Some novel linear regularization methods for a deblurring problem

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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.
[2]	E. O. Brigham, The Fast Fourier Transform, Prentice-Hall, Englewood Cliffs, NJ, 1974.
[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.
[4]	A. S. Carasso, Image restoration and diffusion processes, SPIE Proceedings, 2035 (1993), 255-266.
[5]	A. S. Carasso, Overcoming Hölder continuity in ill-posed continuation problems, SIAM J. Numer. Anal., 31 (1994), 1535-1557. doi: 10.1137/0731080.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[16]	R. C. Gonzalez and P. Wintz, Digital Image Processing, Reading, Mass. -London-Amsterdam, 1977.
[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.
[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.
[19]	D. N. Háo, A mollification method for ill-posed problems, Numer. Math., 68 (1994), 469-506. doi: 10.1007/s002110050073.
[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.
[21]	Y. Huang and M. Ng, Lipschitz and Total-Variational Regularization for Blind Deconvolution, Communications in Computational Physics, 4 (2008), 195-206.
[22]	V. Isakov, Inverse Problems for Partial Differential ations, Springer-Verlag, New York, 1998. doi: 10.1007/978-1-4899-0030-2.
[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.
[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.
[25]	L. S. G. Kovasznay and H. M. Joseph, Image Processing, Proc. IRE., 43 (1955), 560-570. doi: 10.1109/JRPROC.1955.278100.
[26]	R. Lattes and J. L. Lions, Methode de Quasi-Reversibility et Applications, Dunod, Paris, 1967 (English translation R. Bellman, Elsevier, New York, 1969).
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[35]	L. E. Payne, Improperly Posed Problems in Partial Differential ations, SIAM, PHILADELPHIA, 1975.
[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.
[37]	W. K. Pratt, Digital Image Processing, Second ed., John Wiley, New York, 1991.
[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.
[39]	T. I. Seidman, Optimal filtering for the backward heat equation, SIAM J. Numer. Anal., 33 (1996), 162-170. doi: 10.1137/0733010.
[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.
[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.
[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.
[43]	A. N. Tikhonov and V. Y. Arsenin, Solutions of Ill-Posed Problems, Winston and Sons, Washington, D. C., 1977.
[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.
[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.
[46]	H. T. Yura, Imaging in clear ocean water, Applied Optics, 12 (1973), 1061-1066. doi: 10.1364/AO.12.001061.
[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.
[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.

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.
[2]	E. O. Brigham, The Fast Fourier Transform, Prentice-Hall, Englewood Cliffs, NJ, 1974.
[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.
[4]	A. S. Carasso, Image restoration and diffusion processes, SPIE Proceedings, 2035 (1993), 255-266.
[5]	A. S. Carasso, Overcoming Hölder continuity in ill-posed continuation problems, SIAM J. Numer. Anal., 31 (1994), 1535-1557. doi: 10.1137/0731080.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[16]	R. C. Gonzalez and P. Wintz, Digital Image Processing, Reading, Mass. -London-Amsterdam, 1977.
[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.
[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.
[19]	D. N. Háo, A mollification method for ill-posed problems, Numer. Math., 68 (1994), 469-506. doi: 10.1007/s002110050073.
[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.
[21]	Y. Huang and M. Ng, Lipschitz and Total-Variational Regularization for Blind Deconvolution, Communications in Computational Physics, 4 (2008), 195-206.
[22]	V. Isakov, Inverse Problems for Partial Differential ations, Springer-Verlag, New York, 1998. doi: 10.1007/978-1-4899-0030-2.
[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.
[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.
[25]	L. S. G. Kovasznay and H. M. Joseph, Image Processing, Proc. IRE., 43 (1955), 560-570. doi: 10.1109/JRPROC.1955.278100.
[26]	R. Lattes and J. L. Lions, Methode de Quasi-Reversibility et Applications, Dunod, Paris, 1967 (English translation R. Bellman, Elsevier, New York, 1969).
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[35]	L. E. Payne, Improperly Posed Problems in Partial Differential ations, SIAM, PHILADELPHIA, 1975.
[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.
[37]	W. K. Pratt, Digital Image Processing, Second ed., John Wiley, New York, 1991.
[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.
[39]	T. I. Seidman, Optimal filtering for the backward heat equation, SIAM J. Numer. Anal., 33 (1996), 162-170. doi: 10.1137/0733010.
[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.
[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.
[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.
[43]	A. N. Tikhonov and V. Y. Arsenin, Solutions of Ill-Posed Problems, Winston and Sons, Washington, D. C., 1977.
[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.
[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.
[46]	H. T. Yura, Imaging in clear ocean water, Applied Optics, 12 (1973), 1061-1066. doi: 10.1364/AO.12.001061.
[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.
[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.

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.

Method	PSNR
Quasi-boundary regularization	16.5
Modified quasi-boundary regularization	15.9
Tikhonov	14.7

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Method	PSNR
Quasi-boundary regularization	16.5
Modified quasi-boundary regularization	15.9
Tikhonov	14.7

Table 2. behavior of PSNR for the three methods.

Method	PSNR
Quasi-boundary regularization	30.0
Modified quasi-boundary regularization	29.1
Tikhonov	25.2

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Method	PSNR
Quasi-boundary regularization	30.0
Modified quasi-boundary regularization	29.1
Tikhonov	25.2

Table 3. behavior in the Lena image in Fig. 7, Fig. 8, Fig. 9, Fig. 10.

Deblurring method	Parameter $(t=0.01,\, \, \mbox {if, necessary})$	PSNR
Ture Wiener filtering	8.80%	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	$k=1800,\epsilon=410^{-3}, \alpha=610^{-4}, \tau=0.86$	25
FQBM	$s=1.5, \alpha=1*10^{-4}$	27
FMQBM	$s=1.2, \alpha=1*10^{-4}$	27

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Deblurring method	Parameter $(t=0.01,\, \, \mbox {if, necessary})$	PSNR
Ture Wiener filtering	8.80%	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	$k=1800,\epsilon=410^{-3}, \alpha=610^{-4}, \tau=0.86$	25
FQBM	$s=1.5, \alpha=1*10^{-4}$	27
FMQBM	$s=1.2, \alpha=1*10^{-4}$	27

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2017 Impact Factor: 1.465

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2017 Impact Factor: 1.465

American Institute of Mathematical Sciences