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Improved total variation minimization method for fewview computed tomography image reconstruction
BioMedical Engineering OnLine volume 13, Article number: 70 (2014)
Abstract
Background
Due to the harmful radiation dose effects for patients, minimizing the xray exposure risk has been an area of active research in medical computed tomography (CT) imaging. In CT, reducing the number of projection views is an effective means for reducing dose. The use of fewer projection views can also lead to a reduced imaging time and minimizing potential motion artifacts. However, conventional CT image reconstruction methods will appears prominent streak artifacts for fewview data. Inspired by the compressive sampling (CS) theory, iterative CT reconstruction algorithms have been developed and generated impressive results.
Method
In this paper, we propose a fewview adaptive prior image total variation (APITV) algorithm for CT image reconstruction. The prior image reconstructed by a conventional analytic algorithm such as filtered backprojection (FBP) algorithm from densely angularsampled projections.
Results
To validate and evaluate the performance of the proposed algorithm, we carried out quantitative evaluation studies in computer simulation and physical experiment.
Conclusion
The results show that the APITV algorithm can yield images with quality comparable to that obtained with existing algorithms.
Background
In recent years, the tremendous advances in computed tomography (CT) technology and applications have increased the clinical utilization of CT, creating concerns about individual and population doses of ionizing radiation. Studies indicate that dose from CT scans may have a lifetime attributable risk of cancer higher than previously assumed [1–3]. Reducing CT dose has been an area of active research in medical imaging [4]. Fewview CT image reconstruction is of great importance in clinical imaging for its potential to reduce the xray radiation dose to the human subject and the scan time. In fewview CT, less number of projection data than is required to satisfy the Nyquist sampling theorem is used. However, conventional filtered backprojection (FBP) based image reconstruction algorithms will occurs severe streak artifacts. Therefore, it is important to develop new algorithms in order to obtain more accurate images from fewview projections.
In 2006, a new image reconstruction theory, compressive sampling (CS), was proposed by Candes et al. [5–8]. They proved that an image of sparse signals could be satisfactorily reconstructed from far less measurements than what is usually considered necessary. Based on the fact that medical images are usually edgesparse, total variation (TV) minimization is often used for solving incomplete data problems in tomography. Inspired by this work, many iterative algorithms for fewview CT image reconstruction have been proposed and investigated [9–16]. Sidky et al. [9, 10] developed an adaptive steepest descent projection onto convex sets (ASDPOCS) algorithm based on an optimization strategy that minimizes the TV of the estimated image subject to data condition and other constraints. The ASDPOCS algorithm interleaves ART iterations with gradient descent steps for the TV penalty, aiming at the solution of the constrained minimization problem.
However, ASDPOCS algorithm does not directly incorporate a prior image into the reconstruction. In many applications of CT, prior CT projection data of the scanned object may be available. For example, in daily CBCT examinations for accurate patient setup and target localization in imageguided radiation therapy (IGRT), repeated scans have become routine procedures for the following days.
In this work, we develop an adaptive prior image TV (APITV) algorithm for fewview CT image reconstruction. Similar to ASDPOCS, this algorithm is based on an interlaced iteration and is a simple combination of prior image. The APITV algorithm follows the framework of ASDPOCS and has its own benefits. The paper is organized as follows. In section II, we detail describe the proposed APITV algorithm. In Section III, we present results of our numerical studies by using both computersimulation data and experimental data for validation and evaluation of the APITV algorithm, respectively. The conclusion in Section IV.
Methods
For CT image reconstruction from fewview data, the traditional filtered backprojection (FBP) method oftens suffers from serious streaking artifacts due to the ill condition of the number of projections. Recently, compressive sampling (CS) theory has been used to the CT reconstruction problem. In particular, total variation (TV) methods have demonstrated its power in image reconstruction from fewview projection. Sidky et al. presented an adaptive steepest descent projection on orthogonal convex subsets (ASDPOCS) algorithm for fewview reconstruction problem by solving the following constrained optimization problem
where $\tilde{\mathit{g}}$ represents the projection data, f is the discrete image, M is the transfer matrix, and ϵ is the error tolerance parameter. The TV of the tobereconstructed image, i.e. ∥ f ∥_{ TV } , is defined as:
where s, t and v are the indices of the location of the discrete image.
Accordingly, the derivative of ∥ f ∥_{ TV } is:
Here τ is a relax parameter to keep the denominator not equal to zero.
In the proposed APITV algorithm, the prior images can be first obtained from a prior scan. It could be an image reconstructed by a conventional analytic algorithm such as FBP algorithm from fullview projections. In summary, the pseudocode for the presented APITV algorithm is listed as follows:where the α is the control parameter and is selected to be 0.85. When α is set to be 0, the APITV algorithm reduces to the conventional ASDPOCS algorithm. In line 1, an image obtained by a conventional FBP algorithm from fullview projections. In line 2, an initial estimate of the tobereconstructed image is set to be uniform with voxel value of 0. Each outer loop (lines 3–14) is performed by two separated iteration steps, i.e. the POCS (or the ART) (lines 4–10) and the gradient descent for the APITV minimization (lines 11–13). After several general iterations, an accurate image reconstruction is obtained from fewview data samples.

