American Institute of Mathematical Sciences

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August  2013, 7(3): 777-794. doi: 10.3934/ipi.2013.7.777

Wavelet frame based color image demosaicing

Jingwei Liang 1, , Jia Li 2, , Zuowei Shen 2, and  Xiaoqun Zhang 3,

1. 

Department of Mathematics, Shanghai Jiao Tong University, 800 Dongchuan Road, Shanghai, 200240, China
2. 

Department of Mathematics, National University of Singapore, Block S17, 10 Lower Kent Ridge Road, 119076, Singapore, Singapore
3. 

Department of Mathematics, MOE-LSC and Institute of Natural Sciences, Shanghai Jiao Tong University, 800 Dongchuan Road, Shanghai, 200240, China

Received  July 2012 Revised  January 2013 Published  September 2013

Color image demosaicing consists in recovering full resolution color information from color-filter-array (CFA) samples with 66.7% amount of missing data. Most of the existing color demosaicing methods [14, 25, 16, 2, 26] are based on interpolation from inter-channel correlation and local geometry, which are not robust to highly saturated color images with small geometric features. In this paper, we introduce wavelet frame based methods by using a sparse wavelet [8, 22, 9, 23] approximation of individual color channels and color differences that recovers both geometric features and color information. The proposed models can be efficiently solved by Bregmanized operator splitting algorithm [27]. Numerical simulations of two datasets: McM and Kodak PhotoCD, show that our method outperforms other existing methods in terms of PSNR and visual quality.
Keywords: operator splitting method, inpainting., Wavelet frame, color image demosaicing.
Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C3.
Citation: Jingwei Liang, Jia Li, Zuowei Shen, Xiaoqun Zhang. Wavelet frame based color image demosaicing. Inverse Problems & Imaging, 2013, 7 (3) : 777-794. doi: 10.3934/ipi.2013.7.777
References:
[1]

B. E. Bayer, Color imaging array, U.S. Patent, 3971065, 1976. Google Scholar

[2]

A. Buades, B. Coll, J.-M. Morel and C. Sbert, Self-similarity driven color demosaicking, IEEE Transactions on Image Processing, 18 (2009), 1192-1202. doi: 10.1109/TIP.2009.2017171.  Google Scholar

[3]

J. F. Cai, R. Chan, L. Shen and Z. Shen, Simultaneously inpainting in image and transformed domains, Numerische Mathematik, 112 (2009), 509-533. doi: 10.1007/s00211-009-0222-x.  Google Scholar

[4]

J. F. Cai, R. H. Chan and Z. Shen, Simultaneous cartoon and texture inpainting, Inverse Problems and Imaging, 4 (2010), 379-395. doi: 10.3934/ipi.2010.4.379.  Google Scholar

[5]

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

[6]

J. F. Cai, H. Ji, F. Shang and Z. Shen, Inpainting for compressed images, Applied and Computational Harmonic Analysis, 29 (2010), 368-381. doi: 10.1016/j.acha.2010.01.005.  Google Scholar

[7]

P. L. Combettes and V. R. Wajs, Signal recovery by proximal forward-backward splitting, Multiscale Model. Simul., 4 (2005), 1168-1200. doi: 10.1137/050626090.  Google Scholar

[8]

I. Daubechies, "Ten Lectures on Wavelets," CBMS-NSF Regional Conference Series in Applied Mathematics, 61. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1992. xx+357 pp. doi: 10.1137/1.9781611970104.  Google Scholar

[9]

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

[10]

B. Dong, H. Ji, J. Li, Z. Shen and Y. Xu, Wavelet frame based blind image inpainting, Applied and Computational Harmonic Analysis, 32 (2012), 268-279. doi: 10.1016/j.acha.2011.06.001.  Google Scholar

[11]

B. Dong and Z. Shen, MRA based wavelet frames and applications, IAS Lecture Notes Series, Summer Program on "The Mathematics of Image Processing," Park City Mathematics Institute}, 2010. Google Scholar

[12]

J. W. Glotzbach, R. W. Schafer, and K. Illgner, A method of color fillter array interpolation with alias cancellation properties, IEEE Int. Conf. Image Processing, 1 (2001), 141-144. doi: 10.1109/ICIP.2001.958973.  Google Scholar

[13]

T. Goldstein and S. Osher, The split bregman algorithm for l1 regularized problems, SIAM Journal on Imaging Sciences, 2 (2009), 323-343. doi: 10.1137/080725891.  Google Scholar

[14]

