Computational and Mathematical Methods in Medicine

Volume 2015, Article ID 354869, 8 pages

http://dx.doi.org/10.1155/2015/354869

## CT Image Reconstruction from Sparse Projections Using Adaptive TpV Regularization

School of Biomedical Engineering, Southern Medical University, Guangzhou 510515, China

Received 4 August 2014; Revised 12 September 2014; Accepted 18 September 2014

Academic Editor: Liang Li

Copyright © 2015 Hongliang Qi 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

Radiation dose reduction without losing CT image quality has been an increasing concern. Reducing the number of X-ray projections to reconstruct CT images, which is also called sparse-projection reconstruction, can potentially avoid excessive dose delivered to patients in CT examination. To overcome the disadvantages of total variation (TV) minimization method, in this work we introduce a novel adaptive TpV regularization into sparse-projection image reconstruction and use FISTA technique to accelerate iterative convergence. The numerical experiments demonstrate that the proposed method suppresses noise and artifacts more efficiently, and preserves structure information better than other existing reconstruction methods.

#### 1. Introduction

X-ray computed tomography (CT), as an important medical imaging protocol, has been widely used in clinical applications. However, the involved X-ray radiation dose delivered to patients may potentially increase the probability of causing cancer [1–4]. In this sense, reducing radiation dose without significantly losing image quality is highly required.

Radiation dose in CT examination can be reduced by decreasing the number of projections. However, conventional filtered back-projection (FBP) reconstruction algorithm suffers from systematic geometric distortion and streak artifacts when the measured projection data is not sufficient [5–7]. Iterative methods have been proposed to overcome this problem. Recently, compressed sensing (CS) theory [8] has been applied in CT image reconstruction. It is possible to reconstruct high-quality images from sparse-projection data under the frame of CS. Many optimization methods have been studied following such concepts. Among these optimization methods, total variation (TV) minimization has been widely used. The most famous reconstruction model with TV is ART-TV, firstly proposed by Sidky et al. [9, 10]. This method consists of two steps: ART reconstruction and TV minimization. However, TV is based on an assumption that the signal is piecewise smooth, so this makes TV algorithm suffer from oversmoothing in image edges. To solve this problem, many improved TV methods have been proposed. Tian et al. proposed a TV-based edge preserving (EPTV) model [11]. This model can preserve edges by bringing in different weights in the TV term from edges and constant areas of the to-be-estimated image. Different from the EPTV model, Liu et al. considered the anisotropic edge property of an image and proposed a novel adaptive-weighted TV (AwTV) model [12] for low-dose CT image reconstruction from sparse-sampled projection data. Zhang et al. used a high-order norm coupled within TV to overcome the disadvantages of traditional TV minimization [13]. Chang et al. proposed a few-view reweighed sparsity hunting (FRESH) method for CT image reconstruction [14]. Sidky et al. replaced norm with norm in the minimization function and investigated image reconstruction by minimizing the norm of the image gradient magnitude or the so-called total -variation (TpV) [15]. Chen et al. proposed a CT reconstruction algorithm based on regularization, where norm is used as the regularization norm and gradient as the sparse conversion [16]. However, the TpV and regularization methods choose value as a constant in the whole image without identifying edges and constant areas. The disadvantage is that larger value can oversmooth edges and sometimes produce blocky artifacts, while smaller value can preserve edges well but enhance blocky artifacts in constant areas when the projection data is noisy (as shown in Figure 3(f) in [16]). The blocky artifacts are introduced by the noise in the projections whenever is less than 1 or . Although regularization is the sparest and most ideal regularization norm, -norm minimization problem is known to be NP-hard, and it is difficult to solve equations. Theoretically, a regularization, which is closer to norm, could obtain higher-quality CT images in CT reconstruction. It should be noted that TV is the norm of gradient image. Traditional TpV is sparser than TV, and the success of traditional TpV is sharpening image edges, but blocky artifacts still exist in homogeneous regions due to the noisy projection data. The same disadvantage of TV and TpV is their tendency to uniformly penalize the image gradient irrespective of the underlying image structures.

In this study, to deal with the trade-off between smoothing nonedge part and preserving edge part of the image, we propose a CT reconstruction algorithm using adaptive TpV regularization wherein each pixel in reconstructed image corresponds to one value determined by the pixel’s gradient magnitude. From our experiments, one can see that the low-contrast features can be reconstructed better than other methods and blocky artifacts are reduced much to a certain extent. The rest of the paper is organized as follows. In Section 2, ART-TV, traditional TpV, adaptive TpV regularization, and the proposed CT reconstruction algorithm are introduced, respectively. In Section 3, quantitative and qualitative experimental results are shown. Section 4 concludes the paper.

