Olhede, “Multiple multidimensional morse wavelets

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 eigen-functions 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 re-gion 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 eigen-functions 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, orienta-tion and phase structure of locally oscillatory signals. The full set of quaternionic wavelet functions are component by component or-thogonal; 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 es-timators across wavelets, a substantial reduction in the variance is achieved. Index Terms—Analytic signal, image analysis, monogenic signal, Riesz transform, scalogram, wavelets. I