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December  2019, 13(6): 1259-1282. doi: 10.3934/ipi.2019055

## A parallel domain decomposition algorithm for large scale image denoising

 1 Shenzhen Institutes of Advanced Technology, Chinese Academy of Sciences, Shenzhen, Guangdong 518055, China 2 LSEC, ICMSEC, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China 3 Department of Computer Science, University of Colorado Boulder, Boulder, CO 80309, USA

* Corresponding author: Xiao-Chuan Cai

Received  October 2018 Revised  July 2019 Published  October 2019

Total variation denoising (TVD) is an effective technique for image denoising, in particular, for recovering blocky, discontinuous images from noisy background. The problem is formulated as an optimization problem in the space of bounded variation functions, and the solution is obtained by solving the associated Euler–Lagrange equation defined on the domain occupied by the entire image. The method offers high quality results, but is computationally expensive for large images, especially for three-dimensional problems. In this paper, we introduce a highly parallel version of the algorithm which formulates the problem as multiple overlapping, but independent, optimization problems, and each is defined on a portion of the image domain. This approach is similar to the overlapping Schwarz type domain decomposition method, but is non-iterative, for solving partial differential equations, and is highly scalable, without using any coarse grids, for parallel computers with a large number of processors. We show by a theory and also by some two- and three-dimensional numerical experiments that the new approach has similar numerical accuracy as the classical TVD approach, but is much more efficient on parallel computers.

Citation: Rongliang Chen, Jizu Huang, Xiao-Chuan Cai. A parallel domain decomposition algorithm for large scale image denoising. Inverse Problems & Imaging, 2019, 13 (6) : 1259-1282. doi: 10.3934/ipi.2019055
##### References:
 [1] A. Asaduzzaman, A. Martinez and A. Sepehri, A time-efficient image processing algorithm for multicore/manycore parallel computing, Southeastcon, IEEE, (2015).  doi: 10.1109/SECON.2015.7132924.  Google Scholar [2] G. Aubert and P. Kornprobst, Mathematical Problems in Image Processing: Partial Differential Equations and the Calculus of Variations, Applied Mathematical Sciences, 147. Springer, New York, 2006.  Google Scholar [3] S. Balay, S. Abhyankar, M. Adams, J. Brown, P. Brune, K. Buschelman, L. Dalcin, A. Dener, V. Eijkhout, W. Gropp, B. Smith, D. Karpeyev, D. Kaushik and et al., PETSc Web page, (2019), https://www.mcs.anl.gov/petsc. Google Scholar [4] M. Basu, Gaussian-based edge-detection methods-a survey, IEEE Transactions on Systems, Man, and Cybernetics, Part C (Applications and Reviews), 32 (2002), 252-260.  doi: 10.1109/TSMCC.2002.804448.  Google Scholar [5] P. Blomgren, T. F. Chan, P. Mulet, L. Vese and W. L. Wan, Variational PDE models and methods for image processing, Numerical analysis, Chapman and Hall CRC Research Notes in Mathematics, Boca Raton, FL, 420 (2000), 43-67.   Google Scholar [6] A. Buades, B. Coll and J. M. Morel, Image denoising methods, a new nonlocal principle, SIAM Review, 52 (2010), 113-147.  doi: 10.1137/090773908.  Google Scholar [7] X.-C. Cai, M. Dryja and M. Sarkis, Restricted additive Schwarz preconditioners with harmonic overlap for symmetric positive definite linear systems, SIAM Journal on Numerical Analysis, 41 (2003), 1209-1231.  doi: 10.1137/S0036142901389621.  Google Scholar [8] X.-C. Cai, W. D. Gropp, D. E. Keyes, R. G. Melvin and D. P. Young, Parallel Newton-Krylov-Schwarz algorithms for the transonic full potential equation, SIAM Journal on Scientific Computing, 19 (1998), 246-265.  doi: 10.1137/S1064827596304046.  Google Scholar [9] A. Chambolle and P.