BioMed Research International

Volume 2017 (2017), Article ID 6753831, 10 pages

https://doi.org/10.1155/2017/6753831

## Combining Acceleration Techniques for Low-Dose X-Ray Cone Beam Computed Tomography Image Reconstruction

^{1}Medical Physics Research Center, Institute for Radiological Research, Chang Gung University and Chang Gung Memorial Hospital, Taoyuan 33302, Taiwan^{2}Department of Nuclear Medicine and Neuroscience Research Center, Chang Gung Memorial Hospital, Taoyuan 33302, Taiwan^{3}Department of Medical Imaging and Radiological Sciences and Healthy Aging Research Center, College of Medicine, Chang Gung University, Taoyuan 33302, Taiwan

Correspondence should be addressed to Ing-Tsung Hsiao

Received 23 December 2016; Accepted 9 May 2017; Published 5 June 2017

Academic Editor: Kwang Gi Kim

Copyright © 2017 Hsuan-Ming Huang and Ing-Tsung Hsiao. 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

*Background and Objective*. Over the past decade, image quality in low-dose computed tomography has been greatly improved by various compressive sensing- (CS-) based reconstruction methods. However, these methods have some disadvantages including high computational cost and slow convergence rate. Many different speed-up techniques for CS-based reconstruction algorithms have been developed. The purpose of this paper is to propose a fast reconstruction framework that combines a CS-based reconstruction algorithm with several speed-up techniques.* Methods*. First, total difference minimization (TDM) was implemented using the soft-threshold filtering (STF). Second, we combined TDM-STF with the ordered subsets transmission (OSTR) algorithm for accelerating the convergence. To further speed up the convergence of the proposed method, we applied the power factor and the fast iterative shrinkage thresholding algorithm to OSTR and TDM-STF, respectively.* Results*. Results obtained from simulation and phantom studies showed that many speed-up techniques could be combined to greatly improve the convergence speed of a CS-based reconstruction algorithm. More importantly, the increased computation time (≤10%) was minor as compared to the acceleration provided by the proposed method.* Conclusions*. In this paper, we have presented a CS-based reconstruction framework that combines several acceleration techniques. Both simulation and phantom studies provide evidence that the proposed method has the potential to satisfy the requirement of fast image reconstruction in practical CT.

#### 1. Introduction

Based on the theory of compressive sensing (CS) [1, 2], near optimal computed tomography (CT) image can be reconstructed from very few projection data. This new methodology indicates a potential for substantially reducing radiation dose. Over the past decade, many CS-based image reconstruction methods have been shown to improve image quality in low-dose CT [3–8]. Such methods are often referred to as total variation (TV). To solve the TV problem in the field of CT reconstruction, a two-step alternating minimization framework is commonly used. In the first step, a general iterative reconstruction algorithm is performed to reduce data discrepancy. In the second step, the TV of the reconstructed image is minimized by a gradient descent method. Despite a great improvement of image quality with CS-based reconstruction methods, both high computational load and slow convergence rate limit their practical use.

Recently, the one-step minimization scheme such as first-order gradient-projection backtracking-line search method [9] and gradient-projection Barzilai-Borwein method was found to converge faster than two-step alternating minimization scheme [9–11]. However, there are various ways to improve the convergence rate of the two-step alternating minimization scheme. For example, image reconstruction based on ordered subsets (OS) of projection data is a common acceleration technique used in emission tomography [12] and CT [13–15]. Using OS acceleration [12], data discrepancy can be reduced rapidly compared with non-OS methods. In addition to OS-type simultaneous algebraic reconstruction technique (SART) [15], other faster methods such as ordered subsets transmission (OSTR) algorithm [13] and ordered subsets convex (OSC) algorithm [14] were derived previously. It was also reported that OS-type reconstruction methods can be further accelerated using a bigger step size or a power factor [16–18]. Previous studies showed that accelerated OS-type algorithms using a power factor can converge three or even four times faster than conventional OS-type algorithms [16–18]. These methods, although originally used in emission tomography, can be directly applied for the CT reconstruction. Besides reducing data discrepancy, many different minimization techniques, including fast iterative shrinkage thresholding algorithm (FISTA) [19], TV minimization with dual dictionaries [20], anisotropic TV minimization [21], total difference (TD) with soft-threshold filtering (STF) [6, 8], weighted TD (WTD) minimization with STF [22], and TV minimization with half-threshold filtering [23], can be introduced to improve the convergence of the two-step alternating minimization scheme.

Although many acceleration techniques have been developed previously, a combination of acceleration techniques has not been completely studied. To investigate this, we present a reconstruction framework that applies some above-mentioned acceleration techniques to the two-step alternating minimization. Specifically, we implemented the TD minimization with STF (TDM-STF), which is one type of TV [6, 8, 24]. In the TDM-STF, the OSTR algorithm was chosen to accelerate the convergence. To further speed up the convergence of the proposed method, we applied the power factor [16–18] and the FISTA algorithm [19] to OSTR and TDM-STF, respectively. This study is different from our recent work [25] that investigated the feasibility of using the power factor [16–18] to accelerate the TV-based reconstruction [5]. The purpose of this paper is to study whether combining these techniques can further accelerate the convergence of the two-step alternating minimization. We used simulation and phantom data to evaluate the performance of the proposed algorithm.

