American Institute of Mathematical Sciences

Advanced
February  2017, 11(1): 65-85. doi: 10.3934/ipi.2017004

Reducing spatially varying out-of-focus blur from natural image

Faming Fang 1, , Fang Li 2, and  Tieyong Zeng 3,,

1. 

Shanghai Key Laboratory of Multidimensional Information Processing, Department of Computer Science, East China Normal University, Shanghai, China
2. 

Department of Mathematics, East China Normal University, Shanghai, China
3. 

Department of Mathematics, Hong Kong Baptist University, Kowloon Tong, Hong Kong, China

* Corresponding author: Tieyong Zeng

Received  December 2015 Revised  September 2016 Published  January 2017

In this paper, we focus on the challenging problem of removing the spatially varying out-of-focus blur from a single natural image. We first propose an effective method to estimate the blur map by the total variation refinement on Hölder coefficient, then discuss the properties of the corresponding kernel matrix. A tight-frame based energy functional, whose minimizer is related to the optimal defocus result, is thus built. For tackling functional more efficiently, we describe the numerical procedure based on an accelerated primal-dual scheme. To verify the effectiveness of our method, we compare it with some state-of-the-art schemes using both synthesized and natural images. Experimental results demonstrate that the proposed method performs better than the compared methods.

Keywords: Blur map, debluring, framelet, out-of-focus, spatially-varying, variational method.
Mathematics Subject Classification: Primary: 68U10, 65K10, 35A15, 91-08; Secondary: 62H35.
Citation: Faming Fang, Fang Li, Tieyong Zeng. Reducing spatially varying out-of-focus blur from natural image. Inverse Problems & Imaging, 2017, 11 (1) : 65-85. doi: 10.3934/ipi.2017004
References:
[1]

S. Bae and F. Durand, Defocus magnification, in Computer Graphics Forum, 26 (2007), 571-579.  doi: 10.1111/j.1467-8659.2007.01080.x.  Google Scholar

[2]

L. Bar, N. Sochen and N. Kiryati, Restoration of images with piecewise space-variant blur, in In Proceedings of the First International Conference on Scale Space Methods and Variational Methods in Computer Vision, 4485 (2007), 533-544.  doi: 10.1007/978-3-540-72823-8_46.  Google Scholar

[3]

G. Blanchet and L. Moisan, An explicit sharpness index related to global phase coherence, in 2012 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), (2012), 1065-1068.  doi: 10.1109/ICASSP.2012.6288070.  Google Scholar

[4]

C. Boor, A Practical Guide to Splines vol. 27, 1st edition, New York: Springer, 1987. Google Scholar

[5]

S. Boyd, N. Parikh, E. Chu, B. Peleato and J. Eckstein, Distributed optimization and statistical learning via the alternating direction method of multipliers, Foundations and Trends in Machine Learning, 3 (2011), 1-122.  doi: 10.1561/2200000016.  Google Scholar

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C. Byrne, Bounds on the largest singular value of a matrix and the convergence of simultaneous and block-iterative algorithms for sparse linear systems, International Transactions in Operational Research, 16 (2009), 465-479.  doi: 10.1111/j.1475-3995.2009.00692.x.  Google Scholar

[7]

J. Cai, R. Chan and Z. Shen, A framelet-based image inpainting algorithm, Applied Computational Harmonic Analysis, 24 (2008), 131-149.  doi: 10.1016/j.acha.2007.10.002.  Google Scholar

[8]

J. Cai, H. Ji, C. Liu and Z. Shen, Framelet-based blind motion deblurring from a single image, IEEE Transactions on Image Processing, 21 (2012), 562-572.  doi: 10.1109/TIP.2011.2164413.  Google Scholar

[9]

J. Cai, S. Osher and Z. Shen, Linearized bregman iterations for frame-based image deblurring, SIAM Journal on Imaging Sciences, 2 (2009), 226-252.  doi: 10.1137/080733371.  Google Scholar

