American Institute of Mathematical Sciences

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April  2017, 11(2): 403-426. doi: 10.3934/ipi.2017019

Some novel linear regularization methods for a deblurring problem

Xiangtuan Xiong *,, , Jinmei Li and  Jin Wen

Department of Mathematics, Northwest Normal University, Lanzhou, Gansu 730070, China

* Corresponding author: xiongxt@gmail.com; xiongxt@fudan.edu.cn.

*The first author is supported by NSF grant of China 11661072

Received  November 2012 Revised  November 2016 Published  March 2017

In this article, we consider a fractional backward heat conduction problem (BHCP) in the two-dimensional space which is associated with a deblurring problem. It is well-known that the classical Tikhonov method is the most important regularization method for linear ill-posed problems. However, the classical Tikhonov method over-smooths the solution. As a remedy, we propose two quasi-boundary regularization methods and their variants. We prove that these two methods are better than Tikhonov method in the absence of noise in the data. Deblurring experiment is conducted by comparing with some classical linear filtering methods for BHCP and the total variation method with the proposed methods.

Keywords: Ill-posed problems, image deblurring, regularization, error estimate.
Mathematics Subject Classification: Primary: 35S10, 65M12; Secondary: 65M32.
Citation: Xiangtuan Xiong, Jinmei Li, Jin Wen. Some novel linear regularization methods for a deblurring problem. Inverse Problems & Imaging, 2017, 11 (2) : 403-426. doi: 10.3934/ipi.2017019
References:
[1]

K. A. Ames, W. C. Gordon, J. F. Epperson and S. F. Oppenhermer, A Comparison of Regularizations for an Ill-Posed Problem, Math. Comput., 67 (1998), 1451-1471.  doi: 10.1090/S0025-5718-98-01014-X.  Google Scholar

[2]

E. O. Brigham, The Fast Fourier Transform, Prentice-Hall, Englewood Cliffs, NJ, 1974. Google Scholar

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A. Carasso, Error bounds in the final value problem for the heat equation, SIAM J. Math. Anal., 7 (1976), 195-199.  doi: 10.1137/0507015.  Google Scholar

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A. S. Carasso, Image restoration and diffusion processes, SPIE Proceedings, 2035 (1993), 255-266.   Google Scholar

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A. S. Carasso, Overcoming Hölder continuity in ill-posed continuation problems, SIAM J. Numer. Anal., 31 (1994), 1535-1557.  doi: 10.1137/0731080.  Google Scholar

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A. S. Carasso, The APEX method in image sharpening and the use of low exponent Lévy stable laws, SIAM J. Appl. Math., 63 (2002), 593-618.  doi: 10.1137/S0036139901389318.  Google Scholar

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A. S. Carasso, Singluar integrals, image smoothness, and the reconvery of texture in image deblurring, SIAM J. Appl. Math., 64 (2004), 1749-1774.  doi: 10.1137/S0036139903428306.  Google Scholar

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A. Chambolle, An algorithm for total variation minimization and applications, J. Math. Imaging and Vision, 20 (2004), 89-97.  doi: 10.1023/B:JMIV.0000011321.19549.88.  Google Scholar

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T. F. Chan and C. K. Wong, Total variation blind deconvolution, IEEE Trans. Image Process, 7 (1998), 370-375.  doi: 10.1109/83.661187.  Google Scholar

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R. H. Chan, A. Lanza, S. Morigi and F. Sgallari, An adaptive strategy for the restoration of textured images using fractional order regularization, Numer. Math. Theor. Meth. Appl., 6 (2013), 276-296.   Google Scholar

[11]

R. H. Chan and K. Chen, A multilevel algorithm for simultaneously denoising and deblurring images, SIAM J. Sci. Comput., 32 (2010), 1043-1063.  doi: 10.1137/080741410.  Google Scholar

[12]

R. R. Coifman and A. Sowa, Combining the calculus of variations and wavelets for image enhancement, Appl. Comput. Harmon. Anal., 9 (2000), 1-18.  doi: 10.1006/acha.2000.0299.  Google Scholar

[13]

