December  2019, 13(6): 1161-1188. doi: 10.3934/ipi.2019052

Nonlinear fractional diffusion model for deblurring images with textures

School of Mathematics, Harbin Institute of Technology, Harbin, Heilongjiang 150001, China

* Corresponding author: Jiebao Sun

Received  June 2018 Revised  July 2019 Published  October 2019

Fund Project: The work of the authors was partially supported by the National Natural Science Foundation of China (11971131, U1637208, 61873071, 51476047, 11871133) and the Natural Science Foundation of Heilongjiang Province (LC2018001, A2016003).

It is a long-standing problem to preserve fine scale features such as texture in the process of deblurring. In order to deal with this challenging but imperative issue, we establish a framework of nonlinear fractional diffusion equations, which performs well in deblurring images with textures. In the new model, a fractional gradient is used for regularization of the diffusion process to preserve texture features and a source term with blurring kernel is used for deblurring. This source term ensures that the model can handle various blurring kernels. The relation between the regularization parameter and the deblurring performance is investigated theoretically, which ensures a satisfactory recovery when the blur type is known. Moreover, we derive a digital fractional diffusion filter that lives on images. Experimental results and comparisons show the effectiveness of the proposed model for texture-preserving deblurring.

Citation: Zhichang Guo, Wenjuan Yao, Jiebao Sun, Boying Wu. Nonlinear fractional diffusion model for deblurring images with textures. Inverse Problems and Imaging, 2019, 13 (6) : 1161-1188. doi: 10.3934/ipi.2019052
References:
[1]

R. A. Adams, Pure and Applied Mathematics, Ser., Monographs and Textbooks, Sobolev Spaces, New York, Academic, 1975.

[2]

O. P. Agrawal, Fractional variational calculus in terms of Riesz fractional derivatives, J. Phys. A, 40 (2007), 6287-6303.  doi: 10.1088/1751-8113/40/24/003.

[3]

J. Bai and X. C. Feng, Fractional-order anisotropic diffusion for image denoising, IEEE Trans. Image Process., 16 (2007), 2492-2502.  doi: 10.1109/TIP.2007.904971.

[4]

A. BjörnJ. BjörnU. Gianazza and M. Parviainen, Boundary regularity for degenerate and singular parabolic equations, Calc. Var. Partial Differential Equations, 52 (2015), 797-827.  doi: 10.1007/s00526-014-0734-9.

[5]

J. F. CaiB. Dong and Z. W. Shen, Image restoration: A wavelet frame based model for piecewise smooth functions and beyond, Appl. Comput. Harmon. Anal., 41 (2016), 94-138.  doi: 10.1016/j.acha.2015.06.009.

[6]

A. Chambolle and T. Pock, A first-order primal-dual algorithm for convex problems with applications to imaging, J. Math. Imaging. Vis., 40 (2011), 120-145.  doi: 10.1007/s10851-010-0251-1.

[7]

D. ChenS. SunC. ZhangY. Chen and D. Xue, Fractional-order TV-L2 model for image denoising, Cent. Eur. J. Phys., 11 (2013), 1414-1422.  doi: 10.2478/s11534-013-0241-1.

[8]

D. Chen, Y. Chen and D. Xue, Three fractional-order TV-L2 models for image denoising, J. Comput. Inf. Syst., (2013).

[9]

D. Q. Chen and L. Z. Cheng, Alternative minimization algorithm for nonlocal total variational image deblurring, IET Image Processing, 4 (2010), 353-364. 

[10]

R. R. Coifman and D. L. Donoho, Translation-invariantde-noising, Wavelet and Statistics, A. Antoniadis and G. Oppenheim, Eds., Springer-Verlag, New York, (1995), 125–150.

[11]

K. DabovA. FoiV. Katkovnik and K. Egiazarian, Image denoising by sparse 3-d transform-domain collaborative filtering, IEEE Trans. Image Process, 16 (2007), 2080-2095.  doi: 10.1109/TIP.2007.901238.

[12]

K. Dabov, A. Foi, V. Katkovnik and K. Egiazarian, Image restoration by sparse 3-D transform-domain collaborative filtering, SPIE Electronic Imaging, 2008.

