We examined the effect of perceived orientation on the perceived color of matte surfaces in rendered three-dimensional scenes illuminated by a blue diffuse light and a yellow punctate light. On each trial, observers first adjusted the color of a matte test patch, placed near the center of the scene, until it appeared achromatic, and then estimated its orientation by adjusting a monocular gradient probe. The orientation of the test patch was varied from trial to trial by the experimental program, effectively varying the chromaticity of the light mixture from the two light sources that would be absorbed and reemitted by a neutral test patch. We found that observers’ achromatic settings varied with perceived orientation but that observers only partially discounted orientation in making achromatic settings. We developed an equivalent illuminant model for our task in which we assumed that observers discount orientation using possibly erroneous estimates of the chromaticities of the light sources and/or their spatial distribution. We found that the observers’ failures could be explained by two factors: errors in estimating the direction to the punctate light source and errors in estimating the chromaticities of the two light sources. We discuss the pattern of errors in estimating these factors across observers.

*λ*is denoted by

*E*

_{P}(

*λ*) and the spectral power distribution of the diffuse light by

*E*

_{D}(

*λ*). The angle between the punctate light direction and the surface normal

**n**is denoted by

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

*v*. In the Lambertian model, the intensity of emitted light does not depend on the direction to the viewer, so long as the viewer and the light source are on the same side of the surface. When this condition is satisfied, the intensity of the light reflected from an achromatic Lambertian surface at any wavelength

*λ*. is given where

*α*is the wavelength independent albedo (reflectance) of the achromatic surface. The spectral power distribution of the effective illuminant

*E*(

*λ*) is a weighted mixture of the spectral power distributions of the diffuse and punctate sources and the mixture changes as the orientation of the test patch changes. If the diffuse and punctate sources differ in chromaticity, then the chromaticity of the mixture will also change as a function of angle.

*coplanar ratio hypothesis*).

*ψ*,

*φ*,

*r*) to specify a simulated scene (Figure 4). This coordinate system has the origin at the center of the test patch. The spherical coordinate system (

*ψ*,

*φ*,

*r*) and the Cartesian coordinate system underlying it are explained in the figure legend.

*E*

^{R},

*E*

^{G}, and

*E*

^{B}, respectively, and refer to the tristimulus coordinates (

*E*

^{R},

*E*

^{G},

*E*

^{B}that describe the light at a particular location on the monitor as an RGB code.1 In making an achromatic setting, the observer in effect selects the RGB code for the test patch that makes it appear to be an achromatic surface, as described in more detail below. We report the u’v’ chromaticities (Wyszecki & Stiles, 1982, p. 165) of the guns (and, therefore, of the primaries) in the “Calibration” section below.

*E*

^{R},

*E*

^{G},

*E*

^{B}) strikes a Lambertian surface with RGB code

*S*

^{R},

*S*

^{G},

*S*

^{B}), the light emitted from the surface is assigned the RGB code (

*E*

^{R}×

*S*

^{R},

*E*

^{G}, ×

*S*

^{G},

*E*

^{B}×

*S*

^{B}, scaled by a factor that takes into account the orientation of the surface with respect to the light (see the discussion leading up to Equation 1). Yang and Maloney 2001; Maloney, 1999) point out that this rendering interpretation (“the RGB heuristic” in Maloney, 1999) does not always lead to accurate simulation of light-surface interactions.

*α*,

*α*,

*α*), then typical rendering packages will simulate light-surface interaction correctly. So long as our chromatic lights interact with only neutral surfaces, the resulting RGB codes assigned to the light reemitted will be accurate. There are other surfaces in our scenes that are rendered, but the RGB codes of these surfaces are assigned at random and change from trial to trial. Consequently, errors in rendering, due to using the RGB heuristic, are of no consequence. The intended random color assigned to a surface is just replaced by a different random color.

