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

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 $\{x\in {\mathbb{R}}^2:\mathrm{\mid }x\mathrm{\mid }<1\}$ 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.