Computational and Mathematical Methods in Medicine

Volume 2015, Article ID 161797, 16 pages

http://dx.doi.org/10.1155/2015/161797

## Accelerated Compressed Sensing Based CT Image Reconstruction

^{1}Institute of Biomaterials and Biomedical Engineering, University of Toronto, Toronto, ON, Canada M5S 3G9^{2}Department of Electrical and Computer Engineering, Ryerson University, Toronto, ON, Canada M5B 2K3^{3}Rambus Inc., Sunnyvale, CA 94089, USA^{4}Joint Department of Medical Imaging, Toronto General Hospital, University Health Network, Toronto, ON, Canada M5G 2C4

Received 26 November 2014; Revised 12 May 2015; Accepted 20 May 2015

Academic Editor: Hugo Palmans

Copyright © 2015 SayedMasoud Hashemi 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

In X-ray computed tomography (CT) an important objective is to reduce the radiation dose without significantly degrading the image quality. Compressed sensing (CS) enables the radiation dose to be reduced by producing diagnostic images from a limited number of projections. However, conventional CS-based algorithms are computationally intensive and time-consuming. We propose a new algorithm that accelerates the CS-based reconstruction by using a fast pseudopolar Fourier based Radon transform and rebinning the diverging fan beams to parallel beams. The reconstruction process is analyzed using a maximum-a-posterior approach, which is transformed into a weighted CS problem. The weights involved in the proposed model are calculated based on the statistical characteristics of the reconstruction process, which is formulated in terms of the measurement noise and rebinning interpolation error. Therefore, the proposed method not only accelerates the reconstruction, but also removes the rebinning and interpolation errors. Simulation results are shown for phantoms and a patient. For example, a 512 × 512 Shepp-Logan phantom when reconstructed from 128 rebinned projections using a conventional CS method had 10% error, whereas with the proposed method the reconstruction error was less than 1%. Moreover, computation times of less than 30 sec were obtained using a standard desktop computer without numerical optimization.

#### 1. Introduction

Compared to conventional radiography, CT results in a relatively large radiation dose to patients, which is of serious long-term concern in its potential for increasing the risk of developing cancer [1, 2]. As a result, low dose CT imaging that maintains the resolution and achieves good contrast to noise ratio has been the goal of many CT developments over the past decade. However, low dose CT images reconstructed with conventional filtered back projection (FBP), which directly calculates the image in a single reconstruction step, suffer from low contrast to noise ratios. Iterative reconstruction approaches, namely, Algebraic Reconstruction Technique (ART) [3–5] and statistical iterative reconstruction (SIR) [6, 7], have been proposed to improve the reconstruction quality and to decrease image artifacts. The iterative algorithms improve the quality by considering more accurate models for the CT images and geometries. However, they significantly increase the computational complexity, compared to the FBP based methods.

Iterative reconstruction methods have progressed with the introduction of compressed sensing (CS) [8, 9]. Such methods are capable of reconstructing high quality images from a substantially smaller number of views than those needed in FBP [10], thereby permitting the use of a much lower dose scanning protocol than that needed in conventional reconstruction methods. However, conventional CS-based CT reconstructions are computationally expensive and the statistics of CT measurements are not usually incorporated in the problem formulation [11–16].

In this paper, we propose a fast weighted CS-based CT reconstruction algorithm, the weights of which are direct consequences of the geometry and the CT statistics. The first part of this paper leads to the proposed weighted CS formulation, which is solved by a computationally efficient method discussed in the second part.

#### 2. CS-Based CT Reconstruction and Its Challenges

Compressed sensing prescribes solving the optimization problem that can be represented byor other similar forms, for example, the norm of the image gradient, such asto recover a sparse signal from few samples. In these equations, acts as a regularization parameter specifying a trade-off between the image prior model and the fidelity to observations, is the measurement matrix, is the column vector representation of the desired image (), is the measured data, is a sparsifying transform, , and denotes the total variation , where and are the first derivatives in the and directions of the desired image.

The main challenge in solving these optimization problems within a reasonable amount of time arises from the size of the measurement matrix . Currently, in most available CS-based reconstruction methods, the measurement matrix is a projection matrix which models the rays going through the patient. To reconstruct a 512 × 512 pixel image from 900 sensors and 1200 projection angles, would be a 1080000 × 262144 matrix. Although this matrix is sparse, each iteration typically requires two multiplications by and , resulting in a very significant increase in the computation burden for reconstructing a 512 × 512 image [11, 12] as compared to FBP based methods. To enable the CS-based CT reconstruction to be done in a reasonable computation time, GPU based algorithms have been proposed [17].

##### 2.1. Complexity Reduction Using the Pseudopolar Fourier Transform (PPFT)

To reduce the computational burden on the Radon transform, the central slice theorem (CST) or direct Fourier reconstruction (DFR) has been used [18]. This relates the 1D Fourier transform of the projections to the 2D Fourier transform of the image. Such a method requires the interpolation of polar data onto a Cartesian grid followed by an inverse FFT on the same grid to reconstruct the CT image. Since interpolation does not have a known analytical adjoint, its use in iterative algorithms is not a practical option. In addition, inclusion of a gridding and regridding step at each iteration increases the overhead computational complexity. This problem has been extensively studied in non-Cartesian magnetic resonance imaging reconstruction algorithms [19].

An equally sloped tomography (EST) method was originally proposed for electron beam tomography [20–22] to improve the DFR-based algorithms. EST is an iterative method that makes use of the pseudopolar Fourier transform (PPFT) [23]. It calculates the Fourier coefficients of an image directly on pseudopolar grids, which contain two types of samples: basically horizontal (BH) and basically vertical (BV), as can be seen in Figure 1. To reconstruct an image from its PPFT coefficients, samples are needed ( samples on equally sloped radial lines). A fast algorithm has been proposed by Averbuch et al. [23] to calculate the PPFT and its adjoint with complexity of . This algorithm can then be used to implement a fast and efficient 2D Radon transform on the equally sloped radial lines. The PPFT has three important properties which makes it a good alternative to conventional DFR methods: (1) it is closer to a polar (equiangular line) grid than to a Cartesian grid, which significantly decreases the gridding error, (2) it has both a fast forward and a fast backward calculation algorithm [23], which enables our proposed algorithm to avoid the regridding step used in iterative non-Cartesian Fourier based reconstruction methods, and (3) it has an analytical adjoint function. As a result, it can efficiently be used in iterative algorithms, including compressed sensing [24, 25]. However, it should be noted that Fourier-based reconstruction algorithms, for example, ESR and DFR, are only valid for parallel X-ray projections.