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Image restoration from noisy incomplete frequency data by alternative iteration scheme

  • * Corresponding author: Prof. Dr. Jijun Liu

    * Corresponding author: Prof. Dr. Jijun Liu
Abstract Full Text(HTML) Figure(6) / Table(2) Related Papers Cited by
  • Consider the image restoration from incomplete noisy frequency data with total variation and sparsity regularizing penalty terms. Firstly, we establish an unconstrained optimization model with different smooth approximations on the regularizing terms. Then, to weaken the amount of computations for cost functional with total variation term, the alternating iterative scheme is developed to obtain the exact solution through shrinkage thresholding in inner loop, while the nonlinear Euler equation is appropriately linearized at each iteration in exterior loop, yielding a linear system with diagonal coefficient matrix in frequency domain. Finally the linearized iteration is proven to be convergent in generalized sense for suitable regularizing parameters, and the error between the linearized iterative solution and the one gotten from the exact nonlinear Euler equation is rigorously estimated, revealing the essence of the proposed alternative iteration scheme. Numerical tests for different configurations show the validity of the proposed scheme, compared with some existing algorithms.

    Mathematics Subject Classification: Primary: 65M32, 68U10; Secondary: 65K10, 65T50.


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  • Figure 1.  Object images: (A) circles under black background; (B) circles under gray background; (C) a phantom from Matlab; (D) an MRI chest image

    Figure 2.  Masks: (A) random sampling with 20 rows and 20 columns; (B) random sampling with 40 rows and 40 columns; (C) radial sampling with 22 lines; (D) radial sampling with 44 lines

    Figure 3.  Reconstructions by random band sampling. From left to right: exact images, images by back projections, images by DM, images by RecPF, images by C-SMRM, images by H-SMRM

    Figure 5.  Reconstructions by radial sampling. From left to right: exact images, images by back projections, images by DM, images by RecPF, images by C-SMRM, images by H-SMRM

    Figure 4.  Errors by random band sampling: $ \|\mathbf{f}^{(k+1)}-\mathbf{f}^{(k)}\|_{l^2} $ (top line); $ \|\mathbf{f}^{(k+1)}-\mathbf{f}^*\|_{l^2} $ (bottom line). The first column is for C-SMRM method, while the second column is for H-SMRM method

    Figure 6.  Error distributions by radial sampling. $ \|\mathbf{f}^{(k+1)}-\mathbf{f}^{(k)}\|_{l^2} $ (top line), $ \|\mathbf{f}^{(k+1)}-\mathbf{f}^*\|_{l^2} $ (bottom line). The first column is for C-SMRM, the second column is for H-SMRM

    Table 1.  Computational costs for random band sampling

    image scheme ISNR(dB) ReErr(%) CPU time(s) IterNum
    circles direct 3.0437 17.0221 1.5734 40
    RecPF 14.6056 1.0492 0.2680 27
    C-SMRM 14.2219 1.1462 1.9248 100
    H-SMRM 14.7205 1.0265 1.3517 100
    grayscale direct 1.8183 5.7418 9.6594 40
    RecPF 11.9131 0.5245 1.0943 21
    C-SMRM 11.5221 0.5627 1.9255 100
    H-SMRM 13.5264 0.2621 39.1639 100
    phantom direct 1.7383 8.7405 2.1778 40
    RecPF 11.3009 1.9514 0.3689 38
    C-SMRM 11.4490 1.8190 1.2895 100
    H-SMRM 14.4623 0.8047 50.9566 100
    chest direct 2.6484 3.8296 2.1683 40
    RecPF 11.4839 1.0310 0.2189 21
    C-SMRM 11.5464 0.9734 1.7518 100
    H-SMRM 17.4996 0.0164 45.6364 100
     | Show Table
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    Table 2.  Computational costs for radial sampling

    image scheme ISNR(dB) ReErr(%) CPU time(s) IterNum
    phantom direct 3.8014 11.1883 2.4127 40
    RecPF 12.3966 2.7976 0.4137 36
    C-SMRM 12.0596 3.0233 1.3909 100
    H-SMRM 13.0561 2.2250 45.5967 100
    chest direct 4.0305 4.3061 2.2478 40
    RecPF 2.6372 4.5132 0.2848 22
    C-SMRM 11.4866 0.6724 1.3651 100
    H-SMRM 12.5369 3.5845 37.5576 100
     | Show Table
    DownLoad: CSV
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