Wavelet tight frame and prior image-based image reconstruction from limited-angle projection data

Chengxiang Wang; Li Zeng; Yumeng Guo; Lingli Zhang

doi:10.3934/ipi.2017043

Article Contents

2017, Volume 11, Issue 6: 917-948. Doi: 10.3934/ipi.2017043

This issue Previous Article Next Article Multiplicative noise removal with a sparsity-aware optimization model

Wavelet tight frame and prior image-based image reconstruction from limited-angle projection data

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

¹The corresponding author: drlizeng@cqu.edu.cn
¹The corresponding author: drlizeng@cqu.edu.cn

Received: November 30, 2015

Revised: July 31, 2017

Published: November 2017

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Abstract

The limited-angle projection data of an object, in some practical applications of computed tomography (CT), are obtained due to the restriction of scanning condition. In these situations, since the projection data are incomplete, some limited-angle artifacts will be presented near the edges of reconstructed image using some classical reconstruction algorithms, such as filtered backprojection (FBP). The reconstructed image can be fine approximated by sparse coefficients under a proper wavelet tight frame, and the quality of reconstructed image can be improved by an available prior image. To deal with limited-angle CT reconstruction problem, we propose a minimization model that is based on wavelet tight frame and a prior image, and perform this minimization problem efficiently by iteratively minimizing separately. Moreover, we show that each bounded sequence, which is generated by our method, converges to a critical or a stationary point. The experimental results indicate that our algorithm can efficiently suppress artifacts and noise and preserve the edges of reconstructed image, what's more, the introduced prior image will not miss the important information that is not included in the prior image.

Keywords:

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

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Figure 1. Scanning geometry of limited-angle CT. $\textrm{S}$ denotes the X-ray source, $\textrm{o}$ denotes the rotation center of object, $\textrm{D}$ denotes the detector, and $\theta$ denotes the rotation angle which is less than $180^{0}$ plus a fan-angle

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Figure 2. Reconstructed result of NCAT phantom using FBP algorithm for the scanning angle $[0,120^{0}]$. The limited-angle artifacts are labelled by the red rectangles

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Figure 3. (a) and (b) are the reconstructed results using FBP algorithm for scanning range $[0,360^{0}]$ and $[0,120^{0}]$, respectively

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Figure 4. The wavelet transform of (a) of Figure 3 under B-spline frame

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Figure 5. The wavelet transform of (b) of Figure 3 under B-spline frame

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Figure 6. NCAT phantom and prior image. The first column is NCAT phantom, the second column is the prior image and the third column is the absolute value of the difference between phantom and prior image. The prior image is the same with NCAT phantom

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Figure 7. The reconstructed results for different scanning ranges using FBP algorithm, PICCS algorithm and our algorithm. The display window is $[0,255]$. The noise levels are $0.5\%\|g\|_{\infty}$

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Figure 8. The reconstructed results for different scanning ranges using FBP algorithm, PICCS algorithm and our algorithm. The display window is $[0,255]$. The noise levels are $1\%\|g\|_{\infty}$

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Figure 9. Phantom, prior image, and the absolute value of the difference between phantom and prior image. Three circles are labelled by red rectangles

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Figure 10. The reconstructed results for the different scanning ranges using FBP algorithm, PICCS algorithm and our algorithm. Three circles are labelled by red rectangles. The display window is $[0,255]$

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Figure 11. Phantom and prior image. The first column is the phantom, the subsequent columns are the prior image and the absolute value of the difference between phantom and prior image

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Figure 12. The reconstructed results using PICCS algorithm and our algorithm. The display window for the first row is $[0,1]$ and for the second row is $[0.8, 0.9]$

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Figure 13. The reconstructed results for the different scanning ranges using $\ell_{1}-\ell_{0}$ algorithm and $\ell_{2}-\ell_{0}$ algorithm. Three circles are labelled by red rectangles. The display window is $[0,255]$

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Figure 14. The reconstructed results of simulation phantom for the different scanning ranges using $\ell_{1}-\ell_{0}$ algorithm and $\ell_{2}-\ell_{0}$ algorithm. The display window for the first row is $[0,1]$ and for the second row is $[0.8, 0.9]$

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Figure 15. The reconstructed results of gear using FBP algorithm and our algorithm. ROI is labelled by red rectangle. The display window is $[0, 0.0096]mm^{-1}$

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Figure 16. The zoom-in view of ROI of Figure 15

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Table 1. Geometrical scanning parameters of simulated CT system

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

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Table 2. Quantitatively characterize the reconstruction quality

