It has recently been shown that an increase of the relief height of a glossy surface positively correlates with the perceived level of gloss (Y.-H. Ho, M. S. Landy, & L. T. Maloney, 2008). In the study presented here we investigated whether this relation could be explained by the finding that glossiness perception correlates with the skewness of the luminance histogram (I. Motoyoshi, S. Nishida, L. Sharan, & E. H. Adelson, 2007). First, we formally derived a general relation between the depth range of a Lambertian surface, the illumination direction and the associated image intensity transformation. From this intensity transformation we could numerically simulate the relation between relief stretch and the skewness statistic. This relation predicts that skewness increases with increasing surface depth. Furthermore, it predicts that the correlation between skewness and illumination can be either positive or negative, depending on the depth range. We experimentally tested whether changes in the depth range and illumination direction alter the appearance. We indeed find a convincingly strong illusory gloss effect on stretched Lambertian surfaces. However, the results could not be fully explained by the skewness hypothesis. We reinterpreted our results in the context of the bas-relief ambiguity (P. N. Belhumeur, D. J. Kriegman, & L. Yuille, 1999) and show that this model qualitatively predicts illusory highlights on locations that differ from actual specular highlight locations with increasing illumination direction.

*λ*of the relief

*x*,

*y*) =

*λz*(

*x*,

*y*), the associated image transform for each image pixel intensity

*I*(

*x*,

*y*) is

*λ*(red solid lines). This transformation, which we call the lambda transformation, resembles the gamma transformation (

_{ γ }(

*I*) =

*I*

^{ γ }) that is used to linearize screen luminance but can also be used to alter the skewness of the histogram (see for example the supplementary notes of Motoyoshi et al., 2007). The resemblance between the two transformations can be seen in Figure 1a where the dashed blue lines denote the gamma transformation. The gray area between the gamma and lambda transformation denotes the difference. To quantify the effect of the lambda transformation on a luminance histogram, it was applied to a beta distribution, which can be seen in Figure 1b. While Equation 1 only deals with frontal illumination (

*σ*= 0), we also derived a relation for non-zero polar angle. As can be read in 1, this relation relies on a simplification of the 3D surface and is this less generic than Equation 1. We solved Equation A9 (see 1) numerically to calculate the results of the non-zero polar angle model, using the same zero skew beta distribution as reference distribution. The results are presented in Figures 1c and 1d.

*σ*= 0) would result in a considerable skew change. However, when the illumination direction increases, the relation between skewness and relief stretch changes from positive to negative. In Figure 1d the same skewness data is plotted but now as a function of illumination direction for various relief stretches. Note that when an image has zero skew (in this case for

*λ*= 1), it is invariant under illumination change. The range of skewness values in these figures is of comparable magnitude as the range that previously resulted in percepts ranging from matte to glossy (Motoyoshi et al., 2007). On the basis of those psychophysical results, it can be predicted that stretching a Lambertian (i.e. matte) surface about 8 times, would change the appearance substantially towards glossy. We wanted to test this hypothesis that a relief stretch transformation can change the material appearance from matte to glossy.

*λ*-range around 1. However, for the psychophysical stimuli that we will describe below, we could not predict the absolute skew. Therefore, the

*λ*-ranges are relative and we attributed the shallowest relief the arbitrary value of 1. As will become clear in the Results section (Figure 3b), our stimuli were in fact approximately distributed around zero skewness for frontal illumination.

*f*

^{2}, which defines a Brownian random surface (Saupe, 1988). We used Brownian surfaces because their geometry is close to natural surfaces (Mandelbrot, 1983). As such they can be used as generic natural (but random) surface stimuli.

*λ*= 1, 2, 4, 8. Within each group the surfaces were rendered with a collimated light source that varied in polar angle from 0 (frontal illumination) to 60 degrees (grazing illumination). The rendering procedure was programmed in

*C*and consisted of two phases. First, at each pixel location the inner product of the surface normal and illumination direction (Lambertian shading) was calculated. The illumination strength was set at one (white), so surface patches oriented in the direction of illumination were white. No tone mapping was applied. Second, the image was multiplied with a cast and body shadow map so that all pixels located in body or cast shadows were rendered black. Inter-reflections were discarded. The azimuthal angle was uniformly, randomly distributed between 1 and 360 degrees. The reflectance was Lambertian. A black circular aperture with a diameter of 488 pixels was superimposed on the rendering. Examples of the stimuli can be seen in Figure 2.

*λ*= 4. This is a free parameter since we did not know the skewness of the image of a Brownian surface in advance. The predictions were all relative to a zero skew image. When the stretch is lower, the relation between the polar angle and skewness is positive while for larger values the relation becomes negative. Lastly, we highlighted the near-frontal illumination results by the red symbols. This positive relation is (trivially) similar for with and without shadows calculations and is in line with our prediction.

*z*) was confirmed by a multiple linear regression on the independent variables stretch (

*x*) and illumination direction (

*y*):

*z*=

*a*

_{0}+

*a*

_{ stretch }

*x*+

*a*

_{ illumination }

*y*(

*F*(297, 2) = 957.934,

*p*< 0.001;

*a*

_{ stretch }:

*t*(296) = 34.075,

*p*< 0.001;

*a*

_{ illumination }:

*t*(296) = −27.881,

*p*< 0.001).

