in all directions or the concentration of light scattering in a direction near the specular direction for glossy

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2 Virtual Gonio-spectrophotometer for Validation of BRDF Designs in all directions or the concentration of light scattering in a direction near the specular direction for glossy materials. Besides empirical models, another class of BRDF models exists. These models apply basic principles of physics to the surface microscopic structure, for example the Cook-Torrance s model [5]. Manufacturers in industry often need to characterize the appearance of their products (eg. paint finish). Because of the complexness of full BRDF acquisition, it is more reasonable to relay on simple measurements. One of the proposed reflection evaluation of internal building materials by the empirical method of qualitative appreciation is described in [4]. To measure the appearance attributes in industry, industrial spot measurement devices, such as colorimeters or glossmeters has been developed [6]. Glossmeters are build according to industrial standards ISO 2813 [1] or ASTM D523 [2], that define the geometry of the specular gloss measurement including the incidence angles of light, and optics [1]. Gloss is the appearance attribute of surfaces that causes them to have appearance from shiny to mat. We can measure it by focusing on the light reflected off the surface and not by focusing on the surface. Perception of the gloss is influenced by physiological and psychological aspects of the human vision. The psychophysically-based models of surface gloss have been proposed based on multi-dimensional scaling algorithms and the subjective evaluation of the images [11, 16]. Our goal is to design and implement a methodology devoted to the virtual measurement of gloss and spectral reflectance on samples represented by BRDF. Industry uses the real measurements performed by industrial devices on real material samples. During the coarse of this work we consider the measurements on virtual samples with the paint surface represented by abstract BRDF. The measurements on virtual samples can be compared with actual measurements on real samples and then the precision of virtual representation based on BRDF can be evaluated. The simulation of the light reflection enables us also to predict the appearance parameters of samples, to design desired appearance [9]. Virtual gonio-spectrophotometer can also be used to find correspondence between the parameters of designed reflectance model and industrial measurement scale for the albedo and gloss used by glossmeters and colorimeters. The correspondence can be found by several standardized measurements using virtual gonio-spectrophotometer on multiple virtual samples. Appearance of virtual samples is usually represented by an analytical reflectance model with multiple controlling parameters. Analytical model defines the scattering profile at a single point. Unfortunately, the real measurement devices are unable to produce single ray or perform measurement at single point of the sample, but rather they perform the measurement on a small spot using a beam of rays. Therefore an integration over certain areas, with dimensions defined in industrial standards, is required [15]. Virtual gonio-spectrophotometer consists of a standard light source and luminous flux sensor (standard observer). Sample appearance is represented as a BRDF. The area and the location of the light source and the standard observer is adjustable with respect to the sample surface. These settings allow to simulate several measurement devices and set up the different standards. Numerical calculation of the luminous flux is done by splitting the light source and detector aperture into small finite area samples. 2

