Mathematical insights into the original Retinex algorithm for image enhancement

The Retinex theory, originally developed by Land and McCann as a computation model of the human color sensation, has become, with time, a pillar of digital image enhancement. In this area, the Retinex algorithm is widely used to improve the quality of any input image by increasing the visibility of its content and details, enhancing its colorfulness, and weakening, or even removing, some undesired effects of the illumination. The algorithm was originally described by its creators in terms of a sequence of image processing operations and was not fully formalized mathematically. Later, works focusing on aspects of the original formulation and adopting some of its principles tried to frame the algorithm within a mathematical formalism: this yielded every time a partial rendering of the model and resulted in several interesting model variants. The purpose of the present work is to fill a gap in the Retinex-related literature by providing a complete mathematical formalization of the original Retinex algorithm. The overarching goals of this work are to provide mathematical insights into the Retinex theory, promote awareness of the use of the model within image enhancement, and enable a better appreciation of differences and similarities with later models based on Retinex principles. For this purpose, we compare our model with others proposed in the literature, paying particular attention to the work published in 2005 by Provenzi and others.