Compressed Sensing from Several Sets of Downsampled Fourier Values using Only FFTs

Reconstruction of signals or images from a few Discrete Fourier Transform (DFT) values has applications in MRI and SAR. Compressed sensing is the reconstruction from a reduced set of observations of a signal or image that can be sparsified. Many realworld signals and images may be sparsified by convolution with a differencing operator, such as a wavelet; this multiplies the given DFT values by its known frequency response. We present a simple procedure for reconstructing the signal or image from a few sets of downsampled DFT values, using only the Fast Fourier Transform (FFT) algorithm and some multiplications. Three examples and Matlab programs are provided. Keywords— Sparse reconstruction Phone: 734-763-9810. Fax: 734-763-1503. Email: [email protected]. EDICS: 2-REST.