A geometric model of brightness perception and its application to color images correction.

*(English)*Zbl 1437.94007Summary: Human perception involves many features like contours, shapes, textures, and colors to name a few. Whereas several geometric models for contours, shapes and textures perception have been proposed, the geometry of color perception has received very little attention, possibly due to the fact that our perception of colors is still not fully understood. Nonetheless, there exists a class of mathematical models, gathered under the name Retinex, which aim at modeling the color perception of an image, which are inspired by psychophysical/physiological knowledge about color perception, and which can geometrically be viewed as the averaging of perceptual distances between image pixels. Some of the Retinex models turn out to be associated with an efficient image processing technique for the correction of camera output images. The aim of this paper is to show that this image processing technique can be improved by including more properties of the human visual system. To that purpose, we first present a generalization of the perceptual distance between image pixels by considering the parallel transport map associated with a covariant derivative on a vector bundle, from which can be derived a new image processing model for color images correction. Then, we show that the family of covariant derivatives constructed in [the first author and N. Sochen, J. Math. Imaging Vis. 48, No. 3, 517–543 (2014; Zbl 1378.94003)] can model some color appearance phenomena related to brightness perception. Finally, we conduct experiments in which we show that the image processing techniques induced by these covariant derivatives outperform the original approach.

##### MSC:

94A08 | Image processing (compression, reconstruction, etc.) in information and communication theory |

##### Keywords:

differential geometry; variational model; contrast enhancement; brightness perception; human visual system; retinex
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\textit{T. Batard} and \textit{M. Bertalmío}, J. Math. Imaging Vis. 60, No. 6, 849--881 (2018; Zbl 1437.94007)

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##### References:

[1] | Batard, T; Bertalmío, M, A class of nonlocal variational problems on a vector bundle for color image local contrast reduction/enhancement, Geom. Imaging Comput., 2, 187-236, (2015) · Zbl 1344.49070 |

[2] | Batard, T; Sochen, N, A class of generalized Laplacians devoted to multi-channel image processing, J. Math. Imaging Vis., 48, 517-543, (2014) · Zbl 1378.94003 |

[3] | Batard, T., Bertalmío, M.: Duality Principle for Image Regularization and Perceptual Color Correction Models. Proc. 5th Int. Conf. Scale-Space and Variational Methods in Computer Vision, J.F. Aujol et al. Eds LNCS 9087, 449-460 (2015) |

[4] | Ben-Shahar, O; Zucker, SW, Hue geometry and horizontal connections, Neural Netw., 17, 753-771, (2004) |

[5] | Bertalmío, M; Caselles, V; Provenzi, E; Rizzi, A, Perceptual color correction through variational techniques, IEEE Trans. Image Process., 16, 1058-1072, (2007) |

[6] | Bertalmío, M; Caselles, V; Provenzi, E, Issues about retinex theory and contrast enhancement, Int. J. Comput. Vis., 83, 101-119, (2009) |

[7] | Bertalmío, M.: Image Processing for Cinema. Chapman & Hall, Boca Raton (2014) · Zbl 1285.68002 |

[8] | Chambolle, A; Pock, T, A first-order primal-dual algorithm for convex problems with applications to imaging, J. Math. Imaging Vis., 40, 120-145, (2011) · Zbl 1255.68217 |

[9] | Cooper, TJ; Baqai, FA, Analysis and extensions of the frankle-mccann retinex algorithm, J. Electronic Imaging, 13, 85-92, (2004) |

[10] | Dorst, L., Fontijne, D., Mann, S.: Geometric Algebra for Computer Science: An Object-Oriented Approach to Geometry. Morgan Kaufmann, Burlington (2009) |

[11] | Fairchild, MD; Pirrotta, E, Predicting the lightness of chromatic objects colors using CIELAB, Color Res. Appl., 16, 385-393, (1991) |