[10]

J. Cai, S. Osher and Z. Shen, Split bregman method and frame based image restoration, Notices of the American Mathematical Society, 8 (2009), 337-369.   Google Scholar

[11]

A. Chai and Z. Shen, Deconvolution: A wavelet frame approach, Numerische Mathematik, 106 (2007), 529-587.  doi: 10.1007/s00211-007-0075-0.  Google Scholar

[12]

A. Chakrabarti, T. Zickler and W. Freeman, Analyzing spatially-varying blur, in Computer Vision and Pattern Recognition (CVPR), 2010 IEEE Conference on, (2010), 2512-2519.  doi: 10.1109/CVPR.2010.5539954.  Google Scholar

[13]

A. Chambolle and A. Pock, A first-order primal-dual algorithm for convex problems with applications to imaging, Journal of Mathematical Imaging and Vision, 40 (2011), 120-145.  doi: 10.1007/s10851-010-0251-1.  Google Scholar

[14]

R. Chan, S. Riemenschneider, L. Shen and Z. Shen, Tight frame: an efficient way for high-resolution image reconstruction, Applied Computational Harmonic Analysis, 17 (2004), 91-115.  doi: 10.1016/j.acha.2004.02.003.  Google Scholar

[15]

S. Chan and T. Nguyen, Single image spatially variant out-of-focus blur removal, in Image Processing (ICIP), 2011 18th IEEE International Conference on, (2011), 677-680.   Google Scholar

[16]

T. Chan, Theory and Computation of Variational Image Deblurring Lecture notes series, WSPC, 2005. Google Scholar

[17]

Y. Chen, G. Lan and Y. Ouyang, Optimal primal-dual methods for a class of saddle point problems, SIAM Journal on Optimization, 24 (2014), 1779-1814.  doi: 10.1137/130919362.  Google Scholar

[18]

I. Daubechies, B. Han, A. Ron and Z. Shen, Framelets: Mra-based constructions of wavelet frames, Applied and Computational Harmonic Analysis, 14 (2003), 1-46.  doi: 10.1016/S1063-5203(02)00511-0.  Google Scholar

[19]

I. Ekeland and R. Témam, Convex Analysis and Variational Problems Classics in Applied Mathematics (Book 28), Society for Industrial and Applied Mathematics, Philadelphia, 1999.  Google Scholar

[20]

L. Evans, Partial Differential Equations, vol. 19, chapter Holder-spaces, 456–461, American Mathematical Society, 1998. Google Scholar

[21]

F. Fang, G. Zhang, F. Li and C. Shen, Framelet based pan-sharpening via a variational method, Neurocomputing, 129 (2014), 362-377.  doi: 10.1016/j.neucom.2013.09.022.  Google Scholar

[22]

E. Faramarzi, D. Rajan and M. Christensen, Unified blind method for multi-image super-resolution and single/multi-image blur deconvolution, IEEE Transactions on Image Processing, 22 (2013), 2101-2114.  doi: 10.1109/TIP.2013.2237915.  Google Scholar

[23]

T. Goldstein and S. Osher, The split bregman method for L1-regularized problems, SIAM Journal on Imaging Sciences, 2 (2009), 323-343.  doi: 10.1137/080725891.  Google Scholar

[24]

R. Gonzalez and R. Woods, Digital Image Processing 3rd edition, Prentice-Hall, Inc. , Upper Saddle River, NJ, USA, 2006. Google Scholar

[25]

A. Gupta, N. Joshi, L. Zitnick, M. Cohen and B. Curless, Single image deblurring using motion density functions, in ECCV '10: Proceedings of the 10th European Conference on Computer Vision, 6311 (2010), 171-184.  doi: 10.1007/978-3-642-15549-9_13.  Google Scholar

[26]

R. Horn and C. Johnson, Matrix Analysis Cambridge University Press, Cambridge, 1985.  Google Scholar

[27]

