Reduced complexity Retinex algorithm via the variational approach

Abstract

Retinex theory addresses the problem of separating the illumination from the reflectance in a given image, and thereby compensating for non-uniform lighting. In a previous paper ( Kimmel et al., 2003), a variational model for the Retinex problem was introduced. This model was shown to unify previous methods, leading to a new illumination estimation algorithm. The main drawback with the above approach is its numerical implementation. The computational complexity of the illumination reconstruction algorithm is relatively high, since in the obtained Quadratic Programming (QP) problem, the whole image is the unknown. In addition, the process requirements for obtaining the optimal solution are not chosen a priori based on hardware/software constraints. In this paper we propose a way to compromise between the full fledged solution of the theoretical model, and a variety of efficient yet limited computational methods for which we develop optimal solutions. For computational methods parameterized linearly by a small set of free parameters, it is shown that a reduced size QP problem is obtained with a unique solution. Several special cases of this general solution are presented and analyzed: a Look-Up-Table (LUT), linear or nonlinear Volterra filters, and expansion using a truncated set of basis functions. The proposed solutions are sub-optimal compared to the original Retinex algorithm, yet their numerical implementations are much more efficient. Results indicate that the proposed methodology can enhance images for a reduced computational effort.