In computed tomography (CT), there are many situations where reconstruction has to be performed with sparseview data. In sparseview CT imaging, strong streak artifacts may appear in conventionally reconstructed images due to limited sampling rate that compromises image quality. Compressed sensing (CS) algorithm has shown potential to accurately recover images from highly undersampled data. In the past few years, totalvariation(TV) based compressed sensing algorithms have been proposed to suppress the streak artifact in CT image reconstruction. In this paper, we propose an efficient compressed sensingbased algorithm for CT image reconstruction from fewview data where we simultaneously minimize three parameters: the
Xray computed tomography (CT) is extensively used clinically to evaluate patients with a variety of conditions. However, by its nature, CT scans expose the patients to high Xray radiation doses which can result in an increased lifetime risk of cancer [
Recently a number of strategies have been proposed to decrease radiation dose in CT scans. One approach to lower the total Xray radiation dose is to simply reduce the dose level mAs/view in data acquisition protocols. This approach typically results in an insufficient number of Xray photons received by the detectors, increasing the noise level on the sinograms produced. The noisecontaminated sinogram data will degrade the quality of reconstructed CT images when a conventional FBP algorithm is used [
Since analytical reconstruction methods, such as FBP, cause such serious streaking artifacts in the resulting reconstructed CT images, iterative algorithms have been proposed and investigated as a means to eliminate these defects. One approach is algebraic and is based upon solving a system of linear equations. This scheme is often referred to as algebraic reconstruction technique (ART) [
In the past few years, compressed sensing (CS) algorithms [
A parallel beam CT scanning system uses an array of equally spaced sources of Xray beams and an array of detectors. Let
In this study, our proposed algorithm will be compared with the stateoftheart methods, including filtered backprojection (FBP) [
Consider the parallel beam of rays intersecting an object as shown in Figure
A parallel beam projection through
Using a Dirac delta function, we have an alternate representation:
FBP begins by filtering the projection data with a high pass filter, which in reality is implemented by the RamLak filter or SheppLogan filter, then takes the integral over 0 to
ART considers the CT imaging process as a linear system of equations as in (
The reason for calling the methods “simultaneous” is that all the equations are used at the same time in one iteration. The general form of simultaneous iterative reconstruction technique (SIRT) is
ARTtype methods are known to have better performance than FBP algorithms in suppressing streak artifacts and noise in sparseview CT imaging.
The problem of sparseview CT image reconstruction actually leads to an underdetermined system of linear equations (equation (
INPUTS:
OPTIONAL PARAMETERS:
Tol: stopping criteria by gradient magnitude (default 10^{−4})
Iter: stopping criteria by number of iterations (default 100)
OUTPUTS:
% Initialization
% Iterations
{
% Backtracking linesearch
and
In a discrete version, (
The two regularization factors
Since (
The conjugate gradient requires the computation of
As the
In this section, we present our experimental results. There are four sets of experiments. In the first two experiments, true CT images and simulated projections were used to study the performance of our algorithm under ideal and degraded conditions. The third and fourth experiments used real data collected using the Canadian Light Source (
Reconstructions were quantitatively evaluated in terms of relative root mean square error (RRMSE), streak indicator (SI), and structural similarity (SSIM) index. The relative root mean square error (RRMSE) is defined as
The lower the value of SI is, the less the streaking artifacts are present in the reconstructed image.
The structural similarity (SSIM) index is highly effective for measuring the structural similarity between two images [
In order to find the optimum number of iteration, we have conducted another experiment using simulated phantom. The results are shown in Figure
Analysis to find the optimum number of iterations for different methods: (a) ART and SART, (b) TV and the proposed scheme.
Moreover, the reconstruction accuracy depends on the selection of optimum regularization parameters for both TV method and the proposed method. We have used a real dataset (such as rat dataset as described later in Section
Optimum parameter selections for each dataset.
Data  TV algorithm  Proposed algorithm  



 
Phantom without noise  0.0005  0.0005  0.0005 
Phantom with noise  0.0015  0.001  0.0006 
Human bone  0.001  0.001  0.001 
Rat  0.0005  0.001  0.0005 
Analysis to find the optimum regularization parameters (for rat dataset): (a)
For the proposed algorithm, there are two parameters. We alternately plotted the reconstruction error against one parameter keeping the other fixed. We started by setting
The first experiment was performed using nodule phantom image and simulated projection without any noise purposely added. This data is provided free of charge by the National Cancer Institute (NCI) [
The reconstruction results of the nodule phantom using 50 projections. (a) The ground truth image, (b) the result obtained using FBP algorithm, (c) the ART algorithm, (d) the SART algorithm, (e) the TV algorithm, and (f) the proposed CS algorithm.
However, we can still see some residual streak artifacts in the TV reconstruction. The image reconstructed from our proposed method shows the least level of streaking artifacts. One possible reason for that is, in wavelet domain, the noise is uniformly spread throughout the coefficients while mostly the image information is concentrated in the few largest coefficients [
Reconstruction results using phantom image.
Reconstruction methods  RRMSE  SI  SSIM 

