October  2017, 11(5): 783-798. doi: 10.3934/ipi.2017037

A wavelet frame approach for removal of mixed Gaussian and impulse noise on surfaces

College of Science, Hohai University, No.8 Focheng West Road, Jiangning, Nanjing, Jiangsu Province, China, 211100

* Corresponding author: Jianbin Yang

Received  August 2016 Revised  May 2017 Published  June 2017

Fund Project: The first author is supported by NSF grant #11101120 and the Fundamental Research Funds for the Central Universities 2015B19514, China; The second author is supported by the Fundamental Research Funds for the Central Universities 2015B38014

Surface denoising is a fundamental problem in geometry processing and computer graphics. In this paper, we propose a wavelet frame based variational model to restore surfaces which are corrupted by mixed Gaussian and impulse noise, under the assumption that the region corrupted by impulse noise is unknown. The model contains a universal $\ell_1 + \ell_2$ fidelity term and an $\ell_1$-regularized term which makes additional use of the wavelet frame transform on surfaces in order to preserve key features such as sharp edges and corners. We then apply the augmented Lagrangian and accelerated proximal gradient methods to solve this model. In the end, we demonstrate the efficacy of our approach with numerical experiments both on surfaces and functions defined on surfaces. The experimental results show that our method is competitive relative to some existing denoising methods.

Citation: Jianbin Yang, Cong Wang. A wavelet frame approach for removal of mixed Gaussian and impulse noise on surfaces. Inverse Problems & Imaging, 2017, 11 (5) : 783-798. doi: 10.3934/ipi.2017037
References:
[1]

P. AlfeldM. Neamtu and L. L. Schumaker, Fitting scattered data on sphere-like surfaces using spherical splines, J. Comput. Appl. Math., 73 (1996), 5-43. doi: 10.1016/0377-0427(96)00034-9.

[2]

M. Botsch, L. Kobbelt, M. Pauly, P. Alliez and B. Lévy, Polygon Mesh Processing A. K. Peters, Ltd, 2010. doi: 10.1201/b10688.

[3]

J. F. CaiS. Osher and Z. Shen, Split bregman methods and frame based image restoration, Multiscale Model. Simul.: A SIAM Interdiscip. J., 8 (2009), 337-369. doi: 10.1137/090753504.

[4]

J. F. Cai and Z. Shen, Framelet based deconvolution, J. Comput. Math., 28 (2010), 289-308. doi: 10.4208/jcm.2009.10-m1009.

[5]

U. ClarenzU. Diewald and M. Rumpf, Anisotropic geometric diffusion in surface processing, Proceedings of the Conference on Visualization 2000, (2000), 397-405. doi: 10.1109/VISUAL.2000.885721.

[6]

M. DesbrunM. MeyerP. Schröder and A. H. Barr, Implicit fairing of arbitrary meshes using diffusion and curvature flow, Proceedings of SIGGRAPH 1999, (1999), 317-324.

[7]

M. DesbrunM. MeyerP. Schröder and A. H. Barr, Anisotropic feature-preserving denoising of height fields and bivariate data, Proceedings of the Graphics Interface 2000 Conference, (2000), 145-152.

[8]

B. Dong, Sparse representation on graphs by tight wavelet frames and applications, Appl. Comput. Harmon. Anal., 42 (2017), 452-479. doi: 10.1016/j.acha.2015.09.005.

[9]

B. DongQ. T. JiangC. Q. Liu and Z. Shen, Multiscale representation of surfaces by tight wavelet frames with applications to denoising, Appl. Comput. Harmon. Anal., 41 (2016), 561-589. doi: 10.1016/j.acha.2015.03.005.

[10]

B. Dong and Z. Shen, MRA-based wavelet frames and applications, IAS/Park CityMathematics Series, 19 (2010), 9-158.

[11]

M. Elsey and S. Esedoglu, Analogue of the total variation denoising model in the context of geometry processing, Multiscale Model. Simul., 7 (2009), 1549-1573. doi: 10.1137/080736612.

[12]

Z. T. FanH. Ji and Z. Shen, Dual Graminan analysis: Duality and unitary extension principle, Math. Comp., 85 (2016), 239-270. doi: 10.1090/mcom/2987.

[13]

S. FleishmanI. Drori and D. Cohen-Or, Bilateral mesh denoising, ACM Trans. Graph., 22 (2003), 950-953. doi: 10.1145/1201775.882368.

[14]

Z. Gong, Augmented Lagrangian Based Algorithms for Convex Optimization Problems with Non-separable $\ell_1$-regularization Ph. D. thesis, National University of Singapore, 2013.

[15]

Z. GongZ. Shen and K. Toh, Image restoration with mixed or unknown noises, Multiscale Model. Simul., 12 (2014), 458-487. doi: 10.1137/130904533.