*cd*/

*m*

^{2}. To test the linear additivity for a monitor, first we measured the isolated spectrum of each gun alone, set to about half of its maximum intensity. Then we measured the spectra of each pair of guns simultaneously set to half of their maximum intensities and compared it to the sum of the isolated spectra for each gun in the pair. Last, we measured the spectrum with all three guns set to half of their maximum intensity and compared it to the sum of the isolated spectra for all three guns. We plot the results of this last test in Figure 5 for both monitors. The red, green, and blue solid lines are the isolated spectra, the gray solid line is the sum of the three isolated spectra, and the black dashed line is the measured spectra when all three guns were simultaneously set to half of their maximum intensities. The curves agree to within 7% or better at each point in the spectrum, for both monitors. The test of additivity for pairs of guns also agreed within 7% or less. The u’v’ chromaticity coordinates (Wyszecki & Stiles, 1982, p. 165) for the three primaries are: red (.409,.519), green (.117,.565) and blue (.157,.196) for the left monitor, and red (.430,.528), green (.115,.564), and blue (.160,.189) for the right monitor. The u’v’ chromaticity coordinate for the mixture of all three guns at half intensity was (.176,.460) for the left monitor and (.172,.455) for the right monitor.

*E*

_{D}/

*E*

_{P}, the diffuse-punctate ratio. The r- and g-chromaticities of the yellow punctate source were always equal, as were those of the diffuse source. The values used in rendering were

*π*

_{B}= 0,

*δ*

_{B}= 0.66, and and δ = 0.37. In other words, the punctate source had no blue component (

*π*

^{B}= 0), the diffuse source was mostly blue (

*δ*

^{B}= 0.66), and the ratio of the intensity of the diffuse source to the intensity of the punctate source was 0.37 (δ = 0.37). The punctate source was always behind and above the observer, and either to his RIGHT or to his LEFT at (

*ψ*

_{P},

*φ*

_{P},

*r*

_{P}) = (±15°, 30°, 670

*cm*) (

*ψ*

_{P}= +15° for RIGHT,

*ψ*

_{P}= −15° for LEFT; Figure 4). The position of the punctate source was varied only from session to session, but in a single session, its position was kept constant. The punctate source was sufficiently far from the test patch so as to treat its light rays collimated. The vector

**p**= (cos

*φ*

_{P}sin

*ψ*

_{P}, sin

*φ*

_{P}, cos

*φ*

_{P}cos

*ψ*

_{P}is a unit vector pointing from the test patch toward the punctate light source (Figure 4).

*ψ*direction (

*ψ — rotation*) or in only the

*φ*direction (

*φ — rotation*). The test patch measured 4.8 cm by 3.6 cm; its center was always 70 cm away from the observer along the observer’s line of sight. The orientation of the test patch was specified by (

*ψ*

_{T},

*φ*

_{T}), and its surface normal was

**n**= (cos

*φ*

_{T}sin

*ψ*

_{T}, sin

*φ*

_{T}, cos

*φ*

_{T}cos

*ψ*

_{T}). After a

*ψ — rotation*(

*φ*

_{T}= 0),

*φ*

_{T}could take any of the values {−60°, −45°, −15°, 15°, 45°} when the punctate source was on the LEFT, and any of the values {−45°, −15°, 15°, 45°, 60°} when the punctate source was on the RIGHT. After a

*φ — rotation*(

*ψ*

_{T}= 0),

*φ*

_{T}could take any of the values {0°, 15°, 30°, 45°, 60°}. Figure 6 shows a schematic drawing of the two kinds of rotations.

*ψ — rotation*trial, the probe could rotate only in the

*ψ*direction; if it was a

*ψ — rotation*, it could rotate only in the

*φ*direction. Observers reported no difficulty with setting the probe and were unaware that it was visible in only the right eye. Once the observer was satisfied with the setting, he or she clicked the left button on the computer mouse to finalize the task. The purpose of this task was to control for the possibility that observers’ perceptions of orientation of the test patch were so different from its actual orientation that it would affect the interpretation of the results. We assume that the observer is using the same cues to test patch orientation during the achromatic setting task as in the orientation task. In these scenes, these cues include binocular disparity and linear perspective. See Landy, Maloney, Johnston, and Young (1995) for a review of cue-combination models for depth and slant.

