Retinal Correspondence and the Theoretical Horopter

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Disparity is usually considered a 2D vector representing deviation from retinal correspondence, and visualized as the theoretical horopter, the set of locations projecting onto correspondence pairs. Disparity is assumed to decompose naturally into two orthogonal components, horizontal and vertical disparity, processed in fundamentally different ways by the visual system. But when disparity is considered to be relative to a non-identical correspondence pattern, simple definitions break down. In general, scalar horizontal and vertical disparities and the disparity vector composed of them are not well defined entities. Retinally, a binocular target is represented by one 2D position vector per eye, or four dimensions total. If disparity is assumed to be the difference between these projection vectors and a retinal correspondence pattern, the resulting entity has eight degrees of freedom twice that of a retinal disparity vector. Only when empirical correspondence obeys certain constraints is disparity reducible to such a vector. But even then it can not be simply split into retinal horizontal and vertical components, as eye movements change epipolar projection geometry. Additionally, we demonstrate that the accepted model forms of the correspondence patterns of the Hering-Hillebrandt deviation and the Helmholtz shear are not compatible across a 2D retina.