Gradient-based Image Recovery Methods from Incomplete Fourier Measurements

A major problem in imaging applications such as Magnetic Resonance Imaging (MRI) and Synthetic Aperture Radar (SAR) is the task of trying to reconstruct an image with the smallest possible set of Fourier samples, every single one of which has a potential time and/or power cost. The theory of Compressive Sensing (CS) points to ways of exploiting inherent sparsity in such images in order to achieve accurate recovery using sub- Nyquist sampling schemes. Traditional CS approaches to this problem consist of solving total-variation minimization programs with Fourier measurement constraints or other variations thereof. This paper takes a different approach: Since the horizontal and vertical differences of a medical image are each more sparse or compressible than the corresponding total-variational image, CS methods will be more successful in recovering these differences individually. We develop an algorithm called GradientRec that uses a CS algorithm to recover the horizontal and vertical gradients and then estimates the original image from these gradients. We present two methods of solving the latter inverse problem: one based on least squares optimization and the other based on a generalized Poisson solver. After a thorough derivation of our complete algorithm, we present the results of various experiments that compare the effectiveness of the proposed method against other leading methods.