**Abstract**:

**Abstract**
**All images are highly ambiguous, and to perceive 3-D scenes, the human visual system relies on assumptions about what lighting conditions are most probable. Here we show that human observers' assumptions about lighting diffuseness are well matched to the diffuseness of lighting in real-world scenes. We use a novel multidirectional photometer to measure lighting in hundreds of environments, and we find that the diffuseness of natural lighting falls in the same range as previous psychophysical estimates of the visual system's assumptions about diffuseness. We also find that natural lighting is typically directional enough to override human observers' assumption that light comes from above. Furthermore, we find that, although human performance on some tasks is worse in diffuse light, this can be largely accounted for by intrinsic task difficulty. These findings suggest that human vision is attuned to the diffuseness levels of natural lighting conditions.**

*light field*, a function

*F*(

*x?*,

*v?*) giving the luminance in each direction

*v?*at each point

*x?*throughout a region of space, and the

*light probe*, a sample

*E*(

*v?*) =

*F*(

*x?*

_{0},

*v?*) from the light field at a single location.

*plenopter*, to measure low-pass light probes. They used the plenopter to investigate the low-pass structure of lighting in natural scenes. Mury et al. (2009b) measured light probes in several environments, and consistent with their previous work (Mury et al., 2007), they found that a lighting model with a coarse description of the scene layout accounted for the structure of the measured light probes.

^{1}(Metzger, 1936/2006; Morgenstern, Murray, & Harris, 2011; Ramachandran, 1988). An equally important property of lighting, though, is its

*diffuseness*, the extent to which light comes mostly from a single direction as on a sunny day or from all directions as on a cloudy day (Langer & Bülthoff, 2000; Tyler, 1998). Lighting diffuseness has a large effect on object appearance, and the assumptions that observers make about diffuseness can have a correspondingly strong influence on their perception of 3-D scenes. Five recent psychophysical studies have investigated the assumptions that observers make about lighting diffuseness when estimating shape and reflectance, and they found that observers tend to assume high levels of diffuseness—often higher than the actual diffuseness of the light in the scene being viewed (Bloj et al., 2004; Boyaci, Doerschner, & Maloney, 2004, 2006; Boyaci, Maloney, & Hersh, 2003; Schofield, Rock, & Georgeson, 2011). This suggests that observers may have a prior for highly diffuse lighting.

*illuminance contrast energy (ICE)*, which allows us to describe our lighting measurements and the results of previous psychophysical studies in a common language. Let

*E*(

*?*,

*?*) be the illuminance pattern over the surface of a unit sphere illuminated by a light probe, described in spherical coordinates where

*?*is the declination from the north pole (i.e., 0° at the north pole, 90° at the equator, 180° at the south pole) and

*?*is the azimuth. We define the ICE of the light probe to be the coefficient of variation of the illuminance over the sphere, i.e., its standard deviation divided by its mean: Here

*?*is the mean illuminance over the sphere. Under diffuse light, illuminance is largely constant across surface orientations whereas under directional light the illuminance depends on the orientation of a surface relative to the dominant light sources. ICE is a measure of the variation of illuminance across orientations, and hence of diffuseness. ICE ranges from 0 for a completely uniform, ambient light source to 1.29 for a distant point light source (Morgenstern, 2011). Santa Clara (2009) independently developed an equivalent measure of lighting diffuseness.

^{2}The light probe

*L*(

*?*,

*?*) is a function that reports the luminance in direction (

*?*,

*?*). The illuminance pattern

*E*(

*?*,

*?*) generated by the light probe on a sphere is the cosine-weighted integral of the light probe over a full hemisphere of directions: Here

*w*(

*?*,

*?*,

*?*?,

*?*?) is the cosine of the angle between directions (

*?*,

*?*,) and (

*?*?,

*?*?), half-wave rectified so that negative values of the cosine are clipped to zero. As a result, the illuminance pattern

*E*(

*?*,

*?*) is much more low pass than the light probe itself. Even a rich and detailed lighting environment creates a simple, smoothly varying illuminance pattern (Figure 2). Ramamoorthi and Hanrahan (2001) and Basri and Jacobs (2003) describe a simple method of using spherical harmonics to convert the luminance of a light probe to the illuminance it generates on a sphere.

