CURVELET-WAVELET REGULARIZED SPLIT BREGMAN ITERATION FOR COMPRESSED SENSING
Compressed sensing is a new concept in signal processing. Assuming that a signal can be represented or approximated by only a few suitably chosen terms in a frame expansion, compressed sensing allows one to recover this signal from much fewer samples than the Shannon-Nyquist theory requires. Many images can be sparsely approximated in expansions of suitable frames as wavelets, curvelets, wave atoms and others. Generally, wavelets represent point-like features while curvelets represent line-like features well. For a suitable recovery of images, we propose models that contain weighted sparsity constraints in two different frames. Given the incomplete measurements f = Phi u vertical bar c with the measurement matrix Phi is an element of R(KxN), K << N, we consider a jointly sparsity-constrained optimization problem of the form argmin(u) {parallel to Lambda(c) Psi (c)u parallel to 1 + parallel to Lambda(w) Psi (w)u parallel to 1 + 1/2 parallel to f - Phi u parallel to(2)(2)}. Here Psi(c) and Psi(w) are the transform matrices corresponding to the two frames, and the diagonal matrices Lambda(c), Lambda(w) contain the weights for the frame coefficients. We present efficient iteration methods to solve the optimization problem, based on Alternating Split Bregman algorithms. The convergence of the proposed iteration schemes will be proved by showing that they can be understood as special cases of the Douglas-Rachford Split algorithm. Numerical experiments for compressed sensing-based Fourier-domain random imaging show good performances of the proposed curvelet-wavelet regularized split Bregman (CWSpB) methods, where we particularly use a combination of wavelet and curvelet coefficients as sparsity constraints.
curvelets compressed sensing douglas-rachford split algorithm compressive sampling alternating split bregman iteration iterative shrinkage/thresholding (ist) remote sensing cs imaging