We examined how observers discount perceived surface orientation in estimating perceived albedo (lightness). Observers viewed complex rendered scenes binocularly. The orientation of a test patch was defined by depth cues of binocular disparity and linear perspective. On each trial, observers first estimated the orientation of the test patch in the scene by means of a gradient probe and then matched its perceived albedo to a reference scale. We found that observers’ perception of orientation was nearly veridical and that they substantially discounted perceived orientation in estimating perceived albedo.

*E*

^{P}and the intensity of the diffuse light source by

*E*

^{D}. The angle between the direction to the punctate source and the surface normal

**N**at a point is denoted by

*ϑ*and the angle between the surface normal and the direction to the viewer is denoted by

*ν*.

*θ*, ν < 90°), the luminance emitted from the surface along the direction to the viewer is, where

*α*is the

*albedo*of the surface, the fraction of absorbed light that is re-emitted. Note that in Equation 1,

*E*

^{P}and

*E*

^{D}are in units of luminance. In the Lambertian model, the luminance does not depend upon the direction to the viewer

*ν*but only the angle

*ϑ*between the surface normal

**N**and the direction to the light. When the punctate light is behind the plane of the surface patch (0° ≤

*v*< 90,

*θ*90°) the surface receives only diffuse illumination. Equation 1 becomes, . For convenience, we define

*E*=

*E*

^{P}+

*E*

^{D}to be the

*total light intensity*and

*π*=

*E*

^{P}/

*E*to be the

*relative punctate intensity*, the proportion of punctate light intensity in the total light intensity

*E*.

*π*is 0 (the light is perfectly diffuse) or

*ϑ*is 0° (the direction to the punctate light is orthogonal to the surface patch), the geometric discounting function is 1. When

*ϑ*is outside the range 0° to 90°, the punctate light is behind the surface and the geometric discounting function reflects only the diffuse component of the light. As we vary the angle

*ϑ*, the right hand side reaches a maximum when

*ϑ*is 0 and the light from the punctate source falls perpendicularly onto the surface. We solve the equation above for

*α*to get .

*ϑ*

^{R}= 0°). The angle between the surface normal of the other (the test surface) and the direction to the punctate light source is

*ϑ*

^{T}. The luminance of the test surface is set to a fixed luminance

*L*

^{T}. The observer is asked to adjust the luminance

*L*

^{R}of the reference surface until the perceived albedo

*α*

^{R}of this patch matches the perceived albedo

*α*

^{R}of the test surface. This task is an example of an asymmetric lightness matching task.

*ϑ*

^{R}= 0°, we have Γ(

*θ*) = 1 the previous equation can be rearranged as . If we repeat this asymmetric lightness setting task for many values of

^{R}, π*ϑ*

^{T}, then the

*setting ratio*of the reference and test luminances, Λ =

*L*

^{R}/

*L*

^{T}, plotted against

*ϑ*

^{T}, traces the curve Γ(

*ϑ*

^{R},

*π*).

*observer’s geometric discounting function*. By means of the asymmetric lightness task just described, we can estimate the observer’s geometric discounting function and compare it to the theoretical form for a Lambertian surface in Equation 3. In the experiment below, though, we will not assume that the observer has access to the correct values of the relative punctate intensity

*π*, or the angle between the test surface and the direction to the punctate light source,

*ϑ*

^{T}. We will also not assume that observers perceive the reference patch as belonging to the scene and illuminated by the same light sources (see the discussion on local versus global frameworks in Gilchrist et al., 1999). We will show that we can still estimate the observer’s geometric discounting function up to an unknown positive scaling factor by measuring the setting ratio, Λ.

*ϑ*and the relative punctate intensity

*π*. They found that observers’ estimates of the lightness (perceived albedo) of the surface differed in the two viewing conditions. The difference was in the direction consistent with Equation 5, but much smaller in magnitude than the equations would predict. Hochberg and Beck state that when there were no cubes, the effect of viewing condition disappeared, consistent with the claim that observers used information about the direction of the punctate light source and its relative intensity in estimating surface albedo.

*coplanar ratio hypothesis*: local contrast plays a role in lightness judgment only if the regions of interest are coplanar and at the same depth. Gilchrist credits this hypothesis to Kardos (1934). Therefore, he concluded, lightness must be intimately related to the perceived geometric layout of the scene.

*central cube*. A

*test patch*was attached to the central cube. We varied its orientation and albedo from trial to trial as described below. The center of the test patch was always in the same position in the scene. Each scene also contained a number of small objects that were randomly varied from trial to trial. These objects could be shiny, matte, or partly shiny and matte.

