We report the results of three experiments in which observers judged the albedo of surfaces at different locations in rendered, three-dimensional scenes consisting of two rooms connected by a doorway. All surfaces composing the rooms were achromatic and Lambertian, and a gradient of illumination increased with depth. Observers made asymmetric albedo matches between a standard surface placed in the rooms at different depths along the line of sight and an adjustable surface at a fixed location. In Experiment 1, gradients of intensity on the walls, floor, and ceiling of the scene, as well as its three-dimensional structure, provided information about variations in the intensity of illumination across depth (the illumination profile). In Experiment 2, specular spheres provided an additional veridical cue to the illumination profile. We sought to determine whether observers would make use of this additional cue. They did: all observers exhibited a greater degree of lightness constancy in Experiment 2 than in Experiment 1. In Experiment 3, the specular spheres reflected an illumination profile in conflict with that signaled by the other cues in the scene. We found that observers chose albedo matches consistent with an illumination profile that was a mixture of the illumination profiles signaled by the specular spheres and by the remaining cues.

*Lambertian,*so that it absorbs the light which falls upon it and reradiates this light uniformly in all directions (Lambert, 1760). If the surface changes location and orientation within the scene and the light falling upon it changes, the luminance radiated toward the observer will also change. From the perceptual point-of-view of the observer, how will the surface's appearance change? Will the gray surface appear darker or lighter or remain the same?

**ν**(Figure 2). The plenoptic function of Adelson & Bergen (1991),

**υ**. We can write the total light incident upon the Lambertian surface as

**ν**. In limiting the area of integration to the hemisphere

*illumination profile*of the scene.

**ν**.

**ν**, but not surface location,

**ν**held constant?

^{1}The patchwork designs of the near and far Mondrians were identical except that the far Mondrian's surfaces were 500% more luminous than corresponding surfaces in the near Mondrian.

^{2}

^{3}in depth along the line of sight in rendered scenes, viewed binocularly. In all experiments, the illumination was akin to that realized in Figure 1—intense in the far room, dim in the near room—and gradients of intensity on the Lambertian walls, floor, and ceiling of the scene, as well as its three-dimensional structure, provided cues to the illumination.

^{4}We achieved a three-dimensional percept by rendering the scene twice using two viewpoints corresponding to the locations of the observer's eyes in the experimental apparatus (Figure 8; details to follow). The entire stimulus subtended 8.6 degrees of visual angle (dva).

*Radiance*(Larson & Shakespeare, 1996), an open-source software package that uses 6-point ray tracing to generate models of illuminated scenes. Given a set of surfaces and sources of illumination,

*Radiance*computes a three-dimensional model of the scene, which the experimenter may render from any viewpoint. In our scene, all surfaces were Lambertian and all light sources were diffuse. The output of

*Radiance*was a specification of normalized luminance at each pixel in the left and right images of the stereo pair.

*Radiance*to the reference surface. We repeated this procedure with the reference surface placed at 19, equally spaced depths along the line of sight. We denote these relative luminance values by

*d*. Figure 4b plots these values,

^{5}normalized so that the value at the far wall is 1, as a function of relative depth. We defined the depth at the scene's edge nearest to the observer as zero and the depth at the far wall as 10. We refer to the function of illumination over depth (the solid black curve in Figure 4b) as the relative illumination profile along the line of sight. Starting at the far wall, the relative illumination profile,

^{6}

^{2}) of the reference surface at each depth using a Photo Research PR-650 SpectraScan spectrophotometer. We placed the spectrophotometer at the location of the observer's right eye and measured the light that would reach the eye from the center of the reference surface within the scene on the right monitor (Figure 4c). (Recall that the observer never saw the reference surfaces during the experiment.) We thrice repeated this measurement for each of seven different depths (marked in Figures 4a–b) assumed by the reference surface. We then repeated these measurements for the left eye. In Figure 4c, we plotted the measurements of the seven reference surfaces in the right eye against the relative luminance values assigned to them by

*Radiance.*The points fall along a straight line through the origin, indicating that the luminance values of regions on the screen were a fixed scaling of those assigned by

*Radiance*. The results for the left eye were equivalent.