I. Daubechies and G. Teschke, Variational image restoration by means of wavelets: Simultaneous decomposition, deblurring, and denoising, Appl. Comput. Harmon. Anal., 19 (2005), 1-16.  doi: 10.1016/j.acha.2004.12.004.  Google Scholar

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Y. Gousseau and J. M. Morel, Are natural images of bounded variation?, SIAM J. Math. Anal., 33 (2001), 634-648.  doi: 10.1137/S0036141000371150.  Google Scholar

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M. E. Hochstenbach and L. Reichel, Fractional Tikhonov regularization for linear discrete Ill-posed problems, BIT, 51 (2011), 197-215.  doi: 10.1007/s10543-011-0313-9.  Google Scholar

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Y. Huang and M. Ng, Lipschitz and Total-Variational Regularization for Blind Deconvolution, Communications in Computational Physics, 4 (2008), 195-206.   Google Scholar

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V. Isakov, Inverse Problems for Partial Differential ations, Springer-Verlag, New York, 1998. doi: 10.1007/978-1-4899-0030-2.  Google Scholar

[23]

H. Ji, J. Li, Z. Shen and K. Wang, Image deconvolution using a characterization of sharp images in wavelet domain, Appl. Comput. Harmon. Anal., 32 (2012), 295-304.  doi: 10.1016/j.acha.2011.09.006.  Google Scholar

[24]

M. Jourhmane and N. S. Mera, An iterative algorithm for the backward heat conduction problem based on variable relaxtion factors, Inverse Probl. in Engn., 10 (2002), 293-308.   Google Scholar

[25]

L. S. G. Kovasznay and H. M. Joseph, Image Processing, Proc. IRE., 43 (1955), 560-570.  doi: 10.1109/JRPROC.1955.278100.  Google Scholar

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R. Lattes and J. L. Lions, Methode de Quasi-Reversibility et Applications, Dunod, Paris, 1967 (English translation R. Bellman, Elsevier, New York, 1969).  Google Scholar

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M. M. Lavrentév, V. G. Romanov and S. P. Shishat·sk1iĭ, Ill-posed Problems of Mathematical Physics and Analysis, A M S, Providence, Rhode Island, 1986.  Google Scholar

[28]

J. Lee and D. Sheen, F. John's stability conditions versus A. Carasso's SECB constraint for backward parabolic problems Inverse Probl., 25 (2009), 055001, 12pp. doi: 10.1088/0266-5611/25/5/055001.  Google Scholar

[29]

M. Li and X. Xiong, On a fractional backward heat conduction problem: Application to deblurring, Comput. Math. Appl., 64 (2012), 2594-2602.  doi: 10.1016/j.camwa.2012.07.003.  Google Scholar

[30]

C. S. Liu, Group preserving scheme for backward heat conduction problems, International Journal of Heat and Mass Transfer, 47 (2004), 2567-2576.  doi: 10.1016/j.ijheatmasstransfer.2003.12.019.  Google Scholar

[31]

F. Malgouyres, A framework for image deblurring using wavelet packet bases, Appl. Comput. Harmon. Anal., 12 (2002), 309-331.  doi: 10.1006/acha.2002.0379.  Google Scholar

[32]

N. S. Mera, L. Elliott, D. B. Ingham and D. Lesnic, An iterative boundary element method for solving the one dimensional backward heat conduction problem, International Journal of Heat and Mass Transfer, 44 (2001), 1937-1946.  doi: 10.1016/S0017-9310(00)00235-0.  Google Scholar

[33]

K. Miller, Stabilized quasireversibility and other nearly best possible methods for non-well-posed problems, Symposium on Non-Well-Posed Problems and Logarithmic Convexity, Lecture Notes in Mathematics, 316 (1973), 161-176.   Google Scholar

[34]

S. M. Mohapatra, Transfer function measurement and analysis for a magnetic resonance imager, Medical Physics, 18 (1991), 1141-1144.  doi: 10.1118/1.596622.  Google Scholar

[35]

L. E. Payne, Improperly Posed Problems in Partial Differential ations, SIAM, PHILADELPHIA, 1975.  Google Scholar

[36]