*Ψ,ϕ,r*) to specify a simulated scene (Figure 3A). This coordinate system (Figure 3A) has its origin at the center of the test patch and is most easily specified if we first set up a Cartesian coordinate system (

*x, y, z*) with the same origin. The

*z*-axis lies along the observer’s line of sight to the center of the test patch. The

*y*-axis is vertical, in the fronto-parallel plane. The

*x*-axis is horizontal, also in the fronto-parallel plane. The positive half of the

*x*-axis,

*y*-axis, and

*z*-axis are to the observer’s right, upward, and toward the observer, respectively. If we represent a point as a vector (

*x, y, z*), the angle

*Ψ*is the angle between the positive

*z*-axis and the projection of the point into the

*xz*-plane, and it ranges from −180 to 180 degrees. The angle

*ϕ*is the angle between the point and the

*xz*-plane, and it ranges between −90 and +90 degrees. Any direction away from the origin can be specified by coordinates (

*Ψ,ϕ*). The positive

*x*-axis, for example, is (90°,0°), the positive

*z*-axis is (0°,0°), the direction toward the observer. The third coordinate

*r*is the radial distance from the origin to a point. We report radial distances and other measurements in centimeters, at the size that the simulated objects were presented to the observer.

*Ψ*

^{T},

*ϕ*

^{T}):

*Ψ*

^{T}could take on any of the values {−50°, −40°, −30°, 0, 30°, 40°, 50°} while

*ϕ*

^{T}was always 0°. When

*Ψ*

^{T}= −30°, the test patch was orthogonal to the face of the central cube to which it was attached. Figure 3B shows a schematic of the scene, seen from above, with the seven possible orientations of the test patch marked. For reference, the large cube in the center of the scene is shown. The test patch was rendered with one of two slightly different albedos, DARK and LIGHT. The DARK patch had albedo

*α*= 0.55. We included two different albedos to encourage observers to make fine lightness discriminations in the lightness task described below.

*Ψ*

^{P},

*ρ*

^{P},

*r*

^{P}) = (−21.44°, 14.89°, 155.64 cm). In Cartesian coordinates, the punctate light source was 70 cm behind, 55 cm to the left, and 40 cm above the observer’s viewpoint. The punctate source was sufficiently far from the scene so that we could treat the punctate light source as collimated across the extent of the test patch. The direction to the punctate source is specified by the angles, (

*Ψ*

^{P},

*ϕ*

^{P}) = (−21.44°, 14.89°). It was never varied. The diffuse-punctate balance was always

*π*= 0.62.

^{2}. The stereoscope was contained in a box 124 cm on a side. The front face of the box was missing and that is where the observer sat in a chin/head rest. The interior of the box was coated with black flocked paper (Edmund Scientific, Tonawanda, NY) to absorb stray light. Only the stimuli on the screens of the monitors were visible to the observer. The casings of the monitors and any other features of the room were hidden behind the nonreflective walls of the enclosing box.

*Ψ*. The elevation was always set to the correct value,

*ϕ*= 0°. Observers reported no difficulty with setting the probe and were unaware that it was visible in only the right eye.

*L*

^{R}/

*L*

^{T}of the luminance of the reference surface to the luminance of the test surface that was matched to it. In particular, if the setting ratio is unaffected by perceived orientation, then the observer is not discounting perceived orientation from estimates of perceived albedo. We will also look for an effect of true orientation (as opposed to perceived orientation) on perceived albedo, but, as we will see in a moment, there is little difference between perceived and true orientations in these scenes. We first report the orientation estimates and then the effect of this perceived orientation on setting ratios Λ.

*p*values for observers JJG, CBC, EPB, LR, MM, and NB were

*p*= 0.095, 0.260, 0.975, 0.520, 0.596, 0.779 respectively. There was no significant main effect of albedo (the smallest

*p*value was

*p*= 0.021 for observer MM, for observers CBC, EPB, JJG, LR, and NB the

*p*values are

*p*= 0.996,0.783,0.430,0.105,0.137). The main effect of orientation was significant for all observers (

*p*< 0.001). For all observers, the slopes were significantly less than 1 (

*p*< 0.05). However, the deviations from veridical were modest: all slopes but one were between 0.84 and 0.96, the exception being 0.63 for observer NB.