-L. Lions, Image recovery via total variation minimization and related problems, Numerische Mathematik, 76 (1997), 167-188.  doi: 10.1007/s002110050258.  Google Scholar [10] T. F. Chan and J. H. Shen, Image Processing and Analysis: Variational, PDE, Wavelet, and Stochastic Methods, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2005. doi: 10.1137/1.9780898717877.  Google Scholar [11] T. F. Chan, H. M. Zhou and R. H. Chan, Continuation Method for Total Variation Denoising Problems, Department of Mathematics, University of California, Los Angeles, 1995. Google Scholar [12] H. B. Chang, X.-C. Tai, L.-L. Wang and D. P. Yang, Convergence rate of overlapping domain decomposition methods for the Rudin-Osher-Fatemi model based on a dual formulation, SIAM Journal on Imaging Sciences, 8 (2015), 564-591.  doi: 10.1137/140965016.  Google Scholar [13] H. B. Chang, X. Q. Zhang, X.-C. Tai and D. P. Yang, Domain decomposition methods for nonlocal total variation image restoration, Journal of Scientific Computing, 60 (2014), 79-100.  doi: 10.1007/s10915-013-9786-9.  Google Scholar [14] R. L. Chen and X.-C. Cai, Parallel one-shot Lagrange-Newton-Krylov-Schwarz algorithms for shape optimization of steady incompressible flows, SIAM Journal on Scientific Computing, 34 (2012), B584-B605.  doi: 10.1137/110830769.  Google Scholar [15] R. L. Chen, Y. Q. Wu, Z. Z. Yan, Y. B. Zhao and X.-C. Cai, A parallel domain decomposition method for 3D unsteady incompressible flows at high Reynolds number, Journal of Scientific Computing, 58 (2014), 275-289.  doi: 10.1007/s10915-013-9732-x.  Google Scholar [16] D. L. Collins, A. P. Zijdenbos, V. Kollokian, J. G. Sled, N. J. Kabani, C. J. Holmes and A. C. Evans, Design and construction of a realistic digital brain phantom, IEEE Transactions on Medical Imaging, 17 (1998), 463-468.  doi: 10.1109/42.712135.  Google Scholar [17] P. Coupé, P. Yger, S. Prima, P. Hellier, C. Kervrann and C. Barillot, An optimized blockwise nonlocal means denoising filter for 3-D magnetic resonance images, IEEE Transactions on Medical Imaging, 27 (2008), 425-441.   Google Scholar [18] X. M. Deng, X.-C. Cai and J. Zou, A parallel space-time domain decomposition method for unsteady source inversion problems, Inverse Problems and Imaging, 9 (2015), 1069-1091.  doi: 10.3934/ipi.2015.9.1069.  Google Scholar [19] D. L. Donoho and I. M. Johnstone, Adapting to unknown smoothness via wavelet shrinkage, Journal of the American Statistical Association, 90 (1995), 1200-1224.  doi: 10.1080/01621459.1995.10476626.  Google Scholar [20] A. Eklund, P. Dufort, D. Forsberg and S. M. LaConte, Medical image processing on the GPU-past, Present and Future, Medical Image Analysis, 17 (2013), 1073-1094.   Google Scholar [21] L. C. Evans, Partial Differential Equations, Graduate Studies in Mathematics, 19. American Mathematical Society, Providence, RI, 1998.  Google Scholar [22] M. Fornasier, A. Langer and C.-B. Schönlieb, A convergent overlapping domain decomposition method for total variation minimization, Numerische Mathematik, 116 (2010), 645-685.  doi: 10.1007/s00211-010-0314-7.  Google Scholar [23] M. Fornasier and C.-B. Schönlieb, Subspace correction methods for total variation and $L_1$-minimization, SIAM Journal on Numerical Analysis, 47 (2009), 3397-3428.  doi: 10.1137/070710779.  Google Scholar [24] C. Gerhardt, Global regularity of the solutions to the capillarity problem, Annali della Scuola Normale Superiore di Pisa-Classe di Scienze, 3 (1976), 157-175.   Google Scholar [25] M. Hintermüller and A. Langer, Non-overlapping domain decomposition methods for dual total variation based image denoising, Journal of Scientific Computing, 62 (2015), 456-481.  doi: 10.1007/s10915-014-9863-8.  Google Scholar [26] M. Hintermüller and G. Stadler, An infeasible primal-dual algorithm for total bounded variation-based inf-convolution-type image restoration, SIAM Journal on Scientific Computing, 28 (2006), 1-23.  