[16]

H. HoppeT. DeroseT. DuchampM. HalsteadH. JinJ. McdonaldJ. Schweitzer and W. Stuetzle, Piecewise smooth surface reconstruction, Proceedings of the 21st annual conference on Computer graphics and interactive techniques, (1994), 295-302. doi: 10.1145/192161.192233.

[17]

T. JonesF. Durand and M. Desbrun, Non-iterative, feature-preserving mesh smoothing, ACM Trans. Graph., 22 (2003), 943-949. doi: 10.1145/1201775.882367.

[18]

J. LiZ. ShenR. J. Yin and X. Q. Zhang, A reweighted $\ell^2$ method for image restoration with poisson and mixed poisson-gaussian noise, Inverse Probl. Imaging., 9 (2015), 875-894. doi: 10.3934/ipi.2015.9.875.

[19]

S. Mallat, A Wavelet Tour of Signal Processing: The Sparse Way Academic press, 2009.

[20]

Y. Nesterov, A method of solving a convex programming problem with convergence rate $O(1/k^2)$, Soviet Math. Dokl, 27 (1983), 372-376.

[21]

R. T. Rockafellar, Augmented Lagrangians and applications of the proximal point algorithm in convex programming, Math. Oper. Res., 1 (1976), 97-116. doi: 10.1287/moor.1.2.97.

[22]

A. Ron and Z. Shen, Affine systems in $L_2(\mathbb{R}^d)$: The analysis of the analysis operator, J. Funct. Anal., 148 (1997), 408-447. doi: 10.1006/jfan.1996.3079.

[23]

L. I. RudinS. Osher and E. Fatemi, Nonlinear total variation based noise removal algorithms, Phys. D, 60 (1992), 259-268. doi: 10.1016/0167-2789(92)90242-F.

[24]

R. B. RusuZ. C. MartonN. BlodowM. Dolha and M. Beetz, Towards 3D Point cloud based object maps for household environments, Rob. Auton. Syst., 56 (2008), 927-941. doi: 10.1016/j.robot.2008.08.005.

[25]

Y. ShenB. Han and E. Braverman, Removal of mixed gaussian and impulse noise using directional tensor product complex tight framelets, J. Math. Imaging Vis., 54 (2016), 64-77. doi: 10.1007/s10851-015-0589-5.

[26]

X. F. SunP. L. RosinR. R. Martin and F. C. Langbein, Fast and effective feature-preserving mesh denoising, IEEE Trans. Vis. Comput. Graphics, 13 (2007), 925-938. doi: 10.1109/TVCG.2007.1065.

[27]

T. TasdizenR. WhitakerP. Burchard and S. Osher, Geometric surface processing via normal maps, ACM Trans. Graph., 22 (2003), 1012-1033. doi: 10.1145/944020.944024.

[28]

C. WuJ. ZhangY. Duan and X. C. Tai, Augmented lagrangian method for total variation based image restoration and segmentation over triangulated surfaces, J. Sci. Comput., 50 (2012), 145-166. doi: 10.1007/s10915-011-9477-3.

[29]

M. Yan, Restoration of images corrupted by impulse noise and mixed gaussian impulse noise using blind inpainting, SIAM J. Imaging Sci., 6 (2013), 1227-1245. doi: 10.1137/12087178X.

[30]

J. YangD. Stahl and Z. Shen, An analysis of wavelet frame based scattered data reconstruction, Appl. Comput. Harmon. Anal., 42 (2017), 480-507. doi: 10.1016/j.acha.2015.09.008.

[31]

H. ZhangC. WuJ. Zhang and J. Deng, Variational mesh denoising using total variation and piecewise constant function space, IEEE Trans. Vis. Comput. Graphics, 21 (2015), 873-886. doi: 10.1109/TVCG.2015.2398432.

show all references

References:
[1]

P. AlfeldM. Neamtu and L. L. Schumaker, Fitting scattered data on sphere-like surfaces using spherical splines, J. Comput. Appl. Math., 73 (1996), 5-43. doi: 10.1016/0377-0427(96)00034-9.

[2]

M. Botsch, L. Kobbelt, M. Pauly, P. Alliez and B. Lévy, Polygon Mesh Processing A. K. Peters, Ltd, 2010. doi: 10.1201/b10688.

[3]

J. F. CaiS. Osher and Z. Shen, Split bregman methods and frame based image restoration, Multiscale Model. Simul.: A SIAM Interdiscip. J., 8 (2009), 337-369. doi: 10.1137/090753504.

[4]

J. F. Cai and Z. Shen, Framelet based deconvolution, J. Comput. Math., 28 (2010), 289-308. doi: 10.4208/jcm.2009.10-m1009.

