Iterative thresholding compressed sensing MRI based on contourlet transform

Reducing the acquisition time is important for clinical magnetic resonance imaging (MRI). Compressed sensing has recently emerged as a theoretical foundation for the reconstruction of magnetic resonance images from undersampled k-space measurements, assuming those images are sparse in a certain transform domain. However, most real-world signals are compressible rather than exactly sparse. For example, the commonly used two-dimensional wavelet for compressed sensing MRI (CS-MRI) does not sparsely represent curves and edges. In this article, we introduce a geometric image transform, the contourlet, to overcome this shortage. In addition, the improved redundancy provided by the contourlet can successfully suppress the pseudo-Gibbs phenomenon, a tiresome artefact produced by undersampling of k-space, around the singularities of images. For numerical calculation, a simple but effective iterative thresholding algorithm is employed to solve l1 norm optimization for CS-MRI. Considering the recovered information and image features, we introduce three objective criteria, which are the peak signal-to-noise ratio (PSNR), mutual information and transferred edge information, to evaluate the performance of different image transforms. Simulation results demonstrate that contourlet-based CS-MRI can better reconstruct the curves and edges than traditional wavelet-based methods, especially at low k-space sampling rate.