doi: 10.1137/040613263.  Google Scholar [27] T. Kohlberger, Variational Domain Decomposition For Parallel Image Processing, PhD thesis, Universitat Mannheim, 2007. Google Scholar [28] F. Kong and X.-C. Cai, A scalable nonlinear fluid-structure interaction solver based on a Schwarz preconditioner with isogeometric unstructured coarse spaces in 3D, Journal of Computational Physics, 340 (2017), 498-518.  doi: 10.1016/j.jcp.2017.03.043.  Google Scholar [29] C.-O. Lee and C. M. Nam, Primal domain decomposition methods for the total variation minimization, based on dual decomposition, SIAM Journal on Scientific Computing, 39 (2017), B403-B423.  doi: 10.1137/15M1049919.  Google Scholar [30] F. Malgouyres, Minimizing the total variation under a general convex constraint for image restoration, IEEE Transactions on Image Processing, 11 (2002), 1450-1456.  doi: 10.1109/TIP.2002.806241.  Google Scholar [31] S. Osher, M. Burger, D. Goldfarb, J. J. Xu and W. T. Yin, An iterative regularization method for total variation-based image restoration, Multiscale Model. Simul., 4 (2005), 460-489.  doi: 10.1137/040605412.  Google Scholar [32] G. Pratx and L. Xing, GPU computing in medical physics: A review, Medical Physics, 38 (2011), 2685-2697.   Google Scholar [33] L. I. Rudin, S. Osher and E. Fatemi, Nonlinear total variation based noise removal algorithms, Physica D: Nonlinear Phenomena, 60 (1992), 259-268.  doi: 10.1016/0167-2789(92)90242-F.  Google Scholar [34] Y. Saad, Iterative Methods for Sparse Linear Systems, Society for Industrial and Applied Mathematics, Philadelphia, PA, 2003. doi: 10.1137/1.9780898718003.  Google Scholar [35] R. Shams, P. Sadeghi, R. A. Kennedy and R. I. Hartley, A survey of medical image registration on multicore and the GPU, IEEE Signal Processing Magazine, 27 (2010), 50-60.  doi: 10.1109/MSP.2009.935387.  Google Scholar [36] J. Spruck, On the existence of a capillary surface with prescribed contact angle, Communications on Pure and Applied Mathematics, 28 (1975), 189-200.  doi: 10.1002/cpa.3160280202.  Google Scholar [37] N. Ural'tseva, Solvability of the problem of capillaries, Vestn. Leningr. Univ, (1973), 54-64.   Google Scholar [38] L. A. Vese and T. F. Chan, A multiphase level set framework for image segmentation using the Mumford and Shah model, International Journal of Computer Vision, 50 (2002), 271-293.   Google Scholar [39] C. R. Vogel and M. E. Oman, Iterative methods for total variation denoising, SIAM Journal on Scientific Computing, 17 (1996), 227-238.  doi: 10.1137/0917016.  Google Scholar [40] J. Xu, X.-C. Tai and L.-L. Wang, A two-level domain decomposition method for image restoration, Inverse Problem and Imaging, 4 (2010), 523-545.  doi: 10.3934/ipi.2010.4.523.  Google Scholar

show all references

##### References:
 [1] A. Asaduzzaman, A. Martinez and A. Sepehri, A time-efficient image processing algorithm for multicore/manycore parallel computing, Southeastcon, IEEE, (2015).  doi: 10.1109/SECON.2015.7132924.  Google Scholar [2] G. Aubert and P. Kornprobst, Mathematical Problems in Image Processing: Partial Differential Equations and the Calculus of Variations, Applied Mathematical Sciences, 147. Springer, New York, 2006.  Google Scholar [3] S. Balay, S. Abhyankar, M. Adams, J. Brown, P. Brune, K. Buschelman, L. Dalcin, A. Dener, V. Eijkhout, W. Gropp, B. Smith, D. Karpeyev, D. Kaushik and et al., PETSc Web page, (2019), https://www.mcs.anl.gov/petsc. Google Scholar [4] M. Basu, Gaussian-based edge-detection methods-a survey, IEEE Transactions on Systems, Man, and Cybernetics, Part C (Applications and Reviews), 32 (2002), 252-260.  doi: 10.1109/TSMCC.2002.804448.  