Binocular correspondence and visual direction

Two classic theories of direction vision, one by Hering, the other by Wells, are expressed in mathematical form and compared. The Hering disparity field differs considerably from the Wells disparity field, but if both are scaled for the change of acuity with eccentricity their differences are much more subtle. This explains why it is hard to determine which theory predicts direction perception best, although the tests favour Hering's theory. It is proved that Wells's construction (his rule 3) follows directly from his first two rules and Aguillonius's assumption that the horopter in the fixation plane is a frontoparallel line. Wells's theory is clearly outdated and does not mesh well with modern three-dimensional geometry of binocular vision, which Hering's theory does. Moreover, Wells inextricably mixes distance and direction vision right from the start, whereas Hering properly treats the two-dimensional manifold of directions and the depth-gauging principles separately. The use of terms such as 'Wells-Hering' rules should be discouraged and both Wells and Hering should be remembered separately for their clearly distinct and independent contributions. The work of Hering is still relevant to modern theory and praxis of binocular vision. The extension of Hering's approach to vertical disparities is treated for stimuli in frontoparallel planes. It is shown that acuity-scaled vertical-disparity information sampled at a single glance is below resolution beyond about arm's length. It can only be used if eye movements are allowed. Throughout, the simplest derivations of the geometrical relations that it was possible to find are given, so that the review of binocular geometry might also be of some didactical use. Finally it is indicated in which direction it might be necessary to modernise the concept of binocular correspondence.