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

Code

• github.com/nmarshallf/fle_2d
  OfficialIn papernone★ 7

Abstract

We present a fast and numerically accurate method for expanding digitized L L images representing functions on [-1,1]^2 supported on the disk 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 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.

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