*?*is the angle of the surface normal relative to the point source direction,

*E*is the maximum illuminance from the point source, and

_{P}*E*is the illuminance from the ambient source.

_{A}*E*/

_{A}*E*from the previous psychophysical studies. We used Equation 3 to convert these ratios to ICE.

_{P}*equivalent illumination model*(Brainard & Maloney, 2011), could account for Ripamonti et al.'s (2004) findings. According to this model, the observer assumes that the scene is illuminated by a PA light source described by Equation 2, and the observer estimates the point illuminance

*E*and ambient illuminance

_{P}*E*. The model assumes that the observer perceives the orientation and luminance of the test patch accurately. Using this information, the observer uses a Lambertian shading model to calculate the reflectance of the test patch in a physically realistic way. Bloj et al. showed that the pattern of observers' lightness matches across test patch orientations is just what one would expect if observers followed this model but overestimated the amount of ambient light

_{A}*E*in the scene.

_{A}Mean ICE | Standard deviation | Standard error of mean | |

Indoor | 0.4132 | 0.1498 | 0.0122 |

Rural sunny | 0.6514 | 0.1909 | 0.0208 |

Rural cloudy | 0.6197 | 0.0984 | 0.0100 |

Urban shade | 0.4478 | 0.0790 | 0.0109 |

Urban sunny | 0.6598 | 0.0752 | 0.0079 |

Urban cloudy | 0.5071 | 0.0659 | 0.0068 |

Time of day | 0.4424 | 0.0830 | 0.0115 |

Bloj et al. (2004), expts 1, 2 | 0.4526 | 0.1922 | 0.0533 |

Bloj et al. (2004), expt 3 | 0.3828 | 0.1580 | 0.0304 |

Schofield et al. (2011) | 0.5281 | 0.2314 | 0.0546 |

Schofield et al. (2011), revised | 0.3481 | 0.1989 | 0.0469 |

Boyaci et al. (2003) | 0.0942 | 0.0369 | 0.0165 |

Boyaci et al. (2004) | 0.6139 | 0.4886 | 0.1847 |

Boyaci et al. (2006), expt 1 | 0.0581 | 0.0302 | 0.0135 |

Boyaci et al. (2006), expt 2 | 0.0632 | 0.0034 | 0.0019 |

*prior*on diffuseness as their experiments were run in the dark and their stimulus was simply a sine wave grating, which is highly ambiguous with regard to lighting conditions; Bloj et al.'s and Boyaci et al.'s (2003, 2004, 2006) scenes were more complex and contained cues to lighting diffuseness that may have affected observers' estimates of the lighting conditions. Appendix B shows ICE values based on a refinement of Schofield et al.'s lighting model. The revised ICE values are slightly lower but still well within the range of natural lighting.

^{3}However, there are good reasons to think that their results are biased toward high estimates of diffuseness (i.e., low ICE). Boyaci et al.'s observers' lightness constancy was poor: Their lightness matches were partway between matching the luminance of image patches on the computer screen and matching the reflectance of the surface patches they depicted, with a strong bias toward matching luminance (e.g., Boyaci et al. [2003] figure 8). The “equivalent illuminant” model that Boyaci et al. used to infer observers' diffuseness assumptions attributes such failures of lightness constancy to assumptions of high diffuseness. However, there are many reasons why lightness constancy can fail besides observers assuming unrealistically high levels of diffuseness. Unlike Bloj et al.'s (2004) scenes, Boyaci et al.'s scenes were computer-generated, and if observers did not see them as completely realistic, then they may have been biased toward matching screen luminance instead of matching the depicted surface reflectance. Supporting this view, Lee and Brainard (2014) found that a computer-generated replication of Gilchrist's (1977) paper-based lightness perception experiments led to much weaker constancy. Furthermore, failures of lightness constancy occur when observers judge lightness in scenes that have dark backgrounds, small frameworks, and low articulation (Gilchrist, 2006, p. 276), which are all factors consistent with weak lightness constancy in Boyaci et al.'s experiments. Thus, Boyaci et al.'s estimates of observers' assumptions about diffuseness were probably biased, and we do not see them as persuasive evidence against a diffuseness prior that matches natural lighting. (Furthermore, it was not Boyaci et al.'s goal to estimate observers' diffuseness priors. Their main goal was to test the equivalent illuminant model of lightness perception.)