[5]

U. ClarenzU. Diewald and M. Rumpf, Anisotropic geometric diffusion in surface processing, Proceedings of the Conference on Visualization 2000, (2000), 397-405. doi: 10.1109/VISUAL.2000.885721.

[6]

M. DesbrunM. MeyerP. Schröder and A. H. Barr, Implicit fairing of arbitrary meshes using diffusion and curvature flow, Proceedings of SIGGRAPH 1999, (1999), 317-324.

[7]

M. DesbrunM. MeyerP. Schröder and A. H. Barr, Anisotropic feature-preserving denoising of height fields and bivariate data, Proceedings of the Graphics Interface 2000 Conference, (2000), 145-152.

[8]

B. Dong, Sparse representation on graphs by tight wavelet frames and applications, Appl. Comput. Harmon. Anal., 42 (2017), 452-479. doi: 10.1016/j.acha.2015.09.005.

[9]

B. DongQ. T. JiangC. Q. Liu and Z. Shen, Multiscale representation of surfaces by tight wavelet frames with applications to denoising, Appl. Comput. Harmon. Anal., 41 (2016), 561-589. doi: 10.1016/j.acha.2015.03.005.

[10]

B. Dong and Z. Shen, MRA-based wavelet frames and applications, IAS/Park CityMathematics Series, 19 (2010), 9-158.

[11]

M. Elsey and S. Esedoglu, Analogue of the total variation denoising model in the context of geometry processing, Multiscale Model. Simul., 7 (2009), 1549-1573. doi: 10.1137/080736612.

[12]

Z. T. FanH. Ji and Z. Shen, Dual Graminan analysis: Duality and unitary extension principle, Math. Comp., 85 (2016), 239-270. doi: 10.1090/mcom/2987.

[13]

S. FleishmanI. Drori and D. Cohen-Or, Bilateral mesh denoising, ACM Trans. Graph., 22 (2003), 950-953. doi: 10.1145/1201775.882368.

[14]

Z. Gong, Augmented Lagrangian Based Algorithms for Convex Optimization Problems with Non-separable $\ell_1$-regularization Ph. D. thesis, National University of Singapore, 2013.

[15]

Z. GongZ. Shen and K. Toh, Image restoration with mixed or unknown noises, Multiscale Model. Simul., 12 (2014), 458-487. doi: 10.1137/130904533.

[16]

H. HoppeT. DeroseT. DuchampM. HalsteadH. JinJ. McdonaldJ. Schweitzer and W. Stuetzle, Piecewise smooth surface reconstruction, Proceedings of the 21st annual conference on Computer graphics and interactive techniques, (1994), 295-302. doi: 10.1145/192161.192233.

[17]

T. JonesF. Durand and M. Desbrun, Non-iterative, feature-preserving mesh smoothing, ACM Trans. Graph., 22 (2003), 943-949. doi: 10.1145/1201775.882367.

[18]

J. LiZ. ShenR. J. Yin and X. Q. Zhang, A reweighted $\ell^2$ method for image restoration with poisson and mixed poisson-gaussian noise, Inverse Probl. Imaging., 9 (2015), 875-894. doi: 10.3934/ipi.2015.9.875.

[19]

S. Mallat, A Wavelet Tour of Signal Processing: The Sparse Way Academic press, 2009.

[20]

Y. Nesterov, A method of solving a convex programming problem with convergence rate $O(1/k^2)$, Soviet Math. Dokl, 27 (1983), 372-376.

[21]

R. T. Rockafellar, Augmented Lagrangians and applications of the proximal point algorithm in convex programming, Math. Oper. Res., 1 (1976), 97-116. doi: 10.1287/moor.1.2.97.

[22]

A. Ron and Z. Shen, Affine systems in $L_2(\mathbb{R}^d)$: The analysis of the analysis operator, J. Funct. Anal., 148 (1997), 408-447. doi: 10.1006/jfan.1996.3079.

[23]

L. I. RudinS. Osher and E. Fatemi, Nonlinear total variation based noise removal algorithms, Phys. D, 60 (1992), 259-268. doi: 10.1016/0167-2789(92)90242-F.

[24]

R. B. RusuZ. C. MartonN. BlodowM. Dolha and M. Beetz, Towards 3D Point cloud based object maps for household environments, Rob. Auton. Syst., 56 (2008), 927-941. doi: 10.1016/j.robot.2008.08.005.

[25]

Y. ShenB. Han and E. Braverman, Removal of mixed gaussian and impulse noise using directional tensor product complex tight framelets, J. Math. Imaging Vis., 54 (2016), 64-77. doi: 10.1007/s10851-015-0589-5.

