The light reflected from a glossy surface depends on the reflectance properties of that surface as well as the flow of light in the scene, the *light field*. We asked four observers to compare the glossiness of pairs of surfaces under two different real-word light fields, and used this data to estimate a transfer function that captures how perceived glossiness is remapped in changing from one real-world light field to a second. We wished to determine the form of the transfer function and to test whether for any set of three light fields the transfer function from light field 1 to light field 2 and the transfer function from light field 2 to light field 3 could be used to predict the glossiness transfer function from light field 1 to light field 3. Observers' estimated glossiness transfer functions for three sets of light fields were best described by a linear model. The estimated transfer functions exhibited the expected transitivity pattern for three out of four observers. The failure of transitivity for one observer, while significant, was less than 12.5% of the gloss range.

^{1}collected by Debevec (1998). However matching performance did not indicate that perception of gloss was independent of choice of light field (gloss constancy). Surfaces appeared less glossy under simple light fields generated by a small number of point sources than under real world light fields as those sampled by Debevec.

*glossiness transfer function*.

*isogloss contour*. We refer to this contour as the glossiness transfer function Γ

_{1→2}. It describes how perceived glossiness is remapped in changing from one real world light field to a second. That is, if a surface has perceived glossiness

*g*

_{1}under LP 1 then it has perceived glossiness

*g*

_{2}= Γ

_{1→2}(

*g*

_{1}) under LP 2.

_{1→2}is the glossiness transfer function from LP 1 to LP 2, Γ

_{2→3}is the glossiness transfer function from LP 2 to LP 3, Γ

_{1→3}is the glossiness transfer function from LP 1 to LP 3, and ‘∘’ denotes composition of functions. That is, for any surface with perceived glossiness

*g*

_{1}under LP 1, we test whether

*g*

_{3}= Γ

_{2→3}(Γ

_{1→2}(

*g*

_{1})) across the range of surfaces considered.

*ρ*

_{s}(Ward, 1992) from 0.02 to 0.139, while holding the diffuse component (

*ρ*

_{d}) and surface roughness (

*α*) constant. The chosen values of

*ρ*

_{s}do not correspond to perceptually equally spaced values of gloss. Furthermore we did not perform any compression of the luminance values, nor introduced artificial glare to our renderings. If a luminance value was greater than 1, as it is often encountered with high dynamic range illumination maps it was cut off.

^{2}. The stereoscope was contained in a box 124 cm on a side. The front face of the box was open 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) 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 non-reflective walls of the enclosing box.

*Galileo, RNL*(taken in a eucalyptus grove at the University of California at Berkley), and

*St. Peter's*. The first and third light probes were recorded indoors and the second is sampled from an outdoors scene in a sparse forest. We will refer to these three light probes as LP 1, LP 2 and LP 3, in that order, when convenient. We estimated glossiness transfer functions for the three pairings of these three LP. On any given trial the stimuli corresponding to one of the 30 staircases would be picked at random and displayed side by side. The two stimuli on each trial were randomly placed on the left or right side of the display.

*n*is denoted

*g*

_{ n}. It could take on any of the glossiness levels{1, 2, ⋯, 10}. The stimulus level on the following trials

*g*

_{ n + 1}was computed as

*S*

_{ n}is the step size and

*R*

_{ n}is the response (0 or 1) on the

*n*'th trial. The second part of the computation ensures that the stimulus level

*g*

_{ n+1}remains within the range {1, ⋯, 10}. The step size on the next trial is reduced by a factor of 2 but could not be less than 1:

*S*

_{1}= 8 and after the first three trials the step size is 1 and the staircase behaves as a one-up, one-down staircase for the remaining 47 trials in the staircase.

*power model λ*

_{2}was obtained by fitting a power function of the form

*c, a*and

*b*were free to vary (Figure 7: gray dashed curves). The second,

*linear model*is nested within the first with parameter

*b*set to 1,

*λ*

_{1}. The third,

*identity model*(with

*a*= 1,

*c*= 0,

*b*= 1) is nested within the other two,

*λ*

_{0}. If an observer's performance is predicted by the identity model then his glossiness matches are independent of the choice of real-world light field, a form of glossiness constancy.

*X*

_{ i+1}= 2(

*λ*

_{ i+1}−

*λ*

_{ i}). If the model with fewer parameters is the correct model then this test statistic is asymptotically distributed as a

*χ*

^{2}random variable with degrees of freedom equal to the difference in the number of parameters in the two models under comparison (Mood et al., 1974, p. 440). Accordingly we compared

*X*

_{2}to the 95th percentile of a

*χ*

_{1}

^{2}distribution to test the power model against the linear model and we compared

*X*

_{1}to the 95th percentile of a

*χ*

_{2}

^{2}.