K. S. Pentlow, Quantitative imaging of I-124 using positron emission tomography with applications to radioimmunodiagnosis and radioimmunotheraphy, Medical Physics, 18 (1991), 357-366.   Google Scholar

[37]

W. K. Pratt, Digital Image Processing, Second ed., John Wiley, New York, 1991. Google Scholar

[38]

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

[39]

T. I. Seidman, Optimal filtering for the backward heat equation, SIAM J. Numer. Anal., 33 (1996), 162-170.  doi: 10.1137/0733010.  Google Scholar

[40]

R. E. Showalter, The final value problem for evolution equations, J. Math. Anal. Appl., 47 (1974), 563-572.  doi: 10.1016/0022-247X(74)90008-0.  Google Scholar

[41]

U. Tautenhahn and T. Schröter, On optimal regularization methods for the backward heat equation, Z. Anal. Anw., 15 (1996), 475-493.  doi: 10.4171/ZAA/711.  Google Scholar

[42]

U. Tautenhahn, Optimality for linear ill-posed problems under general source conditions, Numer. Funct. Anal. Optim., 19 (1998), 377-398.  doi: 10.1080/01630569808816834.  Google Scholar

[43]

A. N. Tikhonov and V. Y. Arsenin, Solutions of Ill-Posed Problems, Winston and Sons, Washington, D. C., 1977.  Google Scholar

[44]

X. T. Xiong, A regularization method for a Cauchy problem of the Helmholtz equation, J. Comput. Appl. Math., 233 (2010), 1723-1732.  doi: 10.1016/j.cam.2009.09.001.  Google Scholar

[45]

X. T. Xiong, J. X. Wang and M. Li, An optimal method for fractional heat conduction problem backward in time, Appl. Anal., 91 (2012), 823-840.  doi: 10.1080/00036811.2011.601455.  Google Scholar

[46]

H. T. Yura, Imaging in clear ocean water, Applied Optics, 12 (1973), 1061-1066.  doi: 10.1364/AO.12.001061.  Google Scholar

[47]

X. Zhao, F. Wang and M. Ng, A new convex optimization model for multiplicative noise and blur removal, SIAM Journal on Imaging Sciences, 7 (2014), 456-475.  doi: 10.1137/13092472X.  Google Scholar

[48]

G. H. Zheng and T. Wei, Two regularization methods for solving a Riesz-Feller space-fractional backward diffusion problem Inverse Probl., 26 (2010), 115017, 22pp. doi: 10.1088/0266-5611/26/11/115017.  Google Scholar

show all references

References:
[1]

K. A. Ames, W. C. Gordon, J. F. Epperson and S. F. Oppenhermer, A Comparison of Regularizations for an Ill-Posed Problem, Math. Comput., 67 (1998), 1451-1471.  doi: 10.1090/S0025-5718-98-01014-X.  Google Scholar

[2]

E. O. Brigham, The Fast Fourier Transform, Prentice-Hall, Englewood Cliffs, NJ, 1974. Google Scholar

[3]

A. Carasso, Error bounds in the final value problem for the heat equation, SIAM J. Math. Anal., 7 (1976), 195-199.  doi: 10.1137/0507015.  Google Scholar

[4]

A. S. Carasso, Image restoration and diffusion processes, SPIE Proceedings, 2035 (1993), 255-266.   Google Scholar

[5]

A. S. Carasso, Overcoming Hölder continuity in ill-posed continuation problems, SIAM J. Numer. Anal., 31 (1994), 1535-1557.  doi: 10.1137/0731080.  Google Scholar

[6]

A. S. Carasso, The APEX method in image sharpening and the use of low exponent Lévy stable laws, SIAM J. Appl. Math., 63 (2002), 593-618.  doi: 10.1137/S0036139901389318.  Google Scholar

[7]

A. S. Carasso, Singluar integrals, image smoothness, and the reconvery of texture in image deblurring, SIAM J. Appl. Math., 64 (2004), 1749-1774.  doi: 10.1137/S0036139903428306.  Google Scholar

[8]

A. Chambolle, An algorithm for total variation minimization and applications, J. Math. Imaging and Vision, 20 (2004), 89-97.  doi: 10.1023/B:JMIV.0000011321.19549.88.  Google Scholar