Google Scholar [5] P. Blomgren, T. F. Chan, P. Mulet, L. Vese and W. L. Wan, Variational PDE models and methods for image processing, Numerical analysis, Chapman and Hall CRC Research Notes in Mathematics, Boca Raton, FL, 420 (2000), 43-67.   Google Scholar [6] A. Buades, B. Coll and J. M. Morel, Image denoising methods, a new nonlocal principle, SIAM Review, 52 (2010), 113-147.  doi: 10.1137/090773908.  Google Scholar [7] X.-C. Cai, M. Dryja and M. Sarkis, Restricted additive Schwarz preconditioners with harmonic overlap for symmetric positive definite linear systems, SIAM Journal on Numerical Analysis, 41 (2003), 1209-1231.  doi: 10.1137/S0036142901389621.  Google Scholar [8] X.-C. Cai, W. D. Gropp, D. E. Keyes, R. G. Melvin and D. P. Young, Parallel Newton-Krylov-Schwarz algorithms for the transonic full potential equation, SIAM Journal on Scientific Computing, 19 (1998), 246-265.  doi: 10.1137/S1064827596304046.  Google Scholar [9] A. Chambolle and P.-L. Lions, Image recovery via total variation minimization and related problems, Numerische Mathematik, 76 (1997), 167-188.  doi: 10.1007/s002110050258.  Google Scholar [10] T. F. Chan and J. H. Shen, Image Processing and Analysis: Variational, PDE, Wavelet, and Stochastic Methods, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2005. doi: 10.1137/1.9780898717877.  Google Scholar [11] T. F. Chan, H. M. Zhou and R. H. Chan, Continuation Method for Total Variation Denoising Problems, Department of Mathematics, University of California, Los Angeles, 1995. Google Scholar [12] H. B. Chang, X.-C. Tai, L.-L. Wang and D. P. Yang, Convergence rate of overlapping domain decomposition methods for the Rudin-Osher-Fatemi model based on a dual formulation, SIAM Journal on Imaging Sciences, 8 (2015), 564-591.  doi: 10.1137/140965016.  Google Scholar [13] H. B. Chang, X. Q. Zhang, X.-C. Tai and D. P. Yang, Domain decomposition methods for nonlocal total variation image restoration, Journal of Scientific Computing, 60 (2014), 79-100.  doi: 10.1007/s10915-013-9786-9.  Google Scholar [14] R. L. Chen and X.-C. Cai, Parallel one-shot Lagrange-Newton-Krylov-Schwarz algorithms for shape optimization of steady incompressible flows, SIAM Journal on Scientific Computing, 34 (2012), B584-B605.  doi: 10.1137/110830769.  Google Scholar [15] R. L. Chen, Y. Q. Wu, Z. Z. Yan, Y. B. Zhao and X.-C. Cai, A parallel domain decomposition method for 3D unsteady incompressible flows at high Reynolds number, Journal of Scientific Computing, 58 (2014), 275-289.  doi: 10.1007/s10915-013-9732-x.  Google Scholar [16] D. L. Collins, A. P. Zijdenbos, V. Kollokian, J. G. Sled, N. J. Kabani, C. J. Holmes and A. C. Evans, Design and construction of a realistic digital brain phantom, IEEE Transactions on Medical Imaging, 17 (1998), 463-468.  doi: 10.1109/42.712135.  Google Scholar [17] P. Coupé, P. Yger, S. Prima, P. Hellier, C. Kervrann and C. Barillot, An optimized blockwise nonlocal means denoising filter for 3-D magnetic resonance images, IEEE Transactions on Medical Imaging, 27 (2008), 425-441.   Google Scholar [18] X. M. Deng, X.-C. Cai and J. Zou, A parallel space-time domain decomposition method for unsteady source inversion problems, Inverse Problems and Imaging, 9 (2015), 1069-1091.  doi: 10.3934/ipi.2015.9.1069.  Google Scholar [19] D. L. Donoho and I. M. Johnstone, Adapting to unknown smoothness via wavelet shrinkage, Journal of the American Statistical Association, 90 (1995), 1200-1224.  doi: 10.1080/01621459.1995.10476626.  Google Scholar [20] A. Eklund, P. Dufort, D. Forsberg and S. M. LaConte, Medical image processing on the GPU-past, Present and Future, Medical Image Analysis, 17 (2013), 1073-1094.   Google Scholar [21] L. C. Evans, Partial Differential Equations, Graduate Studies in Mathematics, 19. American Mathematical Society, Providence, RI, 1998.  Google Scholar [22] M. Fornasier, A. Langer and C.-B. Schönlieb, A convergent overlapping domain decomposition method for total variation minimization, Numerische Mathematik, 116 (2010), 645-685.  