**geometric b-chromaticity function**. In Equation 5,

*E*(

*θ*) is the total intensity of the light emitted from the test patch.

*E*

^{B}(

*θ*) is the blue component of the RGB code of the light emitted from the test patch, as defined earlier. The last term in Equation 5 is gotten by substituting Equation 1 and Equation 2 into the middle term. Equation 5 is the relative intensity of the blue primary in the light emitted from the test patch. We define a geometric g-chromaticity function Λ

^{G}(

*θ*) and a geometric r-chromaticity function Λ

^{R}(

*θ*), analogously, and refer to them collectively as geometric chromaticity functions. Note that when the light sources have the same b-, r- and g-chromaticities, the geometric chromaticity functions are all constant, independent of

*θ*.

*θ*. The observer is asked to adjust the chromaticity of the test patch without changing the total intensity until it looks achromatic. We denote this achromatic setting as a function of

*θ*, by (). Note that his setting is always constrained so that , and it is convenient to express the achromatic setting in terms of chromaticities.

*ψ*and

*φ*direction had an effect on observers’ orientation settings by separate ANOVAs for each observer. We rejected the hypothesis that the mean orientation setting did not vary with orientation for all observers, for both directions (

*p*< .0001 in both

*ψ*and

*φ*directions). With the exception of subject MM in the

*ψ*direction, we found no significant interaction between perceived test patch orientation and punctate light source position (LEFT or RIGHT) for both directions (

*ψ*direction:

*p*= .206, .637, and .304 for BH, MD, and RG, respectively,

*p*= .01 for MM;

*φ*direction:

*p*= .852, .39, .07, and .928 for BH, MD, MM, and RG, respectively). This implies that for all but one observer the position of the light source (LEFT or RIGHT) had no significant effect on how observers made their orientation settings.

*ψ*and

*φ*( vs.

*ψ*

_{T}and vs.

*φ*

_{T}) directions. We have plotted the best-fitting regression lines to BH’s settings in Figure 8. We report the regression coefficients for similar fits for all observers in Table 1. The estimated regression coefficient

*a*(intercept) is in units of degrees; the regression coefficient

*b*(slope) is unitless. We report

*p*values for hypothesis tests against the corresponding veridical value (0 for

*a*, 1 for

*b*). In the

*φ*direction, slopes were significantly different from 1 for ject MD (punctate on RIGHT:

*p*< .001) and ject MM (RIGHT:

*p*< .001). All other subjects’ slopes in this direction were not significantly different from 1 for both punctate source positions (LEFT:

*p*= .934, .037, and .148 for BH, MM, and RG, respectively; RIGHT:

*p*= .834, .01, and .125 for BH, MD, and RG, respectively). The intercepts in the

*φ*direction were significantly different from 0 for all observers (

*p*< .001) except for observer RG (LEFT:

*p*= .782, RIGHT:

*p*= .042).

Punctate source position: LEFT | Punctate source position: RIGHT | |||||||
---|---|---|---|---|---|---|---|---|

A | b | A | b | a | b | a | b | |

Veridical | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 |

BH | 3.33* | 1.06 | 4.83* | 0.98 | 3.18* | 1.06 | 7.42* | 0.96 |

p < .001 | p = .009 | p < .001 | p = .934 | p < .001 | p = .005 | p < .001 | p = .834 | |

MD | −2.33* | 1.14* | 4.72* | 1.14* | -2.54* | 1.1* | 3.55* | 1.08 |

p = .002 | p < .001 | p < .001 | p < .001 | p = .002 | p < .001 | p = .003 | p = .001 | |

MM | −0.55 | 0.83* | −6.59* | 0.95 | −1.76 | 0.76* | −4.6* | 0.87* |

p = .398 | p < .001 | p < .001 | p = .037 | p = .038 | p < .001 | p < .001 | p < .001 | |

