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

The visually perceived vertical is biased by the orientation content of the visual field.

Li and Matin (2005a, 2005b) reported three regularities of this induced vertical (VPV): a roll-tilted peripheral line shifts it about linearly with orientation; two lines combine about linearly; and symmetric tilts cancel.

We show all three follow from one principle.

The induced vertical is half the argument of the first circular moment of stimulus orientation in the doubled-angle domain, $\phi=\frac{1}{2}\arg c_1$ with $c_1=\sum_j A_j e^{i2\gamma_j}$ (each line counted at twice its angle).

Equivalently it is the principal axis of the orientation structure tensor, or an orientation population vector; we call it the orientation order-parameter model (PLUMB).

The angle-doubling is forced: orientation is a director ($\theta\equiv\theta+\pi$), so a circular-mean readout must be the doubled-angle one, and linear tracking, combination, and cancellation are signatures of any such readout, not separate findings.

Li and Matin's 2-, 3-, and 4-line combination coefficients are then closely matched with no per-configuration free parameters in the angles; the magnitudes come from one mass-action length function (their Fig.

5, three constants).

The data refute complete summation but do not separate the model from simple averaging for long lines; the decisive parameter-free tests are the short-line regime and a $|\cos 2\theta|$ strength law with a null at $45^\circ$.

Read as a sum and a difference across the two hemifields, the same order parameter yields perceived vertical and eye level, recovering the reversed rules of Shavit, Li and Matin (2013); their 30-observer trial data confirm the predicted sub-additive cross-field combination.

The account is stimulus-side and image-computable, linking induced vertical and eye level to classical image-orientation descriptors.