B. Gunturk, Y. Altunbasak and R. M. Mersereau, Color plane interpolation using alternating projections, IEEE Transactions on Image Processing, 11 (2002), 997-1013. Google Scholar

[15]

A. Haar, Zur theorie der orthogonalen funktionensysteme, Mathematische Annalen, 69 (1910), 331-371. doi: 10.1007/BF01456326.  Google Scholar

[16]

J. Hamilton Jr and J. Adams Jr, Adaptive color plan interpolation in single sensor color electronic camera, U.S. Patent, 5 (1997), 629-734. Google Scholar

[17]

C. A Laroche and M. A Prescott, Apparatus and method for adaptively interpolating a full color image utilizing chrominance gradients, December 13 1994., US Patent 5, ().   Google Scholar

[18]

W. Lu and Y. P. Tan, Color filter array demosaicking: New method and performance measures, IEEE Transactions on Image Processing, 12 (2003), 1194-1210. Google Scholar

[19]

H. S Malvar, L.-W. He, and R. Cutler, High-quality linear interpolation for demosaicing of bayer-patterned color images, In "Acoustics, Speech, and Signal Processing," 2004. Proceedings.(ICASSP'04). IEEE International Conference on, 3 (2004), iii-485. doi: 10.1109/ICASSP.2004.1326587.  Google Scholar

[20]

S. Osher, M. Burger, D. Goldfarb, J. Xu and W. Yin, An iterative regularization method for total variation-based image restoration, Multiscale Model. Simul., 4 (2005), 460-489. doi: 10.1137/040605412.  Google Scholar

[21]

D. Paliy, V. Katkovnik, R. Bilcu, S. Alenius and K. Egiazarian, Spatially adaptive color filter array interpolation for noiseless and noisy data, International Journal of Imaging Systems and Technology, 17 (2007), 105-122. doi: 10.1002/ima.20109.  Google Scholar

[22]

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

[23]

Z. Shen, Wavelet frames and image restorations, Proceedings of the International Congress of Mathematicians, IV (2010), 2834-2863, Hindustan Book Agency, New Delhi.  Google Scholar

[24]

X. Wu and N. Zhang, Primary-consistent soft-decision color demosaicking for digital cameras (patent pending), Image Processing, IEEE Transactions on, 13 (2004), 1263-1274. doi: 10.1109/TIP.2004.832920.  Google Scholar

[25]

L. Zhang and X. Wu, Color demosaicking via directional linear minimum mean square-error estimation, IEEE Transactions on Image Processing, 14 (2005), 2167-2178. Google Scholar

[26]

L. Zhang, X. Wu, A. Buades and X. Li, Color demosaicking by local directional interpolation and non-local adaptive thresholding, Journal of Electronic Imaging, 20 (2011), 023016. Google Scholar

[27]

X. Zhang, M. Burger, X. Bresson and S. Osher, Bregmanized nonlocal regularization for deconvolution and sparse reconstruction, SIAM Journal on Imaging Sciences, 3 (2010), 253-276. doi: 10.1137/090746379.  Google Scholar

[28]

X. Zhang, M. Burger and S. Osher, A unified primal-dual algorithm framework based on bregman iteration, Journal of Scientific Computing, 46 (2010), 20-46. doi: 10.1007/s10915-010-9408-8.  Google Scholar

show all references

References:
[1]

B. E. Bayer, Color imaging array, U.S. Patent, 3971065, 1976. Google Scholar

[2]

A. Buades, B. Coll, J.-M. Morel and C. Sbert, Self-similarity driven color demosaicking, IEEE Transactions on Image Processing, 18 (2009), 1192-1202. doi: 10.1109/TIP.2009.2017171.  Google Scholar

[3]

J. F. Cai, R. Chan, L. Shen and Z. Shen, Simultaneously inpainting in image and transformed domains, Numerische Mathematik, 112 (2009), 509-533. doi: 10.1007/s00211-009-0222-x.  Google Scholar

[4]

J. F. Cai, R. H. Chan and Z. Shen, Simultaneous cartoon and texture inpainting, Inverse Problems and Imaging, 4 (2010), 379-395. doi: 10.3934/ipi.2010.4.379.  Google Scholar

[5]

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

[6]

J. F. Cai, H. Ji, F. Shang and Z. Shen, Inpainting for compressed images, Applied and Computational Harmonic Analysis, 29 (2010), 368-381. doi: 10.1016/j.acha.2010.01.005.  Google Scholar

[7]

P. L. Combettes and V. R. Wajs, Signal recovery by proximal forward-backward splitting, Multiscale Model. Simul., 4 (2005), 1168-1200. doi: 10.1137/050626090.  Google Scholar