3 Andrej Mihálik, Roman Ďurikovič The first part of this paper is devoted to the definition of the virtual gonio-spectrophotometer. We also introduce mathematical principles of the light-surface interaction used in our concept of the virtual material reflectance measurements. In the second part we present utilization of our virtual device to perform standardized gloss measurements. We introduce requirements needed to fulfill standards that prescribes gloss measurements. The results and the discussion is subject of the last part of this paper. 2. Materials and Methods When incident light hits the perfectly polished surface, incident ray is redirected into its mirror image, although certain amount of incident light may be refracted into the material volume. Due to material composition and surface roughness, light may be scattered into the multiple directions. Scattered light that leaves the surface in each direction is typically modeled with Lambertian reflectance (with color considered constant with respect to the viewing angle) [13]. However, goniochromatic materials such as metallic and pearlescent paints change color with viewing angle. Glossy materials exhibit more light reflected in mirror (specular) direction then in other directions (off-specular). In the following sections we will focus mostly on the glossy materials. To measure surface reflectance due to incident light in dependence of incident angle, a gonio-spectrophotometer can be utilized. The gonio-spectrophotometer is defined as an instrument that measures spectral power as a function of illumination and observation. The light flux incident on the material surface comes from the light source aperture. The light viewed by the detector is delimited by its aperture. Both the direction of illumination Θ i (in spherical coordinate, where θ is inclination and ϕ is azimuth: Θ i = (θ i, ϕ i)) and viewing direction Θ r = (θ r, ϕ r) can vary independently within the hemisphere above the material sample. Hemisphere is subdivided into small patches to obtain discretized apertures. A patch belongs to the source or detector aperture (see Figure 1) if its position is close (within adjustable tolerance which represents dimensions of apertures) to direction of the illumination Θ i or observation Θ r, respectively. In the following text, patches belonging to source aperture are indexed as i k and detector aperture are indexed as r l. Total flux reflected from the sample surface is computed by the sum of incident flux from the source aperture patches (from each source aperture patch in direction Θ ik = (θ ik, ϕ ik )) reflected toward the detector aperture patches (to each detector aperture patch in direction Θ rl = (θ rl, ϕ rl )). Thus total flux captured by the detector aperture is computed by the numerical integration over the source and detector patches (adding the energy reflected from each patch i k toward each patch r l ). Proposed virtual gonio-spectrophotometer requires two inputs, a description of the spectrum of the illuminant and a description of the reflectance of the virtual surface by using the BRDF. 3

4 Virtual Gonio-spectrophotometer for Validation of BRDF Designs Figure 1. Geometry set up and hemisphere representation. Left: Device geometry setting of the position and angles of a source and detector. Center: Special subdivision of a source and detector over the hemisphere. Right: Geometry of incident angles relative to the sampled angles Standard illuminant Light sources are described by their spectral power distribution (SPD). SPD is a representation of the radiant power emitted by a light source as a function of wavelength. The International Commission on Illumination (CIE) has defined a set of standard illuminants to be used for colorimetry [1]. Illuminant D 65 represents the noon daylight with a correlated color temperature of approximately 65 K. In the Figure 7, the red curve represents relative SPD of the standard illuminant D 65 normalized about 56nm sampled at 5 nm intervals used in our inplementation Bidirectional reflectance distribution function To measure radiant flux reflected off the surface we need a description of how surface reflects the light in dependence of the angle of the illumination. Such distribution could be ensured by BRDF particularly Cook-Torrance and Ward BRDF models. Proposed virtual gonio-spectrophotometer can also perform measurement using table instead of analytical function where data from real gonio-spectrophotometer are depicted. BRDF describes how incident light energy is distributed by a material with respect to position, direction, and wavelength. BRDF is based on the radiant flux redirected by the surface from the source toward the detector. Radiant flux Φ is defined as the energy Q transferred trough some time period t: Φ = Q t. (1) The average flux leaving per unit area, or radiant exitance M, is the total flux leaving divided by surface area A, or M = Φ A. (2) The radiant energy per unit time and area arriving at a surface is called the irradiance E. It is defined in the same manner as M, with the only difference being whether the radiant energy is approaching or leaving the surface. To include directional effects, we should consider how the point of view affects perception of an area. When viewing direction is perpendicular to a patch then perceived area of this path is greater then perceived area of a 4

