Mathematical Problems in Engineering

Volume 2015, Article ID 615439, 18 pages

http://dx.doi.org/10.1155/2015/615439

## Compressed Sensing MRI Reconstruction from Highly Undersampled -Space Data Using Nonsubsampled Shearlet Transform Sparsity Prior

^{1}School of Information Science & Engineering, Lanzhou University, Lanzhou 730000, China^{2}Department of Radiology, Xinhua Hospital, Shanghai Jiao Tong University School of Medicine, Shanghai 200092, China

Received 25 September 2014; Revised 12 February 2015; Accepted 20 February 2015

Academic Editor: Alessandro Gasparetto

Copyright © 2015 Min Yuan et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Compressed sensing has shown great potential in speeding up MR imaging by undersampling -space data. Generally sparsity is used as a priori knowledge to improve the quality of reconstructed image. Compressed sensing MR image (CS-MRI) reconstruction methods have employed widely used sparsifying transforms such as wavelet or total variation, which are not preeminent in dealing with MR images containing distributed discontinuities and cannot provide a sufficient sparse representation and the decomposition at any direction. In this paper, we propose a novel CS-MRI reconstruction method from highly undersampled -space data using nonsubsampled shearlet transform (NSST) sparsity prior. In particular, we have implemented a flexible decomposition with an arbitrary even number of directional subbands at each level using NSST for MR images. The highly directional sensitivity of NSST and its optimal approximation properties lead to improvement in CS-MRI reconstruction applications. The experimental results demonstrate that the proposed method results in the high quality reconstruction, which is highly effective at preserving the intrinsic anisotropic features of MRI meanwhile suppressing the artifacts and added noise. The objective evaluation indices outperform all compared CS-MRI methods. In summary, NSST with even number directional decomposition is very competitive in CS-MRI applications as sparsity prior in terms of performance and computational efficiency.

#### 1. Introduction

Magnetic resonance imaging (MRI) is a widely used noninvasive imaging modality for clinical diagnosis. However, relatively slow imaging speed of MRI remains a great challenge for clinical application. An effective way to speed up MR imaging is -space undersampling; in the course, only a small subset of -space is measured. However, undersampling often violates the Nyquist sampling criterion, resulting in aliasing artifacts in linear reconstruction, inadequate image resolution, and excessive Gibbs ringing artifacts.

Compressed sensing (CS) [1–3] proposed by Candès et al. and Donoho is a novel signal acquisition and compression theory. As a promising method, CS has been introduced to MR image reconstruction, so-called compressed sensing MRI (CS-MRI) [4–6]. CS-MRI allows high quality reconstruction from undersampled -space data by solving a constrained minimization problem using nonlinear optimization algorithm by enforcing the sparsity of images in a certain predefined sparsifying transform, such as the traditional 2D separable wavelet transform [4], total variation (TV) [4, 7, 8], contourlet [9, 10], sharp frequency localization contourlet (SFLCT) [11, 12], dual-tree complex wavelet transform (DT-CWT) [13, 14], and complex double-density dual-tree DWT [15]. The structured sparsity such as Gaussian scale mixture (GSM) model [16, 17] and wavelet tree sparsity [18, 19] for exploiting the dependencies between wavelet coefficients has been introduced to CS-MRI reconstruction. In terms of the restricted isometry property (RIP) condition in CS, incorporating a prior knowledge to enhance the sparsity into the reconstruction can reduce the reconstruction error and improve the reconstruction of details effectively. In the past several years these analytical sparsifying transforms have been successfully applied to CS-MRI and demonstrate high quality reconstructions from undersampled data. Some of these transforms have been combined to further improve the reconstruction [20–23]. Dictionary learning enables adaptive sparser representation of MR images than the general sparsifying transforms, which has been applied in CS-MRI [24].