#### 2. Materials and Methods

##### 2.1. CT Image Reconstruction Problem

According to the idea of CS [1, 2] and the TDM algorithm proposed by Yu and Wang [6], the CT image reconstruction problem is to solve the constrained convex optimization problem of the following form:where is the image estimate, is the system matrix, is the measured projection data, is a regularization factor, and TD() denotes the total difference of the image estimate defined as [6] in (2) is called a discrete difference transform (DDT) [6, 8]. Similar to the TV problem [3–5], the TD problem in (1) can be solved iteratively using a two-step alternating minimization scheme [6, 8]. The TDM-STF algorithm [6] was a two-step alternating minimization algorithm. In this study, we used the TDM-STF algorithm [6] to minimize (1). Our aim is to improve the convergence rate of the TDM-STF algorithm [6]. The TDM-STF algorithm and its accelerated algorithm are summarized in the following three sections, respectively. Finally, the implementation of the proposed reconstruction algorithm is summarized in a pseudocode.

##### 2.2. TDM-STF and TDM-STF-FISTA

The STF algorithm proposed by Daubechies et al. [26] was originally developed to solve the linear inverse problems regularized by a sparsity constraint. Yu and Wang [6] adapted the STF method for CT image reconstruction and developed the TDM-STF algorithm to minimize (1). In brief, the TDM-STF method involves three steps. In the first step, the data-fidelity term (i.e., ) was minimized via a typical iterative reconstruction algorithm. In the second step, a soft-threshold filtration was performed on the DDT of the current image estimate (i.e., ). In the third step, the inverse of DDT was computed to obtain image estimate. However, DDT is not invertible [6]. Instead, the second and third steps of the TDM-STF algorithm can be performed based on a pseudoinverse of DDT [6]:where , , and denote the three-dimensional location of the voxel and where is a threshold value. As pointed out by Liu et al. [8], the TDM-STF method can be further accelerated by using a portion of FISTA algorithm [19], which is performed by the following steps:This accelerated technique is also called Nesterov’s momentum algorithms [27, 28]. Note that and when .

##### 2.3. Acceleration of OSTR Using a Power Factor

In the first step of the TDM-STF algorithm, the data-fidelity term was minimized via a typical iterative reconstruction algorithm. To speed up the data-fidelity minimization, we used the OSTR algorithm [13]. The OSTR algorithm is chosen because it is a fast, efficient, and easily implemented algorithm [13]. Moreover, the OSTR algorithm [13] can be accelerated by a power factor [16–18]. The accelerated OSTR (AOSTR) algorithm can be expressed as follows: where is the estimated attenuation coefficient at voxel and at the th iteration and th subiteration, is the measured projection data at detector bin , is the blank scan at detector bin , , and is the number of subsets. Note that the AOSTR algorithm becomes the OSTR algorithm when power factor . Using the Taylor series expansion, (9) can be approximated as follows: More details can be found in [16–18]. Interestingly, the AOSTR algorithm is the same as the OSTR algorithm with a fixed step size. However, in order to preserve the total counts of the forward projections [16–18], the reconstructed image updated by (10) is rescaled by the following equation: Note that the solution for (1) from the OSTR algorithm and its accelerated version is not exact, but approximate [29]. An initial condition of uniform image () with a value of 0.0002 was used for all reconstructions.

##### 2.4. AOSTR-TDM-STF-FISTA

In summary, we applied the AOSTR reconstruction rather than the conventional iterative reconstruction method in the first step of TDM-STF with FISTA, and the proposed method was called AOSTR-TDM-STF-FISTA. In fact, other combinations of OSTR, AOSTR, and FISTA into the TDM-STF method are possible. For example, the accelerations of OSTR-TDM-STF using a power factor on OSTR and FISTA on TDM-STF are called AOSTR-TDM-STF and OSTR-TDM-STF-FISTA, respectively. In addition to the proposed AOSTR-TDM-STF-FISTA, difference combinations of the acceleration methods are also explored in this study.

##### 2.5. Implementation of AOSTR-TDM-STF-FISTA

Note that the implementation of AOSTR-TDM-STF-FISTA requires considerably more computation per iteration as compared with OSTR. This is due to the computation of the rescaling factor (i.e., (11)) and the TDM step (i.e., (3)) at each subiteration. To reduce the computation time, the TDM step was applied after the last subiteration of each iteration, indicating that STF is performed only once at each iteration of the AOSTR algorithm. Because of this implementation, the rescaling step can be combined with the forward projection of the next subiteration except for the last subiteration [16–18]. This means that the rescaling step is computed only at the end of each iteration (i.e., the last subiteration). Similarly, the FISTA algorithm is performed once per iteration rather than once per subiteration. Such modifications make the proposed method an efficient approach for CT reconstruction. The final practical and efficient implementation of AOSTR-TDM-STF-FISTA can be summarized in Pseudocode 1.