Existence and convergence analysis of   $\ell_{0}$  and   $\ell_{2}$  regularizations for limited-angle CT reconstruction

Chengxiang Wang; Li Zeng; Wei Yu; Liwei Xu

doi:10.3934/ipi.2018024

Article Contents

2018, Volume 12, Issue 3: 545-572. Doi: 10.3934/ipi.2018024

This issue Previous Article Recovering a large number of diffusion constants in a parabolic equation from energy measurements Next Article Mathematical imaging using electric or magnetic nanoparticles as contrast agents

Existence and convergence analysis of $\ell_{0}$ and $\ell_{2}$ regularizations for limited-angle CT reconstruction

1.
School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu 611731, China
2.
College of Mathematics and Statistics, Chongqing University, Chongqing 401331, China
3.
Engineering Research Center of Industrial Computed Tomography, Nondestructive Testing of the Education Ministry of China, Chongqing University, Chongqing 400044, China
4.
School of Biomedical Engineering, Hubei University of Science and Technology, Xianning 437100, China

* Corresponding author: drlizeng@cqu.edu.cn
* Corresponding author: drlizeng@cqu.edu.cn

Received: March 31, 2017

Revised: November 30, 2017

Published: March 2018

Li Zeng was supported by the National Natural Science Foundation of China (No.61771003) and the National Instrumentation Program of China (2013YQ030629). Liwei Xu was supported in part by a Key Project of the Major Research Plan of NSFC (No.91630205) and by the National Natural Science Foundation of China (No.11771068). Wei Yu was supported by the National Natural Science Foundation of China (No.61701174), the Hubei Provincial Natural Science Foundation of China (No.2017CFB168) and the Ph.D. start-up Fund of HBUST (No. BK1527).

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Abstract

In some practical applications of computed tomography (CT) imaging, the projections of an object are obtained within a limited-angle range due to the restriction of the scanning environment. In this situation, conventional analytic algorithms, such as filtered backprojection (FBP), will not work because the projections are incomplete. An image reconstruction algorithm based on total variation minimization (TVM) can significantly reduce streak artifacts in sparse-view reconstruction, but it will not effectively suppress slope artifacts when dealing with limited-angle reconstruction problems. To solve this problem, we consider a family of image reconstruction model based on $\ell_{0}$ and $\ell_{2}$ regularizations for limited-angle CT and prove the existence of a solution for two CT reconstruction models. The Alternating Direction Method of Multipliers (ADMM)-like method is utilized to solve our model. Furthermore, we prove the convergence of our algorithm under certain conditions. Some numerical experiments are used to evaluate the performance of our algorithm and the results indicate that our algorithm has advantage in suppressing slope artifacts.

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Mathematics Subject Classification: Primary: 90C90, 15A29, 44A12; Secondary: 94A08.

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Figure 1. The scanning geometry of limited-angle CT. $T$ denotes the objective table, $\textrm{O}^{'}$ denotes the scanned object, $\textrm{D}$ denotes the detector array, $\textrm{o}$ denotes the rotation center of X-ray source, $\textrm{S}$ denotes X-ray source that rotates anticlockwise around $\textrm{o}$ with $\textrm{D}$ synchronously and $\theta$ denotes the rotation angle which is less than $180^{0}$ plus a fan-angle

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Figure 4. The reconstructed results of the NCAT phantom. The image on the first row is the original image (or reference image). The subsequent rows are the results reconstructed for the scan ranges $[0,120^{0}]$ and $[0,140^{0}]$, respectively. The images from left to right in each column are the results reconstructed using the SART algorithm, the ASD-POCS algorithm and our algorithm. The location of red arrows present slope artifacts. The grey-scale display window is $[0.2, 0.85]$

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Figure 6. The reconstructed results of the practical gear data. The images on the left column are the results reconstructed using the SART algorithm. The subsequent columns are the ASD-POCS algorithm and our algorithm, respectively. The images from top to bottom in each row present the results reconstructed from scanning ranges $[0,100^{0}]$, $[0,120^{0}]$, $[0,140^{0}]$, and $[0,160^{0}]$. The location of red arrows present slope artifacts. The display window is $[0.005, 0.0068]$ $mm^{-1}$

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Figure 2. The transform results of image under the one level piecewise-constant linear B-spline framelets transform