Y. Huang, D. Lu and T. Zeng, A two-step approach for the restoration of images corrupted by multiplicative, SIAM Journal on Scientific Computing, 35 (2013), 2856-2873.  doi: 10.1137/120898693.  Google Scholar

[28]

S. Jaffard, Wavelet techniques in multifractal analysis, in In Proceedings of symposia in pure mathematics, 72 (2004), 91-152.   Google Scholar

[29]

S. Kim, I. Eom and Y. Kim, Image Interpolation Based on Statistical Relationship Between Wavelet Subbands in Multimedia and Expo, 2007 IEEE International Conference on, 2007. doi: 10.1109/ICME.2007.4285002.  Google Scholar

[30]

S. Kindermann, S. Osher and P. Jones, Deblurring and denoising of images by nonlocal functionals, SIAM Multiscale Modeling and Simulation, 4 (2005), 1091-1115.  doi: 10.1137/050622249.  Google Scholar

[31]

P. Legrand and J. Vehel, Local regularity-based image denoising, in Image Processing, 2003. ICIP 2003. Proceedings. 2003 International Conference on, 3 (2003), 377-380.   Google Scholar

[32]

Y. Lou, E. Esser, H. Zhao and J. Xin, Partially blind deblurring of barcode from out-of-focus blur, SIAM Journal on Imaging Sciences, 7 (2014), 740-760.  doi: 10.1137/130931254.  Google Scholar

[33]

Y. Lou, X. Zhang, S. Osher and A. Bertozzi, Image recovery via nonlocal operators, Journal of Scientific Computing, 42 (2010), 185-197.  doi: 10.1007/s10915-009-9320-2.  Google Scholar

[34]

J. Muzy, E. Bacry and A. Arneodo, The multifractal formalism revisited with wavelets, International Journal of Bifurcation and Chaos, 4 (1994), 245-302.  doi: 10.1142/S0218127494000204.  Google Scholar

[35]

J. Nagy and D. O'Leary, Restoring images degraded by spatially variant blur, SIAM Journal on Scientific Computing, 19 (1998), 1063-1082.  doi: 10.1137/S106482759528507X.  Google Scholar

[36]

J. Oliveira, M. Figueiredo and J. Bioucas-Dias, Parametric blur estimation for blind restoration of natural images: Linear motion and out-of-focus, IEEE Transactions on Image Processing, 23 (2014), 466-477.  doi: 10.1109/TIP.2013.2286328.  Google Scholar

[37]

A. Ron and Z. Shen, Affine system in $l_2(\mathbb{R}^d)$: The analysis of the analysis operator, Journal of Functional Analysis, 148 (1997), 408-447.  doi: 10.1006/jfan.1996.3079.  Google Scholar

[38]

L. 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]

C. Shen, W. Hwang and S. Pei, Spatially-varying out-of-focus image deblurring with l1-2 optimization and a guided blur map, in Acoustics, Speech and Signal Processing (ICASSP), 2012 IEEE International Conference on, (2012), 1069-1072.  doi: 10.1109/ICASSP.2012.6288071.  Google Scholar

[40]

Y. Tai and M. Brown, Single image defocus map estimation using local contrast prior, in Proceedings of the 16th IEEE International Conference on Image Processing, ICIP'09, (2009), 1777-1780.   Google Scholar

[41]

H. Trussell and S. Fogel, Identification and restoration of spatially variant motion blurs in sequential images, IEEE Transactions on Image Processing, 1 (1992), 123-126.  doi: 10.1109/83.128039.  Google Scholar

[42]

M. Unser and T. Blu, Mathematical properties of the {JPEG2000} wavelet filters, IEEE Transactions on Image Processing, 12 (2003), 1080-1090.  doi: 10.1109/TIP.2003.812329.  Google Scholar

[43]

J. Véhel, Fractals in Multimedia, vol. 132 of The IMA Volumes in Mathematics and its Application, chapter Signal Enhancement Based on Hölder Regularity Analysis, 197–209, Springer, New York, 2002. Google Scholar