5 Andrej Mihálik, Roman Ďurikovič Figure 2. The apparent size of surface A is larger when it is projected in the direction of its surface normal, than when it is projected in a direction at an angle θ to the normal. patch viewed at a more glancing angle, see Figure 2. Such projection is achieved by multiplication of the area by cos θ. Radiance L at point x in a particular direction Θ = (θ, ϕ) is defined as the radiant flux per unit solid angle and unit area projected in a direction at an angle θ. θ (, π ) is the angle between the surface normal and direction 2 Θ. Therefore radiance L: L(x, Θ) = ddφ(x, Θ) cos θ da dω. (3) Term ddφ(x, Θ) in above formula is fragment of radiant flux emitted by fragment of surface area da at surface coordinates x and exited unit hemisphere through solid angle dω. To express radiance as a function of wavelength, we consider the quantity at a value of λ within a small band of wavelengths between λ and λ + dλ. By associating a dλ with each value, we can integrate spectral values over the whole spectrum. We express the spectral radiance as: L(λ, x, Θ) = dddφ(λ, x, Θ) cos θ da dω dλ, (4) where dddφ(λ, x, Θ) is fragment of radiant flux ddφ(λ, x, Θ) that is within a band of wavelengths between λ and λ + dλ. To describe how a surface reflects light, we consider light incident on the surface at x with radiance L(λ, x Θ i) within a differential solid angle dω i given by direction Θ i. The irradiance de i on the surface that will be either absorbed or redirected is: de i(λ, x Θ i) = L(λ, x Θ i)cos θ dω i. (5) The cos θ i term appears because the radiance L measures energy per unit area da in the direction of travel Θ i, and that direction projected into the orientation of the surface we are considering is cos θ i da. The dω i term enters in because we want to know the effect of energy coming from a single direction representing a differential section of all the possible directions above the surface. The effect of a material reflecting light is then given by a function that is the ratio of the radiance reflected in a particular direction Θ r = (θ r, ϕ r) as a result of the total incident flux per unit area from another direction Θ i = (θ i, ϕ i) as shown in Figure 3. This ratio is referred to as the BRDF. It is defined by f r: f r(λ, Θ i Θ r) = dlr(λ, x Θr) de i(λ, x Θ i). (6) 5

6 Virtual Gonio-spectrophotometer for Validation of BRDF Designs Figure 3. Flux of the light entered unit sphere through the differential solid angle dω i given by direction Θ i and reflected from the area da through the differential solid angle dω r given by direction Θ r. Note that fragment of the flux emerged in numerator of f r is fragment of the flux of light with wavelength band between λ and λ + dλ, that enters the unit sphere through dω i and is reflected from da and leaves the unit sphere through dω r. Denominator is the emerged flux of light with wavelength band between λ and λ + dλ, that enters unit sphere through dω i and is just incident with da Fresnel reflection The Fresnel equations [8] describe the amount of light reflected and refracted at a perfectly smooth surface between two media. For parallel polarized light (p-polarized, the electric field oscillates in the plane of incidence) reflection coefficient is defined as: r = n2 cos θi n1 cos θt n 2 cos θ i + n 1 cos θ t. (7) For perpendicular polarized light (s-polarized, electric field perpendicular to plane of incident) it is: r = n1 cos θi n2 cos θt n 1 cos θ i + n 2 cos θ t. (8) Here θ i and θ t are the angles between the surface normal and the directions of the incident and transmitted beams (see Figure 5), and n 1 and n 2 are the indices of refraction of the media on the incident and transmitted side of the surface. For unpolarized light the fraction of light that is refracted is F r = r2 + r 2. (9) 2 Real materials are not perfect insulators, therefore, their index of refraction is defined in complex form ˆn = n+ik, where n is the refractive index indicating the phase velocity and extinction coefficient k indicates the amount of absorption loss when the electromagnetic wave propagates through the material. Both n and k depend on the wavelength. The example of copper refractive index is shown in Figure 4. The copper refractive index is later used in calculation of Fresnel reflection (Equation 1) to evaluate the gloss appearance of virtual sample made of copper. 6