1.
Obtain prior image f _{ p }

2.
Initial: ${\mathit{f}}_{\mathit{s},\mathit{t},\mathit{v}}^{\left(0\right)}=0$, s, t, v = 1, 2, …, N

3.
repeat main loop

4.
for ART iterations i=1:I
5.${\mathit{f}}_{\mathit{s},\mathit{t},\mathit{v}}^{\left(\mathit{i}\right)}=$ART_iteration (${\mathit{f}}_{\mathit{s},\mathit{t},\mathit{v}}^{\left(\mathit{i}1\right)}$); s, t, v = 1, …, N

6.
end ART

7.
POCS: enforce positivity

8.
if ${\mathit{f}}_{\mathit{s},\mathit{t},\mathit{v}}^{\left(\mathit{I}\right)}>0$, ${\mathit{f}}_{\mathit{s},\mathit{t},\mathit{v}}^{\left(\mathit{I}\right)}={\mathit{f}}_{\mathit{s},\mathit{t},\mathit{v}}^{\left(\mathit{I}\right)}$; s, t, v = 1, …, N

9.
else ${\mathit{f}}_{\mathit{s},\mathit{t},\mathit{v}}^{\left(\mathit{I}\right)}=0$; s, t, v = 1, …, N

10.
end POCS

11.
for TV gradient descent iterations k=1:K

12.
$${\mathit{f}}_{\mathit{s},\mathit{t},\mathit{v}}^{\left(\mathit{k}\right)}\approx \left(\mathit{\alpha}\cdot {\nabla}_{\mathit{f}}{\u2225\mathit{f}{\mathit{f}}_{\mathit{p}}\u2225}_{\mathit{TV}}^{\left(\mathit{k}1\right)}+\left(1\mathit{\alpha}\right)\cdot {\nabla}_{\mathit{f}}{\u2225\mathit{f}\u2225}_{\mathit{TV}}^{\left(\mathit{k}1\right)}\right)$$

13.
end TV gradient descent

14.
end if stop criterion is satisfy
To evaluate the differences between the results from ASDPOCS and APITV, we performed computer simulations and experimental study in the following section.
Results
Simulation study
In this section, we validate our APITV algorithm for conebeam fewview reconstruction from noisefree and additive noisy projections. The high contrast SheppLogan phantom of an array size of 256 by 256 was used as the true image. The simulated data were generated from SheppLogan phantom with circular conebeam scan trajectory. The radius of the circular trajectory was 500 mm and the sourcetodetector distance was 1000 mm. The simulated detector had a flatpanel geometry of 512 × 512 with detection elements of size 1 × 1 mm^{2}. A total of 60 projection views were simulated evenly over 360 degrees.

A.
Noisefree cases The images were reconstructed from 60 projection views selected. Figure 1 shows a comparison study of the reconstructed images by conventional ART, ASDPOCS and APITV algorithms in the noisefree cases. It can be observed that the images reconstructed by the ASDPOCS and APITV are visually much better than the results of conventional ART. The difference between the images from the ASDPOCS and APITV algorithms can be observed in Figure 1(c) and (d), respectively. It can be observed that the proposed algorithms can produce high quality images with much less streak artifacts than the ASDPOCS results.To further visualize the difference between the two approaches in the cases of 60 projection views, their horizontal profiles are given in Figure 2 for a further illustration. The horizontal profiles of the resulting images were drawn across the 128th row for each approach and are shown in Figure 2, where the corresponding profile from the true phantom image is given for reference. It can be seen that the APITV algorithm can achieve better profiles matching with the ideal ones than the ASDPOCS algorithm.

B.
Noisy cases In this section, image reconstruction from noisy data (about 10% white noise) was performed to analyze the robustness to noise of the APITV algorithm. The parameters are the same as the noisefree situation. Figure 3 shows that the ASDPOCS images have more artifacts as compared to the images reconstructed by the APITV algorithm from 60 projection views of the noisy data. Compared to the ASDPOCS algorithm, the APITV algorithm preserved more edge details.Numerical simulation results with additive noisy projection data (about 10% white noise) are shown in Figure 4. The horizontal profiles of the images reconstructed in the cases of ASDPOCS and APITV algorithms of noisy data along the 128th row are shown, respectively, with the corresponding profile of the true phantom image as a reference. These profiles also show that the APITV algorithm preserved the edge details better than the ASDPOCS algorithm in the noisy cases for fewview reconstruction. These noisy simulation studies were consistent with our previous observations in the noisefree cases, and further concurred with the advantage of using the adaptive prior image for edge preservation in the APITV model as compared to the conventional TV model. The results show the robustness of our algorithm for inconsistent data due to the presence of noise.