[9]

T. F. Chan and C. K. Wong, Total variation blind deconvolution, IEEE Trans. Image Process, 7 (1998), 370-375.  doi: 10.1109/83.661187.  Google Scholar

[10]

R. H. Chan, A. Lanza, S. Morigi and F. Sgallari, An adaptive strategy for the restoration of textured images using fractional order regularization, Numer. Math. Theor. Meth. Appl., 6 (2013), 276-296.   Google Scholar

[11]

R. H. Chan and K. Chen, A multilevel algorithm for simultaneously denoising and deblurring images, SIAM J. Sci. Comput., 32 (2010), 1043-1063.  doi: 10.1137/080741410.  Google Scholar

[12]

R. R. Coifman and A. Sowa, Combining the calculus of variations and wavelets for image enhancement, Appl. Comput. Harmon. Anal., 9 (2000), 1-18.  doi: 10.1006/acha.2000.0299.  Google Scholar

[13]

I. Daubechies and G. Teschke, Variational image restoration by means of wavelets: Simultaneous decomposition, deblurring, and denoising, Appl. Comput. Harmon. Anal., 19 (2005), 1-16.  doi: 10.1016/j.acha.2004.12.004.  Google Scholar

[14]

D. L. Donoho, Nonlinear Solution of Linear Inverse Problems by Wavelet-Vaguelette Decomposition, Appl. Comput. Harmon. Anal., 2 (1995), 101-126.  doi: 10.1006/acha.1995.1008.  Google Scholar

[15]

H. W. Engl, M. Hanke and A. Neubauer, Regularization of Inverse Problems, Kluwer Academic Publisher, Dordrecht Boston London, 1996. doi: 10.1007/978-94-009-1740-8.  Google Scholar

[16]

R. C. Gonzalez and P. Wintz, Digital Image Processing, Reading, Mass. -London-Amsterdam, 1977.  Google Scholar

[17]

Y. Gousseau and J. M. Morel, Are natural images of bounded variation?, SIAM J. Math. Anal., 33 (2001), 634-648.  doi: 10.1137/S0036141000371150.  Google Scholar

[18]

P. C. Hansen, J. G. Nagy and D. P. O'Leary, Deblurring Images Matrices, Spectra, and Filtering, Fundamentals of Algorithms, SIAM, Philadelphia, 2006. doi: 10.1137/1.9780898718874.  Google Scholar

[19]

D. N. Háo, A mollification method for ill-posed problems, Numer. Math., 68 (1994), 469-506.  doi: 10.1007/s002110050073.  Google Scholar

[20]

M. E. Hochstenbach and L. Reichel, Fractional Tikhonov regularization for linear discrete Ill-posed problems, BIT, 51 (2011), 197-215.  doi: 10.1007/s10543-011-0313-9.  Google Scholar

[21]

Y. Huang and M. Ng, Lipschitz and Total-Variational Regularization for Blind Deconvolution, Communications in Computational Physics, 4 (2008), 195-206.   Google Scholar

[22]

V. Isakov, Inverse Problems for Partial Differential ations, Springer-Verlag, New York, 1998. doi: 10.1007/978-1-4899-0030-2.  Google Scholar

[23]

H. Ji, J. Li, Z. Shen and K. Wang, Image deconvolution using a characterization of sharp images in wavelet domain, Appl. Comput. Harmon. Anal., 32 (2012), 295-304.  doi: 10.1016/j.acha.2011.09.006.  Google Scholar

[24]

M. Jourhmane and N. S. Mera, An iterative algorithm for the backward heat conduction problem based on variable relaxtion factors, Inverse Probl. in Engn., 10 (2002), 293-308.   Google Scholar

[25]

L. S. G. Kovasznay and H. M. Joseph, Image Processing, Proc. IRE., 43 (1955), 560-570.  doi: 10.1109/JRPROC.1955.278100.  Google Scholar

[26]

R. Lattes and J. L. Lions, Methode de Quasi-Reversibility et Applications, Dunod, Paris, 1967 (English translation R. Bellman, Elsevier, New York, 1969).  Google Scholar

