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FAST EXPANSION INTO HARMONICS ON THE DISK: A STEERABLE BASIS WITH FAST RADIAL CONVOLUTIONS.

Nicholas F Marshall1, Oscar Mickelin2, Amit Singer3

  • 1Department of Mathematics, Oregon State University, Corvallis, OR 97330 USA.

SIAM Journal on Scientific Computing : a Publication of the Society for Industrial and Applied Mathematics
|August 15, 2024
PubMed
Summary
This summary is machine-generated.

We developed a Fast Disk Harmonics Transform (FDHT) for image analysis on disks. This method efficiently expands images in the Fourier-Bessel basis, enabling faster computations and image rotations.

Keywords:
33C1042-0465D1865R10Fourier–Bessel basisLaplacian eigenfunctionssteerable basis

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Area of Science:

  • * Digital image processing
  • * Applied mathematics
  • * Harmonic analysis

Background:

  • * Functions on disks are challenging to represent efficiently.
  • * Existing methods lack speed and numerical accuracy for disk-supported images.
  • * The Fourier-Bessel basis offers advantageous properties like orthogonality and frequency ordering.

Purpose of the Study:

  • * To introduce a fast and numerically accurate method for expanding digitized images on a disk.
  • * To leverage the properties of the Fourier-Bessel basis for computational efficiency.
  • * To enable efficient image rotation and radial function convolution.

Main Methods:

  • * Development of the Fast Disk Harmonics Transform (FDHT).
  • * Expansion of digitized images on a disk using Dirichlet Laplacian eigenfunctions (Fourier-Bessel basis).
  • * Utilization of diagonal transforms for coefficient manipulation.

Main Results:

  • * The FDHT achieves computational complexity of O(N log N) operations.
  • * Demonstrated efficient image rotation through diagonal transforms on coefficients.
  • * Showcased efficient computation of convolutions with radial functions via diagonal transforms.

Conclusions:

  • * The FDHT provides a computationally efficient and accurate tool for analyzing disk-supported images.
  • * The method's steerability and efficient convolution capabilities offer significant advantages in image processing.
  • * This work advances the field of harmonic analysis on disks with practical computational benefits.