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A nonlocal low rank model for poisson noise removal

This work is supported by NSFC Grant No. 11871210, 11971215, the Construct Program of the Key Discipline in Hunan Province, the SRF of Hunan Provincial Education Department (No.17A128), the Hunan Province Graduate Research and Innovation Project (No. CX20190336), the HKRGC GRF 12306616, 12200317, 12300218 and 12300519, and HKU 104005583
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  • Patch-based methods, which take the advantage of the redundancy and similarity among image patches, have attracted much attention in recent years. However, these methods are mainly limited to Gaussian noise removal. In this paper, the Poisson noise removal problem is considered. Unlike Gaussian noise which has an identical and independent distribution, Poisson noise is signal dependent, which makes the problem more challenging. By incorporating the prior that a group of similar patches should possess a low-rank structure, and applying the maximum a posterior (MAP) estimation, the Poisson noise removal problem is formulated as an optimization one. Then, an alternating minimization algorithm is developed to find the minimizer of the objective function efficiently. Convergence of the minimizing sequence will be established, and the efficiency and effectiveness of the proposed algorithm will be demonstrated by numerical experiments.

    Mathematics Subject Classification: 94A08, 68U10, 65K10.

    Citation:

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  • Figure 1.  The test images, from left to right, top to bottom: Monarch, Cameraman, House, Peppers, Lena, Man

    Figure 2.  The distribution of the residual patches $ E_{\ell} $, fitting Gaussian distribution for Monarch image

    Figure 3.  $ b = 1 $, $ P = 15 $. From left to right, top to bottom, original image and noise image, restored images by methods: PIDSplit, PD, NLM, KSVD, WNNM, LRPD, LRS, LRH

    Figure 4.  $ b = 1 $, $ P = 15 $. From left to right, top to bottom, original image and noise image, restored images by methods: PIDSplit, PD, NLM, KSVD, WNNM, LRPD, LRS, LRH

    Figure 5.  $ b = 1 $, $ P = 15 $. From left to right, top to bottom, original image and noise image, restored images by methods: PIDSplit, PD, NLM, KSVD, MC-WNNM, LRPD, LRS, LRH

    Figure 6.  $ b = 1 $, $ P = 15 $. From left to right, top to bottom, original image and noise image, restored images by methods: PIDSplit, PD, NLM, KSVD, MC-WNNM, LRPD, LRS, LRH

    Table 1.  SNR and SSIM for restoration image under Poisson noise. The best results are in bold