RG | −4.79* | 0.71* | 0.38 | 0.95 | −5.7* | 0.66* | −1.68 | 0.97 |

p < .001 | p < .001 | p = .782 | p = .148 | p < .001 | p < .042 | p < .125 |

*ψ*direction slopes were significantly different from 1 for all observers (

*p*< .001), except observer BH (LEFT:

*p*= .009, RIGHT:

*p*= .005). The intercepts in the

*ψ*direction were significantly different from 0 for all observers (

*p*< .002) except MM (LEFT:

*p*= .398, RIGHT:

*p*= .038).

*π*

_{b},

*δ*

_{B}and δ were defined above. An observer’s visual system can compute what the b-chromaticity of a gray surface should be if estimates of the parameters in Equation 7 are available. However, if the observer’s estimates of the parameters are in error, the achromatic settings would differ from the predicted ones. Let , and denote the observer’s estimates of the parameters in Equation 5, then The observer’s estimate of the angle of incidence depends on his or her estimates of the orientation of the test patch () and the direction to the punctate source () through Equation 4. Note that the observers explicitly estimated the orientation of the test patch by performing the orientation task.

*φ*

_{T}as we do in the experiment. Figure 9a shows the geometric b-chromaticity function Λ

^{B}plotted with respect to the angles

*ψ*

_{T}and

*φ*

_{T}assuming the veridical values of the lighting parameters, Θ. Now suppose that the estimates of the lighting parameters are in error, what kind of distortions would those errors introduce? Misestimating the direction to the punctate source shifts both curves without much effect on their curvatures (Figure 9b). When the test patch is oriented such that it faces the punctate source as directly as possible, that is

*ψ*

_{T}=

*ψ*

_{P}(after a

*ψ*rotation) or (after a

*φ*rotation), it receives the maximum possible amount of light from the yellow punctate source. However the blue content of the mixture of light falling on it remains fixed, hence the b-chromaticity, λ

^{B}, assumes its minimum.

^{B}are found by taking its derivative with respect to ψ

^{B}and

*φ*

_{T}and then equating it to zero, which (For

*ψ*

_{P}= ±15,

*φ**

_{T}= 30.87°, only slightly different from

*φ*

_{P}= 30°.) Note that

*ψ**

_{T}and

*φ**

_{T}correspond to minima for

*π*

^{B}<

*δ*

^{B}, and to maxima for

*π*

^{B}>

*δ*

^{B}(see Equation 13 below). If the model observer misestimated the punctate source direction, his or her achromatic settings would reveal this because the corresponding geometric b-chromaticity function would shift and have its minimum at roughly the estimated direction to the punctate source ().

*π*

^{B},

*δ*

^{B}, and δ would shift the curves up or down and increase or decrease their curvatures (Figure 10). If the observer’s estimates of the b-chromaticity of the punctate source and diffuse source were the same (), then the geometric b-chromaticity function would be a constant, because changing the orientation of the test patch would not affect the overall chromatic balance of the light reaching the patch. would be constant also when or , that is, if the observer estimates that the scene is illuminated either by only a punctate source or by only a diffuse source. However, because veridical values are such that

*π*

^{B}≠

*δ*

^{B}and Δ is not 0 or infinity, should we find that the observer’s geometric b-chromaticity function is constant, then the implication is that the observer does not discount the perceived orientation of the test patch for its color.

^{B}. On the other hand, if the observer were completely ignoring the orientation of the test patch in his or her achromatic judgment, then the ratio would be constant.

^{B}curves in Figure 9 and Figure 10 suggests that observers make settings that are indistinguishable from those of a Lambertian color constant observer who discounts the perceived orientation for estimating color, but who does so using incorrect estimates of the lighting parameters Θ.

*ψ*

^{T}and

*φ*

^{T}.) These estimates are reported in Table 2.