[13]

A. DanielyanV. Katkovnik and K. Egiazarian, BM3D frames and variational image deblurring, IEEE Trans. Image Process., 21 (2012), 1715-1728.  doi: 10.1109/TIP.2011.2176954.

[14]

E. DiBenedettoU. Gianazza and V. Vespri, Harnack estimates for quasi-linear degenerate parabolic differential equations, Acta Math., 200 (2008), 181-209.  doi: 10.1007/s11511-008-0026-3.

[15]

E. DiBenedettoU. Gianazza and V. Vespri, Liouville-type theorems for certain degenerate and singular parabolic equations, C. R. Math. Acad. Sci. Paris., 348 (2010), 873-877.  doi: 10.1016/j.crma.2010.06.019.

[16]

E. DiBenedetto, Degenerate Parabolic Equations, Universitext, Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4612-0895-2.

[17]

F. F. Dong and Y. M. Chen, A fractional-order derivative based variational framework for image denoising, Inverse Probl. Imaging., 10 (2016), 27-50.  doi: 10.3934/ipi.2016.10.27.

[18]

M. Elad and M. A. T. Figueiredo, On the role of sparse and redundant representations in image processing, Proc. IEEE., 98 (2010), 972-982. 

[19]

G. Gilboa and S. Osher, Nonlocal operators with applications to image processing, Multiscale Model. Simul., 7 (2008), 1005-1028.  doi: 10.1137/070698592.

[20]

P. GuidottiY. Kim and J. Lambers, Image restoration with a new class of forward-backward-forward diffusion equations of Perona-Malik type with applications to satellite image enhancement, SIAM J. Imaging Sci., 6 (2013), 1416-1444.  doi: 10.1137/120882895.

[21]

P. Guidotti, A new nonlocal nonlinear diffusion of image processing, J. Differ. Equ., 246 (2009), 4731-4742.  doi: 10.1016/j.jde.2009.03.017.

[22]

P. Guidotti and J. V. Lambers, Two new nonlinear nonlocal diffusions for noise reduction, J. Math. Imaging Vis., 33 (2009), 25-37.  doi: 10.1007/s10851-008-0108-z.

[23]

O. Honigman and Y. Y. Zeevi, Enhancement of textured images using complex diffusion incorporating Schrodinger's Potential, Proc. IEEE Int. Conf. Acoust. Speech Signal Process, 26 (2006), 633-636.  doi: 10.1109/ICASSP.2006.1660422.

[24]

Y. M. HuangNg. Michael K and Y. W. Wen, A fast total variation minimization method for image restoration, Multiscale Model. Simul., 7 (2008), 774-795.  doi: 10.1137/070703533.

[25]

M. JanevS. PilipovićT. AtanackovićR. Obradović and N. Ralević, Fully fractional anisotropic diffusion for image denoising, Math. Comput. Model., 54 (2011), 729-741.  doi: 10.1016/j.mcm.2011.03.017.

[26]

D. Kundur and D. Hatzinakos, Blind image deconvolution, IEEE Signal Process. Mag., 13 (1996), 43-64.  doi: 10.1109/79.489268.

[27]

X. Li, Fine-granularity and spatially-adaptive regularization for projection-based image deblurring, IEEE Trans. Image Process., 20 (2011), 971-983.  doi: 10.1109/TIP.2010.2081681.

[28]

J. LiuZ. HuanH. Huang and H. Zhang, An adaptive method for recovering image from mixed noisy data, Int. J. Comput. Vis., 85 (2009), 182-191.  doi: 10.1007/s11263-009-0254-9.

[29]

C. Louchet and L. Moisan, Total variation as a Local Filter, SIAM J. Imaging Sci., 4 (2011), 651-694.  doi: 10.1137/100785855.

[30]

M. Meerschaert and C. Tadjeran, Finite difference approximations for fractional advection-dispersion flow equations, J. Comput. Appl. Math., 172 (2004), 65-77.  doi: 10.1016/j.cam.2004.01.033.

[31]

V. Papyan and M. Elad, Multi-scale patch-based image restoration, IEEE Trans. Image Processing, 25 (2016), 249-261.  doi: 10.1109/TIP.2015.2499698.

[32]

P. Perona and J. Malik, Scale-space and edge detection using anisotropic diffusion, IEEE Trans. Pattern Anal. Machine Intell., 12 (1990), 629-639.  doi: 10.1109/34.56205.

