Fast expansion into harmonics on the disk: a steerable basis with fast radial convolutions

TL;DR

This paper introduces the Fast Disk Harmonics Transform (FDHT), a fast and accurate method for expanding images on the disk in harmonic bases, enabling efficient rotation and convolution operations.

Contribution

The paper presents a novel $O(L^2 \, \log L)$ algorithm for expanding images in the Fourier-Bessel basis, with steerability and fast radial convolution capabilities.

Findings

Runs in $O(L^2 \log L)$ operations

Enables efficient rotation of images via diagonal coefficient transforms

Allows fast convolution with radial functions

Abstract

We present a fast and numerically accurate method for expanding digitized $L \times L$ images representing functions on $[-1,1]^2$ supported on the disk $\{x \in \mathbb{R}^2 : |x|<1\}$ in the harmonics (Dirichlet Laplacian eigenfunctions) on the disk. Our method, which we refer to as the Fast Disk Harmonics Transform (FDHT), runs in $O(L^2 \log L)$ operations. This basis is also known as the Fourier-Bessel basis, and it has several computational advantages: it is orthogonal, ordered by frequency, and steerable in the sense that images expanded in the basis can be rotated by applying a diagonal transform to the coefficients. Moreover, we show that convolution with radial functions can also be efficiently computed by applying a diagonal transform to the coefficients.