FBP  0.1282  44.9556  0.6110 
ART [ 
0.1120  25.5737  0.7681 
SART [ 
0.1198  25.0023  0.7663 
TV [ 
0.0715  20.0115  0.8716 
Proposed method  0.0609  18.0646  0.9310 
A detailed section of Figure
The second experiment was performed using noisy simulated data. Additive Gaussian white noise
Reconstruction results using phantom image (with noise).
Methods  RRMSE  SI  SSIM 

FBP  0.2908  127.4656  0.3284 
ART [ 
0.1197  28.7409  0.7260 
SART [ 
0.1324  28.0063  0.7344 
TV [ 
0.0891  24.1023  0.7693 
Proposed method  0.0687  21.2074  0.8967 
Simulated reconstruction of noisy phantom from 50 noisy projections over 180°: (a) the true image, (b) FBP, (c) ART, (d) SART, (e) TV, and (f) the proposed method.
Pixelintensity profiles of reconstructed images compared with ground truth (GT): (a) FBP, (b) ART, (c) TV, and (d) the proposed method.
In the third and fourth experiments, we used real data collected from the Canadian Light Source facility and from a desktop Bruker SkyScan 1172 MicroCT system with two datasets: human femoral cortical bone and the hindpaw of a normal Wistar rat. For the human bone, microCT scanning was performed at the BioMedical Imaging and Therapy Bending Magnet Beamline (BMITBM; 05B11). Projections were collected with a Hamamatsu C9300 (Hamamatsu Photonics, Hamamatsu, Japan) CCD camera fitted with a beam monitor with a 10
The FBP reconstruction of the complete dataset. The image has a large smooth region, so to better demonstrate the details, a region of interest (ROI) is selected.
The ROI reconstruction results restricted to 50 views for the human cortical bone image are shown in Figure
The ROI reconstructions of human bone. (a) The image reconstructed by FBP with 1800 projections, (b) the result obtained using FBP algorithm, (c) the ART algorithm, (d) the SART algorithm, (e) the TV algorithm, and (f) the proposed CS algorithm, all using 50views.
In the ART and SART images, the streaking artifacts and noise are reduced, but residual artifacts can be seen and the noise is still pervasive. Besides this, they suffer from edge blurring artifacts and many low contrast structures are lost. The edges of the vascular canals are no longer able to be precisely distinguished, an important feature for characterizing their shape and size. The streaking artifacts in the TV reconstruction are less conspicuous than they are in FBP, ART, and SART, but we can clearly see some relatively low frequency patchy structures present in nonedge regions. In clinical practice, these patchy structures may mimic low contrast lesions and obscure the presence of small details. By comparison, our proposed method provides reconstruction of high fidelity, as presented in Figure
To further quantify the reconstruction accuracy and streaking artifacts, the RRMSEs, SIs, and SSIMs values of the given ROI by these methods are shown in Table
Reconstruction results using real dataset.
Reconstruction methods  RRMSE  SI  SSIM 

FBP  0.5102  97.325  0.3040 
ART [ 
0.1525  22.9236  0.6893 
SART [ 
0.1412  20.0544  0.6955 
TV [ 
0.0783  6.7528  0.7983 
Proposed method  0.0557  4.1120  0.8642 
(a) The FBP reconstruction of human bone, (b) reconstruction using the proposed CS algorithm.
Now let us look at the adult Wistar rat hindpaw image. This image shows a transverse slice through the bones of the paw, with the bottom bone showing trabecular bone and the other four bones showing cortical bone and marrow cavities. The experimental results of the rat are displayed in Figure
Reconstruction results of the hindpaw image of the adult rat. (a) FBP reconstruction using 900 projections, (b) FBP algorithm with 50 projections, (c) ART algorithm, (d) SART algorithm, (e) TV algorithm, and (f) the proposed CS algorithm, all using 50 views.
Although the TV method can suppress the noise and streak artifacts considerably, it is still a great challenge to reconstruct the trabecular bone, the fine structure in the bottom righthand corner of the image as indicated by the red arrows in Figure
For a comprehensive comparison, the RRMSEs, SIs, and SSIMs of the reconstructed images are also plotted against the number of projections in Figure
Plots of relative root mean square error (RRMSE), streak indicator (SI), and structural similarity (SSIM) for rat dataset.
The convergence speed of an algorithm is a crucial factor for all iterative methods in clinical practice. To investigate the convergence speed of the proposed method, the plot of cost function value
Convergence curve (cost function values versus number of iterations) for the proposed method applied to phantom dataset.
In this work, we have investigated a novel compressed sensingbased algorithm for sparseview CT image reconstruction, in which wavelet transform is used in the reconstruction procedure. Results show that the proposed method is able to suppress streak artifacts and noise caused by incomplete and noisy projection data without visible oversmoothing of fine structure details in the images. The proposed CSbased algorithm has potential to reduce the dose in clinical computed tomography imaging techniques.
This research work was supported by Natural Sciences and Engineering Research Council of Canada (NSERC).