doi: 10.1007/s00211-010-0314-7.  Google Scholar [23] M. Fornasier and C.-B. Schönlieb, Subspace correction methods for total variation and $L_1$-minimization, SIAM Journal on Numerical Analysis, 47 (2009), 3397-3428.  doi: 10.1137/070710779.  Google Scholar [24] C. Gerhardt, Global regularity of the solutions to the capillarity problem, Annali della Scuola Normale Superiore di Pisa-Classe di Scienze, 3 (1976), 157-175.   Google Scholar [25] M. Hintermüller and A. Langer, Non-overlapping domain decomposition methods for dual total variation based image denoising, Journal of Scientific Computing, 62 (2015), 456-481.  doi: 10.1007/s10915-014-9863-8.  Google Scholar [26] M. Hintermüller and G. Stadler, An infeasible primal-dual algorithm for total bounded variation-based inf-convolution-type image restoration, SIAM Journal on Scientific Computing, 28 (2006), 1-23.  doi: 10.1137/040613263.  Google Scholar [27] T. Kohlberger, Variational Domain Decomposition For Parallel Image Processing, PhD thesis, Universitat Mannheim, 2007. Google Scholar [28] F. Kong and X.-C. Cai, A scalable nonlinear fluid-structure interaction solver based on a Schwarz preconditioner with isogeometric unstructured coarse spaces in 3D, Journal of Computational Physics, 340 (2017), 498-518.  doi: 10.1016/j.jcp.2017.03.043.  Google Scholar [29] C.-O. Lee and C. M. Nam, Primal domain decomposition methods for the total variation minimization, based on dual decomposition, SIAM Journal on Scientific Computing, 39 (2017), B403-B423.  doi: 10.1137/15M1049919.  Google Scholar [30] F. Malgouyres, Minimizing the total variation under a general convex constraint for image restoration, IEEE Transactions on Image Processing, 11 (2002), 1450-1456.  doi: 10.1109/TIP.2002.806241.  Google Scholar [31] S. Osher, M. Burger, D. Goldfarb, J. J. Xu and W. T. Yin, An iterative regularization method for total variation-based image restoration, Multiscale Model. Simul., 4 (2005), 460-489.  doi: 10.1137/040605412.  Google Scholar [32] G. Pratx and L. Xing, GPU computing in medical physics: A review, Medical Physics, 38 (2011), 2685-2697.   Google Scholar [33] L. I. Rudin, S. Osher and E. Fatemi, Nonlinear total variation based noise removal algorithms, Physica D: Nonlinear Phenomena, 60 (1992), 259-268.  doi: 10.1016/0167-2789(92)90242-F.  Google Scholar [34] Y. Saad, Iterative Methods for Sparse Linear Systems, Society for Industrial and Applied Mathematics, Philadelphia, PA, 2003. doi: 10.1137/1.9780898718003.  Google Scholar [35] R. Shams, P. Sadeghi, R. A. Kennedy and R. I. Hartley, A survey of medical image registration on multicore and the GPU, IEEE Signal Processing Magazine, 27 (2010), 50-60.  doi: 10.1109/MSP.2009.935387.  Google Scholar [36] J. Spruck, On the existence of a capillary surface with prescribed contact angle, Communications on Pure and Applied Mathematics, 28 (1975), 189-200.  doi: 10.1002/cpa.3160280202.  Google Scholar [37] N. Ural'tseva, Solvability of the problem of capillaries, Vestn. Leningr. Univ, (1973), 54-64.   Google Scholar [38] L. A. Vese and T. F. Chan, A multiphase level set framework for image segmentation using the Mumford and Shah model, International Journal of Computer Vision, 50 (2002), 271-293.   Google Scholar [39] C. R. Vogel and M. E. Oman, Iterative methods for total variation denoising, SIAM Journal on Scientific Computing, 17 (1996), 227-238.  doi: 10.1137/0917016.  Google Scholar [40] J. Xu, X.-C. Tai and L.-L. Wang, A two-level domain decomposition method for image restoration, Inverse Problem and Imaging, 4 (2010), 523-545.  doi: 10.3934/ipi.2010.4.523.  Google Scholar
