Multiple Multidimensional Morse Wavelets

Abstract

Abstract-This paper defines a set of operators that localize a radial image in space and radial frequency simultaneously. The eigenfunctions of the operator are determined and a nonseparable orthogonal set of radial wavelet functions are found. The eigenfunctions are optimally concentrated over a given region of radial space and scale space, defined via a triplet of parameters. Analytic forms for the energy concentration of the functions over the region are given. The radial function localization operator can be generalised to an operator localizing any 2 ( 2 ) function. It is demonstrated that the latter operator, given an appropriate choice of localization region, approximately has the same radial eigenfunctions as the radial operator. Based on a given radial wavelet function a quaternionic wavelet is defined that can extract the local orientation of discontinuous signals as well as amplitude, orientation and phase structure of locally oscillatory signals. The full set of quaternionic wavelet functions are component by component orthogonal; their statistical properties are tractable, and forms for the variability of the estimators of the local phase and orientation are given, as well as the local energy of the image. By averaging estimators across wavelets, a substantial reduction in the variance is achieved.

Wavelet Steerability and the Higher-Order Riesz Transform

“…Felsberg used the transform to define the "monogenic signal" as a 2-D generalization of the analytic signal representation [30], [31]. This pioneering work inspired Metikas and Olhede to specify a monogenic version of the continuous wavelet transform which is a fully redundant signal representation [32]. At about the same time as Felsberg, Larkin independently introduced a complexified version of the 2-D Riesz transform in optics-calling it the spiral quadrature phase transform-and applied it to the demodulation of interferograms and the analysis of fringe patterns [33], [34].…”

Multiresolution Monogenic Signal Analysis Using the Riesz–Laplace Wavelet Transform

“…This combined with the fact that the polyharmonic splines for provide a valid multiresolution analysis of yields the theorem below, which is a slight extension of previously published results. Specifically, Bacchelli et al [35], [38] investigated the dyadic, nonfractional case (i.e., and ), while Van De Ville et al [34] (23) which are termed primal or dual depending on the type of reconstruction wavelets. The complementary wavelet functions are the unique duals of in the sense that they satisfy the biorthogonality property: with .…”

“…The monogenic signal is also closely linked to the spiral quadrature phase transform in optics which constitutes a powerful tool for the processing of fringe patterns and the demodulation of interferograms [20]- [22]. Recently, Metikas and Olhede have transposed the approach to the wavelet-domain, albeit within the framework of the continuous wavelet transform which is an overly-redundant representation [23]. In practice, the monogenic analysis is often performed on some bandpass-filtered versions of the input signal which also leads to the idea of multiresolution [15], [24].…”

Hyperanalytic Denoising

“…The RTs have been used in combination with the continuous wavelet transform (CWT) by Metikas and Olhede [16], [17]. The RTs of , denoted by and are calculated by convolving with the Riesz kernels , for 1, 2, given with and by…”

“…If the signal locally takes the form of (13), then the -magnitude given in Definition 3.1, is (16) where , is an error term depending on the leakage of the wavelet filters in the frequency domain, and the follows due to the approximation of the DFT.…”