### Retinex Poisson Equation: a Model for Color Perception

Nicolas Limare, Ana Belén Petro, Catalina Sbert, Jean-Michel Morel
2011 Image Processing On Line
Overview In 1964 Edwin H. Land (ref. 4) formulated the Retinex theory, the first attempt to simulate and explain how the human visual system perceives color. His theory and an extension, the "reset Retinex" (ref. 5) were further formalized by Land and McCann in 1971. Several Retinex algorithms have been developed ever since. These color constancy algorithms modify the RGB values at each pixel to give an estimate of the physical color independent of the shading. The Retinex original method was
more » ... mplex and imprecise. Indeed, this algorithm computes at each pixel an average of a very large and unspecified set of paths on the image. For this reason, Retinex has received several interpretations and implementations which, among other aims, attempt to tune down its excessive complexity. But, as shown in ref. 1, the original Retinex algorithm can be formalized as a (discrete) partial differential equation. More precisely, it can be shown that if the Retinex paths are interpreted as symmetric random walks, then Retinex is equivalent to a Neumann problem for a Poisson equation. This result gives a fast algorithm involving just one parameter, also present in the original theory. The Retinex Poisson equation (given below) is very similar to Horn's (ref. 6) and Blake's (ref. 7) equations, which were proposed as alternatives to Retinex. It also is one of the "Poisson editing" equations proposed in Perez et al. (ref. 3). The final principle of the algorithm is extremely simple. Given a color image I, its small gradients (those with magnitude lower than a threshold t) in each channel are replaced by zero. The resulting vector field is no more the gradient of a function, but the Poisson equation reconstructs an image whose gradient is closes for the quadratic distance t to this vector field. Thus, a new image is obtained, where small details and shades of the original have been eliminated. The elimination of the shades creates more homogeneous colors. This fact, according to Land and McCann, models the property of our perception to perceive constant colors regardless of their shading. The formalization proved in ref. 1 yields a fast implementation of the Land-McCann original theory using only two DFT's. You can test the theory on line on your own color images .