From understanding of color perception to dynamical systems by manifold learning.

Comprehending Color Let us start with a seemingly unrelated field to that described in the article by Yair et al. (1) in PNAS. The field of psychophysics deals with the relationships between physical stimuli and mental phenomena. An excellent example is the scientific community’s early efforts to study the human perception of color. Scientists have been intrigued by visual awareness of colors, trying to understand our interpretation of colors and attempting to quantify human perception with simple equations. Roughly speaking, one could divide these efforts into axiomatic ones that gave birth to the Young, Maxwell, Helmholtz, and, later on, Schrödinger so-called “inductive color line elements” and the empirical color arc-lengths that reflected the effort to virtually embed measurements of human color perception into a simple, often Euclidean, domain. In fact, the latter school of thought, of treating the problem empirically rather than axiomatically, is probably one of the earliest attempts to apply a manifold learning technique to study a psychophysical phenomenon. The outcome was the insightful observation that human color perception is 3D, while most birds probably have (and most dinosaurs probably had) a color perception manifold of higher dimensions and most other mammals share a lower dimensional space for the (lack of) perception of color. One of the analysis tools used to arrive at this important observation is known as multidimensional scaling (MDS), and is related to the famous principal component analysis machinery that is commonly used in big data representation, for which various modern generalizations exist. While axiomatic realizations of studying the color receptors in the eye lead Maxwell (2) to the understanding that color images could be synthesized by a linear combination of three monochromatic colors, the space in which one should operate and the ways by which anchor (basis) colors should be selected has been the topic of many scientific and industrial explorations leading to the design of modern mobile, computer, and television screens, as well as almost all printing devices. The early empirical analysis of color perception by manifold learning is indeed a remarkable step in our ability to model human behavior and harness this understanding to our benefit. When trying to process images so as to enhance them and improve their quality, the color line element should obviously come into play. Geometry modeling of image formation indeed led researchers to the introduction of a new manifold that marries the color line element with the image coordinates, giving rise to a 2D manifold (the image) embedded in a 5D space, where three of these dimensions are an exact result of our understanding of color perception. Indeed, these geometric observations are incorporated, in one way or another, into most modern color image processing tools. The Beltrami filter, bilateral filter, and, in fact, most color processing methodologies exploit our understanding of the fundamental geometry behind color perception in one way or another.