[44]

Y. Wang, W. Yin and Y. Zhang, A Fast Algorithm for Image Deblurring with Total Variation Regularization CAAM technical reports, 2007. Google Scholar

[45]

Z. Wang, A. Bovik, H. Sheikh and E. Simoncelli, Image quality assessment: from error visibility to structural similarity, IEEE Transactions on Image Processing, 13 (2004), 600-612.  doi: 10.1109/TIP.2003.819861.  Google Scholar

[46]

O. Whyte, J. Sivic, A. Zisserman and J. Ponce, Non-uniform deblurring for shaken images, Int. J. Comput. Vis., 98 (2012), 168-186.  doi: 10.1007/s11263-011-0502-7.  Google Scholar

[47]

C. Wu and X. Tai, Augmented lagrangian method, dual methods and split bregman iteration for {ROF} model, in Scale Space and Variational Methods in Computer Vision, (2009), 502-513.   Google Scholar

[48]

J. Xu, H. Chang and J. Qin, Domain decomposition method for image deblurring, Journal of Computational and Applied Mathematics, 271 (2014), 401-414.  doi: 10.1016/j.cam.2014.03.030.  Google Scholar

[49]

L. Xu and J. Jia, Two-phase kernel estimation for robust motion deblurring, Computer Vision C ECCV 2010: The series Lecture Notes in Computer Science, 6311 (2010), 157-170.  doi: 10.1007/978-3-642-15549-9_12.  Google Scholar

[50]

L. Xu and J. Jia, Two-phase kernel estimation for robust motion deblurring http://www.cse.cuhk.edu.hk/leojia/projects/robust_deblur/, 2010. doi: 10.1007/978-3-642-15549-9_12.  Google Scholar

[51]

X. Zhu, S. Cohen, S. Schiller and P. Milanfar, Estimating spatially varying defocus blur from a single image, IEEE Transactions on Image Processing, 22 (2013), 4879-4891.  doi: 10.1109/TIP.2013.2279316.  Google Scholar

[52]

S. Zhuo and T. Sim, Defocus map estimation from a single image, Pattern Recognition, 44 (2011), 1852-1858.  doi: 10.1016/j.patcog.2011.03.009.  Google Scholar

show all references

References:
[1]

S. Bae and F. Durand, Defocus magnification, in Computer Graphics Forum, 26 (2007), 571-579.  doi: 10.1111/j.1467-8659.2007.01080.x.  Google Scholar

[2]

L. Bar, N. Sochen and N. Kiryati, Restoration of images with piecewise space-variant blur, in In Proceedings of the First International Conference on Scale Space Methods and Variational Methods in Computer Vision, 4485 (2007), 533-544.  doi: 10.1007/978-3-540-72823-8_46.  Google Scholar

[3]

G. Blanchet and L. Moisan, An explicit sharpness index related to global phase coherence, in 2012 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), (2012), 1065-1068.  doi: 10.1109/ICASSP.2012.6288070.  Google Scholar

[4]

C. Boor, A Practical Guide to Splines vol. 27, 1st edition, New York: Springer, 1987. Google Scholar

[5]

S. Boyd, N. Parikh, E. Chu, B. Peleato and J. Eckstein, Distributed optimization and statistical learning via the alternating direction method of multipliers, Foundations and Trends in Machine Learning, 3 (2011), 1-122.  doi: 10.1561/2200000016.  Google Scholar

[6]

C. Byrne, Bounds on the largest singular value of a matrix and the convergence of simultaneous and block-iterative algorithms for sparse linear systems, International Transactions in Operational Research, 16 (2009), 465-479.  doi: 10.1111/j.1475-3995.2009.00692.x.  Google Scholar

[7]

J. Cai, R. Chan and Z. Shen, A framelet-based image inpainting algorithm, Applied Computational Harmonic Analysis, 24 (2008), 131-149.  doi: 10.1016/j.acha.2007.10.002.  Google Scholar

