Article Contents
Article Contents

# 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
• 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:

• 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 $\bf{{0.9630}}$

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

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 $\bf{{0.9812}}$
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