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International Conference on Computer Analysis of Images and Patterns

CAIP 2019: Computer Analysis of Images and Patterns pp 317–329Cite as

Restoration of Colour Images Using Backward Stochastic Differential Equations with Reflection

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Part of the book series: Lecture Notes in Computer Science ((LNIP,volume 11679))

Abstract

Colour image denoising methods based on the chromaticity-brightness decomposition are well-known for their excellent results. We propose a novel approach for chromaticity denoising using advanced techniques of stochastic calculus. In order to solve this problem we use backward stochastic differential equations with reflection. Our experiments show that the new approach gives very good results and compares favourably with deterministic differential equation methods.

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References

  1. Aharon, M., Elad, M., Bruckstein, A., et al.: K-SVD: an algorithm for designing overcomplete dictionaries for sparse representation. IEEE Trans. Signal Process. 54(11), 4311 (2006)

    Article  MATH  Google Scholar 

  2. Borkowski, D.: Forward and backward filtering based on backward stochastic differential equations. Inverse Prob. Imaging 10(2), 305–325 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  3. Borkowski, D., Jańczak-Borkowska, K.: Image denoising using backward stochastic differential equations. In: Gruca, A., Czachórski, T., Harezlak, K., Kozielski, S., Piotrowska, A. (eds.) ICMMI 2017. AISC, vol. 659, pp. 185–194. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-67792-7_19

    Chapter  Google Scholar 

  4. Buades, A., Lebrun, M., Morel, J.: Implementation of the non-local bayes (NL-Bayes) image denoising algorithm. Image Process. Line 3, 1–44 (2013)

    Article  Google Scholar 

  5. Buades, A., Coll, B., Morel, J.M.: A non-local algorithm for image denoising. In: 2005 IEEE Computer Society Conference on Computer Vision and Pattern Recognition (CVPR 2005), vol. 2, pp. 60–65. IEEE (2005)

    Google Scholar 

  6. Caselles, V., Sapiro, G., Tang, B.: Color image enhancement via chromaticity diffusion. IEEE Trans. Image Process. 10(5), 701–707 (2001)

    Article  MATH  Google Scholar 

  7. Cecil, T., Osher, S., Vese, L.: Numerical methods for minimization problems constrained to \(S^1\) and \(S^2\). J. Comput. Phys. 198(2), 567–579 (2004)

    MathSciNet  MATH  Google Scholar 

  8. Chan, T., Shen, J.: Variational restoration of nonflat image features: models and algorithms. SIAM J. Appl. Math. 61(4), 1338–1361 (2000). (electronic)

    MathSciNet  MATH  Google Scholar 

  9. Chan, T.F., Kang, S.H., Shen, J.: Total variation denoising and enhancement of color images based on the CB and HSV color models. J. Vis. Commun. Image Represent. 12(4), 422–435 (2001)

    Article  Google Scholar 

  10. Deriche, R., Tschumperlé, D.: Diffusion pde’s on vector-valued images: local approach and geometric viewpoint. IEEE Signal Process. Mag. 19(5), 16–25 (2002)

    Article  MATH  Google Scholar 

  11. Di Zenzo, S.: A note on the gradient of a multi-image. Comput. Vis. Graph. Image Process. 33(1), 116–125 (1986)

    Article  MATH  Google Scholar 

  12. Donoho, D.L., Johnstone, J.M.: Ideal spatial adaptation by wavelet shrinkage. Biometrika 81(3), 425–455 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  13. Duran, J., Coll, B., Sbert, C.: Chambolle’s projection algorithm for total variation denoising. Image Process. Line 3, 311–331 (2013)

    Article  Google Scholar 

  14. Duran, J., Moeller, M., Sbert, C., Cremers, D.: On the implementation of collaborative TV regularization; application to cartoon+ texture decomposition. Image Process. Line 6, 27–74 (2016)

    Article  MathSciNet  Google Scholar 

  15. Gegout Petit, A., Pardoux, E.: Equations différentielles stochastiques rétrogrades réfléchies dans un convexe. Stochast. Int. J. Probab. Stoch. Process. 57(1–2), 111–128 (1996)

