Abstract
In this paper, we present an efficient method for reconstructing color images based on the convex optimization. Current convex optimization methods are all applied to one channel images, so as the number of channels increases, the number of convex optimization problems increase. We present a solution to reconstruct color images represented by three color channels RGB by only one convex optimization problem and use a high-performance algorithm BFGS for solving the problem. We also perform several experiments to compare our method with others on reality datasets.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Afonso, M.V., Bioucas-Dias, J.M., Figueiredo, M.A.T.: An augmented lagrangian approach to the constrained optimization formulation of imaging inverse problems. In: IEEE Transactions on Image Processing, vol. 20, no. 3 (2011)
Anaya, J., Barbu, A.: RENOIR - a dataset for real low-light image noise reduction. J. Visual Commun. Image Rep. 51(2), 144–154 (2018)
Boyd, S., Vandenberghe, L.: Convex Optimization. Cambridge University Press, Cambridge (2004)
Clarkson, J.A., Adams, C.R.: On definitions of bounded variation for functions of two variables. Trans. Am. Math. Soc. 35(1933), 824–854 (1933)
Curry, H.B.: The method of steepest descent for non-linear minimization problems. Quart. Appl. Math. 2, 258–261 (1944)
Sundararajan, D.: Edge detection. Digital Image Processing, pp. 257–280. Springer, Singapore (2017). https://doi.org/10.1007/978-981-10-6113-4_9
Huang, T.S., Yang, G.J., Tang, G.Y.: In: IEEE Transactions on Acoustics, Speech, and Signal Processing, vol. 27, no. 1 (1979)
Hamza, A., Luque, P., Martinez, J., Roman, R.: Removing noise and preserving details with relaxed median filters. J. Math. Image Vision 11(2), 161–177 (1999). https://doi.org/10.1023/A:1008395514426
Hardie, R.C., Barner, K.E.: Rank conditioned rank selection filters for signal restoration. IEEE Trans. Image Process. 3, 192–206 (1994)
Jain, A.K.: Fundamentals of Digital Image Processing. Prentice-Hall, New Jersey (1989)
Jain, P., Tyagi, V.: An adaptive edge-preserving image denoising technique using tetrolet transforms. Vis.Comput. 31(5), 657–674 (2014). https://doi.org/10.1007/s00371-014-0993-7
Kim, D., Park, S.W., Kim, D.H., Yoo, M.S., Lee, Y.J.: Feasibility of sinogram reconstruction based on inpainting method with decomposed sinusoid-like curve (S-curve) using total variation (TV) noise reduction algorithm in computed tomography (CT) imaging system: a simulation study. In: Optik, vol. 161, pp.270-277 (2018)
Motwani, M.C., Gadiya, M.C., Motwani, R.C., Harris, F.C.: Survey of image denoising techniques. In: GSPX (2004)
Nocedal, J., Wright, S.J.: Numerical Optimization, 2nd edn., pp. 136–143. Spinger, New York (2006)
Rudin, L.I., Osher, S., Fatemi, E.: Nonlinear total variation based noise removal algorithms. Physica D. 60(1–4), 259–268 (1992)
Vandenberghe, L.: Quasi Newton Methods. http://www.seas.ucla.edu/~vandenbe/236C/lectures/qnewton.pdf. Accessed 1 Sep 2020
Wali, S., Zhang, H., Chang, H., Wu, C.: A new adaptive boosting total generalized variation (TGV) technique for image denoising and inpainting. J. Vis. Commun. Image Represent. 59, 39–51 (2019)
Wright, M.H.: The interior-point revolution in optimization: History, recent developments, and lasting consequences. In: Bulletin of the American Mathematical Society (2004)
Xu, J., Zhang, L., Zhang, D., Feng, X.: Multi-channel Weighted Nuclear Norm Minimization for Real Color Image Denoising. In: International Conference on Computer Vision (ICCV), pp. 1105–1113. Venice, Italy (2017)
Xu, J., Li, H., Liang, Z., Zhang, D., Zhang, L.: Real-world noisy image denoising: a new benchmark. In: IEEE Conference Computer Vision and Pattern Recognition (2018)
Xue, H., Zhang, S., Cai, D.: Depth image inpainting: improving low rank matrix completion with low gradient regularization. IEEE Trans. Image Process. 26(9), 4311–4320 (2017)
Yang, J., Yin, L., Gabbouj, M., Astola, J., Neuvo, Y.: Optimal weighted median filters under structural constraints. IEEE Trans. Signal Process. 43, 591–604 (1995)
Encyclopedia of Mathematics. https://encyclopediaofmath.org/index.php?title=Newton_method. Accessed 1 Sep 2020
Lagrange Multiplier - Wolfram MathWorld. https://mathworld.wolfram.com/LagrangeMultiplier.html. Accessed 1 Sep 2020
Image denoising by FFT - Scipy lecture. http://scipy-lectures.org/intro/scipy/auto_examples/solutions/plot_fft_image_denoise.html
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2020 Springer Nature Singapore Pte Ltd.
About this paper
Cite this paper
Anh, N.T.H., Toan, N.D.V., Trang, L.H. (2020). A Convex Optimization Based Method for Color Image Reconstruction. In: Dang, T.K., Küng, J., Takizawa, M., Chung, T.M. (eds) Future Data and Security Engineering. Big Data, Security and Privacy, Smart City and Industry 4.0 Applications. FDSE 2020. Communications in Computer and Information Science, vol 1306. Springer, Singapore. https://doi.org/10.1007/978-981-33-4370-2_26
Download citation
DOI: https://doi.org/10.1007/978-981-33-4370-2_26
Published:
Publisher Name: Springer, Singapore
Print ISBN: 978-981-33-4369-6
Online ISBN: 978-981-33-4370-2
eBook Packages: Computer ScienceComputer Science (R0)