Scanning ranges	Variances	Algorithm	RMSE	PSNR	MSSIM
$0\sim 360^{0}$	$0.5\% \\|g\\|_{\infty}$	FBP	8.403	29.64	0.9921
	$1\% \\|g\\|_{\infty}$	FBP	12.88	25.93	0.9800
$0\sim 100^{0}$	$0.5\% \\|g\\|_{\infty}$	our algorithm	2.116	41.62	0.9995
		PICCS	3.743	36.67	0.9985
	$1\% \\|g\\|_{\infty}$	our algorithm	4.747	34.60	0.9973
		PICCS	5.953	32.64	0.9953
$0\sim 80^{0}$	$0.5\% \\|g\\|_{\infty}$	our algorithm	2.240	41.12	0.9994
		PICCS	4.087	35.90	0.9984
	$1\% \\|g\\|_{\infty}$	our algorithm	4.258	35.55	0.9978
		PICCS	4.915	32.69	0.9953
$0\sim 60^{0}$	$0.5\% \\|g\\|_{\infty}$	our algorithm	1.843	42.82	0.9996
		PICCS	5.905	32.71	0.9956
	$1\% \\|g\\|_{\infty}$	our algorithm	3.566	37.09	0.9985
		PICCS	6.250	32.21	0.9952

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Table 3. Geometrical scanning parameters of simulated CT system

The distance between source and object center	$981mm$
The angle interval of two adjacent projection views	$0.703^{0}$
The angle interval of two adjacent rays	$0.0005^{0}$
The diameter of field of view	$279.5mm$
Detector numbers	$560$
Pixel size	$0.5\times0.5mm^{2}$
Image size	$512\times512$

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Table 4. Quantitatively characterize the reconstruction quality

Scanning ranges	Algorithm	RMSE	PSNR	MSSIM
$0\sim 360^{0}$	FBP	4.978	34.19	0.9961
$0\sim 120^{0}$	our algorithm	3.607	36.99	0.9979
	PICCS	4.208	35.65	0.9972
$0\sim 100^{0}$	our algorithm	3.901	36.31	0.9976
	PICCS	4.190	35.69	0.9972
$0\sim 80^{0}$	our algorithm	3.693	36.78	0.9979
	PICCS	4.177	35.71	0.9972

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Table 5. The parameters of simulated phantom

$I$	$h$	$v$	$x_{0}$	$y_{0}$	$r$
1	0.74	0.74	0	0	0
-1	0.5	0.5	0	0	0
-1	0.1	0.1	0.43	0.43	0
-1	0.1	0.1	-0.43	-0.43	0
-1	0.1	0.1	-0.43	0.43	0
-1	0.1	0.1	0.43	-0.43	0
-1	0.12	0.006	0.25	0.55	-18
-1	0.08	0.006	0.25	-0.55	-240
-1	0.08	0.006	-0.55	0.3	20

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Table 6. Quantitatively characterize the reconstruction quality

Scanning ranges	Algorithm	RMSE	PSNR	MSSIM
$0\sim 120^{0}$	our algorithm	0.040	27.97	0.9949
	PICCS	0.057	28.26	0.9897
$0\sim 100^{0}$	our algorithm	0.0378	28.45	0.9954
	PICCS	0.0546	28.65	0.9906

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Table 7. Quantitatively characterize the reconstruction quality

Scanning ranges	Algorithm	RMSE	PSNR	MSSIM
$0\sim 120^{0}$	$\ell_{1}-\ell_{0}$	3.725	36.71	0.9978
$0\sim 120^{0}$	$\ell_{2}-\ell_{0}$	3.607	36.99	0.9979
$0\sim 100^{0}$	$\ell_{1}-\ell_{0}$	3.856	36.41	0.9977
$0\sim 100^{0}$	$\ell_{2}-\ell_{0}$	3.901	36.31	0.9976

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Table 8. Quantitatively characterize the reconstruction quality

Scanning ranges	Algorithm	RMSE	PSNR	MSSIM
$0\sim 120^{0}$	$\ell_{1}-\ell_{0}$	0.0404	27.86	0.9948
$0\sim 120^{0}$	$\ell_{2}-\ell_{0}$	0.0400	27.97	0.9949
$0\sim 100^{0}$	$\ell_{1}-\ell_{0}$	0.0383	28.34	0.9954
$0\sim 100^{0}$	$\ell_{2}-\ell_{0}$	0.0378	28.45	0.9954

| Show Table

DownLoad: CSV

Table 9. Quantitatively characterize the reconstruction quality

Scanning ranges	Algorithm	RMSE	PSNR	MSSIM
$0\sim 360^{0}$	FBP	6.559	31.79	0.9965
$0\sim 80^{0}$	our algorithm	4.543	34.98	0.9983

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