7 Andrej Mihálik, Roman Ďurikovič Figure 4. Complex refractive index of the copper as the function of wavelength. Index n is depicted in the left chart, index k in the right chart. Figure 5. The geometry of reflection showing the angle θ i between the surface normal N and the direction of illumination Θ i L. We incorporated Fresnel reflection into our computations in the following form with u = cos θ i: ( ) F r(u) = 1 (a u) 2 + b 2 (a + u 1 u )2 + b 2 2 (a + u) 2 + b 2 (a u + 1 u )2 + b + 1, (1) 2 where k is extinction coefficient, n is the refractive index indicating the phase velocity and a 2 = 1 2 ( (n 2 k 2 + u 2 1) 2 + 4n 2 k 2 + (n 2 k 2 + u 2 1)) b 2 = 1 2 ( (n 2 k 2 + u 2 1) 2 + 4n 2 k 2 (n 2 k 2 + u 2 1)) Cook-Torrance reflectance model The Cook-Torrance model [5] is a physically-based microfacet model that is focussed on specular reflection. This model treats surface as a collection of microscopic facets. The macroscopic optical properties of a surface are then analytically derived from properties of individual facets and statistical distributions of such properties. Although, the surface has a normal N, at a microscopic level the surface has height variations that result in many different surface orientations at a detailed level. At the perfectly flat surface a viewer is able to see light source at the point where halfway vector H is in the direction of average surface normal N. Halfway vector H is a bisector of the angle between the direction of illumination Θ i L and the viewer directionn Θ r V, see Figure 5. 7

8 Virtual Gonio-spectrophotometer for Validation of BRDF Designs Figure 6. Microfacet geometry. Upper row: Surface composed of microfacets with average normal N. Bottom row: Interaction between microfacets, from left to right: interreflection, masking, shadowing. Rather than explicitly model the small geometric features, general reflectance functions use statistical models. Statistical models are used because the variation in surface height is assumed to be irregular and random. A statistical model for surfaces in reflectance models generally takes the form of giving the distribution of facets that have a particular slope. The facet slope distribution function D represents the facets that are oriented in the direction H. The formula is: D = ce θ 2 h m 2. (11) where θ h is the angle between surface and facet normal, c is arbitrary scaling factor and m is the root mean square slope of microfacets parameterizing the surface s roughness. Smaller values of m produces more specular appearance. Assuming V-grooved surface (Figure 6), then we need to count with self shadowing and masking. The geometric attenuation factor G models the geometric effects inter reflection, masking and shadowing between microfacets that occur at larger angles of incidence or reflection. It is defined by the formula: G = min{1, 2(H N)(V N), H V 2(H N)(L N) }. (12) H V Combination of diffuse and specular component of Cook-Torrance s model can be written as: f r(θ i Θ r) = k d π + F rdg ks, (13) π cos θ i cos θ r where k d + k s 1, θ r is the angle between N and V, and F r is Fresnel term given by the Equation 1, where u = L H. We assume mirrorlike microfacets which are reflecting the light from a source to the viewers direction just in constellation where H is microfacet s normal Ward reflectance model Ward s model [14] is an empirical shading model. It uses an exponential function parameterized by an average slope of the microscopic surface roughness. The Ward model in the terms of BRDF as a sum of specular and 8

9 Andrej Mihálik, Roman Ďurikovič diffuse component is: f r(θ i Θ r) = ρ d π + 1 e tan 2 θ h σ ρs 2, (14) cos θi cos θ r 4πσ 2 where θ h is the angle between N and H, see Figure 5. The parameters ρ d and ρ s are spectral reflectance factors, that control the color of the diffuse and specular reflection. The value of σ represents the standard deviation of the microfacet slope, that models surface roughness. Small values of σ (i.e., less that.1) model a very nearly smooth surface Numerical hemispherical integration Proposed virtual-goniospectrophotometer consists of light source and detector aperture. To perform the integration over these apertures, we project them onto unit hemisphere. Then we subdivide hemisphere into the small patches (see Figure 1). Each patch area of the hemisphere serves us as the solid angle. Particular radiant flux is coming through all source patches ω ik, that are incident with the surface. For each detector patch ω rl, the contribution of reflected flux to this patch is computed. This leads to double summation, over the source and over the detector aperture. The amount of energy reflected from the source to the detector is determined by the BRDF. Total flux reflected from the sample patch at a surface area da reaching detector aperture is then computed by following formula: Φ sample = da k L(x Θ ik ) cos(θ ik ) ω ik f r(x, Θ ik Θ rl ) cos(θ rl ) ω rl, (15) l where da is area of a patch at the surface. ω is area of the particular patch on the hemisphere. Index i refers to the source patches and index r refers to the detector patches. Θ is the direction from surface patch to the particular patch on the hemisphere and θ is angle between surface normal and that direction, see Figure 1. Function f r is the BRDF and L is incident radiance (see Equation 4), where fragment of the radiant flux Φ is determined by the spectral energy of the light source. The real measurement devices such as glossmeters can not measure the light reflection at a single point of the sample, but rather they measure reflection in a small region. Inhomogeneous surfaces such as metallic or pearlescent varnishes have varying f r at each surface point. This can happen often when there is a drawing on the surface, for example. It is therefore necessary to divide the sample into smaller parts and make a calculation on each part. 3. Results and Discussion In this section we present results and describe computations used in our experiments to perform gloss measurements of virtual samples represented by the analytical models. These computations are derived from the standard gloss measurements prescribed by the industrial standards. To perform standardized gloss measurements we utilized our virtual gonio-spectrophotometer principle. 9

