Is the DeVries-Rose to Weber transition empirically possible with sine-wave gratings?
Contrary to this summary description, Garcia-Perez and Peli (1997) presented results of a literature review indicating (a) that a decreasing range exists past the Weber range at low frequencies, with a significant presence also at intermediate frequencies in some subjects, and (b) that the increase in sensitivity over the nominal DeVries-Rose range seems to depart significantly from the square-root law. As discussed by Garcia-Perez and Peli, departures from DeVries-Rose behavior (i.e., a slope of 0.5 in log-log coordinates) are clearly apparent in many empirical data bearing on this issue, leading us to question the validity of the DeVries-Rose law for sine-wave gratings.
This paper analyzes empirical constraints from the shape of the contrast sensitivity function (CSF) at several illuminance levels and from the relationship between illuminance and visual acuity, showing that the DeVries-Rose to Weber transition with sine-wave gratings can only hold at very low spatial frequencies.
Sensitivity as a Function of Spatial Frequency and Illuminance
Several attempts have been made to describe mathematically the dependence of sine-wave contrast sensitivity on spatial frequency and illuminance simultaneously (e.g., Daly, 1993; Rovamo, Mustonen, & Nasanen, 1994). Figure 1a plots the function proposed by Rovamo et al. From this function, cross-sectional profiles at several illuminance levels (see the curves drawn on the right projection plane) have the characteristic shape described by empirical CSFs: Sensitivity decreases with decreasing illuminance, and the CSF peaks at a lower frequency and has a lower cutoff as illuminance decreases (compare with, e.g., DeValois, Morgan, & Snodderly, 1974; van Meeteren & Vos, 1972). Also, orthogonal cross-sections of the same surface (see the curves drawn on the left projection plane) have a shape that accommodates DeVries-Rose and Weber's laws, with the transition point also shifting towards higher illuminance with increasing frequency (compare with, e.g., Hess, 1990; Mustonen, Rovamo, & Nasanen, 1993). Yet, this function (as well as that proposed by Daly, 1993) carries with it an incorrect relationship between acuity and illuminance (see the thick line on the base of the plot), as it suggests that the acuity limit increases without discontinuities as illuminance increases. Quite on the contrary, all available empirical data from man and primate (Cavonius & Robbins, 1973; Hendley, 1948; Hess & Nordby, 1986b; Hetch & Mintz, 1939; Shlaer, 1937; Shlaer, Smith, & Chase, 1942; van de Grind, Koenderink, & van Doorn, 2000) indicate that this relationship has two branches and a point of discontinuity which is usually interpreted as indicating the transition from scotopic to photopic vision. The remainder of this section explores whether a sensitivity surface may exist that satisfies these three constraints, namely, cross-sectional profiles like those in the right and left projection planes of Figure 1a and a two-branch relationship between acuity and illuminance instead of the smooth curve in Figure 1a.
We start our exploration with recourse to available data from a single subject (see the thick curves in Figure 1b). These curves represent fits to data from Figure 1 of Hess and Nordby (1986b) and Figure 6 of Hess and Nordby (1986a), which come from the same normal subject. The empirical CSF was obtained at a reportedly optimal illuminance of 2000 scotopic td (optimal in that sensitivity did not increase significantly at higher illuminance levels) and the empirical acuity limit was obtained by determining the highest frequency that could be resolved at a 100% contrast at each illuminance. The curves were fitted as described in the Appendix. Then, these curves represent empirical constraints on the sensitivity surface for a normal subject who, if he/she is representative of the population, should show the characteristics imposed by any general law that applies to the illuminance-frequency sensitivity surface. The surface in Figure 1b was drawn according to DeVries-Rose and Weber constraints: At each frequency and starting at the minimal illuminance for visibility of that frequency (which is beyond the lower bound of the plot at low frequencies), sensitivity was made to increase with a slope of 0.5 up to the illuminance at which the ceiling CSF was reached, and then continued with a slope of 0 from that point on. Quite clearly, slopes of 0.5 fall short of reaching the ceiling CSF for frequencies above about 1 c/deg. Then, DeVries-Rose and Weber behaviors are incompatible with these empirical data except at low spatial frequencies: Slopes should have to be larger than 0.5 at other frequencies in order to join the acuity limit and the ceiling CSF.
