In complex scenes, the light absorbed and re-emitted by one surface can serve as a source of illumination for a second. We examine whether observers systematically discount this secondary illumination when estimating surface color. We asked six naïve observers to make achromatic settings of a small test patch adjacent to a brightly colored orange cube in rendered scenes. The orientation of the test patch with respect to the cube was varied from trial to trial, altering the amount of secondary illumination reaching the test patch. Observers systematically took orientation into account in making their settings, discounting the added secondary illumination more at orientations where it was more intense. Overall, they tended to under-compensate for the added secondary illumination.

*test surface*marked

*T*absorbs light that reaches it directly from the single light source in the scene. It also absorbs light that arrives from the same light source but only after being absorbed and re-emitted from the nearby orange surface marked

*C*(Figure 1B). Part of the light absorbed and re-emitted by the test patch will in turn be absorbed and re-radiated by the orange surface, initiating an infinite series of inter-reflections between the surfaces. If we denote the spectral power distribution of the original illuminant by

*E*

_{(0)}(

*λ*) and the surface reflectance functions of the two surfaces by

*S*

_{C}(

*λ*) (cube) and

*S*

_{T}(

*λ*) (test patch), then the light emitted from any specified small region of the surface toward the observer can be written in the form

*E*(

*λ*)

*S*

_{T}(

*λ*) where is the

*effective illuminant*. It is the weighted sum of the direct illumination,

*E*

_{(0)}(

*λ*), and the inter-reflected illuminants, The

*geometric Factors*

*γ*

_{i}are determined by the sizes and shapes of the two surfaces, their separation, and their orientations with respect to one another and with respect to the primary light source. We will assume that they do not depend on wavelength

*λ*in the electromagnetic spectrum.

*E*(

*λ*) at each point in the scene. There is some evidence that human observers do so, if only partly and imperfectly.

*test patch*) is small, and its chromaticity is under the control of the observer. The observer is asked to set the second surface to be neutral in appearance. As we explain below, the results of this setting task will permit us to assess how accurately human observers discount inter-reflection. First, we describe how light travels through the scenes that we will use as stimuli.

*T*) that is located on a larger dark gray rectangle is rotated toward the side of a brightly colored orange cube at a certain angle

*τ*. All surfaces are Lambertian (we will define precisely what this means in a moment). The scene is illuminated by a neutral punctate light source

*E*

_{(0)}(

*λ*) placed behind the observer. This light source is sufficiently far away from the small test patch to allow us to assume that the distance to the light source and the angle of incidence

*θ*

_{TP}of light from the primary light source on the test patch is constant across the extent of the patch. See Figure 2 for a definition of the angles that are relevant to our discussion.

*θ*

_{TP}determines the flux of light from the punctate source that falls on the test surface. For Lambertian surfaces, luminance decreases as the angle between surface normal and the direction to the light source increases. If, for example, the test patch is rotated toward the light source so that the angle between its surface normal and the incident light ray is 0° it will receive the maximum amount of light, and its luminance will be at its maximum. Away from this position less light will be received by the test patch, and its luminance will decrease.

*θ*

_{TP}

*S*

_{T}(

*λ*)

*E*

_{(0)}(

*λ*), where

*S*

_{T}(

*λ*) is the surface reflectance of the test patch. The expression cos

*S*

_{TP}

*E*

_{(0)}(

*λ*) is the first term (“zero bounce” term) in the summation in Equation 1. Comparing terms, we see that the first geometric factor is When the test patch is neutral (achromatic), we can replace the surface reflectance function

*S*

_{T}(

*λ*) of the neutral test patch by its albedo,

*E*

_{(1)}(

*λ*), the light reflected from the adjacent surface of the cube (the

*cube surface*C). The spectral power distribution of the “zero bounce” light that is emitted from the surface of the cube is cos

*θ*

_{CP}

*S*

_{C}(

*λ*)

*E*

_{(0)}(

*λ*), where

*S*

_{C}(

*λ*) is the surface reflectance of the cube. If the light arriving at the cube’s surface (

*E*

_{(0)}(

*λ*)) is neutral, then the light reflected from the cube would take on the chromaticity of

*S*

_{C}(

*λ*).

