Perception, 1995, volume 24, pages 215-235 Binocular correspondence and visual direction Wim A van de Grind, Casper J ErkelensIT, Alfons C Laan Helmholtz Institute and Department of Comparative Physiology, Padualaan 8, 3584 CH Utrecht, The Netherlands H Helmholtz Institute and Department of Physics of Man, Utrecht University, Princetonplein 5, 3584 CC Utrecht, The Netherlands Based on paper presented at the Conference on Binocular Stereopsis and Optic Flow, Toronto, Canada, 22-26 June 1993 Abstract. 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 Wells's theory is clearly outdated and does not mesh well with modern three-dimensional geometry of binocular vision, which Hering's theory does.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. 1 Introduction Binocular single vision requires that information about left-eye and right-eye oculo- centric directions be combined into a single binocular visual direction. If this combination fails, a single visual element might get two visual-direction labels, a result called diplopia. Single vision might result if the two oculocentric directions are equal [that is, the two-dimensional (2-D) local signs correspond] or if they are only slightly different so that 'fusion' is possible [the three-dimensional (3-D) local signs correspond]. Here we will consider directional vision, as far as possible without delving too much into problems of depth, fusion, rivalry, and the like. Like Ono (1981, 1991) and Ono and Mapp (1995) we also reconsider well-known 'classic' theories on binocular visual direction, namely those of Hering and of Wells, but we plan to deduce mathematical predictions from these theories. Unfortunately both classic theories and most of their modern reformulations confound the 2-D problem of visual direction with depth gauging, although this is much less severe in Hering's theory (1879/1942) than in that of Wells (1792). The theory propounded by Wells was still cited occasionally in the early part of this century (eg Rohr 1923), but it was virtually unknown to modern visual scientists until Ono (1981) revitalised it. It remains to be established whether this was more than an act of historical justice. Was it also a blessing to modern theory formation or is the theory of Wells as outdated as so many others (eg those by Aguillonius and Kepler)? Wells not only developed an