[26]

X. F. SunP. L. RosinR. R. Martin and F. C. Langbein, Fast and effective feature-preserving mesh denoising, IEEE Trans. Vis. Comput. Graphics, 13 (2007), 925-938. doi: 10.1109/TVCG.2007.1065.

[27]

T. TasdizenR. WhitakerP. Burchard and S. Osher, Geometric surface processing via normal maps, ACM Trans. Graph., 22 (2003), 1012-1033. doi: 10.1145/944020.944024.

[28]

C. WuJ. ZhangY. Duan and X. C. Tai, Augmented lagrangian method for total variation based image restoration and segmentation over triangulated surfaces, J. Sci. Comput., 50 (2012), 145-166. doi: 10.1007/s10915-011-9477-3.

[29]

M. Yan, Restoration of images corrupted by impulse noise and mixed gaussian impulse noise using blind inpainting, SIAM J. Imaging Sci., 6 (2013), 1227-1245. doi: 10.1137/12087178X.

[30]

J. YangD. Stahl and Z. Shen, An analysis of wavelet frame based scattered data reconstruction, Appl. Comput. Harmon. Anal., 42 (2017), 480-507. doi: 10.1016/j.acha.2015.09.008.

[31]

H. ZhangC. WuJ. Zhang and J. Deng, Variational mesh denoising using total variation and piecewise constant function space, IEEE Trans. Vis. Comput. Graphics, 21 (2015), 873-886. doi: 10.1109/TVCG.2015.2398432.

Figure 1.  This figure illustrates how the neighboring vertices of a given vertex $V_i(k)$ are ordered
Figure 2.  The elephant model (first row) and the bunny model (second row) are artificially corrupted by mixed Gaussian and impulse noise ($r=20\%, \mu=0, \sigma=0.005$), then smoothed by our approach ($\lambda=11, \nu=0.2$). From left to right: noisy-free surfaces, noisy surfaces, denoising surfaces
Figure 3.  From left to right: noisy-free surfaces, noisy surfaces, denoising surfaces ($r=40\%, \mu=0, \sigma=0.2, \lambda=11, \nu=0.2$)
Figure 4.  This figure shows two images (first row), 'Slope' and 'Eric Cartman', that are mapped to the graph of the unit sphere to form graph data $f_G$ (second row)
Figure 5.  From left to right: noisy-free surfaces, noisy surfaces, denoising surfaces ($r=20\%, \mu=0, \sigma=0.02, \lambda=0.8, \nu=0.035$)
Figure 6.  From left to right: noisy surfaces, Zhang et al.'s results [31], Ours ($r=20\%, \mu=0, \sigma=0.005$). The $2nd$ row and $4th$ row show zoomed view of surfaces
Figure 7.  It shows denoising results. From left to right: noisy surfaces, Zhang et al.'s results, Ours ($r=40\%, \mu=0, \sigma=0.2$). The $2nd$ row and $4th$ row show zoomed view of surfaces
Figure 8.  From left to right: results of $\ell_2$-variational model, results of $\ell_1$-variational model, Ours. The $2nd$ row and $4th$ row show zoomed view of surfaces
Figure 9.  It shows denoising results. From left to right: results of $\ell_2$-variational model, results of $\ell_1$-variational model, Ours. The $2nd$ row and $4th$ row show zoomed view of surfaces
Figure 10.  From left to right: results of $\ell_2$-variational model, results of $\ell_1$-variational model, Ours
Table 1.  SNRs Comparison for removal of mixed Gaussian and impulse noise
Model Zhang et al.'s Ours
FIGURE 6 Row 1 43.735 45.035
FIGURE 6 Row 3 43.985 44.876
FIGURE 7 Row 1 42.833 45.565
FIGURE 7 Row 3 40.658 43.035
Model Zhang et al.'s Ours
FIGURE 6 Row 1 43.735 45.035
FIGURE 6 Row 3 43.985 44.876
FIGURE 7 Row 1 42.833 45.565
FIGURE 7 Row 3 40.658 43.035
Table 2.  SNRs Comparison with other variational models
Model $\ell_2$-variational model $\ell_1$-variational model Ours
elephant 42.858 43.064 45.035
bunny 43.027 43.356 44.876
$\Omega_1$ 42.024 42.410 45.565
$\Omega_2$ 40.674 40.787 43.035
Slope 28.688 29.647 31.231
Eric Cartman 29.063 29.953 31.087
Model $\ell_2$-variational model $\ell_1$-variational model Ours
elephant 42.858 43.064 45.035
bunny 43.027 43.356 44.876
$\Omega_1$ 42.024 42.410 45.565
$\Omega_2$ 40.674 40.787 43.035
Slope 28.688 29.647 31.231
Eric Cartman 29.063 29.953 31.087
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