10 Virtual Gonio-spectrophotometer for Validation of BRDF Designs 3.1. Virtual glossmeter Industrial gloss measurements satisfy the standards such as ISO 2813 or ASTM D523. Measurements are measured in gloss units [12] which describes amount of reflected light related to the amount of reflected light from a black glass standard with refractive index of Those standards usually prescribe the measurement to be taken at angles 2, 6 and 85 to the surface normal (both directions to the light source and detector are in the same plane with the surface normal; angles in the Figure 1 are set to θ i = θ r {2, 6, 85 }), because these degrees of specular gloss measurements offer numerical values which are roughly linearly correlated over a range of values to perceived gloss of high gloss, medium gloss and low gloss surfaces, respectively [7]. These standards also prescribe size of source and detector apertures, see Table 1. Angle of gloss measurement Aperture size (in degrees) Source Detector Table 1. Left: Angular aperture sizes defined from the measured spot and the angles of the gloss measurement. explanation. Right: Sizes To compute gloss values we first perform computation of radiant flux Φ sample using Equation 15 with f r of the measured material. A patch of the hemisphere belongs to source or detector aperture iff the angle between direction of the measurement and direction to this patch (from the sample center, see Figure 1) is within angle specified in the Table 1. The flux reflected off the black glass standard is computed by following formula: Φ standard = da k L(x Θ ik ) cos(θ ik ) ω ik F r(cos θ ik ), (16) where F r is Fresnel term (see Equation 1) with n = and k =. Equation 16 is derived from Equation 15 by incorporating f r of the black glass standard. Gloss units [15] are then computed by GU = 1 Φ sample Φ standard. (17) We utilized our virtual gonio-spectrophotometer to perform these computations. To represent virtual samples s f r, we have chosen Cook-Torrance s and Ward s analytical models. Then our virtual gloss measurement was used to find the correspondence between analytical models and real samples Real measurements We measured real samples using the real industrial glossmeter (see first three columns of Table 2). We have found the parameters of analytical model corresponding to the real samples empirically for Cook-Torance s and Ward s 1

