Fast Sparse 2-D DFT Computation Using Sparse-Graph Alias Codes

We present a novel algorithm, named the 2D-FFAST (Two-dimensional Fast Fourier Aliasing-based Sparse Transform), to compute a sparse 2D-Discrete Fourier Transform (2D-DFT) featuring both low sample and computational complexity. The proposed algorithm is based on diverse concepts from signal processing (sub-sampling and aliasing), coding theory (sparse-graph codes) and number theory (Chinese-remainder-theorem) and generalizes the 1D-FFAST algorithm recently proposed by Pawar and Ramchandran to the 2D setting. Concretely, our proposed 2D-FFAST algorithm computes a k-sparse 2D-DFT, with a uniformly random support, of size N = N x × N y using O(k) noiseless spatial-domain measurements in O(k log k) computational time. Our results are attractive when the sparsity is sub-linear with respect to the signal dimension, that is, when k → ∞ and k/N → 0. For the case when the spatial-domain measurements are corrupted by additive noise, our 2D-FFAST framework extends to a noise-robust version of computing a 2D-DFT using O(k log 3 N) measurements in sub-linear time of O(k log 4 N). Empirically, we show that the 2D-FFAST can compute a k = 3509 sparse 2D-DFT of a 508 × 508-size phantom image using only 4.75k measurements. We also empirically evaluate the 2D-FFAST algorithm on a real-world magnetic resonance brain image using a total of 60.18% of Fourier measurements to provide an almost instant reconstruction with SNR=4.5 dB. This provides empirical evidence that the 2D-FFAST architecture is applicable to a wider class of input signals than analyzed theoretically in the paper.