[12] | Ferradans, S; Bertalmío, M; Provenzi, E; Caselles, V, An analysis of visual adaptation and contrast perception for tone mapping, IEEE Trans. Pattern Anal. Mach. Intell., 33, 2002-2012, (2011) |

[13] | Foster, DH, Color constancy, Vis. Res., 51, 674-700, (2011) |

[14] | Georgiev, T.: Relighting, Retinex theory, and Perceived Gradients. Proceedings of Mirage (2005) |

[15] | Golz, J; MacLeod, DIA, Influence of scene statistics on colour constancy, Nature, 415, 637-640, (2002) |

[16] | Horn, B, Determining lightness from an image, Comput. Graph. Image Process., 3, 277-299, (1974) |

[17] | Jost, J.: Riemannian Geometry and Geometric Analysis. Springer, Berlin (2008) · Zbl 1143.53001 |

[18] | http://r0k.us/graphics/kodak/ |

[19] | Land, E; McCann, JJ, Lightness and retinex theory, J. Optical Soc. of Am, 61, 1-11, (1971) |

[20] | McCann, J.J., Rizzi, A.: The Art and Science of HDR Imaging. Wiley, New York (2011) |

[21] | Marini, D; Rizzi, A, A computational approach to color adaptation effects, Image Vis. Comput., 18, 1377-1388, (2000) |

[22] | Nikolova, M; Steidl, G, Fast hue and range preserving histogram specification: theory and new algorithms for color image enhancement, IEEE Trans. Image Process., 23, 4087-4100, (2014) · Zbl 1374.94283 |

[23] | Palma-Amestoy, R; Provenzi, E; Bertalmío, M; Caselles, V, A perceptually inspired variational framework for color enhancement, IEEE Trans. Pattern Anal. Mach. Intell., 31, 458-474, (2009) |

[24] | Pierre, F; Aujol, J-F; Bugeau, A; Steidl, G; Ta, V-T, Variational contrast enhancement of gray-scale and RGB images, J. Math. Imag. Vis., 57, 99-116, (2017) · Zbl 1425.68443 |

[25] | Provenzi, E; Carli, L; Rizzi, A; Marini, D, Mathematical definition and analysis of the retinex algorithm, J. Opt. Soc. Am. A, 22, 2613-2621, (2005) |

[26] | Rudin, LI; Osher, S; Fatemi, E, Nonlinear total variation based noise removal algorithms, Phys. D Nonlinear Phenom., 60, 259-268, (1992) · Zbl 0780.49028 |

[27] | Shen, J, On the foundations of vision modeling: I. weber’s law and weberized TV restoration, Phys. D Nonlinear Phenom., 175, 241-251, (2003) · Zbl 1006.91057 |

[28] | Sochen, N; Kimmel, R; Malladi, R, A general framework for low level vision, IEEE Trans. Image Process., 7, 310-318, (1998) · Zbl 0973.94502 |

[29] | Stevens, SS, On the psychophysical law, Psychol. Rev., 64, 153-181, (1957) |

[30] | Toland, JF, A duality principle for non-convex optimisation and the calculus of variations, Arch. Ration. Mech. Anal., 71, 4161, (1979) · Zbl 0411.49012 |

[31] | Valeton, J; Norren, D, Light adaptation of primate cones: an analysis based on extracellular data, Vision. Res., 23, 1539-1547, (1983) |

[32] | Wyszecki, G., Stiles, W.S.: Color Science: Concepts and Methods, Quantitative Data and Formulas. Wiley, New York (1982) |

[33] | Yeonan-Kim, J; Bertalmío, M, Analysis of retinal and cortical components of retinex algorithms, J. Electron. Imaging, 26, 031208, (2017) |

[34] | Zosso, D; Tran, G; Osher, SJ, Non-local retinex—a unifying framework and beyond, SIAM J. Imag. Sci., 8, 787-826, (2015) · Zbl 1328.68286 |

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