[8]

J. Cai, H. Ji, C. Liu and Z. Shen, Framelet-based blind motion deblurring from a single image, IEEE Transactions on Image Processing, 21 (2012), 562-572.  doi: 10.1109/TIP.2011.2164413.  Google Scholar

[9]

J. Cai, S. Osher and Z. Shen, Linearized bregman iterations for frame-based image deblurring, SIAM Journal on Imaging Sciences, 2 (2009), 226-252.  doi: 10.1137/080733371.  Google Scholar

[10]

J. Cai, S. Osher and Z. Shen, Split bregman method and frame based image restoration, Notices of the American Mathematical Society, 8 (2009), 337-369.   Google Scholar

[11]

A. Chai and Z. Shen, Deconvolution: A wavelet frame approach, Numerische Mathematik, 106 (2007), 529-587.  doi: 10.1007/s00211-007-0075-0.  Google Scholar

[12]

A. Chakrabarti, T. Zickler and W. Freeman, Analyzing spatially-varying blur, in Computer Vision and Pattern Recognition (CVPR), 2010 IEEE Conference on, (2010), 2512-2519.  doi: 10.1109/CVPR.2010.5539954.  Google Scholar

[13]

A. Chambolle and A. Pock, A first-order primal-dual algorithm for convex problems with applications to imaging, Journal of Mathematical Imaging and Vision, 40 (2011), 120-145.  doi: 10.1007/s10851-010-0251-1.  Google Scholar

[14]

R. Chan, S. Riemenschneider, L. Shen and Z. Shen, Tight frame: an efficient way for high-resolution image reconstruction, Applied Computational Harmonic Analysis, 17 (2004), 91-115.  doi: 10.1016/j.acha.2004.02.003.  Google Scholar

[15]

S. Chan and T. Nguyen, Single image spatially variant out-of-focus blur removal, in Image Processing (ICIP), 2011 18th IEEE International Conference on, (2011), 677-680.   Google Scholar

[16]

T. Chan, Theory and Computation of Variational Image Deblurring Lecture notes series, WSPC, 2005. Google Scholar

[17]

Y. Chen, G. Lan and Y. Ouyang, Optimal primal-dual methods for a class of saddle point problems, SIAM Journal on Optimization, 24 (2014), 1779-1814.  doi: 10.1137/130919362.  Google Scholar

[18]

I. Daubechies, B. Han, A. Ron and Z. Shen, Framelets: Mra-based constructions of wavelet frames, Applied and Computational Harmonic Analysis, 14 (2003), 1-46.  doi: 10.1016/S1063-5203(02)00511-0.  Google Scholar

[19]

I. Ekeland and R. Témam, Convex Analysis and Variational Problems Classics in Applied Mathematics (Book 28), Society for Industrial and Applied Mathematics, Philadelphia, 1999.  Google Scholar

[20]

L. Evans, Partial Differential Equations, vol. 19, chapter Holder-spaces, 456–461, American Mathematical Society, 1998. Google Scholar

[21]

F. Fang, G. Zhang, F. Li and C. Shen, Framelet based pan-sharpening via a variational method, Neurocomputing, 129 (2014), 362-377.  doi: 10.1016/j.neucom.2013.09.022.  Google Scholar

[22]

E. Faramarzi, D. Rajan and M. Christensen, Unified blind method for multi-image super-resolution and single/multi-image blur deconvolution, IEEE Transactions on Image Processing, 22 (2013), 2101-2114.  doi: 10.1109/TIP.2013.2237915.  Google Scholar

[23]

T. Goldstein and S. Osher, The split bregman method for L1-regularized problems, SIAM Journal on Imaging Sciences, 2 (2009), 323-343.  doi: 10.1137/080725891.  Google Scholar

[24]