*L*

^{T}/

*L*

^{T}would equal the Lambertian geometric discounting function Γ (

*ϑ*,

*π*). In deriving this identity, we assumed that the observer made use of accurate estimates of the parameters that describe the orientation and albedo of the test surface and the orientation and lighting of the references surfaces. What happens to Equation 7 if the observer’s estimates of these parameters are in error? Let , etc. denote the estimates of

*ϑ*,

*E*, etc. that an observer substitutes into Equation 5 in order to estimate the albedo of the test surface. Let , etc. denote the corresponding estimates for the references surfaces. We assume that these estimates, although unknown, do not change over the course of the experiment. Let

*L*

^{T}denote the observer’s estimate of the luminance of the test patch on a trial, , the observer’s estimate of the luminance of the reference patch. Then we can represent the luminance of the test patch as and that of the reference patch as . Once the observer has chosen a reference surface whose apparent albedo matches the apparent albedo of the test surface, we have . Then the equation for Λ becomes where

*m*is a multiplicative constant.

*E*, the total light intensity, from asymmetric lightness matches because scaling the overall light intensity by the same factor for the test scene and for the reference surface would likely lead to the same matching behavior. We insert Equation 10 into Equation 1 to obtain, . Because

*ϕ*

^{P}never changes throughout the experiment,

*E*

^{P}is confounded with cos

*ϕ*

^{P}. Changing the elevation

*ϕ*

^{P}of the punctate source is equivalent to scaling its intensity. We replace

*E*

^{P}cos

*ϕ*

^{P}by

*E*

^{P}

_{eq}, the

*equivalent punctate light intensity*, and obtain . Further, we define , the

*equivalent relative punctate source intensity*. The geometric discounting function of the Lambertian observer can then be written as Γ(

*Ψ*;

^{T}− Ψ^{P}*π*). Given an estimate of the luminance that arrives at the eye,

_{eq}*^L*, a visual system that has available estimates of

*E*,

*π*, and

*ϑ*(denoted

*^E, ^π*and

*^θ*, respectively) can compute an estimate of the albedo of the surface by substituting these estimates into Equation 5, . An observer’s estimate

*^θ*depends on his or her estimates of the orientation of the test, denoted , and his or her estimates of the direction to the punctate light source, denoted , through Equation 8. The observer might also misestimate the overall intensity of the light,

*E*. Misestimation of

*E*would simply lead to an overall scaling of perceived albedo but, as explained above, our estimates of perceived albedo are only determined up to an unknown scaling factor. Misestimation of

*E*would simply alter this unknown value. We explicitly estimated in by asking the observer to perform the orientation task.

*Ψ*

^{P},

*π*

_{eq}, and we vary the orientation of the surface by varying

*Ψ*

^{T}, as we do in the experiment. We plot examples of the geometric discounting function Γ(

*Ψ*;

^{T}− Ψ^{P}*π*) as a function of

_{eq}*Ψ*

^{T}for different values of

*Π*

_{eq}and

*Ψ*

^{P}in Figure 7. Notice that changes to the parameter

*Ψ*

^{P}move the curve to the left and right (Figure 7A), whereas changes to the parameter

*π*

_{eq}increase or decrease curvature (Figure 7B). When

*π*

_{eq}= 0, there is no punctate component to the illumination, and the geometric discounting function is a horizontal line. Indeed, the second derivative of the function Γ(

*Ψ*;

^{T}− Ψ^{P}*π*) with respect to surface orientation

_{eq}*Ψ*

^{T}evaluated at its minimum,

*Ψ*=

^{T}*Ψ*, is, . It is evident that, given the curve , we can estimate the parameters (where the function takes on its minimum) and (the curvature at the minimum). For a Lambertian observer with possibly erroneous estimates of and , then, we can recover estimates of both of these parameters from measurements of the luminance setting ratio, Λ(so long as

^{P}*π*

_{eq}> 0).

*ϑ*,

*π*). This curve is plotted in blue in each plot. If, on the other hand, the observer bases his or her lightness estimate solely on luminance of the test patch without taking the orientation into account, then the ratio would always be constant (the horizontal black line).