#### 2. Materials and Methods

##### 2.1. ART-TV Reconstruction

CT reconstruction problem can be converted to a constrained optimization problem where denotes the system matrix, represents the projection data, and is the reconstructed image. is the regularization function.

To solve 1, Sidky et al. proposed famous total variation (TV) based reconstruction method (ART-TV). In their method, in 1 was considered as a norm of the first-order gradient image or the so-called TV norm. In a 2D image with the size , whose pixel values are labeled by , its gradient magnitude with respect to can be expressed as The TV of image is defined as where the parameter is a small positive constant to avoid discontinuities.

The ART-TV method is implemented by performing ART algorithm as the first step and TV minimization using gradient descent method as the second step. One can see [9] for more implementation details.

##### 2.2. Traditional TpV and Adaptive TpV Regularization

For traditional TpV algorithm, the quantity is the -norm of the image gradient magnitude. For a 2D image it can be defined by

When is set to be 1, the reduces to conventional and reconstructed image will suffer from oversmoothing image feature details. When (e.g., in [16]), the structural information can be efficiently preserved, but at the same time the blocky artifacts in nonedge regions will be enhanced when the projection data is noisy.

To overcome this limitation, in this work we propose an adaptive TpV (ATpV) regularization defined by where is determined by pixel in a 2D image. On one hand, if a pixel’s gradient magnitude is large, this pixel is on the edge and corresponds to a small value to avoid oversmoothing edges. On the other hand, if the gradient magnitude of one pixel is small, this pixel is in the nonedge area and corresponds to a large value to suppress noise and artifacts. In this study, we define as where denotes image is filtered by the well-known bilateral filter which does well in denoising and preserving edge information. is gradient operator. Adding bilateral filter on image is to avoid treating noise point as edge point when calculating gradient magnitude. Obviously in 6, a pixel with large gradient magnitude corresponds to a small value and a pixel with small gradient magnitude corresponds to a large value. The “1” in the denominator in 6 may not be the best option and may be correlated with the contrast value of the reconstruction, but in our experiments, the defined 6 can produce good result. How to replace “1” with an optimal value will be an interesting topic in our future study.

In summary, the benefit of the proposed ATpV is that the parameter is dynamically adopted by identifying edges and nonedges, and larger value is chosen to smooth constant areas while smaller value is chosen to preserve edge part, which will improve the reconstruction quality for sparse-view reconstruction.

##### 2.3. CT Reconstruction Algorithm Based on Adaptive TpV Regularization

According to aforementioned methods, in this paper we propose CT image iterative reconstruction using ATpV regularization. The reconstruction is implemented by solving the following constrained minimization problem:

The algorithm implementation can follow ART-TV in [9].

Besides, we apply fast iterative shrinkage/thresholding algorithm (FISTA) [17] to accelerate iterative convergence. In FISTA, the key idea is that initial value of the next iteration is determined by a linear combination of the two previous iterate results. For simplicity, in this study the proposed reconstruction method is termed ART-ATpV without using FISTA or ART-ATpV-FISTA using FISTA.

In summary, the main steps of ART-ATpV-FISTA are as follows.(A)Initialization: , iteration index , .(B)ART reconstruction: (C)Positivity constraint: .(D)ATpV minimization:

calculate for each pixel , minimize using gradient descent algorithm to get updated .(E)FISTA acceleration: (F)Return to (B) until the stopping criterion is satisfied.

In our experimental implementation, the initial to-be-reconstructed image was set to be uniform with pixel values of 0. The relaxation parameter for the ART was fixed at 1.0, and the step-size used in ATpV minimization using the gradient descent was set to be constant 0.2. The parameter in 7 was fixed as 10^{−5}.

#### 3. Experimental Study

##### 3.1. Numerical Simulation

In this section, we study the ART, ART-TV, ART-TpV (), and our proposed algorithm. Numerical experiment results are given. Shepp-Logan phantom is tested in this paper as shown in Figure 1, and the size of phantom image is 256 × 256. Without losing generality, we choose a fan beam imaging geometry to capture the projection data as illustrated in Figure 2. The source to rotation center distance is 40 cm and the detector to rotation center is 40 cm. The image array is 25.6 × 25.6 cm^{2}. The detector whose length is 61.44 cm is modeled as a straight-line array of 512 detector bins. All the tests are performed by MATLAB on a PC with Intel (R) Core (TM) 2 Quad CPU 2.50 GHz and 3.25 GB RAM.