*Radiance*(Ward, 1994). To increase the understanding of how the appearance of a Brownian surface changes under the shape and light transformations we used in our experiments, we present a movie that shows five paths through this shape-light parameter space. Movie 1 is without inter-reflections and Movie 2 is rendered with 2 ‘ambient bounces’. The colors of the frames relate to the five parameter space paths. The gray path indicates the bas-relief ambiguity relation, which will be discussed later. The other paths denote changes in either the illumination or the relief stretch, keeping the other constant. In the last frame of the movie, the blue, gray, and yellow framed images should appear optimally glossy: high relief in combination with near-frontal illumination. As can be seen, it seems that there is no pronounced difference in gloss appearance between the with and without inter-reflections images. Besides renderings, we also photographed two real surfaces in our lab. Two Gaussian surfaces that only differed with respect to a stretch transformation were photographed with a collimated light source (a theatre spot light). The surfaces were computer milled and were approximately matte. As can be seen in Figure 5, the high relief surface (on the right) shows similar illusory gloss as we found in our experiments. Thus, from these illustrations it does not appear that inter-reflections play an important role in the gloss illusion.

*and*non-spatial statistic. On the other hand, the photo-geometric hypothesis is based on a complex inverse-optics scheme. According to Anderson and Kim (2009), the actual geometry of the surface should be known after which the visual system can check whether the highlights are in the ‘correct’ positions. However, this requires surface geometry knowledge which can only be attained by having assumptions about the reflectance and illumination. As Anderson and Kim (2009) write, the shape, reflectance and illumination are all conflated in the 2D image. Precisely this difficulty would be solved if a ‘short-cut’ existed that is purely based on image statistics. An intermediary hypothesis could involve a

*spatial*image statistic that depends on the geometry of the image instead of the geometry of the imaged scene, for example histograms of (multiscale) image structure curvatures or the statistics related to illuminance flow (Pont & Koenderink, 2003).

*μ*and

*ν*are the affine shear components in the x and y direction, respectively. Since we only considered stretch in the viewing direction, these two values are zero. If we now assume a (locally) symmetric shape, i.e.

*f*

_{ x }=

*f*

_{ y }, the equation simplifies to

*λ*> 1 this correction will darken the points on the surface which are directed in the viewing direction (

*f*

_{ x }= 0). This means that if this correction is not performed, these points in the direction of the viewing direction will be highlighted with respect to the unstretched version (

*λ*= 1). Thus, if the illumination is in the same direction as the viewing direction, highlights will appear that would otherwise appear if the surface were glossy instead of Lambertian. If the illumination direction changes away from frontal, the specular highlights of an actual glossy surface would shift accordingly while the illusory highlights on the stretched Lambertian surface will be unaffected since there is no illumination term in the albedo correction Equation 2. In other words, the congruence of illusory (matte) highlights and actual (specular) highlights decreases with increasing illumination direction. Hence, the illusory gloss highlights are in the ‘wrong’ position which, as Anderson and Kim (2009) have shown, should lead to a decrease in apparent gloss (although off-specular reflection is physically possible (Torrance & Sparrow, 1967)). Indeed, this is what our data show. On the basis of our experiments we cannot completely discard the skewness hypothesis but we can conclude that there is more to perceived gloss than this statistic. The appearance of illusory highlights when stretching a relief could be an important factor that qualitatively explains the relation of apparent gloss with illumination direction.

*z*(

*x*,

*y*) with Lambertian reflectance and unit albedo under collimated illumination, the orthographically projected image as seen from the positive z-direction equals:

*σ*and the azimuthal angle

*θ*:

*z*(

*x*,

*y*):

*z*

_{ x }and

*z*

_{ y }denote the partial derivatives (

_{ λ }(

*z*) =

*λz*, where

*λ*is a positive scalar, what is the associated image transformation

_{ λ }(

*I*)? Importantly, we are looking for a solution in which the shape information (

*z*(

*x*,

*y*) and its derivatives) can be eliminated. If this is possible the image transform will be a generic, shape independent transform. Since the surface is isotropic we can set

*θ*= 0 without loss of generality. The original and transformed image can now be written according to Equation A1:

_{ λ }(

*z*) =

*λz*relation is to take the illumination in frontal direction, i.e.

*σ*= 0. This leads to

*f*(

*x*,

*y*) =

*z*

_{ x }

^{2}+

*z*

_{ y }

^{2}, which contains the shape information of the surface. The shape information can be eliminated, which leads to the desired image transform

*σ*is more complicated, and can only be found numerically. The only way to eliminate the shape information from Equations A4–A5 is to set either

*z*

_{ x }or

*z*

_{ y }to zero. This means that in one of these directions the shape is constant. Since we chose the illumination in the x-direction (

*θ*= 0), the variation in this direction should be non-zero. Therefore, we set

*z*

_{ y }= 0. This resulting shape is some (irregular) grating. We have illustrated this for the case of Brownian surfaces in Figure A1. As can be seen in the axes of the figure, the illumination is directed perpendicular with respect to the grating direction.

*z*

_{ x }from both Equations A4–A5 as follows:

_{ λ,σ }(

*I*). Nevertheless, the equation can be numerically solved. The results of this solution are presented in the main text.