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

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  • 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.

    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

    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

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

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

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

    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

    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 $0.5\%\|g\|_{\infty}$

    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 $1\%\|g\|_{\infty}$

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

    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 $[0,255]$

    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

    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 $[0.8, 0.9]$

    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 $[0,255]$

    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 $[0.8, 0.9]$

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

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

    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 $256\times256$
     | Show Table
    DownLoad: CSV

    Table 2.  Quantitatively characterize the reconstruction quality

    Scanning rangesVariancesAlgorithmRMSEPSNRMSSIM
    $0\sim 360^{0}$ $0.5\% \|g\|_{\infty}$FBP8.40329.640.9921
    $1\% \|g\|_{\infty}$FBP12.8825.930.9800
    $0\sim 100^{0}$ $0.5\% \|g\|_{\infty}$our algorithm2.11641.620.9995
    PICCS3.74336.670.9985
    $1\% \|g\|_{\infty}$our algorithm4.74734.600.9973
    PICCS5.95332.640.9953
    $0\sim 80^{0}$ $0.5\% \|g\|_{\infty}$our algorithm2.24041.120.9994
    PICCS4.08735.900.9984
    $1\% \|g\|_{\infty}$our algorithm4.25835.550.9978
    PICCS4.91532.690.9953
    $0\sim 60^{0}$ $0.5\% \|g\|_{\infty}$our algorithm1.84342.820.9996
    PICCS5.90532.710.9956
    $1\% \|g\|_{\infty}$our algorithm3.56637.090.9985
    PICCS6.25032.210.9952
     | Show Table
    DownLoad: CSV

    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 $512\times512$
     | Show Table
    DownLoad: CSV

    Table 4.  Quantitatively characterize the reconstruction quality

    Scanning rangesAlgorithmRMSEPSNRMSSIM
    $0\sim 360^{0}$FBP4.97834.190.9961
    $0\sim 120^{0}$our algorithm3.60736.990.9979
    PICCS4.20835.650.9972
    $0\sim 100^{0}$our algorithm3.90136.310.9976
    PICCS4.19035.690.9972
    $0\sim 80^{0}$our algorithm3.69336.780.9979
    PICCS4.17735.710.9972
     | Show Table
    DownLoad: CSV

    Table 5.  The parameters of simulated phantom

    $I$$h$ $v$ $x_{0}$ $y_{0}$ $r$
    10.740.74000
    -10.50.5000
    -10.10.10.430.430
    -10.10.1-0.43-0.430
    -10.10.1-0.430.430
    -10.10.10.43-0.430
    -10.120.0060.250.55-18
    -10.080.0060.25-0.55-240
    -10.080.006-0.550.320
     | Show Table
    DownLoad: CSV

    Table 6.  Quantitatively characterize the reconstruction quality

    Scanning rangesAlgorithmRMSEPSNRMSSIM
    $0\sim 120^{0}$our algorithm0.04027.970.9949
    PICCS0.05728.260.9897
    $0\sim 100^{0}$our algorithm0.037828.450.9954
    PICCS0.054628.650.9906
     | Show Table
    DownLoad: CSV

    Table 7.  Quantitatively characterize the reconstruction quality

    Scanning rangesAlgorithmRMSEPSNRMSSIM
    $0\sim 120^{0}$$\ell_{1}-\ell_{0}$3.72536.710.9978
    $0\sim 120^{0}$$\ell_{2}-\ell_{0}$3.60736.990.9979
    $0\sim 100^{0}$$\ell_{1}-\ell_{0}$3.85636.410.9977
    $0\sim 100^{0}$$\ell_{2}-\ell_{0}$3.90136.310.9976
     | Show Table
    DownLoad: CSV

    Table 8.  Quantitatively characterize the reconstruction quality

    Scanning rangesAlgorithmRMSEPSNRMSSIM
    $0\sim 120^{0}$$\ell_{1}-\ell_{0}$0.040427.860.9948
    $0\sim 120^{0}$$\ell_{2}-\ell_{0}$0.040027.970.9949
    $0\sim 100^{0}$$\ell_{1}-\ell_{0}$0.038328.340.9954
    $0\sim 100^{0}$$\ell_{2}-\ell_{0}$0.037828.450.9954
     | Show Table
    DownLoad: CSV

    Table 9.  Quantitatively characterize the reconstruction quality

    Scanning rangesAlgorithmRMSEPSNRMSSIM
    $0\sim 360^{0}$FBP6.55931.790.9965
    $0\sim 80^{0}$our algorithm4.54334.980.9983
     | Show Table
    DownLoad: CSV
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