The quality of reconstructed images largely depends on the performance of exploiting the sparsity prior in CS-MRI, which is key to accurate CS reconstruction. Therefore an outstanding sparsifying transform suitable for representing MR images should be adopted as sparsity prior in CS-MRI reconstruction. The motivation of our study comes from the morphology that MR images consist of different components (point-like and curve-like features) in various orientations, which cannot be sparsely represented by existing sparse representation sufficiently, and yet dictionary learning is at the expense of sacrificing time. In this paper we proposed a novel CS-MRI image reconstruction method from highly undersampled -space data to improve the quality of reconstructed MR images by enhancing the sparsity in nonsubsampled shearlet transform (NSST) domain. With the ability to capture intrinsic geometrical features of multidimensional data efficiently and sparsely represent images containing edges optimally, the prominent sparse representation-NSST is adopted as prior knowledge for the regularization in CS-MRI. The numerical computation employs a corresponding iterative NSST thresholding algorithm to solve this inverse optimization problem.

#### 2. Theory and Methods

##### 2.1. Problem Formulation

The common model of data acquisition with incomplete measurements for CS-MRI is given by the following formulation:where is the reconstructed image, is the acquired -space measurement data corrupted with the noise , and is the undersampled Fourier transform operator which directly relies on the -space undersampling scheme. Suppose that is represented as , where represents the sparsity prior associated with the transform under which MR image has a sparse representation or approximation. Then the measured data is given by , where represents the sensing matrix.

According to CS theory, CS-MRI claims to reconstruct MR image from undersampled -space data by enforcing the image sparsity [4]. That is, can be accurately reconstructed from a small subset of -space data by solving the following norm minimization problem:

However, the norm is not convex and the computational complexity of the optimization is NP-hard [7]. It has been proven that, under certain condition, norm problem is equivalent to norm. Thus the reconstruction can be obtained by solving the following constrained convex optimization:where is a statistic describing the magnitude of the error, defined as the noise variance or the maximum allowable error in the approximation. Minimizing the objective function promotes the sparsity of the images. The constraint enforces the fidelity of the reconstruction to the measured -space data. Besides, constrained norm convex optimization problem in (3) can be written in unconstrained Lagrangian form:The second term of (4) is a regularization term that represents prior sparse information of original images. is a regularization parameter governing the tradeoff between the data fidelity and its sparsity.

##### 2.2. Nonsubsampled Shearlet Transform

The shearlet transform, introduced by the authors and their collaborators in [25, 26], is a very recent sibling in the family of geometric image representations. The shearlet representation originally derived from the framework of affine systems with composite dilations [25–28]. The shearlet frame elements associated with shearlet transform are defined at various scales. The shearlet transform is a multiscale directional transform which is especially adapted to localize distributed discontinuities such as edges. Unlike conventional multiresolution analysis tool, the shearlet representation is theoretically optimal in representing directional and anisotropic features in images and has the ability to accurately and efficiently capture the geometric information of multidimensional data at various scales. As a result, this approach provides optimal approximation properties for 2D images.

The shearlet representation forms an affine-like system and has a simpler mathematical construction. In fact, the elements of this system form a Parseval frame and are generated by applying dilations, shear transformations, and translations to a single well-localized window function. Shearlet transform extends naturally to higher dimensions and can be associated with a multiresolution analysis [28]. In addition, this approach has a fast algorithmic implementation and is very competitive for CS-MRI reconstruction.

The discrete shearlet transform, obtained by discretizing the corresponding continuous transform, has different form in the numerical implementation. Both a frequency and time-domain based implementation of the discrete shearlet transform have been developed [25]. The features of each particular representation will have various advantages for specific applications. Refer to [25–28] for more details about the mathematical framework and the implement of discrete shearlet transform. The reason for using nonsubsampled shearlet transform (NSST) as sparsity prior for CS-MRI reconstruction will be discussed in Section 2.3.1 in detail.

##### 2.3. Proposed Method

The aforementioned sparsifying transforms integrated with CS-MRI are acknowledged to have a limited capability in representing these directional information and important details. The properties of each particular representation will have different advantages for specific application. They may not provide sufficient sparse representation for sharp spatial gradients and intrinsic geometrical features contained in MR images. The contourlet transform is an efficient directional multiresolution image representation. However, nonideal filters used in the original contourlet result in substantial amount of aliasing components and blurring artifacts in representing smooth boundaries. To solve this problem, Lu and Do [11] propose a new construction of the contourlet, called sharp frequency localization contourlet (SFLCT). SFLCT has been integrated with CS-MRI reconstruction [12]. Since the combination of LP and 2D directional filter bank (DFB) makes the aliasing problem serious, new multiscale pyramid with different set of low pass and high pass filters for the first level and all other levels is employed. SFLCT alleviates the nonlocalization problem with the same redundancy of the original contourlet. Though SFLCT is sharply localized in the frequency domain, the downsampling of LP and DFBs stages makes it lack shift invariance, which could easily produce Gibbs ringing artifacts around the singularities, for example, edges. Although the contourlet basis is anisotropic, the directional subbands can be decomposed only at directions in terms of directional selectivity, in which denotes the cascade layers of DFBs.