Punctate source position: LEFT | Punctate source position: RIGHT | |||
---|---|---|---|---|

Veridical | (−15,30) | (0, 0.66,0.37) | (15, 30) | (0, 0.66,0.37) |

BH | (−23.1, 36.6) | (0, 0.33, 1.39)* | (18.4, 36.1) | (0.1, 1, 0.12) |

p = .011 | p < .001 | p = .007 | p = .036 | |

MD | (−22.4, 28.8) | (0.1, 0.38, 1.87)* | (15.2, 38.8) | (0, 0.4, 1.6)* |

p = .63 | p < .001 | p = .04 | p < .001 | |

MM | (−15.8, 39.3) | (0.2, 1, 0.03* | (20.3, 37.9) | (0.2, 1, 0.03)* |

p = .734 | p < .001 | p = .426 | p < .001 | |

RG | (−54.4, 39.5) | (0.2, 0.31, 2.15)* | (5.6, 37.4) | (0.2, 1, 0.05)* |

p = .029 | p < .001 | p = .016 | p < .001 |

*λ*

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

*λ*

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

*X*

^{2}

_{2}-variable if We use this result to test whether observers’ estimates were significantly different from the true values. We summarize the values of the observers’ estimates in Table 2 along with the corresponding

*p*values. None of the observers’ punctate source direction estimates were significantly different from their true values under all conditions. Observers’ azimuth and elevation estimates of the punctate source direction were within 11 deg of the true direction with one exception (the azimuth estimate for RG with the light on the LEFT; See Table 2). We replot the azimuth and elevation estimates in Table 2 as Figure 12. It is then readily seen that the estimates are clustered around the true values (with one exception, RG, LEFT) and that the estimates are typically too large. We have, in effect, derived a crude estimate of the position of the light from the observer’s performance.

*p*< .0001 in all cases). We conclude that the observers’ achromatic settings are affected by changes in test patch orientation.

*π*

^{B}= 0,

*π*

^{B}= 0.66 and δ = 0.37. We nested the hypothesis that the parameters were equal to the true values within a model in which they were free to vary. We rejected the hypothesis that for all observers except observer BH for the punctate-on-the-right condition (

*p*= .036; all other

*p*values are reported in Table 2).

*π*

^{B}= 0), and misestimated the b-chromaticity of the diffuse source

*δ*

^{B}and the diffuse-punctate balance Δ. The values are reported in Table 2. It is as if observers are discounting the orientation of the test patch for the achromatic task consistent with the physically correct model but using incorrect estimates of the lighting parameters.

*DI*is a measure of how much the observer’s geometric b-chromaticity function is curved compared to the theoretical one. A complete lack of curvature () corresponds to the case where the observer’s achromatic settings are unaffected by angle. Then

*DI*= 0, and we would conclude that the observer’s color estimates are not affected by perceived surface orientation. If, in contrast, then

*DI*= 1. Note that

*DI*is a composite measure of all the parameters in the parameter space . However, because the estimated parameters were close to veridical, the errors are effectively due only to the misestimated chromaticity balances of the punctate source and diffuse source and the overall diffuse-punctate balance. The values of

*DI*varied between 0.29 and 0.82 (

*DI*= 0: no discount;

*DI*= 1: perfect discount). The discounting indices are reported in Table 3 and in the legends of Figure 11.

Discounting index, DI | ||
---|---|---|

Punctate source position: LEFT | Punctate source position: RIGHT | |

BH | 0.78 | 0.8 |

MD | 0.69 | 0.82 |

MM | 0.29 | 0.27 |

RG | 0.2 | 0.43 |

^{1}We use the term RGB code as a convenient synonym for the tristimulus coordinates based on the three linearized sources (‘guns’) of the monitor (Wyszecki & Stiles, 1982).

^{2}The RGB code is the tristimulus coordinates with respect to the three linearized sources of the monitor and we define the r-, g- and b-chromaticities e.g. b = B/R + G + B) in the usual manner (Wyszecki & Stiles, 1982).

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