[33] I. Podlubny, Fractional Differential Equations, An introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications, Mathematics in Science and Engineering, 198, Academic Press, Inc., San Diego, CA, 1999. 
[34]

Y. Pu et al., Fractional partial differential equation denoising models for texture image, Sci. China Inf. Sci., 57 (2014), 072115, 19 pp. doi: 10.1007/s11432-014-5112-x.

[35]

Y. Pu, Research on Application of Fractional Calculus to Latest Signal Analysis and Processing, Ph. D. dissertation, School Electron. Inf., Sichuan Univ., Sichuan Province, China, 2006.

[36]

P. D. Romero and V. F. Candela, Mathematical models for restoration of Baroque paintings, Lecture Notes in Computer Science, Springer, Berlin, 4179 (2006), 24–34. doi: 10.1007/11864349_3.

[37]

P. D. Romero and V. F. Candela, Blind deconvolution models regularized by fractional powers of the Laplacian, J. Math. Imaging Vis., 32 (2008), 181-191.  doi: 10.1007/s10851-008-0093-2.

[38]

F. Rousseau, A non-local approach for image super-resolution using intermodality priors, Med. Image. Anal., 14 (2010), 594-605.  doi: 10.1016/j.media.2010.04.005.

[39]

L. RudinS. Osher and E. Fatemi, Nonlinear total variation based noise removal algorithms, Experimental mathematics: Computational issues in nonlinear science (Los Alamos, NM, 1991), Phys. D., 60 (1992), 259-268.  doi: 10.1016/0167-2789(92)90242-F.

[40]

L. Rudin and S. Osher, Total variation based image restoration with free local constraints, Proceedings of the 1st IEEE International Conference on Image Processing, 1 (1994), 31–35. doi: 10.1109/ICIP.1994.413269.

[41]

M. E. Taylor, Partial Differential Equations II - Qualitative Studies of Linear Equations, Second edition. Applied Mathematical Sciences, 116. Springer, New York, 2011. doi: 10.1007/978-1-4419-7052-7.

[42] M. E. Taylor, Pseudodifferential Operators, Princeton Mathematical Series, 34, Princeton University Press, Princeton, N.J., 1981. 
[43]

M. E. Taylor, Partial Differential Equations I - Basic Theory, Second edition, Applied Mathematical Sciences, 115, Springer, New York, 2011. doi: 10.1007/978-1-4419-7055-8.

[44]

A. Tikhonov and V. Arsenin, Solution of Ill-Posed Problems, Translated from the Russian, Preface by translation editor Fritz John. Scripta Series in Mathematics, New York-Toronto, Ont.-London, 1977.

[45]

M. WelkD. TheisT. Brox and J. Weickert, PDE-based deconvolution with forward-backward diffusivities and diffusion tensors, Scale Space PDE Methods Comput. Vis., 3459 (2005), 585-597.  doi: 10.1007/11408031_50.

[46]

B. M. WilliamsJ. Zhang and K. Chen, A new image deconvolution method with fractional regularization, J. Algorithms Comput. Technol., 10 (2016), 265-276.  doi: 10.1177/1748301816660439.

[47]

A. P. Witkin, Scale-space filtering, Proc. Eighth International Joint Conference on Artificial Intelligenc., Karlsruhe, West Germany, 2 (1987) 329–332. doi: 10.1016/B978-0-08-051581-6.50036-2.

[48]

M. Xu, J. Yang, D. Zhao and H. Zhao, An image-enhancement method based on variable-order fractional differential operators, Bio-Med. Mater. Eng., 26 (2015), S1325–S1333. doi: 10.3233/BME-151430.

[49]

X. YinS. Zhou and M. A. Siddique, Fractional nonlinear anisotropic diffusion with p-laplace variation method for image restoration, Multimedia Tools and Applications, 75 (2016), 4505-4526.  doi: 10.1007/s11042-015-2488-6.

[50]

X. Q. ZhangM. BurgerX. Bresson and S. Osher, Bregmanized nonlocal regularization for deconvolution and sparse reconstruction, SIAM J. Imaging Sci., 3 (2010), 253-276.  doi: 10.1137/090746379.

