Statistical aspects of space sampling in retinex models

A distinguishing element of the different models in the Retinex family, is the process by which the space surrounding the target pixel is explored. Several models use a process define in terms of repeated sampling, prescribe to compute a synthesis quantity out of each pixel sample and finall to take an average of those quantities to determine the output. For instance, in MI-Retinex each sample is define by a memoryless random walk, connecting a randomly chosen reference pixel to the target, while in RSR the sample is define by a random set of points centered on the target; in both the averaged quantity is a suitably computed sample maximum. Here we discuss the advantages that arise from a probabilistic reformulation of each of those statistical sampling process, with reference to two recently formulated Retinex models: ReMark and QBRIX. In those two algorithms the sampling-computing-and-averaging process is replaced by the direct calculation of the sampling means of the synthesis quantity, out of the whole population of possible samples. In QBRIX - inspired to RSR - this corresponds to computing, for each pixel, the probability that it becomes the maximum of a spray: overall it reduces to the determination of a high quantile of the histogram of pixel intensities. In ReMark - based on MI-Retinex - this corresponds to computing, for each pixel, the probability that it represents the last point of reset, before the random walk meets the target: this entails rethinking the process as a Markov Process and computing its absorption probabilities, by solving a linear system. In this work, we compare informally the two approaches and argue that reasoning in terms of population models can bring new insight into the features of the distinct Retinex variants and highlight connections and differences among mathematical models.