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

07/27/2022

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by   Nicholas F. Marshall, et al.

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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 ∈ℝ^2 : |x|<1} in the harmonics (Dirichlet Laplacian eigenfunctions) on the disk. Our method 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.