Coarse-to-fine interactions in disparity discriminations

Much of the link between disparity tuning and distance from the horopter is captured by the ‘sizedisparity correlation’ (Felton, Richards & Smith, 1972; Marr & Poggio, 1979; Schor & Wood, 1983; Schor, Wood & Ogawa, 1984a; Smallman & MacLeod, 1994). By this notion, mechanisms tuned to low spatial frequencies (i.e., those with large receptive fields) code larger disparities, and a larger range of disparities, than those tuned to high spatial frequencies (small receptive fields). When coupled with scale-dependent resolution, this means that low-frequency mechanisms provide for low-resolution disparity discrimination over a broad range of disparities and high-frequency mechanisms provide for higher-resolution disparity discrimination over a narrower, near-horopter range. The size-disparity correlation is built into phase-offset coding of binocular disparity (Ohzawa, DeAngelis & Freeman, 1990; Fleet, Jepson & Jenkin, 1991; Fleet, Wagner & Heeger, 1996) and also arises from a parsimonious implementation of position-offset coding. Several predictions come out of the size-disparity correlation. By and large, they have not faired especially well when put to the test, yet there are reasons for not hastily rejecting the correlation (see Smallman & MacLeod, 1994). First, the threshold prediction: Stereoacuity (Schor & Wood, 1983; Schor, Wood & Ogawa, 1994a; Siderov & Harwerth, 1993a; Legge & Gu, 1989), the upper disparity limit (Schor & Wood, 1983), and the diplopia threshold (Schor, Wood & Ogawa, 1984b) are closely linked to stimulus frequency when measured in spatial units, as predicted by the size-disparity correlation (see also Petrov, 2002). However,this linkage is confined to low-to-moderate spatial frequencies; disparity thresholds tend to become independent of frequency when modulations are higher than around 2 to 4 c/d (Schor & Wood, 1983; Schor, Wood & Ogawa, 1984a, 1984b). Second, stereo range: A phase-coding implementation of the size-disparity correlation constrains disparity to within plus-and-minus one-half the receptivefield wavelength. Yet diplopia thresholds and reliable perception of depth of bandpass stimuli have been reported at disparities several times this half-cycle limit (Schor & Wood, 1983; Schor, Wood & Ogawa, 1984a; Wilson, Blake & Halpern, 1991; Rohaly & Wilson, 1993; Smallman & MacLeod, 1994; Prince & Eagle, 1999). Unless the spatial-frequency bandwidth is quite narrow, however, responses to the disparity of lower-frequency stimulus components might disambiguate those of the nominal stimulus components of higher-frequencies (Marr & Poggio, 1979; Tsai & Victor, 2003; Mallot, Gillner & Arndt, 1996). The same can be said of orientation bandwidth, with oblique components disambiguating the responses of more vertical components. While position coding can handle arbitrarily large disparities, this comes at the cost of neural profligacy and a heightened susceptibility to false matches. Finally, a third prediction, on increment thresholds: The size-disparity correlation predicts that disparity increment thresholds should vary with pattern spatial frequency, being a steeper function of standing disparity for stimuli with higher frequency. Empirical evidence on this point has been mostly negative (Mayhew & Frisby, 1979; Badcock & Schor, 1985; Rohaly & Wilson, 1993; Siderov & Harwerth, 1993b, 1995), the exception being disparity increment thresholds as a function of the frequency of disparity modulation (Schumer and Julesz, 1984) and contrast thresholds as a function of disparity (Smallman & MacLeod, 1997; but see Prince & Eagle, 1999). Artifacts, particularly fixation disparities, might have contributed to some of the tepidness of support for a size-disparity correlation (Smallman & MacLeod, 1994).