We investigate how human observers make use of three candidate cues in their lightness judgments. Each cue potentially provides information about the spatial distribution of light sources in complex, rendered 3D scenes. The illumination (lighting model) of each scene consisted of a punctate light source combined with a diffuse light source. The cues were (1) cast shadows, (2) surface shading, and (3) specular highlights. Observers were asked to judge the albedo of a matte grayscale test patch that varied in orientation with respect to the punctate light source. We tested their performance in scenes containing only one type of cue and in scenes where all three cue types were present. From the results, we deduced how accurately they had estimated the spatial distribution of light sources in each scene given the cues available. In Experiment 1, we established that all of the individual cues were used in isolation. We showed that the highlight and cast shadow cues in isolation were used by more than half of the observers. We could reject the hypothesis that the observers did not make use of the shading cue for only one observer. In Experiment 2, we showed that the observers combined information from multiple cues when all three cues were presented together.

*ϒ*= (

_{P}*ψ*) (azimuth and elevation), and the ratio of the punctate light to total light

_{P}, ϕ_{P}*ϒ*and

_{P}*π*. If we find that multiple cues are utilized by the visual system in a scene, we wish to test whether the visual system combines them to arrive at less variable estimates of the parameters of the lighting model (Ernst & Banks, 2002; Landy, Maloney, Johnston, & Young, 1995; Oruç, Maloney, & Landy, 2003).

^{1}matte poles and floor were removed (Figure 4A). In the shading condition, everything except the Lambertian cubic objects was removed (Figure 4B). In the highlights condition only the glossy spheres were present (Figure 4C). In the all cues condition, we presented the scene with all cues available (Figure 4D).

^{2}. The computer used in the experimental apparatus was a Dell Workstation with an Nvidia dual-head graphics card that controlled both monitors. The experimental program ran under Fedora Core Linux, Version 3 (for details, see Boyaci et al., 2003).

*ψ*of the normal to the test patch surface (elevation

_{T}*ϕ*= 0 remained constant). The values

_{T}*ψ*could take on were −60, −45, 0, 45, and 60 deg. The luminance of each test patch was kept constant independent of its orientation. Four different luminance values were interleaved randomly to avoid possible artifacts. These luminance values were 0.44, 0.50, 0.56, and 0.61 proportion of the maximum possible luminance for the computer monitors (115 cd/m

_{T}^{2}).

^{2}The distance between the test patch and the punctate source was 670 cm (580 cm front, 335 cm above). The actual direction to the punctate source was

*ϒ*= (

_{P}*ψ*) = (0, 30 deg). The position of the punctate source was never varied during the experiment and its location was sufficiently far from the test patch so that it could be treated as a collimated source across the extent of the test patch. The actual punctate-total light ratio was

_{P}, ϕ_{P}*π*= 0.67.

^{2}).

*M*= [

*E,*

*ϒ*

_{P},*π*] is given by

*L*is its luminance and

*Γ*(

_{M}*ϒ*) is a geometric correction function,

_{T}*θ*(

*ϒ*

_{T},*ϒ*) is the angle between the normal to the surface and direction to the punctate light source. Intuitively, the geometric correction function is the amount that the “ideal albedo observer” must “correct” perceived albedo to compensate for the effect of orientation. Furthermore, cos

_{P}*θ*is given by

*setting ratio*

*C*characterizes the combined effect of the unknown absolute total light

*E*in the scene and the unknown lighting model of the matching chip (for more details, see Boyaci et al., 2003).

*Λ*plotted versus

*ψ*. If the lightness constant Lambertian observer's visual system had perfect estimates of

_{T}*ϒ*and

_{P}*π,*then his or her geometric correction function should have the same shape as the actual geometric function, up to an unknown scaling factor

*C*. Conversely, given the form of the observer's geometric correction function, we can compute the estimates of

*ψ*shift the function to the right or to the left. Misestimates of punctate-total balance,

_{P},*π,*change the curvature of the function (Figure 5). As explained in 2, all analyses were carried out using a transformation of the parameter

*π*. We retain

*π*in the text because it is readily interpreted as the punctate-total ratio defined above.