    MATH  Google Scholar 

  16. Geman, S., Geman, D.: Stochastic relaxation, GIBBS distributions, and the bayesian restoration of images. In: Readings in Computer Vision, pp. 564–584. Elsevier (1987)

    Google Scholar 

  17. Getreuer, P.: Rudin-osher-fatemi total variation denoising using split bregman. Image Process. Line 2, 74–95 (2012)

    Article  Google Scholar 

  18. Gilboa, G., Sochen, N., Zeevi, Y.Y.: Forward-and-backward diffusion processes for adaptive image enhancement and denoising. IEEE Trans. Image Process. 11(7), 689–703 (2002)

    Article  Google Scholar 

  19. Groot, W., Kruithof, A.A.: The colour triangle. Philips Techn. Rev. 12(5), 137–144 (1950)

    Google Scholar 

  20. Katkovnik, V., Danielyan, A., Egiazarian, K.: Decoupled inverse and denoising for image deblurring: variational BM3D-frame technique. In: 2011 18th IEEE International Conference on Image Processing, pp. 3453–3456. IEEE (2011)

    Google Scholar 

  21. Ma, J., Protter, P., San Martin, J., Torres, S., et al.: Numerical method for backward stochastic differential equations. Ann. Appl. Probab. 12(1), 302–316 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  22. Pardoux, E., Răşcanu, A.: Stochastic Differential Equations, Backward SDEs, Partial Differential Equations. SMAP, vol. 69. Springer, Cham (2014). https://doi.org/10.1007/978-3-319-05714-9

    Book  MATH  Google Scholar 

  23. Perona, P., Malik, J.: Scale-space and edge detection using anisotropic diffusion. IEEE Trans. Pattern Anal. Mach. Intell. 12(7), 629–639 (1990)

    Article  Google Scholar 

  24. Rudin, L.I., Osher, S., Fatemi, E.: Nonlinear total variation based noise removal algorithms. Physica D 60(1–4), 259–268 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  25. Słomiński, L.: Euler’s approximations of solutions of sdes with reflecting boundary. Stochast. Proces. Appl. 94(2), 317–337 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  26. Weickert, J.: Theoretical foundations of anisotropic diffusion in image processing. In: Kropatsch, W., Klette, R., Solina, F., Albrecht, R. (eds.) Theoretical Foundations of Computer Vision, vol. 11, pp. 221–236. Springer, Vienna (1996). https://doi.org/10.1007/978-3-7091-6586-7_13

    Chapter  Google Scholar 

  27. Wikipedia contributors: RG chromaticity from Wikipedia, the free encyclopedia (2019). Accessed 01 Feb 2019

    Google Scholar 

  28. Yaroslavsky, L.P., Egiazarian, K.O., Astola, J.T.: Transform domain image restoration methods: review, comparison, and interpretation. In: Nonlinear Image Processing and Pattern Analysis XII, vol. 4304, pp. 155–170. International Society for Optics and Photonics (2001)

    Google Scholar 

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Authors and Affiliations

  1. Faculty of Mathematics and Computer Science, Nicolaus Copernicus University, Chopina 12/18, 87-100, Toruń, Poland

    Dariusz Borkowski

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  1. Dariusz Borkowski
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Correspondence to Dariusz Borkowski .

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Editors and Affiliations

  1. Department of Computer and Electrical Engineering and Applied Mathematics, University of Salerno, Fisciano (SA), Italy

    Mario Vento

  2. Department of Computer and Electrical Engineering and Applied Mathematics, University of Salerno, Fisciano (SA), Italy

    Gennaro Percannella

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Borkowski, D. (2019). Restoration of Colour Images Using Backward Stochastic Differential Equations with Reflection. In: Vento, M., Percannella, G. (eds) Computer Analysis of Images and Patterns. CAIP 2019. Lecture Notes in Computer Science(), vol 11679. Springer, Cham. https://doi.org/10.1007/978-3-030-29891-3_28

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  • DOI: https://doi.org/10.1007/978-3-030-29891-3_28

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-29890-6

  • Online ISBN: 978-3-030-29891-3

  • eBook Packages: Computer ScienceComputer Science (R0)

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