R. Gonzalez and R. Woods, Digital Image Processing 3rd edition, Prentice-Hall, Inc. , Upper Saddle River, NJ, USA, 2006. Google Scholar

[25]

A. Gupta, N. Joshi, L. Zitnick, M. Cohen and B. Curless, Single image deblurring using motion density functions, in ECCV '10: Proceedings of the 10th European Conference on Computer Vision, 6311 (2010), 171-184.  doi: 10.1007/978-3-642-15549-9_13.  Google Scholar

[26]

R. Horn and C. Johnson, Matrix Analysis Cambridge University Press, Cambridge, 1985.  Google Scholar

[27]

Y. Huang, D. Lu and T. Zeng, A two-step approach for the restoration of images corrupted by multiplicative, SIAM Journal on Scientific Computing, 35 (2013), 2856-2873.  doi: 10.1137/120898693.  Google Scholar

[28]

S. Jaffard, Wavelet techniques in multifractal analysis, in In Proceedings of symposia in pure mathematics, 72 (2004), 91-152.   Google Scholar

[29]

S. Kim, I. Eom and Y. Kim, Image Interpolation Based on Statistical Relationship Between Wavelet Subbands in Multimedia and Expo, 2007 IEEE International Conference on, 2007. doi: 10.1109/ICME.2007.4285002.  Google Scholar

[30]

S. Kindermann, S. Osher and P. Jones, Deblurring and denoising of images by nonlocal functionals, SIAM Multiscale Modeling and Simulation, 4 (2005), 1091-1115.  doi: 10.1137/050622249.  Google Scholar

[31]

P. Legrand and J. Vehel, Local regularity-based image denoising, in Image Processing, 2003. ICIP 2003. Proceedings. 2003 International Conference on, 3 (2003), 377-380.   Google Scholar

[32]

Y. Lou, E. Esser, H. Zhao and J. Xin, Partially blind deblurring of barcode from out-of-focus blur, SIAM Journal on Imaging Sciences, 7 (2014), 740-760.  doi: 10.1137/130931254.  Google Scholar

[33]

Y. Lou, X. Zhang, S. Osher and A. Bertozzi, Image recovery via nonlocal operators, Journal of Scientific Computing, 42 (2010), 185-197.  doi: 10.1007/s10915-009-9320-2.  Google Scholar

[34]

J. Muzy, E. Bacry and A. Arneodo, The multifractal formalism revisited with wavelets, International Journal of Bifurcation and Chaos, 4 (1994), 245-302.  doi: 10.1142/S0218127494000204.  Google Scholar

[35]

J. Nagy and D. O'Leary, Restoring images degraded by spatially variant blur, SIAM Journal on Scientific Computing, 19 (1998), 1063-1082.  doi: 10.1137/S106482759528507X.  Google Scholar

[36]

J. Oliveira, M. Figueiredo and J. Bioucas-Dias, Parametric blur estimation for blind restoration of natural images: Linear motion and out-of-focus, IEEE Transactions on Image Processing, 23 (2014), 466-477.  doi: 10.1109/TIP.2013.2286328.  Google Scholar

[37]

A. Ron and Z. Shen, Affine system in $l_2(\mathbb{R}^d)$: The analysis of the analysis operator, Journal of Functional Analysis, 148 (1997), 408-447.  doi: 10.1006/jfan.1996.3079.  Google Scholar

[38]

L. 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]

C. Shen, W. Hwang and S. Pei, Spatially-varying out-of-focus image deblurring with l1-2 optimization and a guided blur map, in Acoustics, Speech and Signal Processing (ICASSP), 2012 IEEE International Conference on, (2012), 1069-1072.  doi: 10.1109/ICASSP.2012.6288071.  Google Scholar

[40]

Y. Tai and M. Brown, Single image defocus map estimation using local contrast prior, in Proceedings of the 16th IEEE International Conference on Image Processing, ICIP'09, (2009), 1777-1780.   Google Scholar

[41]

