**Does human vision show the contrast invariance expected of an ideal stereoscopic system for computing depth from disparity? We used random-dot stereograms to investigate the luminance contrast effect on perceived depth from disparity. The perceived depth of disparity corrugations was measured by adjusting the length of a horizontal line to match the perceived depth of the corrugations at various luminance contrasts. At each contrast, the perceived depth increased with disparity up to a critical value, decreasing with further increases in disparity. Both the maximum perceived depth and the disparity modulation level where this maximum occurred changed as a sigmoid function of luminance contrast. These results show that perceived depth from disparity depends in a complex manner on the luminance contrast in the image, providing significant limitations on depth perception at low contrasts in a lawful manner but that are incompatible with existing models of cortical disparity processing.**

^{2}. At a viewing distance of 100 cm, one pixel on the screen extended 0.019° × 0.019° (about 1.1 arc min per pixel).

_{0}* (1+ C * U(x, y)), where U(x, y) was a random number drawn from a uniform distribution ranging from −1 to 1, L

_{0}was the background luminance, and C was the Weber contrast parameter. The luminance contrast of the test patterns varied from 5% to 80% (or −26 dB to −1.9 dB) in factors of 2, making five contrast levels with equal log spacing.

*x*ranged from −1.72° to 1.72°, in the rectangular stimulus,

*d*, was where

_{x}*D*is the maximum test disparity and

*sf*is the spatial frequency of the stimulus.

*D*was set to values from 0 to ±20 arc min for the pairs of near and far directions. For the near direction, the test patterns contained a crossed disparity in the middle of the test pattern, whereas for the far direction, the test patterns contained an uncrossed disparity in the middle. We used positive signs to indicate test disparities in the near direction and negative signs for test disparities in the far direction.

_{c}) at each luminance contrast level, c, with the first derivative of a Gaussian function for the two sides of the data separately, where

*D*is the parameter from Equation 1 controlling disparity between the left- and right-eye images,

*α*is the scaling factor of the function for contrast level

_{c}*c*, and

*σ*is the variance parameter of the Gaussian for contrast level

_{c}*c*. The least square fits are shown as the smooth curves in Figure 2 and colored the same as the corresponding data points. Notice that none of the observers perceived any depth at the 5%, or −26 dB, luminance contrast level, nor at 10% in several cases. Thus, we simply set the fitted

*α*to zero for these conditions. This first derivative Gaussian function accounts for 90.4% to 97.5% of variance in the data for the three observers.

_{c}*α*and

_{c}*σ*were the same for the crossed and uncrossed disparities. This reduction in fitting parameters significantly reduced the fit of the single-cycle surface conditions,

_{c}*F*(24, 177) = 6.99,

*p*< 0.001, but did not explain significantly less variance for the corrugated surface conditions,

*F*(24, 177) = 0.73,

*p*= 0.81. Thus, the contrast effect was similar between the near and far disparities in corrugated surface stimuli, whereas it was significantly biased toward greater perceived depth for far disparities in the single-cycle stimuli.

*α*at 1 for all contrast levels, still allowing

_{c}*σ*to be a free parameter. That is, the maximum perceived disparity to a pattern is set to be proportional to the physical disparity but not dependent on the luminance contrast. Such a reduced model provides a much worse account to the data,

_{c}*F*(24, 402) = 80.33,

*p*< 0.001. Thus, the luminance contrast has a highly significant effect on perceived depth. Similarly, if we fix

*σ*to be the same for all contrast level with

_{c}*α*as a free parameter, this reduced model also has a worse fit to the data,

_{c}*F*(18, 402) = 6.68,

*p*< 0.001. Because the peak position and bandwidth of the derivative of Gaussian function depends on

*σ*, this comparison shows that the peak position of the perceived depth is significantly dependent on luminance contrast.