*E*

_{(1)}(

*λ*) is the sum of contributions from each area element on C (as shown in Figure 1C). We set up a Cartesian coordinate system (

*x, z*) for the face of the cube (Figure 1) and integrate the contribution from each such element to obtain the total illumination upon the test patch from the surface of the cube. This constitutes the second term in Equation 1 where

*θ*

_{CP}is the angle of incidence of the light arriving from the collimated punctate light source on the surface of the area element at (

*x,z*). We assume that it is independent of the location (

*x,z*) on the cube, because the punctate light source is far away in our scenes;

*θ*

_{TC}(

*x,z*) is the angle of incidence of the light arriving from the area element at (

*x,z*) on the test patch; and

*D*(

*x, z*) is the distance between the area element at (

*x,z*) on the cube and the center of the test patch.1 Because

*E*

_{(0)}(

*λ*)

*S*

_{C}(

*λ*) =

*E*

_{(1)}(

*λ*), the second geometric factor is seen to be where cos

*θ*

_{CP}is moved out of the integral since it is constant across the side of the cube.

*γ*

_{2}and beyond similarly. Expressions for these terms grow rapidly in complexity and, in many scenes, the first two terms of Equation 1 dominate. This, however, need not always be the case. Under a forest canopy, when neither sun nor sky is directly visible, the higher order terms that result from multiple reflections among leaves likely dominate (Endler, 1993).

*γ*

_{1}of the one-bounce light

*E*

_{(1)}(

*λ*) decreases (because the

*θ*

_{TC}term decreases, while the distance

*D*increases), and the geometric factor

*γ*

*E*of the zero-bounce light

*E*

*λ*(

*λ*) also changes (because the angle

*θ*

*τ*changes as the test patch rotates). If we let

*τ*denote the angle between the surface patch and the face of the cube, then, as we change this angle, we can write the effective illumination of the test patch as where we have made it explicit that the coefficients

*γ*

_{0}and

*γ*

_{1}both depend on

*τ*. In Figure 3, we plot

*γ*

_{0}(

*τ*) and

*γ*

_{1}(

*τ*) versus

*τ*for the scene shown in Figure 1 (See “Appendix” for derivations). The function

*γ*

_{0}(

*τ*) reaches a maximum when the test patch

*τ*is facing the punctate light source. As

*τ*increases, the contribution of secondary illumination

*γ*

_{1}(

*τ*) decreases.

*γ*

_{0}(

*τ*) in estimating surface color and albedo. When a secondary (“one-bounce”) illuminant is present, however, the visual system must somehow compensate for changes in both geometric factors

*γ*

_{0}(

*τ*) and

*γ*

_{1}(

*τ*) with changes in

*τ*, or, more generally, scene layout. The second geometric factor depends on many factors and, as it is written in Equation 6, involves a double integration.2 In this study we investigate whether human vision can compensate for secondary illumination that results from inter-reflection between two surfaces as we vary the angle

*τ*between the surfaces.

*test patch*was attached to the side of the cube.

*z*axis was vertical and aligned with the side of the cube, the

*x*axis was horizontal and aligned with the side of the cube, and the

*y*axis was normal to the same side of the cube (see Figure 6). The angle between the cube’s face and the test patch is denoted by

*τ*.

*L*

^{R}). We will also specify the chromaticities of surfaces by the albedo terms,

*α*

^{R},

*α*

^{G},

*α*

^{B}. For a neutral surface, for example,

*α*

^{R},

*α*

^{G},

*α*

^{B}.

