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

arXiv: https://arxiv.org/abs/2207.13674 · HTML: https://mlsanity.com/paper/2207.13674/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 8 of its 49 references that are in the map.
- First posted on arXiv 27 Jul 2022.

## Builds on (most cited first, 8 of 8)

- [Steerable CNNs](https://mlsanity.com/paper/1612.08498/) (2016)
- [Learning Steerable Filters for Rotation Equivariant CNNs](https://mlsanity.com/paper/1711.07289/) (2017)
- [Rotationally Invariant Image Representation for Viewing Direction Classification in Cryo-EM](https://mlsanity.com/paper/1309.7643/) (2013)
- [RotDCF: Decomposition of Convolutional Filters for Rotation-Equivariant Deep Networks](https://mlsanity.com/paper/1805.06846/) (2018)
- [Fast Steerable Principal Component Analysis](https://mlsanity.com/paper/1412.0781/) (2014)
- [cuFINUFFT: a load-balanced GPU library for general-purpose nonuniform FFTs](https://mlsanity.com/paper/2102.08463/) (2021)
- [Denoising and Covariance Estimation of Single Particle Cryo-EM Images](https://mlsanity.com/paper/1602.06632/) (2016)
- [Steerable Principal Components for Space-Frequency Localized Images](https://mlsanity.com/paper/1608.02702/) (2016)

## Built on by (most cited first, 1 of 1)

- [Selective Disk Bispectrum: A Complete and Rotation Invariant Image Descriptor](https://mlsanity.com/paper/2511.19706/) (2025)

## Abstract (arXiv)

We present a fast and numerically accurate method for expanding digitized L × L images representing functions on [-1,1]2 supported on the disk x ∈ ℝ2 : |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(L2 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.