An example of the domain partition. Here dots are image pixels and the image domain is partitioned into $16$ subdomains. The domain with solid red lines is an example of a nonoverlapping subdomain $\Omega_0^k$ and the corresponding overlapping subdomain $\Omega_{\delta}^k$ is marked with dashed black lines. The overlapping size $\delta$ is the distance between the boundaries of $\Omega_0^k$ and $\Omega_{\delta}^k$
Procedure of NiOS. Step 1: Overlapping domain decomposition (A); Step 2: Distribute overlapping subdomain problems to 4 processors and solve these four overlapping sub-problems (B); Step 3: Combine images from non-overlapping subdomains (C). Here the points with different colors are the image pixels distributed to different processor cores
The boat-$1024\times1024$ image denoising results. From left to right: the original image, the noise image with $\delta^2 = 0.04$, and the restored image by NiOS. Here $\alpha = 0.18$, $\beta = 10^{-4}$, and the PSNR of the restored image is $27.5444$
A comparison of the results obtained by NiOS and NKS. The first row: the clean image (left), the noisy image with $\sigma^2 = 0.04$ (right); second row: the local zoom-in of the figures in the first row; third row: the restored images obtained by NiOS (left) and NKS (right); fourth row: the local zoom-in of the figures in the third row. The PSNR of the restored image with NiOS and NKS are $28.536962$ and $28.536417$, respectively
The surface plot of the error between NiOS and NKS: $u_{\mbox{NKS}} - u_{\mbox{NiOS}}$. Here the image size is $1024 \hspace{-0.02in}\times\hspace{-0.02in} 1024$, $\alpha = 0.18$, $\beta = 1.0\times10^{-4}$. The overlapping size is $4$ and the number of processors is $4$
The speedup of the NiOS algorithm for the 2D cameraman image denoising problem
A noisy image (left) and an example of the partition of the image into 8 sub-images (right)
Slice 100 of the 3D MR images. From left to right, the first row: the original T1-w image, the T1-w image with a Racian noise at $9\%$ and the restored image; second row: the detailed partial images of the first row images; third row: the original T2-w images, the T2-w image with a Racian noise at $9\%$, and the restored image. The last row: the detailed partial images of the third row images
The 3D reconstruction of the MR image. From left to right: Top: the clean image, the image with $9\%$ Racian noise, and the restored image; Bottom: the corresponding zoomined images
The comparison of NiOS and NKS. Here $\beta = 1.0\times10^{-4}$, $\alpha = 0.18$, the overlapping size for NiOS is $4$. "Sp" refers to the compute time speedup of the NiOS method compared with the NKS method, which is defined as the time of NKS divides the time of NiOS
 $n_p$ cameraman-$2048\times2048$ cameraman-$4096\times4096$ NKS NiOS Sp NKS NiOS Sp Newton Time Newton Time Newton Time Newton Time 32 24 9.8 22 5.0 2.0 30 50.0 26 23.4 2.1 64 27 6.2 21 2.8 2.2 37 35.6 23 12.6 2.8 128 25 3.0 20 1.2 2.5 38 17.9 23 6.5 2.8 256 25 1.7 19 0.6 2.8 33 8.7 21 2.9 3.0 512 24 0.9 19 0.4 2.3 38 5.3 20 1.4 3.8 1024 23 1.2 19 0.3 4.0 38 3.6 20 0.8 4.5
 $n_p$ cameraman-$2048\times2048$ cameraman-$4096\times4096$ NKS NiOS Sp NKS NiOS Sp Newton Time Newton Time Newton Time Newton Time 32 24 9.8 22 5.0 2.0 30 50.0 26 23.4 2.1 64 27 6.2 21 2.8 2.2 37 35.6 23 12.6 2.8 128 25 3.0 20 1.2 2.5 38 17.9 23 6.5 2.8 256 25 1.7 19 0.6 2.8 33 8.7 21 2.9 3.0 512 24 0.9 19 0.4 2.3 38 5.3 20 1.4 3.8 1024 23 1.2 19 0.3 4.0 38 3.6 20 0.8 4.5
The error of the NiOS method with respect to the parameters $\alpha$ and $\beta$. Here ERROR is the relative error $||u_{\mbox{NiOS}} - u_{\mbox{NKS}}||_2/||u_{\mbox{true}}||_2$, $u_{\mbox{NiOS}}$, $u_{\mbox{NKS}}$, and $u_{\mbox{true}}$ are the solution of NiOS, NKS, and the clean image, respectively
