BioMed Research International

Volume 2016 (2016), Article ID 2180457, 12 pages

http://dx.doi.org/10.1155/2016/2180457

## Smoothed Norm Regularization for Sparse-View X-Ray CT Reconstruction

^{1}Medical Imaging Department, Suzhou Institute of Biomedical Engineering and Technology, Chinese Academy of Sciences, Suzhou 215163, China^{2}PET Center, Huashan Hospital, Fudan University, Shanghai 200235, China

Received 15 June 2016; Revised 19 August 2016; Accepted 24 August 2016

Academic Editor: Wenxiang Cong

Copyright © 2016 Ming Li 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

Low-dose computed tomography (CT) reconstruction is a challenging problem in medical imaging. To complement the standard filtered back-projection (FBP) reconstruction, sparse regularization reconstruction gains more and more research attention, as it promises to reduce radiation dose, suppress artifacts, and improve noise properties. In this work, we present an iterative reconstruction approach using improved smoothed (SL0) norm regularization which is used to approximate norm by a family of continuous functions to fully exploit the sparseness of the image gradient. Due to the excellent sparse representation of the reconstruction signal, the desired tissue details are preserved in the resulting images. To evaluate the performance of the proposed SL0 regularization method, we reconstruct the simulated dataset acquired from the Shepp-Logan phantom and clinical head slice image. Additional experimental verification is also performed with two real datasets from scanned animal experiment. Compared to the referenced FBP reconstruction and the total variation (TV) regularization reconstruction, the results clearly reveal that the presented method has characteristic strengths. In particular, it improves reconstruction quality via reducing noise while preserving anatomical features.

#### 1. Introduction

X-ray computed tomography has been widely used clinically for disease diagnosis, surgical guidance, perfusion imaging, and so forth. However, the massive X-ray radiations during CT exams are likely to induce cancer and other diseases in patients [1, 2]. Therefore, the issue of low-dose computerized tomography reconstruction has been raised and attracted more and more research attention. As far as we know, there are two low-dose strategies widely studied for dose reduction: lowering X-ray tube current values, measured by milliampere (mA) or milliampere-seconds (mAs), or lowering X-ray tube voltage, measured by kilovolt (KV), and lowering the number of sampling views during CT inspection. The strategy of regulation by mA or KV usually produces high noisy projection data. Thus, when the exposure dose is reduced, the images reconstructed using methods such as FBP suffer from increased artifacts and noise [3]. Diagnostic mistakes may appear in this case. The latter approach may also induce image artifacts due to limited sampling angles. As a result, the diagnostic value of the reconstructed images may be greatly degraded if inappropriate reconstruction approaches are applied.

To solve these problems, statistical reconstruction algorithms [4–9] attempt to produce high quality images by better modeling the projection data and the imaging geometry, which have shown superior performance compared to FBP-type reconstructions. Another path has been recently opened by compressed sensing (CS) with existing range of applications in medical imaging, for example, magnetic resonance imaging (MRI), bioluminescence tomography, optical coherence tomography, and low-dose CT reconstruction [10–24]. The CS theory reveals the potential capability of restoring sparse signals even if the Nyquist sampling theorem cannot be satisfied. Although the restricted isometry property (RIP) condition is not often satisfied in practice, CS-based reconstruction can yield more satisfying results than the traditional FBP algorithms in CT reconstruction [25]. Among several choices of sparse transforms, the gradient operator is motivated by the assumption that a preferable solution should be of bounded variation. It is known as total variation (TV) regularization, which favors solutions to be predominantly piecewise constant. TV has been widely used in the CT reconstruction community. However, TV-regularized images may suffer from loss of detail features and contrast, resulting in the staircasing artifacts. It is well known that norm regularization can provide a sparser representation than the TV regularization ( norm) [26, 27]. However, the application of norm in image reconstruction is often a nondeterministic polynomial-time (NP) hard problem. In addition, norm is a nonconvex function in discontinuous form.

norm is defined as the total number of its nonzeros elements and has stronger effects in promoting sparse solutions, but this minimization issue is NP hard to solve in general. Then, a spontaneous question can be whether preferable results will be achieved if we use regularization forms between norm and norm. In this work, we present a smoothed (SL0) norm regularization model for sparse-view X-ray CT reconstruction. This SL0 regularization permits a dynamic regularization modulation and can achieve a good balance between the regularizations based on norm and norm. The paper is organized as follows. In Section 2, the SL0 norm model is firstly described and then the detailed optimization algorithm and the parameters setting are given. Section 3 includes the experiments conducted on the projection data from the Shepp-Logan phantom, the head slice image, and the scanned mouse. The reconstructed results demonstrate that the proposed SL0 regularization produces better images with legible anatomical features and preferable noise characteristic compared to those using TV regularization. Finally, the discussions and conclusions are given at the end of this paper.