*τ*= {70°, 80°, 90°, 120°, 150°, 160°, 170°} after a rotation about the vertical

*z axis*. The plane was fronto-parallel to the observer when

*τ*= 120°. The dark rectangular plane on which the test patch was embedded was rendered with reflectance

*α*

^{R}=

*α*

^{G}=

*α*

^{B}=0.01, and the light test patch3 with

*α*

^{R}=

*α*

^{G}=

*α*

^{B}= 0.55. The choice of a much darker immediate surround to the test patch was in order to eliminate utilization of a simultaneous color contrast strategy by the observers (e.g., Werner & Walraven, 1982). The orange cube was rendered with reflectance

*α*

^{R}= 1,

*α*

^{G}= 0.05,

*α*

^{B}= 0.

*x y z*) = (93.74

*cm*, 117.63

*cm*, 40

*cm*), behind and above the observer, to the right. It was sufficiently far from the cube and the test patch so that we could treat the punctate light source as collimated across the extent of the surface of the cube and the test patch.

*ψ*

_{P},

*ϕ*

_{P}, where

*ψ*

_{P}is the angle between the

*x*axis and the projection of the position of the light on the

*xy plane*,

*ϕ*

_{P}is the angle between the position of the light source and the

*xy plane*(See Figure 7). The surface of the central cube, which is oriented toward the test patch, constitutes a secondary light source in our scene.

*cd*/

*m*

^{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.

*cm*(Figure 8). To minimize any conflict between binocular disparity and accommodation depth cues, the test patches were rendered to be exactly 70 cm in front of the observer. The monocular fields of view were 55 deg × 55 deg of visual angle each. The observer’s eyes were approximately at the same height as the center of the scene being viewed which was also the height of the center of the test patch.

*L*

^{R},

*L*

^{G},

*L*

^{B}]. We constrained these three primaries so that

*L*

^{R}+

*L*

^{G}+

*L*

^{B}was always constant. If, for example, the observer pressed the “left” key of the “left”-“right” (“green”-“red”) key pair, then

*L*

^{G}was increased by a fixed amount

*δ*and

*L*

^{R}decreased by

*δ*so that

*L*

^{R}+

*L*

^{G}+

*L*

^{B}remained constant. Hence, the “red”-“green” direction was simply a tradeoff between

*L*

^{G}and

*L*

^{R}. The “blue”-“yellow” settings involved a tradeoff between

*L*

^{B}and

*L*

^{R}+

*L*

^{G}. A key point is that the observer could precisely cancel the effect of the secondary light arriving from the cube by adjusting primarily

*L*

^{R}vs.

*L*

^{G}. We expected that blue-yellow settings would not change systematically with changes in test patch orientation

*τ*and, if this is so, it would simplify the analysis of the data.

*E*(

*λ*) =

*γ*

_{0}(

*τ*)

*E*

_{(0)}(

*λ*) +

*γ*

_{1}(

*τ*)

*E*

_{(1)}(

*λ*). So, the observer’s setting [

*L*

^{R},

*L*

^{GG},

*L*

^{B}] should be the RGB code corresponding to

*E*(

*λ*) which we compute below in terms of the parameters describing the geometry of the scene. We will compare these predictions to the observer’s actual settings, which, from this point on, we denote . In our analysis we will also allow for the possibility that the observer’s subjective achromatic point could be biased.

*L*(

*λ*)=π

^{−1}

*E*(

*λ*)

*α*

_{T}, and we can rewrite this identity in terms of each of the three primaries. For example, for R we have We next introduce the ideal geometric red discounting function which is the relative amount of red that is reflected from the test patch for a given orientation

*τ*. Using Equations 10 and 7 we obtain where we denoted the total illuminants with and . We can further simplify the above equation. First of all recall that . This yields ; ; , and . Recall that the central cube was rendered with reflectance , , ; therefore, we can neglect compared to . This yields where we defined whose true value is 1/3.

