A novel hybrid total variation minimization algorithm for compressed sensing

Compressed sensing (CS) is a technology to acquire and reconstruct sparse signals below the Nyquist rate. For images, total variation of the signal is usually minimized to promote sparseness of the image in gradient. However, similar to all L1-minimization algorithms, total variation has the issue of penalizing large gradient, thus causing large errors on image edges. Many non-convex penalties have been proposed to address the issue of L1 minimization. For example, homotopic L0 minimization algorithms have shown success in reconstructing images from magnetic resonance imaging (MRI). Homotopic L0 minimization may suffer from local minimum which may not be sufficiently robust when the signal is not strictly sparse or the measurements are contaminated by noise. In this paper, we propose a hybrid total variation minimization algorithm to integrate the benefits of both L1 and homotopic L0 minimization algorithms for image recovery from reduced measurements. The algorithm minimizes the conventional total variation when the gradient is small, and minimizes the L0 of gradient when the gradient is large. The transition between L1 and L0 of the gradients is determined by an auto-Adaptive threshold. The proposed algorithm has the benefits of L1 minimization being robust to noise/approximation errors, and also the benefits of L0 minimization requiring fewer measurements for recovery. Experimental results using MRI data are presented to demonstrate the proposed hybrid total variation minimization algorithm yields improved image quality over other existing methods in terms of the reconstruction accuracy.