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

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.