*τ*. Then we can rewrite Equation 13, emphasizing the dependence on

*τ*as,

*τ*, the observer’s settings would trace a curve . We refer to this plot as the

*observer’s geometric red discounting function*, and define it as where is the observer’s mean achromatic setting for a particular value of

*τ*. The dependent variable in our study was the relative amount of “red” in the observer’s setting of the color of the test patch, , although the term “red” here refers precisely to the chromaticity of the added secondary illuminant, which is precisely the chromaticity of the cube. We were particularly interested in whether the amount of red in the setting is affected by the orientation of the test patch (i.e., whether the observer “discounts” the angle from the perceived color of the test patch). We compared the observers setting to the prediction

*ρ*(

*τ*) of the one-bounce model of inter-reflection between two Lambertian surfaces derived above.

*τ*< 180°). Such a heuristic could permit more accurate surface color estimation without the need to compute the geometric factors (e.g., Equations 3 and 6) explicitly. It is also consistent with results of the experiment of Bloj et al. (1999). With this heuristic we would expect the relative amount of red to be a constant across the different test patch orientations as in Figure 10A (horizontal, red line). If, on the other hand, the observer correctly takes into account the orientation of the test patch when making his or her setting, we would expect the estimates to be close to the curve predicted by the model,

*ρ*(

*τ*), in Figure 10B.

*ρ*(

*τ*), is plotted red in the data graphs.

*ρ*(

*τ*) for the one-bounce model, yet there is evident inter-observer variability. A number of observers have a geometric discounting function that is flatter than the model prediction. This “flattening” is particularly evident for angles greater than 90°. Parameters that influence the shape of the curve are discussed below. For all observers, we verified that there is, as expected, no systematic change in the

*Blue/Green*ratios across angles. We do not plot these results or discuss them further.

*ρ*(

*τ*) and are the result of systematic errors in the observers’ estimates of scene properties. For example, an observer may misperceive the orientation

*τ*of the test patch. We can refit the data allowing for this possibility by adding parameters

*â*and

*^b*where . If

*â*proved to be close to 0, and

*^b*close to 1, then we could not attribute the systematic differences between

*ρ*(

*τ*and to misperception of

*τ*. Conversely, we may be able to account for these differences as the result of a misperception of

*τ*.

*τ*, put too much or too little red (when compared with the ideal model) into her/his setting. The observers’ geometric discounting function becomes and we wish to bring this family of curves into coincidence with the data by choice of the setting of the three parameters. We used maximum likelihood fitting procedures to estimate the values of the parameters for the observers’ data. The estimates are shown in Table 1.

Observer | ŵ | ^B | â | ^b |
---|---|---|---|---|

Veridical | 2.18 | 0.3333 | 0 | 1 |

MD | 1.121* p<0.001 | 0.363* p<0.001 | −0.000006 p=0.849 | 0.978 p=0.849 |

POH | 1.045* p<0.001 | 0.369* p<0.001 | −0.000011 p=0.637 | 1.059 p=0.637 |

AH | 1.947* p<0.001 | 0.338 p=0.387 | −0.000007 p=0.093 | 1.059 p=0.093 |

JT | 0.622* p<0.001 | 0.407* p<0.001 | −0.000016 p=0.706 | 0.922 p=0.706 |

SL | 0.661* p<0.001 | 0.379* p<0.001 | −0.000013 p=0.89 | 0.958 p=0.89 |

TS | 0.677* p<0.001 | 0.401* p<0.001 | −0.000016 p=0.08 | 0.849 p=0.08 |

*ŵ*<

*w*). Furthermore the angle

*τ*between the test patch and cube was, except for two subjects, perceived as slightly compressed. This finding is in agreement with research that maintains that observers tend to perceive the orientation of a rectangular Lambertian patch as slightly compressed in depth (see Boyaci et al., 2003, for discussion). However, none of the six observers’ estimates of

*τ*were significantly different from the veridical values. The amount of bias varied between observers and, overall, it was relatively small (Table 1).

*λ*

_{1}was obtained by fitting the geometric discounting function to the observers’ relative red settings using the method of maximum likelihood (the four parameters described above were free to vary). In the constrained (“nested”) model, we forced the geometric relative red function to be constant (red line in Figure 10A), allowing only the bias

*^B*to vary. We compared the log likelihood ratio to the relevant chi square distribution (

*χ*

_{3}

^{2}). All observers’ relative red settings were significantly different from a constant geometric relative red function (

*p*<.00001).