C.
Quantification based evaluation
In addition to visualization based evaluation, we performed quantitative measure of the image error to evaluate the image quality. The convergence rates of the two algorithms were compared with each other, as quantified by the image error:
where E denoted the image error, f is the reconstructed image, and f^{*} is the true image.As shown in Figure 5, the APITV algorithm became converged after about 30 iterations, whereas the ASDPOCS algorithm took more iteration to converge. After 30 iterations, the image error was reduced to 0.592 using the APITV method, much lower than 0.996 for the ASDPOCS method.
Experimental study

A.
Prototype microCT system
The developed MicroCT prototype system consisted of an xray source, a rotational object holder and a CCD detector. The open xray source (FXE: 160.51, YXLON, Germany) has a minimum focal spot size less than 2 μm and the focal spot size can been adjusted by changing the focusing current. A cooled xray CCD imaging detector (QuadRO: 4320, Princeton Instruments, USA) was used to acquire the images. The CCD imaging detector has a high spatial resolution, large active imaging area and low noise (24 μm pixel size, 2084 × 2084 array and 50 × 50 mm^{2} active area).

B.
Small animal Imaging
In order to validate the proposed algorithm’s performance, we used the laboratory mouse to obtain projection data. Circular conebeam data were acquired from the mouse of 120 projection views distributed over 2π, with the parameters of 80 kVp, 7w, and 8 s per exposure. The sourcetodetector distance (SDD) was held at 710 mm and the sourcetoobject distance (SOD) was set at 530 mm.
A fullview prior image was reconstructed using the FBP algorithm. The fullview scan was designed to take projection data at 360 views on regular interval. The fewview scan was designed to take projection data set at 120 views on regular interval. The fewview scan implies that dose delivered to the scanned object using few scan is roughly 1/3 times of the full scan.In Figure 6, we display images reconstructed from the 120view data by use of the ART, ASDPOCS, and APITV algorithms, within transverse (z = 0 mm), coronal (x = 5.088 mm), and sagittal (y = 5.188 mm) slices. Visual inspection of reconstructions in Figures 6 suggests that the ASDPOCS and APITV algorithms can effectively suppress streak artifacts and noise observed in images obtained with the ART algorithms. In addition, the result of the APITV shows more details on the edges than the result of the ASDPOCS as indicated by the arrows in figures.
Conclusion
In this paper, we introduced a novel adaptive prior image total variation (APITV) minimization model for lowdose CT image reconstruction from fewview projection measurements. We have evaluated and demonstrated the performance of our algorithm in a number of fewview reconstruction problems. Numerical studies on both the computer simulation and microCT imaging experiments were taken to validate our algorithm. In all the situations shown, APITV algorithm leads to significant artifact reduction without visible over smoothing of low contrast areas in the image. All results were compared to a previously proposed implementation of the ASDPOCS algorithm.
A weakness in our proposed APITV algorithm is that it assumes that the patients were at exactly the same position during the repeated scans. This assumption, however, may not always be the case as the patient is frequently repositioned for optimal imaging in imageguided radiation therapy (IGRT) applications. It is therefore unavoidable that mismatched areas between current and prior images may occur in practice. Therefore, a robust implementation of this algorithm will require accurate registration and voxel consistency for each projection.
In the present study, because of the limitations of the experimental conditions, many clinical conditions were not considered. Future work is therefore needed to verify proposed algorithm on real clinical projections. We hope that the present APITV algorithm may be widely used in medical clinic.
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Acknowledgements
This work was supported in part by the Special Fund Project for Development of Strategic Emerging Industry of Shenzhen in China (JCYJ20130401170306796), Basic Research Program of Shenzhen in China (JC201005280581A, JC201105190923A), National Natural Science Foundation of China (61102161), the National Science & Technology Pillar Program of China (2012BAI13B04).
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Authors’ contributions
ZH worked on the algorithm design and implementation, and wrote the paper. HZ contributed to discussion and suggestions throughout this topic. Both authors read and approved the final manuscript.
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Hu, Z., Zheng, H. Improved total variation minimization method for fewview computed tomography image reconstruction. BioMed Eng OnLine 13, 70 (2014). https://doi.org/10.1186/1475925X1370
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Keywords
 Computed tomography
 Fewview
 Adaptive prior image
 Total variation
 Compressive sampling