#### 2. Methods

##### 2.1. Problem Formulation

The idea of SL0 norm originates from the effort of minimizing a concave function that approximates norm [26]. In order to address the discontinuity of norm, we then try to approximate this discontinuous function via a feasible continuous one and minimize it by means of a minimization algorithm for continuous functions (e.g., steepest decent method). The continuous function which is used to approximate norm should have a modulation parameter (say ), which determines approximation degree. Then the family of the cost functions is defined asnoting thator it can be approximately expressed asThen SL0 norm is defined asIn (4), is the length of reconstructed signals. From (2) and (3), we can obviously observe that when , the SL0 norm tends to be equivalent to norm. Therefore, we can find the minimal norm solution via minimizing (subject to ) with a very small value. As can be seen, the value of determines the smoothness of the function . The larger the value of is, the smoother is, resulting in worse approximation to norm; and the smaller the value of is, the closer the performance between and norm is.

Now, we recall the total variation (TV) norm of a 2-dimensional array (), , which is defined as norm of the magnitudes of the discrete gradient:where ; is the attenuation coefficients to be reconstructed. If we use the proposed SL0 norm to enhance the sparsity of the image gradient, then the superior reconstruction behavior may be achieved. Therefore, to reconstruct the discrete X-ray linear attenuation coefficients, we consider the following constrained optimization problem:where is the system matrix, used to model the CT imaging system; is the log-transformed projection measurements; is the tolerance used to enforce the data fidelity constraint, and it refers to X-ray scatter, electronic noise, scanned materials, and a simplified data model. Sidky and Pan [11] have indicated that the best image root-squared-error is achieved when chosen is around the actual error in the projection data. In practice, the real noise level of a system is usually unknown. Therefore, the optimal value of is selected when the reconstructed image with less artifacts and clearer anatomical structures is achieved.

##### 2.2. Optimization Algorithm

In order to address the optimal solution of the proposed minimization problem, we try to assess the optimality of the solutions by analyzing the Karush-Kuhn-Tucker (KKT) conditions of (6) [28], which are the necessary conditions for optimality in nonlinear programming and can be derived through Lagrangian theory:and the partial derivative of the above Lagrangian function can be expressed aswhere the complimentary slackness isand the nonnegativity isIn conclusion, the optimal solutions can be firstly satisfied with the projection data fidelity constraint, and then corresponding should satisfy . Meanwhile, we intend to acquire the nonzero values of , and then corresponding should satisfy . To obtain the solutions meeting the above conditions, we need to solve the following optimization problem:

Sidky and Pan [11] present an optimization approach composed by an iterative projection operator called projection-onto-convex-sets (POCS) and adaptive steepest descent procedure, which is suitable for dealing with large size constrained optimization problems. In this paper, a similar strategy is applied here. We choose POCS to be the iterative operator, which is an efficient iterative algorithm that can find images that satisfy the given convex constraints. POCS combines the ART technique and the image nonnegativity enforcement, and the proposed SL0 regularization is minimized via an iterative gradient descent of the cost function. The images are updated sequentially through the alternation of the POCS and gradient descent until the Karush-Kuhn-Tucker (KKT) conditions are satisfied. In practice, in order to reduce the computation time, we relax the KKT conditions or stop after a predefined iterative number. Under the current version of the proposed reconstruction algorithm, there is no rigidly theoretical proof on the convergence properties of the optimization procedure. However, the reconstructed results in the following experiments show that they are actually close to the optimal solution.

##### 2.3. Parameters Selection

The implementation of the proposed SL0 regularization algorithm involves the choices of a series of parameters shown in Figure 1. The regularization parameter plays a crucial role in improving reconstruction quality. While we take a small value of , the function is highly unsmooth and includes many local minimums; hence finding its minimization is not easy. However, as increases, becomes smoother and includes less local minimums, and hence it is easier to minimize . In general, if we use a larger value of during the whole iterative process, the smoother reconstruction results can be achieved but the tissue details are worse. On the other hand, if we use a smaller value of *σ* during the whole iterative process, the optimization process may get trapped into local minimum, which will lead to artifacts and noisy reconstructions. Hence, our idea is to solve a sequence of optimization problems. At the first step, we solve (6) using a larger value of (such as ). Subsequently, we reduce by multiplying a small factor and then solve (6) again using . This time we initialize the reconstruction acquired in the last iteration. Due to the fact that decreases gradually, for each value of , the minimization algorithm starts with an initial solution close to the previous optimal value of (this is because both and have only slightly varied and consequently the minimization of new is potentially close to previous ). Hence, it is sufficient that the optimization algorithm is capable of escaping from getting trapped into local optimality and reaching the real minimum value for the small values, which offers the proximate norm solution. In our tests, we select and for all cases studied in this work. At the same time, the selection of should satisfy .