[8]

I. Daubechies, "Ten Lectures on Wavelets," CBMS-NSF Regional Conference Series in Applied Mathematics, 61. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1992. xx+357 pp. doi: 10.1137/1.9781611970104.  Google Scholar

[9]

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

[10]

B. Dong, H. Ji, J. Li, Z. Shen and Y. Xu, Wavelet frame based blind image inpainting, Applied and Computational Harmonic Analysis, 32 (2012), 268-279. doi: 10.1016/j.acha.2011.06.001.  Google Scholar

[11]

B. Dong and Z. Shen, MRA based wavelet frames and applications, IAS Lecture Notes Series, Summer Program on "The Mathematics of Image Processing," Park City Mathematics Institute}, 2010. Google Scholar

[12]

J. W. Glotzbach, R. W. Schafer, and K. Illgner, A method of color fillter array interpolation with alias cancellation properties, IEEE Int. Conf. Image Processing, 1 (2001), 141-144. doi: 10.1109/ICIP.2001.958973.  Google Scholar

[13]

T. Goldstein and S. Osher, The split bregman algorithm for l1 regularized problems, SIAM Journal on Imaging Sciences, 2 (2009), 323-343. doi: 10.1137/080725891.  Google Scholar

[14]

B. Gunturk, Y. Altunbasak and R. M. Mersereau, Color plane interpolation using alternating projections, IEEE Transactions on Image Processing, 11 (2002), 997-1013. Google Scholar

[15]

A. Haar, Zur theorie der orthogonalen funktionensysteme, Mathematische Annalen, 69 (1910), 331-371. doi: 10.1007/BF01456326.  Google Scholar

[16]

J. Hamilton Jr and J. Adams Jr, Adaptive color plan interpolation in single sensor color electronic camera, U.S. Patent, 5 (1997), 629-734. Google Scholar

[17]

C. A Laroche and M. A Prescott, Apparatus and method for adaptively interpolating a full color image utilizing chrominance gradients, December 13 1994., US Patent 5, ().   Google Scholar

[18]

W. Lu and Y. P. Tan, Color filter array demosaicking: New method and performance measures, IEEE Transactions on Image Processing, 12 (2003), 1194-1210. Google Scholar

[19]

H. S Malvar, L.-W. He, and R. Cutler, High-quality linear interpolation for demosaicing of bayer-patterned color images, In "Acoustics, Speech, and Signal Processing," 2004. Proceedings.(ICASSP'04). IEEE International Conference on, 3 (2004), iii-485. doi: 10.1109/ICASSP.2004.1326587.  Google Scholar

[20]

S. Osher, M. Burger, D. Goldfarb, J. Xu and W. Yin, An iterative regularization method for total variation-based image restoration, Multiscale Model. Simul., 4 (2005), 460-489. doi: 10.1137/040605412.  Google Scholar

[21]

D. Paliy, V. Katkovnik, R. Bilcu, S. Alenius and K. Egiazarian, Spatially adaptive color filter array interpolation for noiseless and noisy data, International Journal of Imaging Systems and Technology, 17 (2007), 105-122. doi: 10.1002/ima.20109.  Google Scholar

[22]

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

[23]

Z. Shen, Wavelet frames and image restorations, Proceedings of the International Congress of Mathematicians, IV (2010), 2834-2863, Hindustan Book Agency, New Delhi.  Google Scholar

[24]

X. Wu and N. Zhang, Primary-consistent soft-decision color demosaicking for digital cameras (patent pending), Image Processing, IEEE Transactions on, 13 (2004), 1263-1274. doi: 10.1109/TIP.2004.832920.  Google Scholar

[25]

L. Zhang and X. Wu, Color demosaicking via directional linear minimum mean square-error estimation, IEEE Transactions on Image Processing, 14 (2005), 2167-2178. Google Scholar

[26]

L. Zhang, X. Wu, A. Buades and X. Li, Color demosaicking by local directional interpolation and non-local adaptive thresholding, Journal of Electronic Imaging, 20 (2011), 023016. Google Scholar

[27]

X. Zhang, M. Burger, X. Bresson and S. Osher, Bregmanized nonlocal regularization for deconvolution and sparse reconstruction, SIAM Journal on Imaging Sciences, 3 (2010), 253-276. doi: 10.1137/090746379.  Google Scholar

[28]

X. Zhang, M. Burger and S. Osher, A unified primal-dual algorithm framework based on bregman iteration, Journal of Scientific Computing, 46 (2010), 20-46. doi: 10.1007/s10915-010-9408-8.  Google Scholar

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