Non-Iterative Compressed Sensing Using a Minimal Number of Fourier Transform Values

Reconstruction of signals or images from a few discrete Fourier transform (DFT) values has applications in MRI and SAR. Many real-world signals can be sparsified by an invertible transformation, such as wavelets, into a sparse (mostly zero, with K nonzero values at unknown locations) signal. This Ksparse signal can be reconstructed using the K lowestfrequency DFT values using Prony’s method or MUSIC (2K frequencies are required for complex signals or real-valued images). However, this does not work in practice due to poor conditioning caused by the clustering of the locations of the K nonzero values. We use the scaling property of the DFT to uncluster these locations and to spread out the frequencies of known DFT values. We reconstruct a Shepp-Logan phantom using only 2K DFT values, much fewer than the number required by 1 norm minimization, and using less computation than 1 norm minimization. Keywords— Sparse reconstruction Email: [email protected]. EDICS: 2-REST.