[27]

M. M. Lavrentév, V. G. Romanov and S. P. Shishat·sk1iĭ, Ill-posed Problems of Mathematical Physics and Analysis, A M S, Providence, Rhode Island, 1986.  Google Scholar

[28]

J. Lee and D. Sheen, F. John's stability conditions versus A. Carasso's SECB constraint for backward parabolic problems Inverse Probl., 25 (2009), 055001, 12pp. doi: 10.1088/0266-5611/25/5/055001.  Google Scholar

[29]

M. Li and X. Xiong, On a fractional backward heat conduction problem: Application to deblurring, Comput. Math. Appl., 64 (2012), 2594-2602.  doi: 10.1016/j.camwa.2012.07.003.  Google Scholar

[30]

C. S. Liu, Group preserving scheme for backward heat conduction problems, International Journal of Heat and Mass Transfer, 47 (2004), 2567-2576.  doi: 10.1016/j.ijheatmasstransfer.2003.12.019.  Google Scholar

[31]

F. Malgouyres, A framework for image deblurring using wavelet packet bases, Appl. Comput. Harmon. Anal., 12 (2002), 309-331.  doi: 10.1006/acha.2002.0379.  Google Scholar

[32]

N. S. Mera, L. Elliott, D. B. Ingham and D. Lesnic, An iterative boundary element method for solving the one dimensional backward heat conduction problem, International Journal of Heat and Mass Transfer, 44 (2001), 1937-1946.  doi: 10.1016/S0017-9310(00)00235-0.  Google Scholar

[33]

K. Miller, Stabilized quasireversibility and other nearly best possible methods for non-well-posed problems, Symposium on Non-Well-Posed Problems and Logarithmic Convexity, Lecture Notes in Mathematics, 316 (1973), 161-176.   Google Scholar

[34]

S. M. Mohapatra, Transfer function measurement and analysis for a magnetic resonance imager, Medical Physics, 18 (1991), 1141-1144.  doi: 10.1118/1.596622.  Google Scholar

[35]

L. E. Payne, Improperly Posed Problems in Partial Differential ations, SIAM, PHILADELPHIA, 1975.  Google Scholar

[36]

K. S. Pentlow, Quantitative imaging of I-124 using positron emission tomography with applications to radioimmunodiagnosis and radioimmunotheraphy, Medical Physics, 18 (1991), 357-366.   Google Scholar

[37]

W. K. Pratt, Digital Image Processing, Second ed., John Wiley, New York, 1991. Google Scholar

[38]

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

[39]

T. I. Seidman, Optimal filtering for the backward heat equation, SIAM J. Numer. Anal., 33 (1996), 162-170.  doi: 10.1137/0733010.  Google Scholar

[40]

R. E. Showalter, The final value problem for evolution equations, J. Math. Anal. Appl., 47 (1974), 563-572.  doi: 10.1016/0022-247X(74)90008-0.  Google Scholar

[41]

U. Tautenhahn and T. Schröter, On optimal regularization methods for the backward heat equation, Z. Anal. Anw., 15 (1996), 475-493.  doi: 10.4171/ZAA/711.  Google Scholar

[42]

U. Tautenhahn, Optimality for linear ill-posed problems under general source conditions, Numer. Funct. Anal. Optim., 19 (1998), 377-398.  doi: 10.1080/01630569808816834.  Google Scholar

[43]

A. N. Tikhonov and V. Y. Arsenin, Solutions of Ill-Posed Problems, Winston and Sons, Washington, D. C., 1977.  Google Scholar

[44]

X. T. Xiong, A regularization method for a Cauchy problem of the Helmholtz equation, J. Comput. Appl. Math., 233 (2010), 1723-1732.  doi: 10.1016/j.cam.2009.09.001.  Google Scholar

[45]

X. T. Xiong, J. X. Wang and M. Li, An optimal method for fractional heat conduction problem backward in time, Appl. Anal., 91 (2012), 823-840.  doi: 10.1080/00036811.2011.601455.  Google Scholar

[46]