*Λ*, for all four test patches of different luminances at each orientation. The solid line is the best fit obtained by a maximum likelihood estimation analysis. We fit Equation 4 to the observer's data by the method of maximum likelihood (Mood, Graybill, & Boes, 1974, pp. 276 ff) under the assumption that the observer's settings are perturbed by Gaussian error with mean 0 and variance

*σ*

^{2}.

*π*= 0 versus the alternative that

*π*≠ 0 by a nested hypothesis test (Mood et al., 1974, pp. 440–442) as follows. We first obtain maximum likelihood estimates of the parameters

*π*= 0 (the null hypothesis) and without fitting the parameter

*ψ*(in the

_{P}*π*= 0 case, the observer's geometric correction function is a horizontal line, hence there is no minimum, or in other words the azimuth of the punctate source is not defined because there is no punctate source).

*λ*denotes the ratio of the maximum likelihood achieved in fitting the unconstrained and the constrained models, then −2 log

*λ*is approximately distributed as a

*χ*

_{2}

^{2}distribution under the null hypothesis (Mood et al., 1974, pp. 440–442). For a test of size

*p,*we reject the null hypothesis

*π*= 0 if and only if −2 log

*λ*is greater than the 1 −

*p*quantile of the

*χ*

_{2}

^{2}distribution.

*ψ*is marked by a black dotted line segment. The true value

_{P}*π*= 0.67 lies outside the plot.

*π*in all conditions in agreement with the results in previous experiments with rendered scenes (Boyaci et al., 2003, 2004) and real scenes (Ripamonti et al., 2004). It can be seen that the estimates of

*ψ*cluster around the true value in all conditions. Observers perceive the direction to the hidden punctate light source (nearly) accurately but underestimate its intensity relative to the diffuse source.

_{P}*p*values of the nested hypothesis test described above for each condition and subject. If, in any one cue condition, the observers made no use of the cues provided, then these

*p*values would be distributed as uniform random variables in the unit interval [0,1]. We use this fact to test the hypothesis that none of the observers is using a specific cue by computing the logarithm of the geometric mean of the

*p*values (log

_{10}

*p*values for this omnibus test statistic in the last column of Table 1.

Cue condition | Observers | ||||||
---|---|---|---|---|---|---|---|

MS | CD | SH | SRH | KP | SB | Overall | |

All cues | .023 | .0002 | .002 | .000003 | .002 | .58 | <.0001 |

Cast shadows | .006 | .0072 | .02 | .0002 | .08 | .078 | <.0001 |

Shading | .02 | .0572 | .26 | .5 | .233 | .78 | .0495 |

Highlights | .24 | .13 | .001 | .022 | .78 | .44 | .0023 |

*p*values for individual subjects suggests that one subject (SB) made use of no cues and we cannot claim that he or she is correcting albedo estimates for orientation in any of the conditions. It is plausible that some subjects are making use of at least two of the distinct cues, which we further examine in the next experiment.

^{2}). They could take on the same orientations

*ψ*= {−60, −45, 0, 45, 60 deg}.

_{T}*ϒ*= (

_{P}*θ*) = (±45, 30 deg). The position of the punctate source did not vary within a block. Punctate-total ratio was

_{P}, ϕ_{P}*π*= 0.85.

*p*< .0005) against changes in orientation were chosen (it is worth noting that nearly all 20 observers corrected their lightness settings for the test patch orientation in the all cues conditions at

*p*= .05 level). An analysis of cue combination is more powerful for observers who correct their lightness estimates more in the all cues condition. For instance, data from an observer like SB in Experiment 1 would not be suitable to study cue combination because that observer did not use any of the cues. Only subjects who met criterion ran in the remaining experimental (single cue) conditions.

*ψ*

_{P},*π*) for both observers.

*π*= 0 (no correction) is rejected except for IB in the highlights condition under the light from the right and for KD in the highlights condition under the light from the right. We conclude that each of the individual cues and combined cues is used by all observers.

*ψ*and

_{P}*π*that is more reliable than the estimates based on the cues in isolation?

*r*

_{i}=

*σ*

_{i}^{−2}is defined as the reliability of that cue (terminology due to Backus & Banks, 1999). If the cues are combined according to Equation 6, then the reliability of the combined estimate is the sum of individual reliabilities

*ψ*based on multiple cues.