 $\beta$ $\alpha$ boat-$1024\times1024$ cameraman-$2048 \times2048$ Newton GMRES ERROR PSNR Newton GMRES ERROR PSNR $10^{-1}$ 0.01 2 1 $2.1\times10^{-7}$ 15.1 2 1 $4.4\times10^{-9}$ 15.4 $10^{-1}$ 0.05 3 1 $1.4\times10^{-5}$ 17.4 3 1 $2.9\times10^{-7}$ 17.6 $10^{-1}$ 0.1 3 2 $6.3\times10^{-5}$ 19.5 3 2 $7.1\times10^{-6}$ 19.7 $10^{-1}$ 0.15 4 2 $1.3\times10^{-4}$ 21.1 4 2 $1.6\times10^{-5}$ 21.2 $10^{-1}$ 0.2 4 2 $2.0\times10^{-4}$ 22.3 4 2 $3.6\times10^{-5}$ 22.4 $10^{-1}$ 0.25 6 2 $2.7\times10^{-4}$ 23.2 12 3 $6.1\times10^{-5}$ 23.2 $10^{-2}$ 0.01 2 1 $9.3\times10^{-7}$ 15.6 2 1 $7.5\times10^{-7}$ 15.6 $10^{-2}$ 0.05 4 2 $5.7\times10^{-5}$ 18.8 4 2 $3.7\times10^{-6}$ 19.0 $10^{-2}$ 0.1 5 2 $2.3\times10^{-4}$ 22.3 5 2 $4.5\times10^{-5}$ 22.4 $10^{-2}$ 0.15 6 2 $4.3\times10^{-4}$ 24.5 6 2 $1.4\times10^{-4}$ 24.6 $10^{-2}$ 0.2 8 3 $6.3\times10^{-4}$ 25.7 7 3 $2.6\times10^{-4}$ 25.8 $10^{-2}$ 0.25 10 3 $8.1\times10^{-4}$ 26.4 8 3 $3.8\times10^{-4}$ 26.6 $10^{-3}$ 0.01 3 1 $2.2\times10^{-6}$ 15.4 3 1 $1.3\times10^{-7}$ 15.7 $10^{-3}$ 0.05 9 2 $1.3\times10^{-4}$ 19.3 10 2 $1.9\times10^{-5}$ 19.6 $10^{-3}$ 0.1 8 3 $4.8\times10^{-4}$ 23.9 8 3 $1.8\times10^{-4}$ 24.2 $10^{-3}$ 0.15 10 3 $8.7\times10^{-4}$ 26.6 12 3 $5.0\times10^{-4}$ 26.9 $10^{-3}$ 0.2 20 4 $1.2\times10^{-3}$ 27.4 12 4 $8.3\times10^{-4}$ 28.0 $10^{-3}$ 0.25 17 4 $1.5\times10^{-3}$ 27.3 14 4 $1.1\times10^{-3}$ 28.4 $10^{-4}$ 0.01 9 1 $3.2\times10^{-6}$ 15.4 5 2 $5.4\times10^{-7}$ 15.7 $10^{-4}$ 0.05 16 3 $1.9\times10^{-4}$ 19.4 14 3 $5.1\times10^{-5}$ 19.7 $10^{-4}$ 0.1 21 4 $7.1\times10^{-4}$ 24.5 20 3 $4.1\times10^{-4}$ 24.7 $10^{-4}$ 0.15 19 5 $1.3\times10^{-3}$ 27.3 19 5 $1.0\times10^{-3}$ 27.9 $10^{-4}$ 0.2 22 7 $1.7\times10^{-3}$ 27.5 21 6 $1.6\times10^{-3}$ 28.7 $10^{-4}$ 0.25 21 15 $2.0\times10^{-1}$ 27.6 24 8 $2.1\times10^{-3}$ 28.7
 $\beta$ $\alpha$ boat-$1024\times1024$ cameraman-$2048 \times2048$ Newton GMRES ERROR PSNR Newton GMRES ERROR PSNR $10^{-1}$ 0.01 2 1 $2.1\times10^{-7}$ 15.1 2 1 $4.4\times10^{-9}$ 15.4 $10^{-1}$ 0.05 3 1 $1.4\times10^{-5}$ 17.4 3 1 $2.9\times10^{-7}$ 17.6 $10^{-1}$ 0.1 3 2 $6.3\times10^{-5}$ 19.5 3 2 $7.1\times10^{-6}$ 19.7 $10^{-1}$ 0.15 4 2 $1.3\times10^{-4}$ 21.1 4 2 $1.6\times10^{-5}$ 21.2 $10^{-1}$ 0.2 4 2 $2.0\times10^{-4}$ 22.3 4 2 $3.6\times10^{-5}$ 22.4 $10^{-1}$ 0.25 6 2 $2.7\times10^{-4}$ 23.2 12 3 $6.1\times10^{-5}$ 23.2 $10^{-2}$ 0.01 2 1 $9.3\times10^{-7}$ 15.6 2 1 $7.5\times10^{-7}$ 15.6 $10^{-2}$ 0.05 4 2 $5.7\times10^{-5}$ 18.8 4 2 $3.7\times10^{-6}$ 19.0 $10^{-2}$ 0.1 5 2 $2.3\times10^{-4}$ 22.3 5 2 $4.5\times10^{-5}$ 22.4 $10^{-2}$ 0.15 6 2 $4.3\times10^{-4}$ 24.5 6 2 $1.4\times10^{-4}$ 24.6 $10^{-2}$ 0.2 8 3 $6.3\times10^{-4}$ 25.7 7 3 $2.6\times10^{-4}$ 25.8 $10^{-2}$ 0.25 10 3 $8.1\times10^{-4}$ 26.4 8 3 $3.8\times10^{-4}$ 26.6 $10^{-3}$ 0.01 3 1 $2.2\times10^{-6}$ 15.4 3 1 $1.3\times10^{-7}$ 15.7 $10^{-3}$ 0.05 9 2 $1.3\times10^{-4}$ 19.3 10 2 $1.9\times10^{-5}$ 19.6 $10^{-3}$ 0.1 8 3 $4.8\times10^{-4}$ 23.9 8 3 $1.8\times10^{-4}$ 24.2 $10^{-3}$ 0.15 10 3 $8.7\times10^{-4}$ 26.6 12 3 $5.0\times10^{-4}$ 26.9 $10^{-3}$ 0.2 20 4 $1.2\times10^{-3}$ 27.4 12 4 $8.3\times10^{-4}$ 28.0 $10^{-3}$ 0.25 17 4 $1.5\times10^{-3}$ 27.3 14 4 $1.1\times10^{-3}$ 28.4 $10^{-4}$ 0.01 9 1 $3.2\times10^{-6}$ 15.4 5 2 $5.4\times10^{-7}$ 15.7 $10^{-4}$ 0.05 16 3 $1.9\times10^{-4}$ 19.4 14 3 $5.1\times10^{-5}$ 19.7 $10^{-4}$ 0.1 21 4 $7.1\times10^{-4}$ 24.5 20 3 $4.1\times10^{-4}$ 24.7 $10^{-4}$ 0.15 19 5 $1.3\times10^{-3}$ 27.3 19 5 $1.0\times10^{-3}$ 27.9 $10^{-4}$ 0.2 22 7 $1.7\times10^{-3}$ 27.5 21 6 $1.6\times10^{-3}$ 28.7 $10^{-4}$ 0.25 21 15 $2.0\times10^{-1}$ 27.6 24 8 $2.1\times10^{-3}$ 28.7
The error of the NiOS method with respect to the parameters $\alpha$ and $\beta$ changed proportionally, for example, when $\alpha$ is reduced by $\frac{1}{2}$, $\beta$ is reduced by $(\frac{1}{2})^4$. "ERROR" has the same definition as in Table 2