*vector/scalar ratio*, which we describe further in the General discussion. Interestingly, human factors experiments have found that people prefer vector/scalar ratios of 1.2 to 1.8 for lighting of human faces (Cuttle, 2003, p. 88). This corresponds to an ICE range of 0.39 to 0.58 and brackets both the average ICE across all our natural light probes (blue vertical line in Figure 3a) and the average ICE level that the equivalent illuminant model attributes to Bloj et al.'s (2004) and Schofield et al.'s (2011) observers (blue vertical line in Figure 3b).

*strong*observers' diffuseness priors are, i.e., how narrow the priors are as statistical distributions, and whether they usually override diffuseness cues in individual scenes. On the one hand, the large individual differences in observers' diffuseness assumptions as well as the wide range of diffuseness in natural lighting, suggest that the priors may be weak. On the other hand, the fact that Bloj et al.'s (2004) stimuli provided observers with cues to diffuseness and yet observers' diffuseness estimates consistently had lower ICE than the actual lighting suggests that the prior may be strong. Further work is needed to decide this question.

*?*for human and ideal observers under five levels of lighting diffuseness. We calculated human observers' efficiency, the squared ratio of ideal and human thresholds (Tanner & Birdsall, 1958): Efficiency is 1.0 for a human observer who has the same threshold as the ideal observer and less for an observer who has a higher threshold. Efficiency corrects for intrinsic task difficulty, so it gives an effective way of comparing performance across diffuseness conditions.

*p?*and maximum illuminance

*E*and an ambient source with illuminance

_{P}*E*. The luminance of each triangular patch with reflectance

_{A}*?*and surface normal

*n?*was calculated using the Lambertian shading model: Here · is the vector dot product. We report the values of the lighting parameters

*p?*,

*E*, and

_{P}*E*below under “Procedure.”

_{A}*g*(

_{tr}*x*,

*?*,

*?*) with mean 0.6 and standard deviation 0.2. The distribution was truncated at two standard deviations from the mean, so reflectance ranged from 0.2 to 1.0.

^{2}when directly facing the virtual point light source, and the stimuli were shown on a low-luminance background (0.3 cd/m

^{2}).

*E*+

_{P}*E*constant at 304 lux for all ICE values. (This constraint, along with the ICE value and Equation 3, determines

_{A}*E*and

_{P}*E*.) Each ICE condition was shown in five blocks for a total of 25 blocks. Each block contained 150 trials and took approximately 8 min.

_{A}*?*and tilt 0°, and the other showed a hemisphere illuminated from slant 60° ? ?

*?*and tilt 0°. The two slant directions were randomly assigned to the two stimulus intervals. The observer pressed a key to indicate which interval had a lighting direction closer to the line of sight. Auditory feedback indicated whether the response was correct. The next trial began immediately. The task was not trivial because reflectances were assigned randomly in each new stimulus, so the luminance variations that provided information about lighting direction were partly masked by luminance variations caused by the random assignments of reflectance. The perturbation angle ?

*?*varied over trials according to two interleaved staircases converging on 71% and 79% correct performance (Wetherill & Levitt, 1965). The blank screens and the stimulus background were shown at the monitor's lowest luminance (0.3 cd/m

^{2}). For each block, we made a maximum-likelihood fit of a Weibull psychometric function to proportion correct as a function of the angle between the two lighting directions, and we took the angle corresponding to 75% correct performance as the observer's threshold. This gave five thresholds for each observer in each ICE condition.

*?*and slant 60°, and observers pressed a key to indicate which stimulus had lighting from a more clockwise direction.

^{4}

*L*(

*?*,

*?*) as a sum of orthonormal spherical harmonics

*Y*: Basri and Jacobs (2003) and Ramamoorthi and Hanrahan (2001) show that the illuminance pattern

_{lm}*E*(

*?*,

*?*) generated over a sphere by the light probe is Here

*c*are the spherical harmonic coefficients, and

_{lm}*w*are fixed weights assigned to each order of spherical harmonics. The magnitudes of the weights

_{l}*w*represent how important each order is in the mapping from luminance to illuminance. For an exact result, the expansion can be continued beyond order

_{l}*l*= 2, but the higher-order weights

*w*are small, so the higher-order terms can usually be omitted. In Appendix E, we show that substituting this expansion into Equation 1 gives an expression for ICE in terms of spherical harmonic coefficients: Equation 9 shows that ICE is simply the rms energy in the higher-order (

_{l}*l*? 1) harmonics of the illuminance, divided by rms energy in the zero-order component of the illuminance (which is equivalent to the absolute value of the DC coefficient because there is just one term).

