Adaptive continuation based smooth l0-norm approximation for compressed sensing MR image reconstruction.

Purpose: There are a number of algorithms for smooth $l_0$-norm (SL0) approximation. In most of the cases, sparsity level of the reconstructed signal is controlled by using a decreasing sequence of the modulation parameter values. However, predefined decreasing sequences of the modulation parameter values cannot produce optimal sparsity or best reconstruction performance, because the best choice of the parameter values is often data-dependent and dynamically changes in each iteration.

Approach: We propose an adaptive compressed sensing magnetic resonance image reconstruction using the SL0 approximation method. The SL0 approach typically involves one-step gradient descent of the SL0 approximating function parameterized with a modulation parameter, followed by a projection step onto the feasible solution set. Since the best choice of the parameter values is often data-dependent and dynamically changes in each iteration, it is preferable to adaptively control the rate of decrease of the parameter values. In order to achieve this, we solve two subproblems in an alternating manner. One is a sparse regularization-based subproblem, which is solved with a precomputed value of the parameter, and the second subproblem is the estimation of the parameter itself using a root finding technique.

Results: The advantage of this approach in terms of speed and accuracy is illustrated using a compressed sensing magnetic resonance image reconstruction problem and compared with constant scale factor continuation based SL0-norm and adaptive continuation based $l_1$-norm minimization approaches. The proposed adaptive estimation is found to be at least twofold faster than automated parameter estimation based iterative shrinkage-thresholding algorithm in terms of CPU time, on an average improvement of reconstruction performance 15% in terms of normalized mean squared error.

Conclusions: An adaptive continuation-based SL0 algorithm is presented, with a potential application to compressed sensing (CS)-based MR image reconstruction. It is a data-dependent adaptive continuation method and eliminates the problem of searching for appropriate constant scale factor values to be used in the CS reconstruction of different types of MRI data.