*Ψ*

^{T}by asking the observer to perform the orientation task, we are left with possible errors in estimating

*π*

_{eq}and the direction of the light

*Ψ*

^{P}as explanations for patterns observed in the data. We used a maximum likelihood fitting procedure to estimate values of these parameters that best accounted for each observer’s data separately. These estimates are shown in Table 1. First note that all observers apparently underestimate the equivalent relative punctate intensity of the light (whose actual value is

*π*

_{eq}= 0.62). We tested the hypothesis that the observer’s estimate is equal to the true value by means of a nested hypothesis test (Mood, Graybill, & Boes, 1974, pp. 440). We nested the hypothesis that = 0.62 within a model in which was free to vary. We fit both models to the data by the method of maximum likelihood with other parameters allowed to vary freely. The log likelihood of the constrained model (denoted

*λ*

_{0}) must be less than or equal to that of the unconstrained model (denoted

*λ*

_{1}). Under the null hypothesis, twice the difference in log likelihoods is asymptotically distributed as a

*χ*

_{1}

^{2}-variable and we use this result to test whether the observers’ estimates were significantly different from the true value. We separately tested whether (consistent with luminance matching) by a second application of the nested hypothesis test. We tested for each observer separately with the overall Type I error set to 0.05 and a Bonferroni correction for each series of six tests.

Observer | Slope | Discounting index | ||
---|---|---|---|---|

Veridical values | 1 | 0.62 | −21.44 | 1 |

CBC | 0.85* | 0.20* p < .0001 | −29.51 p = .457 | 0.48 |

EPB | 0.89* | 0.15*p < .0001 | −57.86*p = .002 | 0.38 |

JJG | 0.84* | 0.32* p < .001 | −30.16 p = .191 | 0.68 |

LR | 0.84* | 0.30* p < .0001 | −31.46 p = .155 | 0.65 |

MM | 0.96* | 0.15* p < .0001 | −41.97 p = .038 | 0.38 |

NB | 0.63* | 0.29 p = .015 | −28.65 p = .509 | 0.64 |

We report the regression coefficients for the perceived orientation of the test patch in the second column. In columns 3 and 4, maximum likelihood estimations of the punctate to total light ratio, *^π*, and punctate light direction, , are reported. For each observer, we tested the hypothesis that and report exact *p*-values for this test when the values are larger than 0.001. With a Bonferroni correction for six tests, the significant level corresponding to an overall Type I error rate of 0.05 is 0.0083. Values whose corresponding *p*-values fall below this cutoff are marked with an asterisk. The last column reports the discounting index *DI* (Equation 16).

*p*values of the tests are reported in Table 1). We rejected the hypothesis that for all observers (

*p*< .001 in all cases). The latter result implies that observers are not simply matching the luminance values of the reference patch to that of the test patch. The former indicates that (with the exception of one observer) the observers’ estimates of the equivalent relative punctate intensity are in error.

*ψ*= −21.44°, they are all on one side of the true value, suggesting that observers share a common bias in estimating the light direction.

^{P}*π*

_{eq}. If, for example, the observer estimates to be 0, then is always 1 and the observer does not discount orientation at all. We define a geometric discounting index, that measures the match between the true value

*π*

_{eq}and the observer’s estimate . This index ranges from 0 (no discounting of orientation) to 1 (perfect discounting). The

*DI*values for all six observers are reported in Table 1. This index ignores any errors in the observer’s estimate of the punctate light direction and, together with the error in direction , these two indices provide a measure of how accurately the observer is discounting orientation in forming lightness judgments.

*Ψ*

^{P}. In contrast, we found that was significantly smaller than the true value by a factor of two or more for five observers, and smaller (but not significantly so) for the remaining observer. This result indicates that observers systematically underestimated the contribution of the punctate source to the light re-emitted by the test surface.

*observer’s geometrical discounting function*assuming the observers’ internal model of reflection from matte surfaces was the Oren-Nayar model (Nayar & Oren, 1995; Oren & Nayar, 1995). The Oren-Nayar model includes the Lambertian as a special case. We performed nested hypothesis tests for each observer with the null hypothesis corresponding to the Lambertian model and the alternative hypothesis corresponding to any non-Lambertian form of the Oren-Nayar model. We could not reject the Lambertian model for any observer.

*Ψ*

^{P}from observers’ data that was within 22° of the correct direction for five out of six observers. This result suggests that the visual system is effectively estimating information about the spatial organization of the illuminant and using it to arrive at estimates of surface albedo (see Maloney, 2002).

*π*. Recall that and an error in estimating could be the result of misestimating

_{eq}*E*

^{P}or cos

*ϕ*

^{P}or both. Because the estimates of the other angle component of the punctate light direction were not very different than the correct value (for five out of six of the observers), we can conjecture that is close to the correct value, and that is not very different than

*^π*, implying that observers perceive the lighting in the scene to have a larger component of diffuse light than it does. It would be of great interest to determine what cues in the scene influence the estimates of illuminant properties such as punctate source direction and relative punctate intensity.