*vector/scalar ratio*as a measure of diffuseness (Cuttle, 2003, p. 302). To find the “vector” in this ratio, we orient a disk in space so as to create the greatest difference between the illuminance on the two sides of the disk; the direction of the vector is then the surface normal to the side with the greater illuminance, and the magnitude of the vector is the illuminance difference. The “scalar” in this ratio is the mean illuminance over the surface of a sphere. The vector/scalar ratio is defined to be the ratio of the vector magnitude to the scalar. In Appendix F, we show that for the special case of a PA light source, the vector/scalar ratio is simply ICE multiplied by

*w*represent the importance of the spherical harmonic orders

_{l}*l*in the mapping from luminance to illuminance, and they fall off rapidly with

*l*. Higher-order spherical harmonic components of light probes are important in rendering specular surfaces, though, so a modification of ICE in which the weights

*w*fall off less rapidly may be useful in predicting diffuseness judgments in scenes containing non-matte surfaces.

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^{1}Brewster (1826) is often credited with discovering the light-from-above prior. In fact, he mostly elaborated Rittenhouse's (1786) observation that we perceive ambiguous shaded patterns as having a 3-D shape that is consistent with whatever we believe about the lighting direction in the scene being viewed. Neither Rittenhouse nor Brewster suggested that we have a default assumption that light comes from overhead.

^{2}In Morgenstern (2011), we used the term

*Lambertian contrast energy*instead of ICE. The two terms are synonymous.

^{3}We believe that the four unusually high ICE values from Boyaci et al. (2004) are artifacts. The wide range of ICE values for Boyaci et al. (2004) in our Figure 3 reflects their table 2, where ? =

*E*/

_{A}*E*was highly variable. Large differences in ? do not always reflect clear differences in behavior. Consider their observer BH, who had ? = 1.39 for light from the left and ? = 0.12 for light from the right (their table 2). These correspond to ICE values of 0.20 and 0.87, respectively. Their Figure 11 shows observers' chromaticity settings, and BH's settings were not drastically different for lighting from the left and right. Similar comments apply to RG. Furthermore, their figure 10 suggests that MM's chromaticity settings varied less with patch orientation than MD's, and yet their table 2 attributes more direct illumination assumptions to MM. Thus, their estimates of ? may have had a large variance.

_{P}^{4}In order to make the slant and tilt thresholds comparable, we used the actual angle between the two lighting directions when calculating the psychometric function, not the nominal angles 2?

*?*and 2?

*?*. For example, at a slant of 10°, the angle between tilt directions +90° and ?90° is just 20°, not 180°.

*v*= 1/(

*F*+ 1) (their equation 7).

_{A}*F*is the ratio of the illuminance

_{A}*E*from ambient light to the illuminance

_{A}*I*sin(

_{D}*?*)/

_{D}*d*

^{2}from a point source of luminous intensity

*I*at distance

_{D}*d*and elevation

*?*:

_{D}*F*=

_{A}*d*

^{2}

*E*/

_{A}*I*sin(

_{D}*?*) (unnumbered equation below their equation 6). The illuminance that a surface would receive directly facing the illuminant is

_{D}*E*=

_{P}*I*/

_{D}*d*

^{2}, so

*E*/

_{A}*E*=

_{P}*F*sin(

_{A}*?*). In Bloj et al.,

_{D}*?*= 30°. We read

_{D}*?*from their plots using data capture software and calculated

*E*/

_{A}*E*=

_{P}*F*sin(30°) = ((1/

_{A}*?*) ? 1) sin(30°). Equation 3 converts

*E*/

_{A}*E*to ICE.

_{P}*??*=

*E*/(

_{P}*E*+

_{P}*E*) that explains each observer's behavior (their table 1).

