A Geometric Model of Brightness Perception and Its Application to Color Images Correction

Abstract

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 Batard and Sochen (J Math Imaging Vis 48(3):517–543 2014) 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.

Appendix: Expressions of Some Moving Frames

$$\begin{aligned} \left( \begin{array}{ccc} \cos \varphi \sin \theta \, &{} \, -g_{11} \sin \varphi - g_{21} \cos \theta \cos \varphi \, &{} \, -g_{12} \sin \varphi - g_{22} \cos \theta \cos \varphi \\ \\ \sin \varphi \sin \theta &{} g_{11} \cos \varphi - g_{21} \sin \varphi \cos \theta &{} g_{12} \cos \varphi - g_{22} \sin \varphi \cos \theta \\ \\ \cos \theta &{} g_{21}\sin \theta &{} g_{22} \sin \theta \end{array} \right) \end{aligned}$$

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with $g_{11},g_{12},g_{21},g_{22} \in C^1(\varOmega )$ satisfying

$$\begin{aligned}&\left\{ \begin{array}{ccc} g_{11}^2+g_{21}^2 &{} = &{} 1 \\ g_{12}^2+g_{22}^2 &{} = &{} 1\\ g_{11}g_{12}+g_{21}g_{22} &{} = &{} 0 \\ g_{11} \, d g_{12}+g_{21} \, dg_{22} &{} = &{} \cos \theta \, d\varphi \end{array} \right. \nonumber \\&\quad \left( \begin{array}{ccc} \sin \theta \cos \alpha + \cos \theta \cos \beta \sin \alpha &{} - \sin \theta \sin \alpha + \cos \theta \cos \beta \cos \alpha &{} \cos \theta \sin \beta \\ \\ - \cos \theta \cos \alpha + \cos \beta \sin \theta \sin \alpha &{} \cos \theta \sin \alpha + \cos \beta \sin \theta \cos \alpha &{} \sin \theta \sin \beta \\ \\ - \sin \beta \sin \alpha &{} - \sin \beta \cos \alpha &{} \cos \beta \end{array} \right) \nonumber \\ \end{aligned}$$

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