Supplementary material from "Weber’s Law of perception is a consequence of resolving the intensity of natural scintillating light and sound with the least possible error"

Efficient resolution of natural light and sound intensity is essential for organisms, systems and machines that rely on visual and auditory sensory perception to survive or function effectively in their environment. This resolution obeys Weber’s Law when the smallest resolvable change, a just-notable-difference, grows in direct proportion to the stimulus. Here, Weber’s Law is found to be a consequence of attaining the theoretical minimum mean-square error possible, the Cramer–Rao lower bound, in resolving the intensity of naturally scintillating light and sound. The finding is based on statistics from thousands of measurements of naturally scintillating environmental light and sound signals. Remarkably, just-notable-differences in light and sound intensity measured over decades of psychophysical experiments with artificial sources are also found to approximately attain the respective Cramer–Rao lower bounds. Human intensity resolution is in this way optimally adapted to the natural scintillation of light and sound. Pattern recognition by simple matched-filter correlation between measured and hypothetical images cancels natural scintillation. For intensity perception obeying Weber’s Law, this is found to be advantageous and statistically optimal because perceived scintillation is