_{A}*E*/

_{A}*E*= (1/

_{P}*??*) ? 1, and Equation 3 converts

*E*/

_{A}*E*to ICE.

_{P}*?*in their table 1. The first term in their equation 5, in parentheses following (1 ?

*?*), approximates the luminance pattern of a sinusoidal surface under a point source that would create a luminance of 1 ?

*?*on a surface facing it directly. The second term, following

*?*, approximates the luminance pattern of a sinusoidal surface under an ambient source that creates a maximum luminance of 0.5

*?*. Thus

*E*/

_{A}*E*= 0.5

_{P}*?*/(1 ?

*?*), and Equation 3 converts

*E*/

_{A}*E*to ICE.

_{P}*x*? is

*x*rounded to the next higher integer. Andrew Schofield provided us with the values in Table B1 as an improvement on Schofield et al.'s table 1. The values in Table B1 were obtained by fitting Equation B1 (instead of Schofield et al.'s equation 5) to Schofield et al.'s data. The ICE values that we calculated from these revised data are shown in Figure B1 and described with summary statistics in Table B1.

? | ? | ||

Observer 1 | 0.62 | Observer 8 | 0.50 |

Observer 2 | 0.34 | Observer 9 | 0.87 |

Observer 3 | 0.59 | Observer 10 | 0.82 |

Observer 4 | 0.52 | Observer 11 | 0.56 |

Observer 5 | 0.37 | Observer 12 | 0.67 |

Observer 6 | 0.78 | Observer 13 | 0.60 |

Observer 7 | 0.70 | Observer 14 | 0.54 |

*p?*

_{1}and

*p?*

_{2}. The stimulus is two sets of 152 luminances,

*L*=

_{i}*i*= 1,2, corresponding to the two stimulus intervals. The posterior probabilities that the lighting directions are shown in order (

*p?*

_{1},

*p?*

_{2}) or (

*p?*

_{2},

*p?*

_{1}) are The ideal observer finds the probability ratio The two lighting orders are equally probable, so Equation C3 is equivalent to the likelihood ratio The luminances

*l*of different patches are independent, so the numerator in Equation C4 is Equation 5 shows that the distribution of each luminance

_{ij}*l*is a rescaling of the reflectance distribution with constant of proportionality

_{ij}*c*= (

_{kj}*E*max(

_{P}*p?*·

_{k}*n?*, 0) +

_{j}*E*)/

_{A}*?*. If stimulus interval

*i*has lighting direction

*p?*, the probability density of

_{k}*l*is

_{ij}*g*(

_{tr}*l*,

_{ij}*?*,

*?*), where

*g*(

_{tr}*x*,

*?*,

*?*) is the truncated normal distribution of reflectances. Equation C6 becomes The likelihood of the observed luminances under the opposite lighting order is The ideal observer calculates the likelihoods of the luminances under the two lighting orders using Equations C7 and C8 and chooses the order that generates the higher likelihood. Using simulations, we found the ideal observer's 75% thresholds in the slant- and tilt-discrimination tasks under the five diffuseness levels viewed by human observers.

*E*/

_{A}*E*= 5.54 gave the same ideal performance as in the first task. Equation 3 converts this to an ICE of 0.056, shown by the dashed line in Figure 3.

_{P}*E*(

*?*,

*?*) ?

*?*is the illuminance pattern

*E*(

*?*,

*?*) minus its DC component. The spherical harmonics

*Y*are orthonormal, so the integral of this factor squared is the sum of the squared amplitudes of its spherical harmonic coefficients. The DC component of

_{lm}*E*(

*?*,

*?*) ?

*?*is zero, so

*c*

_{00}= 0. Thus, Here we use

*w*and

_{l}*c*as defined in Equations 6 and 7.

_{lm}*E*+

_{P}*E*) ?

_{A}*E*=

_{A}*E*. The mean illuminance over the surface of a sphere is

_{P}*E*from the ambient source and

_{A}*E*/4 from the point source for a total of

_{P}*E*+

_{A}*E*/4. Thus the vector/scalar ratio is ((

_{P}*E*/

_{A}*E*) + 0.25)

_{P}^{?1}. Comparing this to Equation 3, we see that, for a PA light source, the vector/scalar ratio is ICE times