Parametrizing compactly supported orthonormal wavelets by discrete moments

Abstract

We discuss parametrizations of filter coefficients of scaling functions and compactly supported orthonormal wavelets with several vanishing moments. We introduce the first discrete moments of the filter coefficients as parameters. The discrete moments can be expressed in terms of the continuous moments of the related scaling function. To solve the resulting polynomial equations we use symbolic computation and in particular Gröbner bases. The cases of four to ten filter coefficients are discussed and explicit parametrizations are given.

Wavelet kernel learning

“…We want to emphasize that the framework can be applied to other parametrizations like those based on moments [29] or even to continuous wavelet transform [24]. Algorithmically, the only thing that would change is the way a kernel related to a given wavelet is computed.…”

Improved Technique for the Construction of Parametric M-Band Wavelets

“…The parameterization is offered by the two Rotation matrices. The condition imposed on the two parameters is that 4 π = + u t (14) III. BUILDING A PARA-UNITARY MATRIX Having made a convincing approach to the design, the problem, which is now left out, is the construction of Paraunitary matrix of any given order The matrix, which is under context, is the one that renders the power of extending the proposed method to M-Band system.…”

“…Several authors [2,[9][10][11][12][13][14][15] have studied parameterization of compactly supported orthonormal wavelets extensively. Conventionally, parameterization is achieved either by factoring polyphase matrix [16] corresponding to analysis filters using Householder transformation or by employing lifting scheme.…”

Ecuaciones inversas de filtros de reconstrucción perfecta

“…Se han publicado varios FRP como los Daubechies, los Coiflets y los Symlets [14]. Y también se han publicado ecuaciones paramétricas de filtros que permiten generar FRP [11,16,15,13,8,10].…”

“…Además, se han propuesto varias parametrizaciones, así como métodos de parametrización. En[11] se discute la parametrización de coeficientes de funciones de escalamiento (ortogonales a las funciones wavelet) con varios momentos de desvanecimiento. En[16] se presentan parametrizaciones de wavelets ortonormales con soporte en el intervalo [0, 2N − 1) en donde N es un entero par que involucra N − 1 parámetros que varían en el intervalo [0, 2π).En cuanto a aplicaciones de las ecuaciones paramétricas, en[3] se utilizan parametrizaciones de filtros y se presenta una técnica para calcular la mejor wavelet para una imagen dada.…”