[51]

J. ZhangZ. Wei and L. Xiao, Adaptive fractional-order multi-scale method for image denoising, J. Math. Imaging Vis., 43 (2012), 39-49.  doi: 10.1007/s10851-011-0285-z.

[52]

J. Zhang and K. Chen, A total fractional-order variation model for image restoration with nonhomogeneous boundary conditions and its numerical solution, SIAM J. Imaging Sci., 8 (2015), 2487-2518.  doi: 10.1137/14097121X.

[53]

Z. ZhouZ. GuoG. DongJ. SunD. Zhang and B. Wu, A doubly degenerate diffusion model based on the gray level indicator for multiplicative noise removal, IEEE Trans. Image Process., 24 (2015), 249-260.  doi: 10.1109/TIP.2014.2376185.

show all references

References:
[1]

R. A. Adams, Pure and Applied Mathematics, Ser., Monographs and Textbooks, Sobolev Spaces, New York, Academic, 1975.

[2]

O. P. Agrawal, Fractional variational calculus in terms of Riesz fractional derivatives, J. Phys. A, 40 (2007), 6287-6303.  doi: 10.1088/1751-8113/40/24/003.

[3]

J. Bai and X. C. Feng, Fractional-order anisotropic diffusion for image denoising, IEEE Trans. Image Process., 16 (2007), 2492-2502.  doi: 10.1109/TIP.2007.904971.

[4]

A. BjörnJ. BjörnU. Gianazza and M. Parviainen, Boundary regularity for degenerate and singular parabolic equations, Calc. Var. Partial Differential Equations, 52 (2015), 797-827.  doi: 10.1007/s00526-014-0734-9.

[5]

J. F. CaiB. Dong and Z. W. Shen, Image restoration: A wavelet frame based model for piecewise smooth functions and beyond, Appl. Comput. Harmon. Anal., 41 (2016), 94-138.  doi: 10.1016/j.acha.2015.06.009.

[6]

A. Chambolle and T. Pock, A first-order primal-dual algorithm for convex problems with applications to imaging, J. Math. Imaging. Vis., 40 (2011), 120-145.  doi: 10.1007/s10851-010-0251-1.

[7]

D. ChenS. SunC. ZhangY. Chen and D. Xue, Fractional-order TV-L2 model for image denoising, Cent. Eur. J. Phys., 11 (2013), 1414-1422.  doi: 10.2478/s11534-013-0241-1.

[8]

D. Chen, Y. Chen and D. Xue, Three fractional-order TV-L2 models for image denoising, J. Comput. Inf. Syst., (2013).

[9]

D. Q. Chen and L. Z. Cheng, Alternative minimization algorithm for nonlocal total variational image deblurring, IET Image Processing, 4 (2010), 353-364. 

[10]

R. R. Coifman and D. L. Donoho, Translation-invariantde-noising, Wavelet and Statistics, A. Antoniadis and G. Oppenheim, Eds., Springer-Verlag, New York, (1995), 125–150.

[11]

K. DabovA. FoiV. Katkovnik and K. Egiazarian, Image denoising by sparse 3-d transform-domain collaborative filtering, IEEE Trans. Image Process, 16 (2007), 2080-2095.  doi: 10.1109/TIP.2007.901238.

[12]

K. Dabov, A. Foi, V. Katkovnik and K. Egiazarian, Image restoration by sparse 3-D transform-domain collaborative filtering, SPIE Electronic Imaging, 2008.

[13]

A. DanielyanV. Katkovnik and K. Egiazarian, BM3D frames and variational image deblurring, IEEE Trans. Image Process., 21 (2012), 1715-1728.  doi: 10.1109/TIP.2011.2176954.

[14]

E. DiBenedettoU. Gianazza and V. Vespri, Harnack estimates for quasi-linear degenerate parabolic differential equations, Acta Math., 200 (2008), 181-209.  doi: 10.1007/s11511-008-0026-3.

[15]

E. DiBenedettoU. Gianazza and V. Vespri, Liouville-type theorems for certain degenerate and singular parabolic equations, C. R. Math. Acad. Sci. Paris., 348 (2010), 873-877.  doi: 10.1016/j.crma.2010.06.019.

[16]

E. DiBenedetto, Degenerate Parabolic Equations, Universitext, Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4612-0895-2.