_{P}*r*

_{all}denotes the observer's performance with all cues present. This condition (see Oruç et al., 2003) is the minimum condition that an observer's performance should satisfy if we are to claim that s/he is combining cues effectively and not, for example, using one cue on some trials and another on the remainder. We refer to Equation 7 as the effective cue combination hypothesis. This condition is strictly weaker than the optimal cue combination hypothesis, which requires

*r*

_{all}=

*r*

_{0}. We estimated reliabilities for each observer in each cue condition by bootstrap simulation (Efron & Tibshirani, 1993). The resulting estimates are given in Table 2 (for

Cue condition | Observers | |||
---|---|---|---|---|

KD—L | KD—R | IB—L | IB—R | |

All cues | 33.8 | 19.1 | 53.3 | 36.1 |

π^ | 25.1 | 13.4 | 26.6 | 31.4 |

ψ^—Predicted optimum | 59.3 | 23.1 | 57.5 | 51.1 |

Cast shadows | 21.7 | 9.6 | 12.9 | 19.7 |

Shading | 25.1 | 13.4 | 26.6 | 31.4 |

Highlights | 12.5 | 0.02 | 18 | 0.02 |

z score (1) | −4.01 | −5.10 | −8.38 | −2.00 |

z score (2) | 10.04 | 3.30 | 1.25 | 6.08 |

Cue condition | Observers | |||
---|---|---|---|---|

KD—L | KD—R | IB—L | IB—R | |

All cues | .016 | .015 | .027 | .008 |

r^o | .010 | .008 | .009 | .009 |

ψ^P—Predicted optimum | .025 | .013 | .018 | .022 |

Cast shadows | .011 | .008 | .003 | .009 |

Shading | .010 | .004 | .005 | .007 |

Highlights | .005 | .002 | .009 | .005 |

z score (1) | −4.90 | −8.22 | −10.86 | 2.83 |

z score (2) | 8.56 | −2.28 | −5.98 | 19.18 |

*π*and

*ψ*estimates. We compared

_{P}*z*score

*z*scores, we tested the hypothesis that

*z*scores suggest that the

*π*estimates. The

*z*scores are shown in Tables 2 and 3.

*π*estimate.

*π*(punctate-total balance) and

*ϒ*(light source direction). Our argument and analyses would not be affected by a change of parameterization (2).

_{P}*p*value, that a particular observer used a particular cue. We concluded that, in the scenes containing only one cue type, at least some observers used each of the cues to estimate the lighting model in Experiment 1. In contrast, we reject this null hypothesis for the shading cue for only one observer. This outcome may simply indicate that the shading cue elements we provided (“boxes”) are not very good shading cues.

*p*value for a test of the hypothesis that this observer is using the cue(s) available, reported in Table 1. It can be seen that several of these values are very small. If we tested each of these values against the .05 level, we would simply record whether the exact value was less than or equal to .05 (reject) or greater than .05 (not reject). We wanted to combine this information. We made use of the following: if the null hypothesis was true, then the distribution of each of these exact

*p*values would be uniform on the interval [0,1]. That is, we would expect the exact

*p*value to be greater than .5 one half of the time.

*ζ*under the null hypothesis that none of the observers is making use of the cue. If this hypothesis was true for any cue condition, we would expect the computed value

*ζ*to fall above the 5th percentile of the histogram with probability .95. If, however, the

*p*values for the test tend to be small for some of the observers, then the test statistic

*ζ*will tend to be too small, falling in the extreme left tail of the histogram. The value of the test statistic

*ζ*for each cue condition is marked with vertical red lines. We reject the null hypothesis for all cue conditions at the .05 level. We can report the exact

*p*value of the test statistic

*ζ*for each cue condition, just as we can for any test statistic whose distribution under the null hypothesis is known. These values are reported in the rightmost column of Table 1.

*SD*) could include values outside the range [0,1] that are not physically possible. To avoid such difficulties we performed the analysis with the following change of variable

*γ*parameterization and then transformed into the

*π*parameterization by Equation B2. The justification for using these procedures is that maximum likelihood estimation and hypothesis testing based on likelihood ratios are invariant under invertible re-parameterizations such as the ones above (Pawitan, 2001, pp. 43–45). For example, if

*π*in one parameterization, then its transformation

*Helmholtz's treatise on physiological optics, Volume 1*. Helmholtz's treatise on physiological optics, Volume 1. New York: The Optical Society of America (Translated from the 3rd German edition).