To overcome these limitations, on account of the MR images consisting of curve singularities and anisotropic directional features, a more appropriate sparsity prior with highly directional sensitivity and anisotropy should be applied to reconstruction. So we employ a special form of shearlet transform—NSST—as sparsity prior in CS-MRI reconstruction.

###### 2.3.1. Major Advantages of NSST Sparsity Prior

Taking account of measurement noise in -space and the problems of aliasing and shift-variance caused by decimation, in our implementation for CS-MRI, we adopt the particular form of the time-domain based shearlet transform. This will be simply referred to as the nonsubsampled shearlet transform (NSST), which is to use the nonsubsampled Laplacian pyramid (LP) transform with several different combinations of the shearing filters. The idea of using NSST will be represented one by one below.

*(i) Essentially Optimal Sparsity and Approximation Property.* The family of shearlet function forms a tight frame of , and, thus, an image can be represented by . The coefficients are called shearlet coefficients of the image . The shearlet elements form a tight frame of well-localized waveforms, at various scales and directions, and are optimally sparse in representing images with edges. They are compactly supported in the frequency domain and have fast decay in the spatial domain.

To make the statement of optimal sparsity and approximation property more rigorous, it is useful to quantify the approximation performance from the point of view of approximation theory. The asymptotic convergence rate is actually the correct optimal behavior for approximating general smooth objects having discontinuities along piecewise curves. Denoting as the approximation of an image by the largest transform coefficients in the corresponding representation, the resulting approximation error (in -norm square) is . It is very helpful to achieve the best asymptotic decay rate for this error in application. Let be the space of functions that are twice continuously differentiable. If the image is everywhere away from edge curves that are piecewise , the best -term asymptotic approximation error using shearlets has a decay rate of [28], which is essentially optimal in representing 2D images which are piecewise except for discontinuities along curves and greatly outperforms that of wavelet approximations only with the decay rate of [29]. The shearlet transform has very similar asymptotic approximation properties with the curvelets [30] and the contourlets. The corresponding argument about optimal sparsity is proved in [9, 28, 30]. The error decay rate using shearlets is close to the theoretical optimal approximation, where the error decays as [31, 32]. In particular, the shearlet transform has some advantages over the contourlet transform [25]. In this sense, the shearlet representation provides optimally sparse representation of objects with singularities along piecewise edges. Thus through shearlet transform, MR image can be decomposed into various frequency regions whose supports are contained in pair of trapezoidal regions symmetric with respect to the origin and directionally oriented.

For CS-MRI, two most useful features of shearlets are their ability to efficiently approximate signals containing piecewise singularities and allowing for a much less redundant sparse tight frame representation. Consequently shearlet transform can represent images sparsely better than other representations. Moreover NSST can offer shift invariance. These properties are of vital importance for CS-MRI reconstruction because these properties results in MR images that are more compressible and hence more effective to be reconstructed. We can achieve better approximation performance and better sparse representation by shearlet coefficients compared to others. The reconstruction error in CS is proportional to approximation error. By using NSST as the sparse transform for some singularities in MR images, the reconstruction error decreases faster than that of other representations from undersampled -space data. This is the reason why we use NSST as sparsity prior to better reconstruct some crucial features than using other representations in CS-MRI.

*(ii) Highly Directional Sensitivity and Anisotropy.* The collection of shearlets is a Parseval frame (tight frame) for [25]. This indicates that the decomposition is invertible and the transformation is numerically well-conditioned. Details about this construction can be found in [25, 28]. The tiling of the frequency plane induced by the shearlets is illustrated in Figure 1(a).