Abstract
This paper presents a systematic study of the analytic aspects of Fourier–Zernike series of convolutions of functions supported on disks. We then investigate different aspects of the presented theory in the cases of zero-padded functions.
Highlights
The mathematical theory of convolution function algebras plays significant roles in classical harmonic analysis, representation theory, functional analysis, and operator theory; see [1–8] and the references therein
If the functions are compactly supported, their supports are enclosed in a solid cube with dimensions at least twice the size of the support of the functions, and periodic versions of the functions are constructed. Convolution of these periodic functions on the d-torus can be used to replace convolution on d-dimensional Euclidean space. The benefit of this is that the spectrum is discretized, and fast Fourier transform (FFT) methods can be used to compute the convolutions
We present a constructive closed form for Fourier–Zernike coefficients of convolution functions supported on disks
Summary
The mathematical theory of convolution function algebras plays significant roles in classical harmonic analysis, representation theory, functional analysis, and operator theory; see [1–8] and the references therein. The benefit of this is that the spectrum is discretized, and fast Fourier transform (FFT) methods can be used to compute the convolutions This approach is computationally attractive, but in this periodization procedure, the natural invariance of integration on Euclidean space under rotation transformations is lost when moving to the torus. Replacing each of the given functions with a sum of Gaussian distributions allows the convolution of the given functions to be computed as a sum of convolutions of Gaussians, which have simple closed-form expressions as Gaussians The problem with this approach is that the resulting functions are not compactly supported. We present a constructive closed form for Fourier–Zernike coefficients of convolution functions supported on disks. We employ this closed form to present a constructive Fourier–Zernike approximation for convolution of zero-padded functions on R2
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