*λ*

_{1}was obtained as above. For our constrained model

*λ*

_{0}we assigned

*ŵ*,

*â*, and

*^b*their true values: 2.18, 0, and 1, respectively, and let

*^B*vary freely.3 Comparing the resulting log likelihood ratio to the appropriate chi square distribution (

*χ*

_{3}

^{2}) we find that for all observers the relative red settings were significantly different from the predictions of the model (

*p*< .00001), leading to rejection of the null hypothesis.

*ŵ*,

*^B*,

*^b*and

*χ*) was equal to the veridical value (2.18, 0.3333,0, and 1 respectively). The log likelihood of unconstrained model was obtained as described above. In the constrained model, we assigned the parameter in question its veridical value and let the other parameters vary freely. We compared the resulting log likelihood ratio to the chi-square distribution with one degree of freedom (

_{1}

^{2}

*p*). All

*p*values are reported in Table 1.

*â*,

*^b*and constant over- or under-estimation of the amount of red in the scene (

*^B*).

*Influence of*

*â*,

*#x005E;b*: Based on the estimates of parameters

*â*and

*^b*, we conclude that observers perceived the orientation of the test patch nearly veridically, with a slight tendency to underestimate

*τ*. This finding is consistent with the results of Boyaci et al. (2003), who used a similar stimulus configuration.

*Influence*of

*^B*: The observed bias in most of the observers’ model fits is rather small. The bias parameter shifts the entire discounting curve up (when putting too much red in the achromatic setting) or down (this actually never occurred), it can be interpreted as a slight overestimation (or underestimation) of the amount of red in the light source that counts as neutral. This might be brought about, for example, by means of chromatic adaptation to the scene (specifically to the large bright orange cube) or may simply mean that the observer disagrees with our arbitrary choice of “neutral.”

*Influence*of

*ŵ*: The fitted values of

*ŵ*indicate that most observers use less than the optimal area of the cube for their discounting function. A value less than veridical corresponds to a compression of the observer’s geometric relative red function, leading to an overall flattening of the curve. If we attribute this to a failure to choose the correct area of integration, the data suggest that all observers utilize less than the relevant part of the cube.

*ŵ*is confounded with any other factor that would lead the observer to underestimate the overall intensity of the secondary illumination from the cube. One possibility is that the visual system does not use the Lambertian model in computing the intensity of the secondary (“one bounce”) illumination.

*τ*> 90°). In the model, the slope of the curve decreases greatly after 90°, it might be that the observer is not sensitive to this fine gradient, and anchors on a constant setting for wider angles. It is quite possible that observers may compute inter-reflection between two surfaces differently, that is, they assume different models for small and wide angles, adopting a variant of the binary heuristic for angles greater than 90°.

*ŵ*as visual errors in choosing the limits of integration. By making the chromatic surface flat and rectangular, we set up conditions where we could compute the effective illumination with relative ease. Had we picked a curved surface or a surface with an irregular boundary instead, then our computation would have been more difficult. It is presumably this more general problem that the visual system addresses and, consequently, the problem of selecting the proper area of integration and computing the geometric factor

*γ*

_{1}(

*τ*) is plausibly difficult and prone to error. Under this interpretation, the deviations in are less surprising.

*α*=0.01). Further, evidence that local contrast cannot solely account for the achromatic shifts has also been brought by studies by several other researchers (see Delahunt, 2001; Brainard, 1998; Bloj, 1999).