^{2})

^{7}across trials, and we were interested in whether this constant-luminance surface took on different apparent albedos at different depths. We asked observers to set the lightness of the adjustable surface to match that of the standard surface presented at one of seven depths along the line of sight passing through the center of both rooms. The seven locations at which the standard surface could appear (marked locations in Figures 4a–b) were a subset of the 19 locations where we measured the luminance profile using reference surfaces (previous section). We randomized the depth at which the standard surface appeared from trial to trial.

^{2}.

*i*. This outcome would indicate that the observer is simply discounting the same EIP at every depth. We therefore would conclude that the observer does not discount an illumination profile that varies in depth. In other words, while this observer is expert at matching luminance values, she does not use the information about depth provided in the scene. These perfect and no lightness constancy models define two extremes of a continuum of possible models. A third and likeliest possibility is that observers exhibit some amount of lightness constancy. Figure 9 displays a discrete sample of possible models. The dashed line at the

*y*-axis value of 1 is the no lightness constancy model. The solid curve is the true relative illumination profile or perfect lightness constancy model. β is the scaling parameter, and the dotted curves are copies of the perfect lightness constancy model, scaled by β values ranging from 0 to 1. We note that this analysis is similar in spirit to that of Brunswik (1929). (See also Gilchrist, 2006, and Thouless, 1931.). We are approximating the observer's performance as a weighted mixture of a “no constancy” profile and a “perfect constancy” profile. One can interpret a β value as one would interpret a Brunswik ratio.

*p*< .01). We then performed two-tailed

*t*tests for each observer's data for depth conditions 1–6

^{8}versus the null hypothesis that the observer's true normalized EIP falls on the line

*t*tests for each observer at each depth, testing whether the observer's settings were significantly different from the relative illumination profile,

*p*< .01). We performed two-tailed

*t*tests for each observer's data for depth conditions 1–6, versus the null hypothesis that the observer's mean setting is not different from 1. We rejected the null hypothesis (with 99% confidence intervals) for all six depths (again, we excluded the fixed point at the far wall) for all five observers.

*t*tests for each observer at each depth, testing whether the observer's settings were significantly different from the relative illumination profile,

*y*-intercept equal to 1 at the far wall. We used a bootstrap estimation technique to estimate

*R*

^{2}statistic to summarize the variability in the data. An

*R*

^{2}equal to 1 indicates that modeling the data as a scaled copy of the illumination profile can account for all of the variance in the data, whereas an

*R*

^{2}equal to 0 indicates that this model can account for none of the variance in the data.

*R*

^{2}(Table 1) values for five observers for Experiments 1 and 2. Figure 12 plots

*The Mother*'

*s*subtle, complex illumination by concealing the light sources and rendering their reflection from surfaces shiny, like the checkerboard floor tiles, to matte, like the mother's skirts. Arranged in pictorial perspective, these surfaces accurately communicate the illumination profile of the scene to the observer.

^{1}Land & McCann (1971) coined this term, eponymous to Piet Mondrian, the early twentieth-century Dutch painter who experimented with compositions of the most basic elements: line and color.

^{2}We duplicated the lightness condition of the first experiment of Schirillo and Shevell (1993) as a pilot study in designing the experiments reported here. Four naïve observers made lightness matches to stimuli of constant luminance presented at different depths. We found no effect of depth on perceived lightness of these constant luminance stimuli, substantially in agreement with the results of Schirillo and Shevell for naïve observers. We did not ask observers to estimate brightness.

^{5}Note that the relative illumination profile (Figure 4b) has its highest value at the far wall. To those unfamiliar with the properties of Lambertian surfaces, the reason for this phenomenon may be non-intuitive. One may ask why the relative illumination profile does not achieve its highest value in the vicinity of the hidden light sources, and then drop off in the direction of the far wall. The answer lies in the fact that it is not the distance to the light source that parameterizes the relative illumination profile but instead the cosine of the angle between the vector,

**υ**, of the incident light and the vector,

**ν**, of the surface normal. At the far wall, the value of the cosine is maximized and therefore so is the value of the relative illumination profile.

^{6}We designed the luminance profile to be as comparable to the 5:1 luminance ratio of Schirillo & Shevell (1993) as possible.