Storyboard

The resolution of any physical sensor is limited by fundamental laws, not just technology. The Rayleigh criterion sets the minimum angular resolution at _min = 1.22/D: the human eye (D 3 mm, 550 nm) has _min 2×10 rad (1 arcminute), which remarkably coincides with the separation between cones in the fovea biology has reached the physical limit.

Nyquist's theorem states that to sample a signal of maximum frequency f_max, at least 2f_max samples per second are needed. The human ear detects up to 20 kHz and the cochlea samples at 40,000 effective points per second. Exceeding this limit results in aliasing high frequencies perceived as incorrect low frequencies.

Weber's law (/ = k_W) states that the differential threshold is proportional to the background stimulus. k_W varies by modality: brightness vision (k_W 0.02), intensity hearing (k_W 0.05), weight in hand (k_W 0.02), taste (k_W 0.15). This constant fraction implies that biological sensors are optimal for signals that vary on a logarithmic scale.

The Cramér-Rao bound establishes the minimum theoretical limit of the variance of any unbiased estimator: Var() 1/I_F(). The brain acts as a statistical estimator that approximates this optimal limit under laboratory conditions. Fisher information I_F quantifies how much information about stimulus is contained in the distribution of neural responses.

Spatial resolution x = R/(2L) is critical in vision, radar, and sonar: improving resolution at fixed distance R requires increasing the aperture L (larger eyes, longer antennas) or reducing (using higher frequencies). Insect compound eyes trade angular resolution (small L) for very wide field of view a different evolutionary solution to the camera eye.

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