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Stanley A. Klein, Christopher W. Tyler; Paradoxical, quasi-ideal, spatial summation in the modelfest data. Journal of Vision 2005;5(8):478. doi: 10.1167/5.8.478.
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© ARVO (1962-2015); The Authors (2016-present)
The Spatial-Modelfest dataset consists of detection thresholds (defined at d'=1.4) and standard errors for 43 diverse static stimuli on 16 observers. 22 of the stimuli were single Gabor patches of diverse spatial frequencies, sizes and aspect ratios. The variability across the 16 observers was surprisingly small. The rms Weber fraction SE for the 16 stimuli between 1 and 15 c/deg was less than 5.5%, placing strong constraints on models of spatial vision. Here we present two of the surprises revealed by fitting the Modelfest data with a simple area summation model.
1) The seven 4 c/deg Gabor thresholds (with rms Weber SE=4.6% across observers) were well fit (chi square=7.9, df=5) by a straight line (log threshold vs log stimulus area) with a Minkowski pooling exponent of p=2.24±0.12. This relatively low exponent is close to 2, the ideal observer prediction. The low value of p is not compatible with the measured d' function exponent of b>1.5 (where d'=cb), since Pythagorean d' summation would predict p=2b>3. The strong spatial summation may be due to a combination of defining threshold at a high level (d'=1.4) plus a two-limbed d' function with an exponent of b=2 near zero contrast dropping to a unity log-log slope (b=1) above threshold (as is characteristic of a Weibull function). Our high d' would have an effective d' exponent close to 1, accounting for the strong spatial summation.
2) When the fitted data is expanded to the full set of Gabor stimuli a different picture emerges. The 22 Gabor thresholds were fit with 7 parameters: a 5 parameter full-field CSF, a Minkowski spatial pooling exponent, and an aspect ratio parameter (tiger tail Gabors being slightly less visible than baguette Gabors). The fit is remarkably good (chi square = 13.9, df=15) considering that the SEs are so small. The Minkowski pooling exponent is p=2.61±0.07. The discrepancy with the value of p=2.24 for the 4 c/deg data may indicate a limit to the efficiency of spatial summation as the number of cycles gets large.
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