    Image b P PIDSplit PD NLM KSVD WNNM LRPD LRS LRH
    SNR
    Monarch 1 15 17.86 17.92 18.26 18.83 18.71 17.21 $ \bf{{19.56}} $ 19.48
    30 20.13 20.23 20.38 20.65 21.09 17.57 $ \bf{{21.35}} $ 21.27
    45 21.27 21.48 21.40 21.86 22.26 18.07 22.45 $ \bf{{22.47}} $
    60 22.21 22.45 22.17 22.68 23.11 18.71 23.28 $ \bf{{23.38}} $
    5 15 17.27 17.33 17.11 17.34 17.75 17.89 $ \bf{{18.53}} $ 18.25
    30 19.62 19.81 19.81 20.03 20.39 18.71 $ \bf{{20.98}} $ 20.85
    45 20.83 21.14 21.20 21.41 21.89 18.77 22.20 $ \bf{{22.27}} $
    60 21.84 22.19 21.82 22.27 22.70 19.22 22.98 $ \bf{{23.06}} $
    Cameraman 1 15 19.26 19.29 19.72 19.73 19.68 18.46 20.11 $ \bf{{20.59}} $
    30 21.00 21.00 21.54 21.75 21.76 18.61 21.64 $ \bf{{22.03}} $
    45 21.90 22.00 22.43 22.77 22.71 19.19 22.68 $ \bf{{23.12}} $
    60 22.61 22.73 22.97 23.40 23.41 19.87 23.33 $ \bf{{23.82}} $
    5 15 18.28 18.33 18.32 18.87 18.69 18.50 $ \bf{{19.35}} $ 19.08
    30 20.32 20.38 20.95 21.22 21.28 19.70 21.27 $ \bf{{21.50}} $
    45 21.31 21.41 21.98 22.19 22.35 19.86 22.32 $ \bf{{22.52}} $
    60 22.15 22.27 22.62 22.94 23.11 20.32 23.03 $ \bf{{23.34}} $
    SSIM
    Monarch 1 15 0.9867 0.9869 0.9881 0.9888 0.9895 0.9863 $ \bf{{0.9906}} $ 0.9903
    30 0.9788 0.9789 0.9813 0.9823 0.9830 0.9697 $ \bf{{0.9834}} $ 0.9833
    45 0.9724 0.9727 0.9745 0.9763 0.9772 0.9551 0.9773 $ \bf{{0.9774}} $
    60 0.9673 0.9674 0.9698 0.9721 0.9737 0.9448 0.9732 $ \bf{{0.9738}} $
    5 15 0.9854 0.9854 0.9848 0.9856 0.9871 0.9815 $ \bf{{0.9883}} $ 0.9872
    30 0.9768 0.9771 0.9784 0.9795 0.9809 0.9701 $ \bf{{0.9823}} $ 0.9817
    45 0.9699 0.9706 0.9727 0.9741 0.9763 0.9545 $ \bf{{0.9766}} $ $ \bf{{0.9766}} $
    60 0.9644 0.9653 0.9668 0.9696 0.9709 0.9437 0.9713 $ \bf{{0.9715}} $
    Cameraman 1 15 0.9882 0.9883 0.9893 0.9893 0.9894 0.9863 0.9906 $ \bf{{0.9915}} $
    30 0.9776 0.9778 0.9811 0.9816 0.9820 0.9678 0.9801 $ \bf{{0.9823}} $
    45 0.9685 0.9688 0.9727 0.9734 0.9730 0.9530 0.9716 $ \bf{{0.9745}} $
    60 0.9601 0.9598 0.9634 0.9641 0.9635 0.9412 0.9634 $ \bf{{0.9662}} $
    5 15 0.9852 0.9852 0.9844 0.9863 0.9870 0.9818 $ \bf{{0.9887}} $ 0.9878
    30 0.9742 0.9746 0.9781 0.9790 0.9802 0.9692 0.9792 $ \bf{{0.9803}} $
    45 0.9645 0.9652 0.9700 0.9700 0.9713 0.9531 0.9705 $ \bf{{0.9714}} $
    60 0.9560 0.9568 0.9606 0.9611 0.9620 0.9416 0.9618 $ \bf{{0.9630}} $
     | Show Table
    DownLoad: CSV

    Table 2.  SNR and SSIM for restoration image under Poisson noise. The best results are in bold

    Image PIDSplit PD NLM KSVD WNNM LRPD LRS LRH
    SNR
    House 23.91 23.98 24.50 25.14 25.92 22.00 $ \bf{{26.01}} $ 25.98
    Peppers 21.71 21.83 21.57 21.91 22.16 20.53 22.38 $ \bf{{22.49}} $
    Lena 23.29 23.32 23.68 23.63 23.92 22.23 $ \bf{{24.37}} $ 24.01
    Man 21.08 20.98 20.75 20.55 20.81 18.84 $ \bf{{21.47}} $ 21.29
    SSIM
    House 0.9813 0.9812 0.9841 0.9864 0.9870 0.9804 0.9871 $ \bf{{0.9872}} $
    Peppers 0.9783 0.9783 0.9794 0.9805 0.9812 0.9712 0.9808 $ \bf{{0.9814}} $
    Lena 0.9810 0.9808 0.9837 0.9837 0.9842 0.9774 $ \bf{{0.9845}} $ 0.9840
    Man 0.9701 0.9706 0.9714 0.9696 0.9702 0.9562 $ \bf{{0.9733}} $ 0.9723
     | Show Table
    DownLoad: CSV

