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

Abstract

Li and Matin (2005a, 2005b) reported three regularities of the induced visually-perceived vertical (VPV): a roll-tilted line shifts the vertical nearly linearly with its orientation; two lines combine nearly linearly; symmetric tilts cancel. All three follow from one principle. The induced vertical -- a stimulus-side axis, not yet the observer's setting -- is half the argument of the first circular moment of the stimulus orientation distribution in the doubled-angle domain, phi = (1/2) arg(c_1) with c_1 = sum_j A_j exp(i 2 gamma_j). We call this the orientation order-parameter model (PLUMB), and prove c_1 is the unique additive, salience-homogeneous spin-2 axial summary. Director symmetry rules out the odd rotational orders, so the doubling is the lowest available; linear pooling and positive homogeneity then fix c_1 within that rung. Li and Matin's 2-, 3-, and 4-line coefficients are matched with no per-configuration free parameters in the angles, one published saturating length-gain function setting the gain; the orthogonal and square configurations have a first moment of exactly zero, so there is no angle to fit. The data refute complete summation but cannot separate the account from simple averaging for long lines; the decisive tests are short lines and the |cos 2theta| coherence law vanishing at 45 degrees. A further, substantive assumption decomposes the signal per hemifield into sum and difference readouts, making perceived eye level (VPEL) the antisymmetric counterpart of VPV and recovering Shavit, Li, and Matin's (2013) reversed integration rules. Trial-level data (30 observers) confirm the predicted sign and null structure; the sub-additive magnitude is a gain read from the data, not derived. The account is stimulus-side, computed before body/gravity weighting, and image-computable, linking both percepts to classical image-orientation descriptors.

45 pages, 14 figures