Compressed Sensing Shape Estimation of Star-Shaped Objects in Fourier Imaging

Recent theory of compressed sensing informs us that near-exact recovery of an unknown sparse signal is possible from a very limited number of Fourier samples by solving a convex L₁ optimization problem. The main contribution of the present letter is a compressed sensing-based novel nonparametric shape estimation framework and a computational algorithm for binary star shape objects, whose radius functions belong to the space of bounded-variation functions. Specifically, in contrast with standard compressed sensing, the present approach involves directly reconstructing the shape boundary under sparsity constraint. This is done by converting the standard pixel-based reconstruction approach into estimation of a nonparametric shape boundary on a wavelet basis. This results in an L₁ minimization under a nonlinear constraint, which makes the optimization problem nonconvex. We solve the problem by successive linearization and application of one-dimensional L₁ minimization, which significantly reduces the number of sampling requirements as well as the computational burden. Fourier imaging simulation results demonstrate that high quality reconstruction can be quickly obtained from a very limited number of samples. Furthermore, the algorithm outperforms the standard compressed sensing reconstruction approach using the total variation norm.