[17]

F. F. Dong and Y. M. Chen, A fractional-order derivative based variational framework for image denoising, Inverse Probl. Imaging., 10 (2016), 27-50.  doi: 10.3934/ipi.2016.10.27.

[18]

M. Elad and M. A. T. Figueiredo, On the role of sparse and redundant representations in image processing, Proc. IEEE., 98 (2010), 972-982. 

[19]

G. Gilboa and S. Osher, Nonlocal operators with applications to image processing, Multiscale Model. Simul., 7 (2008), 1005-1028.  doi: 10.1137/070698592.

[20]

P. GuidottiY. Kim and J. Lambers, Image restoration with a new class of forward-backward-forward diffusion equations of Perona-Malik type with applications to satellite image enhancement, SIAM J. Imaging Sci., 6 (2013), 1416-1444.  doi: 10.1137/120882895.

[21]

P. Guidotti, A new nonlocal nonlinear diffusion of image processing, J. Differ. Equ., 246 (2009), 4731-4742.  doi: 10.1016/j.jde.2009.03.017.

[22]

P. Guidotti and J. V. Lambers, Two new nonlinear nonlocal diffusions for noise reduction, J. Math. Imaging Vis., 33 (2009), 25-37.  doi: 10.1007/s10851-008-0108-z.

[23]

O. Honigman and Y. Y. Zeevi, Enhancement of textured images using complex diffusion incorporating Schrodinger's Potential, Proc. IEEE Int. Conf. Acoust. Speech Signal Process, 26 (2006), 633-636.  doi: 10.1109/ICASSP.2006.1660422.

[24]

Y. M. HuangNg. Michael K and Y. W. Wen, A fast total variation minimization method for image restoration, Multiscale Model. Simul., 7 (2008), 774-795.  doi: 10.1137/070703533.

[25]

M. JanevS. PilipovićT. AtanackovićR. Obradović and N. Ralević, Fully fractional anisotropic diffusion for image denoising, Math. Comput. Model., 54 (2011), 729-741.  doi: 10.1016/j.mcm.2011.03.017.

[26]

D. Kundur and D. Hatzinakos, Blind image deconvolution, IEEE Signal Process. Mag., 13 (1996), 43-64.  doi: 10.1109/79.489268.

[27]

X. Li, Fine-granularity and spatially-adaptive regularization for projection-based image deblurring, IEEE Trans. Image Process., 20 (2011), 971-983.  doi: 10.1109/TIP.2010.2081681.

[28]

J. LiuZ. HuanH. Huang and H. Zhang, An adaptive method for recovering image from mixed noisy data, Int. J. Comput. Vis., 85 (2009), 182-191.  doi: 10.1007/s11263-009-0254-9.

[29]

C. Louchet and L. Moisan, Total variation as a Local Filter, SIAM J. Imaging Sci., 4 (2011), 651-694.  doi: 10.1137/100785855.

[30]

M. Meerschaert and C. Tadjeran, Finite difference approximations for fractional advection-dispersion flow equations, J. Comput. Appl. Math., 172 (2004), 65-77.  doi: 10.1016/j.cam.2004.01.033.

[31]

V. Papyan and M. Elad, Multi-scale patch-based image restoration, IEEE Trans. Image Processing, 25 (2016), 249-261.  doi: 10.1109/TIP.2015.2499698.

[32]

P. Perona and J. Malik, Scale-space and edge detection using anisotropic diffusion, IEEE Trans. Pattern Anal. Machine Intell., 12 (1990), 629-639.  doi: 10.1109/34.56205.

[33] I. Podlubny, Fractional Differential Equations, An introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications, Mathematics in Science and Engineering, 198, Academic Press, Inc., San Diego, CA, 1999. 
[34]

Y. Pu et al., Fractional partial differential equation denoising models for texture image, Sci. China Inf. Sci., 57 (2014), 072115, 19 pp. doi: 10.1007/s11432-014-5112-x.

[35]

Y. Pu, Research on Application of Fractional Calculus to Latest Signal Analysis and Processing, Ph. D. dissertation, School Electron. Inf., Sichuan Univ., Sichuan Province, China, 2006.

[36]

P. D. Romero and V. F. Candela, Mathematical models for restoration of Baroque paintings, Lecture Notes in Computer Science, Springer, Berlin, 4179 (2006), 24–34. doi: 10.1007/11864349_3.

