The monogenic wavelet transform
IEEE Transactions on Signal Processing
Multiresolution monogenic signal analysis using the Riesz-Laplace wavelet transform
IEEE Transactions on Image Processing
Wavelet steerability and the higher-order Riesz transform
IEEE Transactions on Image Processing
Steerable wavelet frames based on the Riesz transform
IEEE Transactions on Image Processing
Quaternion multiplier inspired by the lifting implementation of plane rotations
IEEE Transactions on Circuits and Systems Part I: Regular Papers
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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 L2(Ropf2) 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