Perceived vertical and eye level as one orientation order parameter: a closed-form account of the Li-Matin rules for egocentric space

What the math gives to ML

The paper supplies a differentiable, symmetry-correct pooling rule for unoriented image structure: represent each orientation as a doubled-angle unit vector and read out the phase of their weighted resultant. This is more than ordinary angle averaging because it is invariant to the arbitrary arrow direction of a line, handles cancellation at orthogonal configurations, and exposes a natural confidence variable given by resultant magnitude. The same complex orientation order parameter can be decomposed across image hemifields into symmetric and antisymmetric channels, suggesting a compact module for global orientation, verticality, and left-right asymmetric structure. The most direct neural-network transfer is an orientation-energy pooling layer or auxiliary loss inserted after steerable filters, retaining Cartesian real/imaginary features rather than an unstable angle near cancellation.