An efficient algorithm for compressed MR imaging using total variation and wavelets

Compressed sensing, an emerging multidisciplinary field involving mathematics, probability, optimization, and signal processing, focuses on reconstructing an unknown signal from a very limited number of samples. Because information such as boundaries of organs is very sparse in most MR images, compressed sensing makes it possible to reconstruct the same MR image from a very limited set of measurements significantly reducing the MRI scan duration. In order to do that however, one has to solve the difficult problem of minimizing nonsmooth functions on large data sets. To handle this, we propose an efficient algorithm that jointly minimizes the ℓ 1 norm, total variation, and a least squares measure, one of the most powerful models for compressive MR imaging. Our algorithm is based upon an iterative operator-splitting framework. The calculations are accelerated by continuation and takes advantage of fast wavelet and Fourier transforms enabling our code to process MR images from actual real life applications. We show that faithful MR images can be reconstructed from a subset that represents a mere 20 percent of the complete set of measurements.