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 × 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.