Universal Scale Laws for Colors and Patterns in Imagery

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.

Figures (16)

• Figure 1: Bottom: Patterns appear during exposure time, cascade $C(k,1)=im/k$ .Top : flag of independent colors in levels of grey. Bottom : im/k, rescaled to 1 octet
• Figure 2: Patterns evolution with $k$, cascade $C(k,1)=\frac{im}{k}$ for the 7 patterns (arbitrary colors). At $k=1$ in most images local patterns resume to 0123 because all local $\sigma_i$ differ. For $k>\max(\text{im})$, $C(k,1)=\left[0\right]$ and patterns resume to 0000. In between, patterns fluctuate with universal maximum $1.74\pm 0.013$ for Shannon entropy and at compliance with the Integral Fluctuation Theorem for entropy production over scales (see text for details). Original image (left) from database nam2019real.
• Figure 3: Entropy $S_{C}$ of $C(k,s)$. Cuprite image $(256\times 256)\times 16$ channels, Fully Colored. $S_{C_k}$ depends only on density of states of $C(1,s)$ by binning into factor k. It does not decrease monotonically with k which contributes to variations in patterns along the cascade.
• Figure 4: Entropy $S_C$ of the cascade (k,s). Hyperion image $(256\times 256)\times 32$ channels, Fully Colored. The linear Law is universal for Fully Colored Images (black line). A lower entropy, curvature at high resolution (low $s$) and bias at low resolution (high $s$).
• Figure 5: Shift to $\log(s)$ as a Entropy of $C(k,s=1)$ with k. Cuprite image $(256\times 256)\times 8$, Fully Colored. FCI images (bottom left), are perfectly linear with slope $a=-2$ while best linear match at lower entropy of $im$ reveals higher order contributions, when the image in near zero, both $a$ and its standard deviation tends to zero.