    Table 3.  SNR and SSIM for color image. The best results are in bold

    Image b P PIDSplit PD NLM KSVD MC-WNNM LRPD LRS LRH
    SNR
    McM1 1 15 17.39 17.24 17.41 17.30 17.54 17.22 $ \bf{{18.16}} $ 17.71
    30 19.29 19.09 19.44 19.37 19.58 17.21 $ \bf{{19.92}} $ 19.35
    45 20.36 20.36 20.50 20.49 20.75 17.47 $ \bf{{20.93}} $ 20.53
    60 21.24 21.21 21.21 21.23 21.54 17.83 $ \bf{{21.66}} $ 21.38
    5 15 16.94 15.91 16.44 16.34 16.78 19.82 $ \bf{{17.66}} $ 17.34
    30 18.87 18.36 18.77 18.63 18.88 18.64 $ \bf{{19.61}} $ 19.41
    45 19.96 19.85 20.02 19.97 20.15 18.35 $ \bf{{20.69}} $ 20.54
    60 20.85 20.77 20.84 20.82 21.07 18.44 $ \bf{{21.46}} $ 21.43
    McM2 1 15 22.35 21.15 22.05 21.81 23.76 21.02 $ \bf{{24.47}} $ 24.24
    30 24.29 23.65 24.39 24.48 26.26 20.93 $ \bf{{26.64}} $ 26.52
    45 25.28 25.30 25.43 25.82 27.56 21.78 27.70 $ \bf{{27.71}} $
    60 26.13 26.23 26.09 26.62 28.45 22.79 28.44 $ \bf{{28.69}} $
    5 15 21.55 19.50 20.94 20.60 22.80 20.13 $ \bf{{23.97}} $ 23.45
    30 23.93 22.83 23.87 23.88 25.63 21.65 $ \bf{{26.34}} $ 26.08
    45 25.05 24.85 25.16 25.38 27.16 22.08 27.48 $ \bf{{27.50}} $
    60 25.99 25.97 25.85 26.37 28.13 22.97 28.24 $ \bf{{28.37}} $
    SSIM
    McM1 1 15 0.9828 0.9836 0.9821 0.9821 0.9828 0.9835 $ \bf{{0.9850}} $ 0.9832
    30 0.9684 0.9687 0.9698 0.9690 0.9699 0.9543 $ \bf{{0.9705}} $ 0.9663
    45 0.9557 0.9584 0.9586 0.9571 0.9581 0.9224 $ \bf{{0.9574}} $ 0.9531
    60 0.9469 $ \bf{{0.9501}} $ 0.9491 0.9468 0.9487 0.8939 0.9465 0.9420
    5 15 0.9812 0.9780 0.9779 0.9776 0.9799 0.9900 $ \bf{{0.9829}} $ 0.9810
    30 0.9650 0.9625 0.9643 0.9623 0.9646 0.9606 $ \bf{{0.9686}} $ 0.9665
    45 0.9510 0.9520 0.9527 0.9512 0.9522 0.9279 $ \bf{{0.9553}} $ 0.9528
    60 0.9417 0.9435 0.9437 0.9411 0.9426 0.8972 $ \bf{{0.9446}} $ 0.9429
    McM2 1 15 0.9901 0.9875 0.9897 0.9894 0.9932 0.9877 $ \bf{{0.9936}} $ 0.9933
    30 0.9823 0.9805 0.9830 0.9832 0.9890 0.9681 $ \bf{{0.9891}} $ 0.9890
    45 0.9752 0.9773 0.9771 0.9782 0.9850 0.9576 0.9846 $ \bf{{0.9852}} $
    60 0.9703 0.9739 0.9717 0.9734 0.9818 0.9529 0.9801 $ \bf{{0.9820}} $
    5 15 0.9880 0.9815 0.9867 0.9862 0.9914 0.9903 $ \bf{{0.9929}} $ 0.9919
    30 0.9809 0.9761 0.9807 0.9807 0.9873 0.9696 $ \bf{{0.9884}} $ 0.9879
    45 0.9743 0.9745 0.9755 0.9763 0.9841 0.9569 0.9839 $ \bf{{0.9847}} $
    60 0.9700 0.9723 0.9702 0.9722 0.9807 0.9521 0.9795 $ \bf{{0.9812}} $
     | Show Table
    DownLoad: CSV
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