Retinex Poisson Equation: a Model for Color Perception

Abstract

In 1964 Edwin H. Land formulated the Retinex theory, the first attempt to simulate and explain how the human visual system perceives color. Unfortunately, the Retinex Land-McCann original algorithm is both complex and not fully specified. Indeed, this algorithm computes at each pixel an average of a very large 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, Morel et al. have shown that the original Retinex algorithm can be formalized as a (discrete) partial differential equation. This article describes the PDE-Retinex, a fast implementation of the Land-McCann original theory using only two DFT's. Source CodeThe source code, the code documentation, and the online demo are accessible at the IPOL web part of this article 1 . In this link an implementation is available for download. Compilation and usage instruction are included in the README.txt file of the compressed archive.Keywords: PDE; Retinex; color OverviewIn 1964 Edwin H. Land [1] 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" [2] 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 complex 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.

Poisson Image Editing

“…Even though the proposed (Fourier) implementation solves the Poisson equation over the entire domain of the image (instead of the selected sub-domain), Morel et al empirically showed that the solutions of both (the exact and the Fourier based ) implementations are perceptually equivalent outside the sub-domain for a varied set of images. In a similar direction, Limare et al [18] proposed an implementation in the Fourier domain of the Retinex 3 algorithm as a Poisson Equation with Neumann boundary conditions. The method Limare et al proposed consists in setting to zero those gradients whose modulus is below certain threshold, and then compute the image that corresponds to the modified gradient map by solving the Poisson equation.…”

“…As in the previous method, we complete the processing step applying a normalization to the output. The normalization is performed by setting the mean and variance of the solution of Poisson equation to match the mean and variance of the original image, this is the same normalization performed in [18], and allows to recover the mean value of the solution (which is undefined when Fourier solver is used) and give a result that can be easily compared with the input image.…”

Selective Contrast Adjustment by Poisson Equation

“…Like in the implementation of the Retinex equation [4], the Neumann boundary condition is implicitly imposed by extending the original image symmetrically across its sides, so that the extended image, which is four times bigger, becomes symmetric and periodic. The discrete Fourier transform (DFT) permits to compute directly the Fourier coefficients of a band limited and (J, L)-periodic function u from its samples u jl on a J × L grid.…”

Obtaining High Quality Photographs of Paintings by Image Fusion

“…This corresponds to Poisson's equation on the rectangle with vanishing Neumann boundary conditions. For the implementation details see, for example, [6]. In the present article we use Poisson equation as a procedure for recovering an image from its gradient and its average value.…”