An algorithm for sparse MRI reconstruction by Schatten p-norm minimization

Abstract

In recent years, there has been a concerted effort to reduce the MR scan time. Signal processing research aims at reducing the scan time by acquiring less K-space data. The image is reconstructed from the subsampled K-space data by employing compressed sensing (CS)-based reconstruction techniques. In this article, we propose an alternative approach to CS-based reconstruction. The proposed approach exploits the rank deficiency of the MR images to reconstruct the image. This requires minimizing the rank of the image matrix subject to data constraints, which is unfortunately a nondeterministic polynomial time (NP) hard problem. Therefore we propose to replace the NP hard rank minimization problem by its nonconvex surrogate — Schatten p-norm minimization. The same approach can be used for denoising MR images as well.

Since there is no algorithm to solve the Schatten p-norm minimization problem, we derive an efficient first-order algorithm. Experiments on MR brain scans show that the reconstruction and denoising accuracy from our method is at par with that of CS-based methods. Our proposed method is considerably faster than CS-based methods.

Introduction

Magnetic resonance imaging (MRI) is a comparatively slow imaging modality. Speeding up the data acquisition time has always been of interest to the MRI research community. Until recently, most of the effort in decreasing the MR scan (data acquisition) time had been dedicated to improving the hardware of the scanner. Once the full K-space data are acquired, reconstructing the image is almost trivial — applying 2D inverse Fourier transform. In recent years, signal processing researchers have shown that it is possible to reduce the scan time by acquiring less K-space data followed by a nonlinear reconstruction technique based on the recent findings of compressed sensing [1], [2], [3], [4], [5], [6], [7]. Compressed sensing (CS)-based methods reconstruct the image by framing a nonlinear optimization problem that exploits the sparsity of the MR image in a transform domain such as wavelet, contourlet or total variation.

Our work has the same interest in mind — reconstructing the MR image from subsampled K-space data. But instead of applying CS-type methods, we will show that similar reconstruction accuracy can be achieved by exploiting the rank deficiency of the image. The way we formulate the reconstruction problem requires solving a nonlinear optimization problem that minimizes the Schatten p-norm (0<p≤1) [8] of the image. This work does not compete against CS-based MR reconstruction techniques; rather, it proposes an alternate approach. Our reconstruction accuracy is similar to standard CS-based methods, but takes considerably less time.

Rank deficiency can also be exploited to denoise MR images from legacy scanners. Current MR scanners sample the full K-space and reconstruct the image by applying an inverse Fourier transform. Since the K-space data is itself corrupted by noise, the reconstructed image is noisy as well. Such images require an additional denoising step. The MR image is assumed to be rank deficient (has a small number of high singular values); however, the noise is not. The noise is full rank but has very small singular values. Thus it is possible to denoise the image by properly thresholding the singular values of the noisy image. However, instead of applying arbitrary thresholding to the singular values noisy image, we propose to denoise it through Schatten p-norm minimization.

The rest of the article is organized into several sections. The following section discusses the CS-based MR image reconstruction. Section 3 formulates the reconstruction problem as one of Schatten p-norm minimization. In Section 4, the formulation behind the denoising problem is provided. The algorithmic development for solving the minimization problem is described in Section 5. The experimental evaluation is performed in Section 6. Finally, in Section 7, the conclusions of the work are discussed.