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Figure 3. Sketch map of slope artifacts correction for the limited-angle CT image reconstruction if some proper parameters are chose. The wavelet tight framlet transform is piecewise-constant linear B-spline framelets transform

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Figure 5. The images from left to right present the Shepp-Logan phantom, the reconstructed result using the $\ell_{0}$ and the reconstructed result $\ell_{0}-\ell_{2}$ regularization. The display window is $[0, 1]$

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Figure 7. The reconstructed image from the projection data of the scanning angular ranges $[0,360^{0}]$ using the FBP algorithm. Metal artifacts are labelled by red rectangles

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Figure 8. The reconstructed image from the projection data of the scanning angular ranges $[0,180^{0}]$ using the ASD-POCS algorithm and our algorithm. ROIs are labelled by red rectangles. The display window is $[0.0051, 0.0068]$ $mm^{-1}$

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Figure 9. The reconstructed image from the projection data of the scanning angular ranges $[0,160^{0}]$ using the ASD-POCS algorithm and our algorithm. ROIs are labelled by red rectangles. The display window is $[0.0051, 0.0062]$ $mm^{-1}$

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Figure 10. The zoomed-in view of the image ROIs of Figure 8 and Figure 9. The upper plane of Figure are the reconstructed results of ROIs for the scan ranges $[0,180^{0}]$, and the bottom plane of Figure are the reconstructed results of ROIs for the scan ranges $[0,160^{0}]$. The Figure on the first and three rows are the results using the ASD-POCS algorithm, and the second and four rows are the results using the our algorithm. The display window is $[0.0051, 0.0068]$ $mm^{-1}$ for the last column, and $[0.005, 0.005007]$ $mm^{-1}$ for the rest

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Table 1. Geometrical scanning parameters for simulated CT imaging system

The distance between source and rotation center	$981mm$
The angle interval of projection views	$1^{0}$
The distance between source and detector	$1200mm$
The diameter of field of view	$143.6222mm$
The detector bin numbers	$256$
The angle interval of rays	$0.00329^{0}$
Pixel size	$0.5632\times0.5632mm^{2}$
Image size	$256\times256$

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Table 2. Characterize quantitatively the reconstruction quality for NCAT

Scanning ranges	Algorithm	RMSE	PSNR	MSSIM
	SART	0.0514	25.78	0.9997
$[0,120^{0}]$	ASD-POCS	0.0287	30.83	0.9999
	our method	0.0250	32.05	0.9999
	SART	0.0429	27.35	0.9998
$[0,140^{0}]$	ASD-POCS	0.0209	33.60	1.0000
	our method	$0.0215$	$33.33$	1.0000

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Table 3. Characterize quantitatively the reconstruction quality for Shepp-Logan phantom

Scanning ranges	Algorithm	RMSE	PSNR	MSSIM
$[0,150^{0}]$	$\ell_{0}$ regularization	0.0628	24.04	0.9676
	$\ell_{0}-\ell_{2}$ regularization	0.0618	24.17	0.9681

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Table 4. Quantitatively characterize the reconstruction quality of gear

Scan ranges	Algorithm	RMSE	PSNR	MSSIM
	SART	40.157	16.056	0.752
$0^{0}\sim100^{0}$	ASD-POCS	17.296	23.372	0.784
	our method	8.000	30.069	0.805
	SART	37.156	16.730	0.776
$0^{0}\sim120^{0}$	ASD-POCS	10.728	27.521	0.787
	our method	7.691	30.412	0.815
	SART	30.425	18.466	0.790
$0^{0}\sim140^{0} $	ASD-POCS	9.096	28.954	0.788
	our method	6.984	31.248	0.814
	SART	19.766	22.212	0.805
$0^{0}\sim160^{0} $	ASD-POCS	9.680	28.413	0.807
	our method	6.570	31.779	0.807

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Table 5. Characterize quantitatively the reconstruction quality for metal lath

	Algorithm	RMSE	PSNR	MSSIM
$0\sim 180^{0}$	ASD-POCS	105.7	7.6481	0.8050
	our method	107.4	7.5139	0.8060
$0\sim 160^{0}$	ASD-POCS	108.5	7.4209	0.8014
	our method	107.9	7.4704	0.8014

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