Universal Scale Laws for Colors and Patterns in Imagery

TL;DR

This paper uncovers universal scale laws governing color and pattern distributions in images, revealing consistent mathematical properties across natural scenes and textures, with implications for neural networks and physics.

Contribution

It introduces new universal laws for color and pattern cascades in images, supported by empirical evidence across various image types and scales.

Findings

Colors follow a linear log-scale law with slope -2.

Discrete patterns exhibit a universal entropy maximum around 1.74.

Patterns adhere to the Integral Fluctuation Theorem at unity.

Abstract

Distribution of colors and patterns in images is observed through cascades that adjust spatial resolution and dynamics. Cascades of colors reveal the emergent universal property that Fully Colored Images (FCIs) of natural scenes adhere to the debated continuous linear log-scale law (slope $-2.00 \pm 0.01$) (L1). Cascades of discrete $2 \times 2$ patterns are derived from pixel squares reductions onto the seven unlabeled rotation-free textures (0000, 0001, 0011, 0012, 0101, 0102, 0123). They exhibit an unparalleled universal entropy maximum of $1.74 \pm 0.013$ at some dynamics regardless of spatial scale (L2). Patterns also adhere to the Integral Fluctuation Theorem ($1.00 \pm 0.01$) (L3), pivotal in studies of chaotic systems. Images with fewer colors exhibit quadratic shift and bias from L1 and L3 but adhere to L2. Randomized Hilbert fractals FCIs better match the laws than basic-to-AI-based simulations. Those results are of interest in Neural Networks, out of equilibrium physics and spectral imagery.