*dσ*around a point

**r**, at a given wavelength

*λ*is (LeGrand, 1957, pp. 18ff), where

*E*(

**r**;

*λ*) is the illumination (surface density of the light flux received) and

*S*(

**r**;

*λ*) is the surface reflectance function at the point

*r*. In the following derivations, wavelength

*λ*is not displayed for simplicity. The total illumination upon the surface element at

**r**satisfies the equation, where

*E*

_{(0)}

**r**is the illumination due to the primary source and Ω the area of integration. The integral is over all surfaces which contribute to the illumination of the surface element at

**r**, and is a geometric factor, where the angle of incidence

*θ*is the angle between the normal vector to the surface at the point

**r**and the vector connecting the points

**r**and

**r**′.

*E*

_{(0)}(

*r*is equal to

*I*

_{(0)}

*d*

^{−2}cos

*θ*where

*I*

_{(0)}is the intensity of the primary source,

*d*is the distance from the point

**r**to the primary source, and

*θ*is the angle of incidence of the primary source on the infinitesimal surface element at

**r**. The first-order (“one-bounce”) approximation to Equation 18 is obtained by inserting

*E*

_{(0)}(

**r**′) in place of

*E*(

**r**′) in the integral Naturally, a better approximation is obtained by inserting the first-order approximation from Equation 20 into the integral in Equation 18 This is the 2

^{nd}order or “two-bounce” approximation. Explicitly writing the second order approximation yields as we improve the approximation by repeating the recursion, the

*n-th*approximation will involve integrals taken over the region of interest once, twice,... and

*n*times. As the order of approximation increases the expression gets more complicated, therefore we symbolically write it as The physical significance of this expression is the following. The first term in Equation 23 is the direct illumination due to the primary source. The second term represents the response of a small area element at the point

**r**′ with area dσ′ around. It acts as an effective source that makes a contribution

*E*

_{(0)}(

**r**′)

*S*(

**r**′)

*G*(

**r−r**′;

**n**(

**r**))

*dσ*′ to the field at another point

**r**. The higher order terms can also be interpreted in the same way as contributions from higher number of scatterings (up to the n-bounce term).

*effective intensity*” of the punctate source as Then, for example, for zero-bounce we obtain Let us next consider the first-order approximation to the illumination of a point

**r**on the test patch. Equation 25 will constitute the first term of the approximation in Equation 20. The region of integration

**r**will be the face of the cube “visible” to the test patch, denoted by

*C*. The zero-th-order illumination of a point

**r**′ on the cube surface is where

*θ*

_{CP}is the angle of incidence of the light from the primary source on the cube’s surface at

**r**′. The surface reflectance of the cube is a constant throughout the face of the cube,

*S*(

**r**′)=

*S*

_{C}. The geometric factor is Note that

*θ*

_{TC}is a function of

**r**′ but

*θ*

_{TP}

*θ*

_{CP}are not (because we assume that the light arriving at the test patch and the cube’s surface is collimated). Inserting Equations 26 and 27 into Equation 20, we obtain the first-order approximation where we also introduced

*E*

_{(1)}=

*E*

_{(0)}

*S*

_{C}and took it out of the integration since it is a constant. Similarly, the second-order approximation is found as follows

*n*-th-order approximation which contains one, two,…,

*n*bounces of the primary light before reaching the test patch. The coefficients

*γ*

_{i}are calculated as in Equations 28 and 29. The

*γ*

_{1}. We will calculate one-bounce illumination only in the center of the test patch. Because the test patch is very small, we will assume that the illumination is roughly constant across its surface. Application of the law of cosines, yields the relevant angles of incidence in Equation 28 where (

*ψ*

_{P},

*ϕ*

_{P}is the direction to the punctate source as described above,

*d*is the distance from the origin to the center of the test patch. With these in place,

*γ*

_{0}and

*γ*

_{1}become where we have assumed that the integration region

*C*is taken as a square of area 2

*w*×

*w*. Note that this integral can be solved analytically, indicating that the computation of the geometric factors of inter-reflection need not always involve explicit integration. The integration yields This is the equation plotted in Figure 3 as a function of

*τ*.

^{1}The 1/

*π*term in Equation 6 stems from calculating the

*radiant emittance in all directions*(after LeGrand, 1957).

^{2}For the conditions of Figure 1 and the experiment reported here, the double integral of Equation 6 can be solved in closed form. We report these results and derive them in the “Appendix.”

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