[37]

P. D. Romero and V. F. Candela, Blind deconvolution models regularized by fractional powers of the Laplacian, J. Math. Imaging Vis., 32 (2008), 181-191.  doi: 10.1007/s10851-008-0093-2.

[38]

F. Rousseau, A non-local approach for image super-resolution using intermodality priors, Med. Image. Anal., 14 (2010), 594-605.  doi: 10.1016/j.media.2010.04.005.

[39]

L. RudinS. Osher and E. Fatemi, Nonlinear total variation based noise removal algorithms, Experimental mathematics: Computational issues in nonlinear science (Los Alamos, NM, 1991), Phys. D., 60 (1992), 259-268.  doi: 10.1016/0167-2789(92)90242-F.

[40]

L. Rudin and S. Osher, Total variation based image restoration with free local constraints, Proceedings of the 1st IEEE International Conference on Image Processing, 1 (1994), 31–35. doi: 10.1109/ICIP.1994.413269.

[41]

M. E. Taylor, Partial Differential Equations II - Qualitative Studies of Linear Equations, Second edition. Applied Mathematical Sciences, 116. Springer, New York, 2011. doi: 10.1007/978-1-4419-7052-7.

[42] M. E. Taylor, Pseudodifferential Operators, Princeton Mathematical Series, 34, Princeton University Press, Princeton, N.J., 1981. 
[43]

M. E. Taylor, Partial Differential Equations I - Basic Theory, Second edition, Applied Mathematical Sciences, 115, Springer, New York, 2011. doi: 10.1007/978-1-4419-7055-8.

[44]

A. Tikhonov and V. Arsenin, Solution of Ill-Posed Problems, Translated from the Russian, Preface by translation editor Fritz John. Scripta Series in Mathematics, New York-Toronto, Ont.-London, 1977.

[45]

M. WelkD. TheisT. Brox and J. Weickert, PDE-based deconvolution with forward-backward diffusivities and diffusion tensors, Scale Space PDE Methods Comput. Vis., 3459 (2005), 585-597.  doi: 10.1007/11408031_50.

[46]

B. M. WilliamsJ. Zhang and K. Chen, A new image deconvolution method with fractional regularization, J. Algorithms Comput. Technol., 10 (2016), 265-276.  doi: 10.1177/1748301816660439.

[47]

A. P. Witkin, Scale-space filtering, Proc. Eighth International Joint Conference on Artificial Intelligenc., Karlsruhe, West Germany, 2 (1987) 329–332. doi: 10.1016/B978-0-08-051581-6.50036-2.

[48]

M. Xu, J. Yang, D. Zhao and H. Zhao, An image-enhancement method based on variable-order fractional differential operators, Bio-Med. Mater. Eng., 26 (2015), S1325–S1333. doi: 10.3233/BME-151430.

[49]

X. YinS. Zhou and M. A. Siddique, Fractional nonlinear anisotropic diffusion with p-laplace variation method for image restoration, Multimedia Tools and Applications, 75 (2016), 4505-4526.  doi: 10.1007/s11042-015-2488-6.

[50]

X. Q. ZhangM. BurgerX. Bresson and S. Osher, Bregmanized nonlocal regularization for deconvolution and sparse reconstruction, SIAM J. Imaging Sci., 3 (2010), 253-276.  doi: 10.1137/090746379.

[51]

J. ZhangZ. Wei and L. Xiao, Adaptive fractional-order multi-scale method for image denoising, J. Math. Imaging Vis., 43 (2012), 39-49.  doi: 10.1007/s10851-011-0285-z.

[52]

J. Zhang and K. Chen, A total fractional-order variation model for image restoration with nonhomogeneous boundary conditions and its numerical solution, SIAM J. Imaging Sci., 8 (2015), 2487-2518.  doi: 10.1137/14097121X.

[53]

Z. ZhouZ. GuoG. DongJ. SunD. Zhang and B. Wu, A doubly degenerate diffusion model based on the gray level indicator for multiplicative noise removal, IEEE Trans. Image Process., 24 (2015), 249-260.  doi: 10.1109/TIP.2014.2376185.

