Papers › Fast expansion into harmonics on the disk: a steerable basis with fast radial convolutions
Fast expansion into harmonics on the disk: a steerable basis with fast radial convolutions
Nicholas F. Marshall, Oscar Mickelin, Amit Singer
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 logL) 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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