H. Trussell and S. Fogel, Identification and restoration of spatially variant motion blurs in sequential images, IEEE Transactions on Image Processing, 1 (1992), 123-126.  doi: 10.1109/83.128039.  Google Scholar

[42]

M. Unser and T. Blu, Mathematical properties of the {JPEG2000} wavelet filters, IEEE Transactions on Image Processing, 12 (2003), 1080-1090.  doi: 10.1109/TIP.2003.812329.  Google Scholar

[43]

J. Véhel, Fractals in Multimedia, vol. 132 of The IMA Volumes in Mathematics and its Application, chapter Signal Enhancement Based on Hölder Regularity Analysis, 197–209, Springer, New York, 2002. Google Scholar

[44]

Y. Wang, W. Yin and Y. Zhang, A Fast Algorithm for Image Deblurring with Total Variation Regularization CAAM technical reports, 2007. Google Scholar

[45]

Z. Wang, A. Bovik, H. Sheikh and E. Simoncelli, Image quality assessment: from error visibility to structural similarity, IEEE Transactions on Image Processing, 13 (2004), 600-612.  doi: 10.1109/TIP.2003.819861.  Google Scholar

[46]

O. Whyte, J. Sivic, A. Zisserman and J. Ponce, Non-uniform deblurring for shaken images, Int. J. Comput. Vis., 98 (2012), 168-186.  doi: 10.1007/s11263-011-0502-7.  Google Scholar

[47]

C. Wu and X. Tai, Augmented lagrangian method, dual methods and split bregman iteration for {ROF} model, in Scale Space and Variational Methods in Computer Vision, (2009), 502-513.   Google Scholar

[48]

J. Xu, H. Chang and J. Qin, Domain decomposition method for image deblurring, Journal of Computational and Applied Mathematics, 271 (2014), 401-414.  doi: 10.1016/j.cam.2014.03.030.  Google Scholar

[49]

L. Xu and J. Jia, Two-phase kernel estimation for robust motion deblurring, Computer Vision C ECCV 2010: The series Lecture Notes in Computer Science, 6311 (2010), 157-170.  doi: 10.1007/978-3-642-15549-9_12.  Google Scholar

[50]

L. Xu and J. Jia, Two-phase kernel estimation for robust motion deblurring http://www.cse.cuhk.edu.hk/leojia/projects/robust_deblur/, 2010. doi: 10.1007/978-3-642-15549-9_12.  Google Scholar

[51]

X. Zhu, S. Cohen, S. Schiller and P. Milanfar, Estimating spatially varying defocus blur from a single image, IEEE Transactions on Image Processing, 22 (2013), 4879-4891.  doi: 10.1109/TIP.2013.2279316.  Google Scholar

[52]

S. Zhuo and T. Sim, Defocus map estimation from a single image, Pattern Recognition, 44 (2011), 1852-1858.  doi: 10.1016/j.patcog.2011.03.009.  Google Scholar