H. T. Yura, Imaging in clear ocean water, Applied Optics, 12 (1973), 1061-1066.  doi: 10.1364/AO.12.001061.  Google Scholar

[47]

X. Zhao, F. Wang and M. Ng, A new convex optimization model for multiplicative noise and blur removal, SIAM Journal on Imaging Sciences, 7 (2014), 456-475.  doi: 10.1137/13092472X.  Google Scholar

[48]

G. H. Zheng and T. Wei, Two regularization methods for solving a Riesz-Feller space-fractional backward diffusion problem Inverse Probl., 26 (2010), 115017, 22pp. doi: 10.1088/0266-5611/26/11/115017.  Google Scholar

Figure 1.  (left): the original image $f(x,y)$, (right): the blurred image $g(x,y)$.
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Figure 2.  Comparison of the quasi-boundary methods and Tikhonov method.(A): quasi-boundary method. (B): modified quasi-boundary method. (C): Tikhonov method.
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Figure 3.  (A): the original image $f(x,y)$, (B): the blurred image $g(x,y)$.
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Figure 4.  Comparison of the quasi-boundary methods and Tikhonov method.(A): quasi-boundary method. (B): modified quasi-boundary method. (C): Tikhonov method.
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Figure 5.  Zooming on the partial region in Figure 4. (A): quasi-boundary method.(B): modified quasi-boundary method. (C): Tikhonov method.
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Figure 6.  Profiles for the convergence rates of quasi-boundary methods.
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Figure 7.  Comparison of different deblurring methods. (A) the original image. (B) the blurred and noisy image. (C)the true Wiener filtering method.
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Figure 8.  Comparison of different deblurring methods. (A) the SECB method. (B) the backward beam method. (C)the Tikhonov method.
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Figure 9.  Comparison of different deblurring methods. (A) the modified quasi-boundary method. (B) the Spectral method 2. (C)the convolution-type method.
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Figure 10.  Comparison of different deblurring methods. (A) the TV method. (B) the FQBM. (C)the FMQBM.
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Table 1.  behavior of PSNR for the three methods.
MethodPSNR
Quasi-boundary regularization16.5
Modified quasi-boundary regularization15.9
Tikhonov14.7
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MethodPSNR
Quasi-boundary regularization16.5
Modified quasi-boundary regularization15.9
Tikhonov14.7
Table 2.  behavior of PSNR for the three methods.
MethodPSNR
Quasi-boundary regularization30.0
Modified quasi-boundary regularization29.1
Tikhonov25.2
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MethodPSNR
Quasi-boundary regularization30.0
Modified quasi-boundary regularization29.1
Tikhonov25.2
Table 3.  behavior in the Lena image in Fig. 7, Fig. 8, Fig. 9, Fig. 10.
Deblurring methodParameter $(t=0.01,\, \, \mbox {if, necessary})$PSNR
Ture Wiener filtering8.80%28
SECB $s^*=0.0016,\,s=0.01,\,K=12.0$27
Backward beam$\rho=7.0$25
Tikhonov $\alpha=8*10^{-7}$26
Modified quasi-boundary $\alpha=0.001$26
Spectral $\alpha=0.01$24
Convolution-type $\epsilon=0.005$22
TV $k=1800,\epsilon=4*10^{-3}, \alpha=6*10^{-4}, \tau=0.86$25
FQBM $s=1.5, \alpha=1*10^{-4}$27
FMQBM $s=1.2, \alpha=1*10^{-4}$27
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Deblurring methodParameter $(t=0.01,\, \, \mbox {if, necessary})$PSNR
Ture Wiener filtering8.80%28
SECB $s^*=0.0016,\,s=0.01,\,K=12.0$27
Backward beam$\rho=7.0$25
Tikhonov $\alpha=8*10^{-7}$26
Modified quasi-boundary $\alpha=0.001$26
Spectral $\alpha=0.01$24
Convolution-type $\epsilon=0.005$22
TV $k=1800,\epsilon=4*10^{-3}, \alpha=6*10^{-4}, \tau=0.86$25
FQBM $s=1.5, \alpha=1*10^{-4}$27
FMQBM $s=1.2, \alpha=1*10^{-4}$27
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