 $\beta$ $\alpha$ boat-$1024\times1024$ cameraman-$2048 \times2048$ Newton GMRES ERROR PSNR Newton GMRES ERROR PSNR $6.25\times10^{-3}$ 0.1 8 2 $2.8\times10^{-4}$ 22.7 6 2 $6.4\times10^{-5}$ 22.9 $3.9\times10^{-4}$ 0.05 11 2 $1.6\times10^{-4}$ 19.3 11 2 $3.0\times10^{-5}$ 19.7 $2.4\times10^{-5}$ 0.025 14 2 $4.1\times10^{-5}$ 16.8 12 2 $6.8\times10^{-6}$ 17.1 $1.5\times10^{-6}$ 0.0125 17 2 $7.6\times10^{-6}$ 15.6 10 2 $1.1\times10^{-6}$ 15.9
 $\beta$ $\alpha$ boat-$1024\times1024$ cameraman-$2048 \times2048$ Newton GMRES ERROR PSNR Newton GMRES ERROR PSNR $6.25\times10^{-3}$ 0.1 8 2 $2.8\times10^{-4}$ 22.7 6 2 $6.4\times10^{-5}$ 22.9 $3.9\times10^{-4}$ 0.05 11 2 $1.6\times10^{-4}$ 19.3 11 2 $3.0\times10^{-5}$ 19.7 $2.4\times10^{-5}$ 0.025 14 2 $4.1\times10^{-5}$ 16.8 12 2 $6.8\times10^{-6}$ 17.1 $1.5\times10^{-6}$ 0.0125 17 2 $7.6\times10^{-6}$ 15.6 10 2 $1.1\times10^{-6}$ 15.9
Parallel performance of the NiOS algorithm for the 2D cameraman image denoising
 $n_p$ $2048\times2048$ $4096 \times4096$ Newton GMRES Time(s) Newton GMRES Time(s) 64 21 4 1.7 23 4 7.2 128 20 4 0.8 22 4 3.3 256 20 4 0.4 21 4 1.6 512 19 4 0.2 20 4 0.8 1024 18 4 0.1 20 4 0.4
 $n_p$ $2048\times2048$ $4096 \times4096$ Newton GMRES Time(s) Newton GMRES Time(s) 64 21 4 1.7 23 4 7.2 128 20 4 0.8 22 4 3.3 256 20 4 0.4 21 4 1.6 512 19 4 0.2 20 4 0.8 1024 18 4 0.1 20 4 0.4
The detail of the sub-problem solving in NiOS with 24 processors. Image size is $2048\times2048$. Here "rank" is the processor core ID
 rank Newton GMRES Time rank Newton GMRES Time 0 11 5 7.3 12 17 5 9.1 1 11 5 7.3 13 14 5 8.4 2 17 5 9.2 14 10 5 6.4 3 19 5 9.8 15 8 5 5.2 4 12 5 7.2 16 14 5 8.0 5 12 5 7.3 17 14 5 8.3 6 12 5 7.3 18 16 5 8.8 7 12 5 7.6 19 16 5 8.8 8 16 5 8.8 20 13 5 7.6 9 17 5 9.0 21 13 5 7.5 10 11 5 6.8 22 16 5 8.5 11 12 5 7.3 23 16 5 6.6
 rank Newton GMRES Time rank Newton GMRES Time 0 11 5 7.3 12 17 5 9.1 1 11 5 7.3 13 14 5 8.4 2 17 5 9.2 14 10 5 6.4 3 19 5 9.8 15 8 5 5.2 4 12 5 7.2 16 14 5 8.0 5 12 5 7.3 17 14 5 8.3 6 12 5 7.3 18 16 5 8.8 7 12 5 7.6 19 16 5 8.8 8 16 5 8.8 20 13 5 7.6 9 17 5 9.0 21 13 5 7.5 10 11 5 6.8 22 16 5 8.5 11 12 5 7.3 23 16 5 6.6
Comparison of the NiOS and NKS for the 3D image denoising. The image size is $181\times217\times181$. "Sp" refers to the compute time speedup of the NiOS method compared with the NKS method
 $n_p$ NKS NiOS Sp Newton GMRES Time(s) Newton GMRES Time(s) 32 35 17 38.5 35 12 22.0 1.8 64 36 17 24.5 31 11 12.4 2.0 128 36 18 12.0 29 10 5.0 2.4 256 36 18 6.6 26 9 3.3 2.0 512 39 18 3.9 24 8 1.4 2.8 1024 35 19 2.4 22 8 0.7 3.4
 $n_p$ NKS NiOS Sp Newton GMRES Time(s) Newton GMRES Time(s) 32 35 17 38.5 35 12 22.0 1.8 64 36 17 24.5 31 11 12.4 2.0 128 36 18 12.0 29 10 5.0 2.4 256 36 18 6.6 26 9 3.3 2.0 512 39 18 3.9 24 8 1.4 2.8 1024 35 19 2.4 22 8 0.7 3.4
The effect of various choices of the overlapping parameter $\delta$ in the NiOS method for the 3D image. The number of processors is $64$ and the image size is $181\times217\times181$. The PSNR of the NKS method is $27.84$ for this image denoising. Here $\text{DIFF} = || \mathbf{u}_{\text{NiOS}} - \mathbf{u}_{\text{NKS}}||_2$ is the difference of the solution of NiOS ($\mathbf{u}_{\text{NiOS}}$) and NKS ($\mathbf{u}_{\text{NKS}}$)
 $\delta$ Newton GMRES Time (s) PSNR DIFF 0 14 5 2.7 27.83 5.89 1 14 5 2.8 27.85 1.20 2 14 5 3.1 27.84 0.36 3 14 5 3.4 27.84 0.07
 $\delta$ Newton GMRES Time (s) PSNR DIFF 0 14 5 2.7 27.83 5.89 1 14 5 2.8 27.85 1.20 2 14 5 3.1 27.84 0.36 3 14 5 3.4 27.84 0.07
Parallel performance of the NiOS-NKS algorithm for the 3D image denosing. The image size is $217\times217\times181$
 $n_p$ First Stage Second Stage Newton GMRES Time Newton GMRES Time 32 15 5 4.48 3 7 2.39 64 14 5 3.5 3 7 1.41 128 15 5 1.71 3 7 0.77 256 14 5 1.11 3 7 0.47 512 14 5 0.45 3 7 0.36 1024 14 5 0.26 3 8 0.42
 $n_p$ First Stage Second Stage Newton GMRES Time Newton GMRES Time 32 15 5 4.48 3 7 2.39 64 14 5 3.5 3 7 1.41 128 15 5 1.71 3 7 0.77 256 14 5 1.11 3 7 0.47 512 14 5 0.45 3 7 0.36 1024 14 5 0.26 3 8 0.42
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