Figure 1.  Blurry image and the deblurred results
Figure 2.  Comparison results: (a) Original signal (b) Result of fractional model (c) Result of integer model (d)-(f) Results in [0.44, 0.56] from (a)-(c)
Figure 3.  Test images: (a) Barbara ("$ 300 \times 300 $"), (b) Boat ("$ 512\times 512 $"), (c) Aerial ("$ 512\times 512 $"), (d) Chimpanzee ("$ 256 \times 256 $"), (e) Synthetic ("$ 256 \times 256 $"), (f) Cameraman ("$ 256 \times 256 $")
Figure 4.  The images in the first row are blurred images with different kernel: (a) Scenario Ⅰ, (b) Scenario Ⅱ, (c) Scenario Ⅲ, (d) Scenario Ⅳ, (e) Scenario Ⅴ. The images in the second row are the corresponding results using our method
Figure 5.  The images in the first row are blurred images with different kernel: (a) Scenario Ⅰ, (b) Scenario Ⅱ, (c) Scenario Ⅲ, (d) Scenario Ⅳ, (e) Scenario Ⅴ. The images in the second row are the corresponding results using our method
Figure 6.  The images in the first row are blurred images with different kernel: (a) Scenario Ⅰ, (b) Scenario Ⅱ, (c) Scenario Ⅲ, (d) Scenario Ⅳ, (e) Scenario Ⅴ. The images in the second row are the corresponding results using our method
Figure 7.  The images in the first row are blurred images with different kernel: (a) Scenario Ⅰ, (b) Scenario Ⅱ, (c) Scenario Ⅲ, (d) Scenario Ⅳ, (e) Scenario Ⅴ. The images in the second row are the corresponding results using our method
Figure 8.  The images in the first row are recovered by the proposed model with different $ k_1 $: (a) $ k_1 = 5 $, (b) $ k_1 = 10 $, (c) $ k_1 = 50 $, (d) $ k_1 = 100 $. The images in the second row are the corresponding results using variational method with: (e) $ k_1 = 5 $, (f) $ k_1 = 10 $, (g) $ k_1 = 50 $, (h) $ k_1 = 100 $
Figure 9.  Deblurring of the Barbara image, scenario Ⅱ. From left to right: blurred, reconstructed by FastTVMM [24], WFBM [5], our method
Figure 10.  Deblurring of the Barbara image, scenario Ⅲ. From left to right: blurred, reconstructed by FastTVMM [24], WFBM [5], our method
Figure 11.  Deblurring of the Aerial image, scenario Ⅱ. From left to right: blurred, reconstructed by FastTVMM [24], WFBM [5], our method
Figure 12.  Deblurring of the Aerial image, scenario Ⅲ. From left to right: blurred, reconstructed by FastTVMM [24], WFBM [5], our method
Figure 13.  The original Barbara image and Aerial image, the marked rectangles are marked for zooming
Figure 14.  Zooming regions
Figure 15.  Deblurring of the Barbara image, scenario Ⅱ. From left to right, zoomed fragments of the following images are presented: blurred, reconstructed by FastTVMM [24], WFBM [5], our method
Figure 16.  Deblurring of the Barbara image, scenario Ⅱ. From left to right, zoomed fragments of the following images are presented: blurred, reconstructed by FastTVMM [24], WFBM [5], our method
Figure 17.  Deblurring of the Barbara image, scenario Ⅲ. From left to right, zoomed fragments of the following images are presented: blurred, reconstructed by FastTVMM [24], WFBM [5], our method
Figure 18.  Deblurring of the Barbara image, scenario Ⅲ. From left to right, zoomed fragments of the following images are presented: blurred, reconstructed by FastTVMM [24], WFBM [5], our method
Figure 19.  Deblurring of the Aerial image, scenario Ⅱ. From left to right, zoomed fragments of the following images are presented: blurred, reconstructed by FastTVMM [24], WFBM [5], our method