_{c}*Amp*), and the peak position (

*PP*) of the fit curves. Based on the derivative of Equation 2, the peaks occurred when the disparity D =

*σ*. Substituting D with

_{c}*σ*in Equation 2, the peak amplitudes are given by

_{c}*Amp*) as a function of luminance contrast for the three observers. For both single-cycle (blue circles and curves) and corrugated stimuli (red circles and curves), the amplitude increases as a sigmoid function of luminance contrast. We formalized this function according to where

*A*is the maximum perceived depth experienced by the observer,

_{max}*p*is an exponent parameter, and

*z*is an additive constant. The exponent

_{1}*p*was much greater than unity (a linear increase in maximum perceived depth with contrast), ranging from 2.76 to 4.84 for the single-cycle stimuli with near disparities, from 2.43 to 5.77 for the single-cycle stimuli with far disparities, and from 3.19 to 8.13 for the corrugated stimuli. The asymptotic amplitude for the single-cycle stimuli was greater than that for the corrugated condition in two observers but not for the third one. The half-height point, represented by

*z*

_{1}, for the single-cycle condition was significantly smaller (shift = 3.1%, or −30 dB, two-tail pair-comparison,

*t*[2] = −4.76,

*p*= 0.02) than that for the corrugated condition. This suggests that the mechanism underlying the depth perception for the single-cycle stimuli is more susceptible to luminance contrast.

*PP*is the peak position,

*P*is the disparity where the maximum perceived depth occurs,

_{max}*q*is an exponent, and

*z*is an additive constant

_{2}**.**The exponent

*q*ranged from 2.13 to 4.25 for the single-cycle stimuli with near disparities, from 2.69 to 26.80 for the single-cycle stimuli with far disparities, and from 4.99 to 9.92 for the corrugated stimuli. At every contrast level, the peak position for the single-cycle was greater than that for the corrugated surface. The scaling parameter

*P*also showed a significant difference between two spatial frequency conditions (two-tail pair-comparison,

_{max}*t*[2] = 7.45,

*p*= 0.01). That is, the corrugated surface does not support as much depth perception as a single-cycle of the same amplitude. The half-height point, represented by

*z*

_{2}, however, showed no significant difference (two-tail pair-comparison

*t*[2] = 0.86,

*p*= 0.24) between the single-cycle and the corrugated conditions.

*S*

_{0}), based on the derivative of Equation 2 at the disparity D = 0 where the slope

*S*

_{0}=

*α*. If the depth impression were veridical, the perceived depth in millimeter, by the geometry of binocular disparity (Cormack & Fox, 1985), should be 2.35 times the disparity in arc min given our experimental setup. Hence, the slope at zero disparity should be 2.35 in all conditions and for all the observers. As shown in Figure 4, the slope at different luminance contrast levels (data points) approximates an initial linear increase with disparity under many conditions but tends to have a steeper than veridical slope at high contrasts (especially for the single-cycle stimulus) and falls to zero slope at very low contrasts.

_{c}*Journal of Neurophysiology*, 86, 2054–2068.

*Data reduction and error analysis for the physical sciences*. Boston: McGraw-Hill.

*Spatial Vision*, 10, 433–436.

*Proceedings. Biological Sciences/The Royal Society*, 268, 509–516.

*Perception & Psychophysics*, 37, 176–178.

*Vision Research*, 31, 2195–2207.

*Treatise on painting*[J. S. Hawkins, Trans.]. London: J. Taylor.

*Vision Research*, 36, 1839–1857.

*American Journal of Optometry and Archives of American Academy of Optometry*, 26, 9–15.

*Signal detection theory and psychophysics*. New York: Wiley.

*Perception*, 17, 483–495.

*Vision Research*, 29, 593–607.

*Perception*, 36, 14.

*Vision Research*, 39, 2729–2737.

*Vision Research*, 29, 989–1004.

*Vision Research*, 12, 1241–1251.

*Vision*. San Francisco: W. H. Freeman and Co.

*A.M.A*.

*Archives of Ophthalmology*, 59, 4–17.

*Current Opinion in Neurobiology*, 8, 509–515.

*Science*, 249, 1037–1041.

*Vision Research*, 25, 397–406.

*Neuron*, 18, 359–368.

*Vision Research*, 37, 1811–1827.

*Vision Neuroscience*, 19, 735–753.

*Vision Research*, 34, 1315–1325.

*Vision Research*, 39, 9–18.

*Perception*, 15, 249–258.

*Nature*, 251, 140–142.