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Partial convolution for total variation deblurring and denoising by new linearized alternating direction method of multipliers with extension step

  • * Corresponding author: Yuan Shen

    * Corresponding author: Yuan Shen 

Research supported by National Natural Science Foundation of China under grant 11401295 and by National Natural Science Foundation Tianyuan Project under grant 11726618 and by Jiangsu Provincial Natural Science Foundation under grant BK20141007 and by Major Program of the National Social Science Foundation of China under Grant 12&ZD114 and by National Social Science Foundation of China under Grant 15BGL58 and by Jiangsu Provincial Social Science Foundation under Grant 14EUA001 and by Qinglan Project of Jiangsu Province

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  • In this paper, we propose a partial convolution model for image deblurring and denoising. We also devise a new linearized alternating direction method of multipliers (ADMM) with an extension step. As the computation of its subproblem is simple enough to have closed-form solutions, its per-iteration cost is low; however, the relaxed parameter condition together with the extra extension step inspired by Ye and Yuan's ADMM enables faster convergence than the original linearized ADMM. Preliminary experimental results show that our algorithm can produce better quality results than some existing efficient algorithms within a similar computation time. The performance advantage of our algorithm is particularly evident at high noise ratios.

    Mathematics Subject Classification: Primary: 90C30, 65K05; Secondary: 94A08.


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  • Figure 1.  From left to right are the results of FTVD, New algorithm, TVAL3, and SNR history plot. First row: $NR$ = 60%, $airplane$; second row: $NR$ = 80%, $lena$

    Figure 2.  Performance v.s. noise ratio. Left: time; right: SNR. First row: $RC = 0.1$, $coins$; second row: $RC = 0$, $westconcordorthophoto$

    Figure 3.  Performance v.s. indices matrix quality. Left: time, right: SNR. First row: $moon$, $NR = 60%$; second row: $tire$, $NR = 80%$

    Figure 4.  From left to right to below are the results of FTVD, New algorithm, and TVAL3, and SNR history plot. First row: "average" kernel with dirichlet boundary condition; second row: "Gaussian" kernel; third row: "Disk" kernel; fourth row: "motion" kernel

    Table 1.  Numerical results of basic test

    Noise Ratio time SNR time SNR time SNR
    60% 3.806 20.894 20.155 23.594 14.212 11.806
    80% 8.174 8.751 19.687 15.226 26.442 7.106
     | Show Table
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