Figure 1c shows the results of relaxing the requirement that slopes are 0.5 at illuminance levels below the Weber range, and at each frequency the minimal illuminance for visibility has simply been linearly connected to the maximal sensitivity at frequencies where the DeVries-Rose constraint would not do. Yet, those linear increases do not solve a problem that was also apparent in Figure 1b, namely, linear increases have the side effect of mapping the discontinuity of the acuity line onto the CSFs (see the profiles on the right projection plane in Figure 1c). Although this characteristic was reported by Patel (1966, his Figure 1) for three subjects, this result does not seem to have been replicated since. In any case, these linear increases require a slope larger than 0.5 and, then, they disobey the DeVries-Rose law.
[FIGURE 1 OMITTED]
The only possibility, then, is that sensitivity at each frequency increases nonlinearly with illuminance in a way that the discontinuity of the acuity line does not map onto CSFs. Not all curvilinear relations produce this characteristic, but Figure 1d shows a plausible case. Note that any smooth CSF surface must have nonlinear cross-sectional profiles along the illuminance axis whose average slope is larger than 0.5 at high frequencies (see the curves on the left projection plane in Figure 1d).
Given the empirical constraints provided by (a) the shape of the acuity versus illuminance curve and (b) the shape of the high-illuminance CSF, the DeVries-Rose law can only hold with sine-wave gratings of low spatial frequency. Yet, at low frequencies, empirical data indicate that there is a further range of decreasing sensitivity beyond the Weber range (see Garcia-Perez & Peli, 1997), and this decreasing range is also observed at higher frequencies in some subjects (Garcia-Perez & Peli, 1999). At intermediate and high frequencies, the increase in sensitivity with illuminance cannot be linear, and the average slope of this nonlinear increase must often be much larger than 0.5.
The presence of DeVries-Rose and Weber ranges has been well documented at threshold and above threshold when the stimulus is a small spot that is flashed briefly
(Aguilar & Stiles, 1954; Barlow, 1957; Brown & Rudd,
1998; Rose, 1948), and quantitative explanations or models for their occurrence have been put forward (DeVries, 1943;
Rudd, 1996; Rudd & Brown, 1996, 1997). When stimuli
are sine-wave gratings, the DeVries-Rose law seems to be more dogma than established fact (see Garcia-Perez & Peli, 1997). Although there is an appealing argument to expect DeVries-Rose and Weber ranges (see Rovamo, Nasanen, & Mustonen, 1997), empirically, these ranges are only distinctly observed at low spatial frequencies. The source of the relationship between sensitivity to sine-wave gratings and retinal illuminance seems to require a new theoretical justification.
Figure 1 in Hess and Nordby (1986b) and Figure 6 in Hess and Nordby (1986a) were digitally scanned at a resolution of 600 dots per inch and the digital versions of these figures were used to extract the coordinates of the data points plotted on them. The empirical data in Figure 1 of Hess and Nordby (1986b) were described by the function
CSF(x) = 188 [x.sup.082] exp [-0.25 x], (A1)
where x is spatial frequency. This mathematical form for the CSF was proposed by Kelly (1975) and the parameters in Equation (A1) were determined by eye yielding the fit shown in Figure 2a. Objective criteria for parameter estimation (e.g., least squares methods) might have yielded different parameter values, but the shape of the resultant function would not have been much different.
Similarly, the empirical data in Figure 6 of Hess and Nordby (1986a) were described by the function
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (A2)
where x is again spatial frequency (x [less than or equal to] 50). For mathematical convenience, Equation (A2) expresses luminance as a function of acuity (and not the other way around) and its parameters were also determined by eye yielding the fit shown in Figure 2b. Note that data points to fit this function were only available for illuminances above about -2 log td. Any extrapolated region below that illuminance level is, then, suspect. Nevertheless, the extrapolation is conservative in the sense that the resultant AVI curve does not threaten the validity of the DeVries-Rose to Weber transition in that region.
Although the fit of each function to the applicable empirical data is not excellent, the approximation is sufficiently good for our purposes here. As is clear in Figures 1b and 1c, alternative expressions that are only minimally different to better accommodate the empirical data will not have any major effect on the main conclusion that these two empirical functions cannot render a smooth contrast sensitivity surface that satisfies the DeVries-Rose to Weber transition.
[FIGURE 2 OMITTED]
Received January 13, 2005
Revision received March 31, 2005
Accepted April 7, 2005
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Miguel A. Garcia-Perez
Universidad Complutense de Madrid
Address correspondence to: Miguel A. Garcia-Perez, Departamento de Metodologia, Facultad de Psicologia, Universidad Complutense, Campus de Somosaguas, 28223 Madrid (Spain). Phone: +34 913 943 061. Fax: +34 913 943 189. E-mail: email@example.com
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|Author:||Garcia-Perez, Miguel A.|
|Publication:||Spanish Journal of Psychology|
|Date:||Nov 1, 2005|
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