*linear model*fit to the data differs significantly from the

*power model*fit. Furthermore for all but one condition (observer 3, transfer function Γ

_{1→3}) we rejected the hypothesis that the

*identity model*is not significantly different from the linear model, i.e. in only one condition did we observe gloss constancy. The parameter estimates (intercepts, slopes, exponents) and the results of the two sets of model tests are shown in Table 1.

Parameter | Transfer | S01 | S02 | S03 | S04 | |
---|---|---|---|---|---|---|

POWER | a | Γ _{1→2} | 0.697 | 3.410 | 0.326 | 4.381 |

Γ _{2→3} | 4.248 | 0.140 | 3.895 | 1.699 | ||

Γ _{1→3} | 0.990 | 0.909 | 0.364 | 0.873 | ||

b | Γ _{1→2} | 1.163 | 0.474 | 1.923 | 0.497 | |

Γ _{2→3} | 0.555 | 1.625 | 0.471 | 0.854 | ||

Γ _{1→3} | 1.050 | 0.830 | 1.379 | 1.129 | ||

c | Γ _{1→2} | 1.482 | 1.854 | 3.001 | −0.637 | |

Γ _{2→3} | −5.881 | 1.687 | −5.422 | −3.971 | ||

Γ _{1→3} | 0.058 | 3.132 | 1.485 | 1.387 | ||

LINEAR | a | Γ _{1→2} | 1.043 | 0.817 | 1.687 | 1.161 |

Γ _{2→3} | 1.123 | 0.651 | 0.604 | 1.117 | ||

Γ _{1→3} | 1.103 | 0.584 | 0.922 | 1.184 | ||

c | Γ _{1→2} | 0.936 | 4.901 | 1.331 | 3.170 | |

Γ _{2→3} | −1.447 | 0.673 | −0.043 | −2.899 | ||

Γ _{1→3} | 0.000 | 3.563 | 0.532 | 0.945 | ||

POWER vs. LINEAR | Γ _{1→2} | 0.236 | 0.877 | 0.067 | 1.000 | |

Γ _{2→3} | 0.011 | 0.011 | 0.028 | 0.616 | ||

Γ _{1→3} | 0.700 | 0.278 | 0.032 | 0.518 | ||

LINEAR vs. IDENTITY | Γ _{1→2} | 0* | 0* | 0* | 0* | |

Γ _{2→3} | 0* | 0* | 0* | 0* | ||

Γ _{1→3} | 0* | 0* | 0.257 | 0* |

_{1→3}

_{1→2}denotes the glossiness transfer function from Galileo to RNL, or equivalently we can say that perceived glossiness under RNL can be measured as a function of perceived glossiness under Galileo. We can express this relation as a linear equation

*c*

_{12}is the intercept and

*a*

_{12}is the slope of Γ

_{1→2}. Similarly we can write Γ

_{2→3}, the glossiness transfer function from RNL to St. Peter's as

_{2→3}∘ Γ

_{1→2}= Γ

_{1→3}. We wish to test whether the estimates in Table 1 of

_{12},

_{12},

_{23}, ⋯,

_{13}are consistent with Equations 11 and 12.

_{1→3}(see previous section) from our data we wanted to compare the measured intercept and slope to the predicted

*c*

_{13}and

*a*

_{13}. We tested the hypothesis that the measured Γ

_{1→3}is significantly different from the predicted one. For the unconstrained log likelihood model

*λ*

_{1}we introduced two parameters Δ

*c*and Δ

*a*. These will be estimated additive constants to the predicted intercept and slope, such that

*c*and Δ

*a*vary freely. For the constrained model

*λ*

_{0}we set Δ

*c*= 0 and Δ

*a*= 0. Comparing the resulting test statistics

*R*= 2(

*λ*

_{2}−

*λ*

_{0}) for each equation to the 95th percentile of a

*χ*

_{2}

^{2}random variable, we found that for three out of four observers the predicted Γ

_{1→3}was not significantly different from the measured one. For plots and p-values see Figure 9. While the failure of transitivity was significant for one subject, the magnitude of the failure was less than 12.5% of the full range of specularity.

_{1→2}from LP 1 to LP 2 and the transfer function Γ

_{2→3}from LP 2 to LP 3 could be used to predict the glossiness transfer function from LP 1 to LP 3 by the relation Γ

_{1→3}= Γ

_{2→3}° Γ

_{1→2}where ‘°’ denotes composition of functions. We refer to this property as

*transitivity*.

*g*) could be expressed as linear transformations. Γ(

*g*) =

*a*+

*bg*: points of equal perceived gloss between two real world light fields fell on a line.