt;title&gt;Cosine transform generalized to lie groups SU(2)xSU(2), O(5), and SU(2)xSU(2)xSU(2): application to digital image processing&lt;/title&gt;

Germain, Mickaël; Patera, J.; Allard, Yannick

doi:10.1117/12.650866

Cited by 5 publications

(10 citation statements)

References 5 publications

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“…Values of E + m (x) on other points are obtained by using the relation (13). The exponential functions E − m (x) with integral m = (m 1 , m 2 , .…”

Section: 2mentioning

confidence: 99%

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(Anti)symmetric multivariate exponential functions and corresponding Fourier transforms

Klimyk¹,

Patera²

2007

J. Phys. A: Math. Theor.

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Abstract. We define and study symmetrized and antisymmetrized multivariate exponential functions. They are defined as determinants and antideterminants of matrices whose entries are exponential functions of one variable. These functions are eigenfunctions of the Laplace operator on corresponding fundamental domains satisfying certain boundary conditions. To symmetric and antisymmetric multivariate exponential functions there correspond Fourier transforms. There are three types of such Fourier transforms: expansions into corresponding Fourier series, integral Fourier transforms, and multivariate finite Fourier transforms. Eigenfunctions of the integral Fourier transforms are found.

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“…Values of E + m (x) on other points are obtained by using the relation (13). The exponential functions E − m (x) with integral m = (m 1 , m 2 , .…”

Section: 2mentioning

confidence: 99%

“…In fact, symmetric and antisymmetric exponential functions are connected with orbit functions corresponding to the Coxeter-Dynkin diagram A n . Discrete orbit function transforms, corresponding to Coxeter-Dynkin diagrams of low order, were detailly studied and it was shown that they are very useful for applications [5]- [13].…”

Section: Introductionmentioning

confidence: 99%

(Anti)symmetric multivariate exponential functions and corresponding Fourier transforms

Klimyk¹,

Patera²

2007

J. Phys. A: Math. Theor.

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“…Its various mathematical applications are found in [4,5,6,7,8,9,10,11,12,13,14,15,16]. For other applications, see [15,16,17,18,19,20]. Existence of the continuous version of the transform is implied there as well.…”

Section: Introductionmentioning

confidence: 99%

“…It is discrete transforms that primarily interest us here because they have a number of practically useful properties [2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20]. In particular, continuous extension of the discrete transforms smoothly interpolate digital data in any dimension and for any lattice symmetry afforded by the structure of the given Lie group G. Many examples show that relative to the amount of available data, these transforms provide much smoother interpolation than the conventional Fourier transform.…”

Section: Introductionmentioning

confidence: 99%

Orthogonality within the Families of C-, S-, and E-Functions of Any Compact Semisimple Lie Group

Moody

Patera

2006

SIGMA

115

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Abstract. The paper is about methods of discrete Fourier analysis in the context of Weyl group symmetry. Three families of class functions are defined on the maximal torus of each compact simply connected semisimple Lie group G. Such functions can always be restricted without loss of information to a fundamental regionF of the affine Weyl group. The members of each family satisfy basic orthogonality relations when integrated overF (continuous orthogonality). It is demonstrated that the functions also satisfy discrete orthogonality relations when summed up over a finite grid inF (discrete orthogonality), arising as the set of points inF representing the conjugacy classes of elements of a finite Abelian subgroup of the maximal torus T. The characters of the centre Z of the Lie group allow one to split functions f onF into a sum f = f 1 + · · · + f c , where c is the order of Z, and where the component functions f k decompose into the series of C-, or S-, or E-functions from one congruence class only.

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“…This discrete transform is useful in many things related to discretization (see [13,14,15,16,17,18,19,20]). Construction of the discrete orbit function transform is fulfilled by means of the results of paper [21].…”

Section: Introductionmentioning

confidence: 99%

Antisymmetric Orbit Functions

Klimyk¹

2007

SIGMA

148

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Abstract. In the paper, properties of antisymmetric orbit functions are reviewed and further developed. Antisymmetric orbit functions on the Euclidean space E n are antisymmetrized exponential functions. Antisymmetrization is fulfilled by a Weyl group, corresponding to a Coxeter-Dynkin diagram. Properties of such functions are described. These functions are closely related to irreducible characters of a compact semisimple Lie group G of rank n. Up to a sign, values of antisymmetric orbit functions are repeated on copies of the fundamental domain F of the affine Weyl group (determined by the initial Weyl group) in the entire Euclidean space E n . Antisymmetric orbit functions are solutions of the corresponding Laplace equation in E n , vanishing on the boundary of the fundamental domain F . Antisymmetric orbit functions determine a so-called antisymmetrized Fourier transform which is closely related to expansions of central functions in characters of irreducible representations of the group G. They also determine a transform on a finite set of points of F (the discrete antisymmetric orbit function transform). Symmetric and antisymmetric multivariate exponential, sine and cosine discrete transforms are given.

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<title>Cosine transform generalized to lie groups SU(2)xSU(2), O(5), and SU(2)xSU(2)xSU(2): application to digital image processing</title>

Cited by 5 publications

References 5 publications

(Anti)symmetric multivariate exponential functions and corresponding Fourier transforms

(Anti)symmetric multivariate exponential functions and corresponding Fourier transforms

Orthogonality within the Families of C-, S-, and E-Functions of Any Compact Semisimple Lie Group

Antisymmetric Orbit Functions

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