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Journal of the Optical Society of America A

Journal of the Optical Society of America A


  • Editor: Franco Gori
  • Vol. 28, Iss. 8 — Aug. 1, 2011
  • pp: 1677–1688

Constrained pseudo-Brownian motion and its application to image enhancement

Roberto Montagna and Graham D. Finlayson  »View Author Affiliations

JOSA A, Vol. 28, Issue 8, pp. 1677-1688 (2011)

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Brownian motion is a random process that finds application in many fields, and its relation to certain color perception phenomena has recently been observed. On this ground, Marini and Rizzi developed a retinex algorithm based on Brownian motion paths. However, while their approach has several advantages and delivers interesting results, it has a high computational complexity. We propose an efficient algorithm that generates pseudo-Brownian paths with a very important constraint: we can guarantee a lower bound to the number of visits to each pixel, as well as its average. Despite these constraints, we show that the paths generated have certain statistical similarities to random walk and Brownian motion. Finally, we present a retinex implementation that exploits the paths generated with our algorithm, and we compare some images it generates with those obtained with the McCann99 and Frankle and McCann’s algorithms (two multiscale retinex implementations that have a low computational complexity). We find that our approach causes fewer artifacts and tends to require a smaller number of pixel comparisons to achieve similar results, thus compensating for the slightly higher computational complexity.

© 2011 Optical Society of America

OCIS Codes
(100.2000) Image processing : Digital image processing
(100.2980) Image processing : Image enhancement

ToC Category:
Image Processing

Original Manuscript: March 14, 2011
Revised Manuscript: June 28, 2011
Manuscript Accepted: June 29, 2011
Published: July 25, 2011

Roberto Montagna and Graham D. Finlayson, "Constrained pseudo-Brownian motion and its application to image enhancement," J. Opt. Soc. Am. A 28, 1677-1688 (2011)

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  1. R. Durret, Brownian Motion and Martingales in Analysis (Wadsworth, 1984).
  2. D. Freedman, Brownian Motion and Diffusion (Holden-Day, 1971).
  3. R. Engbert and R. Kliegl, “Microsaccades keep the eyes’ balance during fixation,” Psychol Sci. 15, 431–436 (2004). [CrossRef] [PubMed]
  4. M. Shelhamer, “Sequence of predictive saccades are correlated over a span of ∼2 s and produce a fractal time series,” J. Neurophysiol. 93, 2002–2011 (2005). [CrossRef]
  5. S. Zeki, A Vision of the Brain (Blackwell Science, 1993).
  6. D. Marini and A. Rizzi, “A computational approach to color adaptation effects,” Image Vis. Comput. 18, 1005–1014 (2000). [CrossRef]
  7. E. Provenzi, L. De Carli, A. Rizzi, and D. Marini, “Mathematical definition and analysis of the Retinex algorithm,” J. Opt. Soc. Am. A 22, 2613–2621 (2005). [CrossRef]
  8. K. Barnard and B. V. Funt, “Analysis and improvement of multi-scale retinex,” in Proceedings of the Fifth Color Imaging Conference (Society for Information Systems and Technology, Society for Information Display, 1997), pp. 221–226.
  9. Z. Rahman, D. J. Jobson, and G. A. Woodell, “Retinex processing for automatic image enhancement,” J. Electron. Imaging 13, 100–110 (2004). [CrossRef]
  10. D. Gadia, A. Rizzi, and D. Marini, “Tuning retinex for HDR images visualization,” in Proceedings of the Second European Conference on Color in Graphics, Imaging and Vision (Society for Imaging Sciences and Technology, 2004), pp. 326–331.
  11. L. Meylan and S. Süsstrunk, “High dynamic range image rendering with a retinex-based adaptive filter,” IEEE Trans. Image Process. 15, 2820–2830 (2006). [CrossRef] [PubMed]
  12. R. Sobol, “Improving the Retinex algorithm for rendering wide dynamic range photographs,” J. Electron. Imaging 13, 65–74(2004). [CrossRef]
  13. J. A. Frankle and J. J. McCann, “Method and apparatus for lightness imaging,” U.S. patent 4,384,336 (17 May 1983).
  14. B. V. Funt, F. Ciurea, and J. J. McCann, “Retinex in MATLAB,” J. Electron. Imaging 13, 48–57 (2004). [CrossRef]
  15. J. J. McCann, “Lessons learned from Mondrians applied to real images and color gamuts,” in Proceedings of the Seventh Color Imaging Conference (Society for Information Systems and Technology, Society for Information Display, 1999), pp. 1–8.
  16. J. M. Morel, A. B. Petro, and C. Sbert, “A PDE formalization of retinex theory,” IEEE Trans. Image Process. 19, 2825–2837(2010). [CrossRef]
  17. D. H. Brainard and B. A. Wandell, “Analysis of the retinex theory of color vision,” J. Opt. Soc. Am. 3, 1651–1661 (1986). [CrossRef]
  18. C. Fredembach and G. D. Finlayson, “Hamiltonian path-based shadow removal,” in Proceedings of the 16th British Machine Vision Conference (British Machine Vision Association, 2005), Vol.  2, pp. 502–511.
  19. C. Fredembach and G. D. Finlayson, “The 1.5D sieve algorithm,” Pattern Recogn. Lett. 29, 629–636 (2008). [CrossRef]
  20. J. Rudnick and G. Gaspari, Elements of the Random Walk(Cambridge University, 2004). [CrossRef]
  21. G. F. Lawler and L. N. Coyle, Lectures on Contemporary Probability, Student Mathematical Library (American Mathematical Society, 2000).
  22. B. B. Mandelbrot, The Fractal Geometry of Nature (Freeman, 1983).
  23. T. H. Cormen, C. E. Leiserson, R. L. Rivest, and C. Stein, Introduction to Algorithms, 2nd ed. (MIT Press, 2001).
  24. S. D. Chen, H. Shen, and R. Topor, “An efficient algorithm for constructing Hamiltonian paths in meshes,” Parallel Comput. 28, 1293–1305 (2002). [CrossRef]
  25. R. Dafner, D. Cohen-Or, and Y. Matias, “Context-based space filling curves,” Comput. Graph. Forum 19, 209–218 (2000). [CrossRef]
  26. D. R. Karger, P. N. Klein, and R. E. Tarjan, “A randomized linear-time algorithm to find minimum spanning trees,” J. Assoc. Comput. Mach. 42, 321–328 (1995). [CrossRef]
  27. R. Gould, Graph Theory (Benjamin/Cummins, 1988).
  28. G. Marsaglia and W. W. Tsang, “The ziggurat method for generating random variables,” J. Stat. Software 5, 1–7 (2000).
  29. Kodak, “Kodak lossless true color image suite,” 2004, retrieved 23 October 2010, http://r0k.us/graphics/kodak/.
  30. E. Provenzi, M. Fierro, A. Rizzi, L. De Carli, D. Gadia, and D. Marini, “Random spray Retinex: a new retinex implementation to investigate the local properties of the model,” IEEE Trans. Image Process. 16, 162–171 (2007). [CrossRef] [PubMed]

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