Figure 20.  Deblurring of the Aerial image, scenario Ⅲ. From left to right, zoomed fragments of the following images are presented: blurred, reconstructed by FastTVMM [24], WFBM [5], our method
Figure 21.  Top row: Noisy blurred images. Bottom row: Recovered images corresponding to the above images. Left two columns: scenario Ⅴ and noise levels $ \sigma $ = 2, 5, respectively. Right two columns: scenario Ⅱ and noise levels $ \sigma $ = 2, 5, respectively
Figure 22.  Top row: Noisy blurred images with scenario Ⅰ and noise levels $ \sigma = 2 $ (left two columns), $ \sigma = 5 $ (right two columns). Bottom row: Recovered images corresponding to the above images
Figure 23.  Left to right: Noisy blurred image (scenario Ⅰ and $ \sigma = 2 $), recovered image from FastTVMM [24], MSPB [31], WFBM [5], BM3D [11], and recovered image with proposed model
Figure 24.  Deblurring results for the Synthetic blurred with scenario Ⅳ and corrupted by noise of standard deviation $ \sigma = 3 $. From left to right: original image; noisy blurred image; MSPB [31] (PSNR = 28.41, elapsed time = 202 seconds) and proposed model (PSNR = 30.39, elapsed time = 6 seconds)
Table 1.  The five different types of blurring kernels used in the experiments
Scenario PSF
[1 4 6 4 1]$ ^T $[1 4 6 4 1]/256
$ 15\times 15 $ Gaussian PSF with standard deviation 1.5
fspecial('motion', 20, 45) is used in Matlab
fspecial('disk', 3) is used in Matlab
[1 1 1 1 1]$ ^T $[1 1 1 1 1]/25
Scenario PSF
[1 4 6 4 1]$ ^T $[1 4 6 4 1]/256
$ 15\times 15 $ Gaussian PSF with standard deviation 1.5
fspecial('motion', 20, 45) is used in Matlab
fspecial('disk', 3) is used in Matlab
[1 1 1 1 1]$ ^T $[1 1 1 1 1]/25
Table 2.  Values of PSNR for the restoration by using the proposed model with $ \alpha = 0.9 $, $ k_1 = 1 $, $ \beta = 1 $, $ \lambda = 5 $
Images $ r=0 $ $ r=0.5 $ $ r=1 $ $ r=1.5 $ $ r=2 $
Barbara 40.24 40.24 40.22 40.20 40.18
Boat 45.95 46.26 46.41 46.50 46.57
Aerial 44.15 44.32 44.38 44.42 44.45
Chimpanzee 44.46 44.46 44.48 44.49 44.51
Synthetic 27.11 27.08 27.06 27.05 27.03
Cameraman 42.45 42.57 42.58 42.58 42.58
Images $ r=0 $ $ r=0.5 $ $ r=1 $ $ r=1.5 $ $ r=2 $
Barbara 40.24 40.24 40.22 40.20 40.18
Boat 45.95 46.26 46.41 46.50 46.57
Aerial 44.15 44.32 44.38 44.42 44.45
Chimpanzee 44.46 44.46 44.48 44.49 44.51
Synthetic 27.11 27.08 27.06 27.05 27.03
Cameraman 42.45 42.57 42.58 42.58 42.58
Table 3.  PSNR values for the experiments shown in Figures 2122
PSNR (noisy) PSNR(recovered)
Figure 21Left two columns$\sigma=2$38.6342.55
$\sigma=5$38.5341.23
Right two columns$\sigma=2$39.8141.74
$\sigma=5$39.6740.55
Figure 22Left two columns$\sigma=2$41.2744.51
$\sigma=2$42.8146.73
Right two columns$\sigma=5$41.0942.63
$\sigma=5$42.6745.03
PSNR (noisy) PSNR(recovered)
Figure 21Left two columns$\sigma=2$38.6342.55
$\sigma=5$38.5341.23
Right two columns$\sigma=2$39.8141.74
$\sigma=5$39.6740.55
Figure 22Left two columns$\sigma=2$41.2744.51
$\sigma=2$42.8146.73
Right two columns$\sigma=5$41.0942.63
$\sigma=5$42.6745.03
Table 4.  Recovery experiment (scenario Ⅰ and additive noise $ \sigma = 2 $). Figure 23 was obtained using different models
Methods PSNR Time(s)
FastTVMM in [24] 42.91 0.44
MSPB in [31] 43.21 220.12
WFBM in [5] 42.32 19.72
BM3D in [11] 42.98 60.85
Proposed model 42.57 1.45
Methods PSNR Time(s)
FastTVMM in [24] 42.91 0.44
MSPB in [31] 43.21 220.12
WFBM in [5] 42.32 19.72
BM3D in [11] 42.98 60.85
Proposed model 42.57 1.45
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