Figure 1.  The spatially-varying blurring images.
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Figure 2.  The distributions of the NHC for in-focus (red) and blur images (blue).
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Figure 1(a) ($800\times 600$ pixels): (a) the rough blur map $\boldsymbol{\tilde{\sigma}}$; (b) the refined blur map $\boldsymbol{{\sigma}}$.">Figure 3.  The blur maps of Figure 1(a) ($800\times 600$ pixels): (a) the rough blur map $\boldsymbol{\tilde{\sigma}}$; (b) the refined blur map $\boldsymbol{{\sigma}}$.
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52], SHP [39] and proposed methods; (f) the ground truth of blur map; (g) corresponding profiles of 1-st line for (c)-(f).">Figure 4.  (a) a sharp image (size:100$\times$ 400); (b) a blur version of (a) (the blur levels are 1 and 4 for 171th-230th and 271th-330th columns, respectively); (c)-(e) the blur maps genrated by ZS [52], SHP [39] and proposed methods; (f) the ground truth of blur map; (g) corresponding profiles of 1-st line for (c)-(f).
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Figure 5.  (a) and (b) are original images; (c) and (d) are two blur maps (range: 1-5); (e): the blurred version of (a) using (c) as blur map; (f): the blurred version of (a) using (d) as blur map; (g): the blurred version of (b) using (c) as blur map; (h): the blurred version of (b) using (d) as blur map.
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Figure 5(e)/(f) by Matlab, XJ [49], CJLS [8], SHP [39], and proposed methods, respectively.">Figure 6.  Synthesised experiments: (A1-A5)/(B1-B5) are the deblurred results (with zoomed regions) of Figure 5(e)/(f) by Matlab, XJ [49], CJLS [8], SHP [39], and proposed methods, respectively.
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Figure 5(g)/(h) by Matlab, XJ [49], CJLS [8], SHP [39], and proposed methods, respectively.">Figure 7.  Synthesised experiments: (A1-A5)/(B1-B5) are the deblurred results (with zoomed regions) of Figure 5(g)/(h) by Matlab, XJ [49], CJLS [8], SHP [39], and proposed methods, respectively.
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Figure 8.  PSNR, SSIM, and SI values corresponding to each figure.
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49], CJLS [8], SHP [39], and the proposed methods.">Figure 9.  Original image and its deblurred results with zoomed regions. (a) The original image (RGB, $296\times 877$ pixels); (b)-(f) the results of Matlab, XJ [49], CJLS [8], SHP [39], and the proposed methods.
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49], CJLS [8], SHP [39], and the proposed methods.">Figure 10.  Original image and its deblurred results with zoomed regions. (a) The original image (RGB, $800\times 600$ pixels); (b)-(f) the results of Matlab, XJ [49], CJLS [8], SHP [39], and the proposed methods.
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49], CJLS [8], SHP [39], and the proposed methods.">Figure 11.  Original image and its deblurred results with zoomed regions. (a) The original image (RGB, $550\times 760$ pixels); (b)-(f) the results of Matlab, XJ [49], CJLS [8], SHP [39], and the proposed methods.
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Table 1.  The MAE value of each blur map generated by ZS [52], SHP [39] and Proposed methods.
ZS [52]SHP [39]Proposed
Figure 4(b)1.37990.64950.5308
Figure 5(e)0.40430.37530.3751
Figure 5(f)0.39590.39500.3684
Figure 5(g)0.48110.44550.3064
Figure 5(h)0.48230.47560.3418
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ZS [52]SHP [39]Proposed
Figure 4(b)1.37990.64950.5308
Figure 5(e)0.40430.37530.3751
Figure 5(f)0.39590.39500.3684
Figure 5(g)0.48110.44550.3064
Figure 5(h)0.48230.47560.3418
Table 2.  Time comparison with other methods (second).
Figure 5(e)Figure 5(f)Figure 5(g)Figure 5(h)Figure 9(a)Figure 10(a)Figure 11(a)
image size300 × 286300 × 286265 × 300265 × 300200 × 300800 × 600534 × 800
Matlab1.181.221.001.172.036.706.00
XJ [49](C)6.456.556.366.3511.4318.0116.95
CJLS [8]86.5983.7488.2789.84194.95780.30693.77
SHP [39]70.2169.4072.2671.55126.37187.01168.68
Proposed38.3337.6034.9836.4693.45137.93141.32
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Figure 5(e)Figure 5(f)Figure 5(g)Figure 5(h)Figure 9(a)Figure 10(a)Figure 11(a)
image size300 × 286300 × 286265 × 300265 × 300200 × 300800 × 600534 × 800
Matlab1.181.221.001.172.036.706.00
XJ [49](C)6.456.556.366.3511.4318.0116.95
CJLS [8]86.5983.7488.2789.84194.95780.30693.77
SHP [39]70.2169.4072.2671.55126.37187.01168.68
Proposed38.3337.6034.9836.4693.45137.93141.32
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