11 Andrej Miha lik, Roman D urikovic model, see Table 2. Then we evaluated and adjusted the representation by comparing real gloss measurements with the virtual ones showing as root mean square error. We performed gloss measurements of several automotive car paintings and the polished glass. We have measured gloss units (GU) of small samples (see Table 2) by portable device Konica MULTI GLOSS268. Measurements were performed in the dark room by applying device onto the flat surface of the small material sample. real measurements 2 [GU] virtual measurements Cook-Torrance [GU] parameter [GU] RMSE 6 85 c m n 2 6 Ward 85 RMSE parameter σ polished glass KKX Rouge Tiziano EZRC Gris Aluminum polished copper Table err err.416 Real and virtual gloss measurements with parameters of BRDF models. Virtual measurements As example of using our virtual gonio-spectrophotometer we have calculated spectral reflectance of the virtual copper sample. Resulting spectrum is depicted in the Figure 7. In this experiment we have used Cook-Torrance s model (see Equation 13), where the root mean square slope of microfacets orientations is m =.4 and c =.8 (see Equation 11), index of refraction was used as shown in Figure 4. Result was calculated using Equation 15 as radiant flux reflected off the surface in the direction of 6 (θs = θd = 6 ; directions to the center of both apertures and the surface normal are in the same plane) to the surface normal as the function of the wavelength. Reflected radiant flux is relative to the spectral power of the standard illuminat D65 that serves us as light source determining incident radiance (specifically radiant flux in Equation 4). Our virtual gonio-spectrophotometer was utilized to perform gloss measurements to find the correspondence between real samples and analytical fr such as Cook-Torrance s or Ward s. We have performed gloss measurements of the real samples using the portable glossmeter device. We have measured a polished glass, two automotive paints and a metal sample, see Table 2. Pearlescent and metallic automotive paint sample was the effect red paint KKX Rouge Tiziano and aluminum paint sample EZRC Gris Aluminum with large aluminum sparkles. Metal sample was the unpolished and the polished copper. For each sample we have estimated parameters of CookTorrance s and Ward s model that approximates particular sample. Parameters of Cook-Torrance s models are scaling factor c, root mean square slope of microfacets m that determines surface roughness (see Equation 11) and refractive index n. Parameter σ of Ward s model determines surface roughness. While all samples exhibit shiny appearance and we aim for gloss measurements, we have used only specular component of these models. Once we had determined function fr we computed gloss units by Equation 17 using our Virtual Gonio-spectrophotometer. Results of our measurements are shown in Table 2. We can compare real and virtual measurements in the 11

12 Virtual Gonio-spectrophotometer for Validation of BRDF Designs Figure 7. Spectral reflectance of the copper represented by Cook-Torrance s BRDF measured under 6 to the surface normal. Blue curve represents reflected flux (as the function of wavelength) off the sample relative to spectral power of standard illuminat D 65 (red curve). table and conclude plausibility of used models. In the Table 2 is computed root mean square error (RMSE) to compare virtual measurements using analytical models with real ones. From our measurements we can see that Ward s BRDF produces small low gloss values (measured under 85 ) and therefore is not very plausible model for approximating surface s BRDF based on the standard gloss measurements. Let us discus root mean square slope parameter m of the distributed sparkles in Cook-Torrance analytical reflectance model. For polished glass m, because there are no sparkles at all. For an affect red paint KKX Rouge Tiziano, the value m is still less then 1 because there are many small sparkles, while in aluminum paint sample EZRC Gris Aluminum the parameter m is larger due to large size of aluminum sparkles. TO DO: zapracovat diskusiu The Cook-Torrance representation for a polished glass has a very narrow and long specular lobe that is apparent from c = 595. We can observe that pure polished metal surfaces have large gloss values then automotive paintings. mozno dat do conlusions: For metallic paints with small aluminum sparkles (1 µm 5 µm) this worked well. 4. Conclusion We have developed Virtual Gonio-spectrophotometer that performs measurements on the virtual material samples represented by the BRDF. The Virtual Gonio-spectrophotometer consists of the standard light source and the detector aperture. Parameters of apertures (eg. position, size) are highly customizable to allow simulations of devices like colorimeter or glossmeter. Our application can evaluate BRDF representation of real paint samples using a few measurements by glossmeter. 12

13 Andrej Mihálik, Roman Ďurikovič Glossmeters retrieve too few data to direct construction of the BRDF, therefore to obtain BRDF we need to use a certain analytical reflectance model such as Cook-Torrance or Ward. We have used our Virtual Gonio-spectrophotometer to perform standardized gloss measurements using these analytical representations and evaluated and compared results with the real measurements of real samples. As a result we found out that Cook-Torrance model is more suitable for the gloss reflectance modeling then Ward model. In future we would like to perform standardized measurement of the diffuse color. This measurement requires a hemispherical light source over the measured spot of the sample. The detector is situated above the spot and angle between surface normal and viewing direction is 8. Standard colorimeter retrieves values in CIE La b color representation. Acknowledgments This research was partially supported by a VEGA 1/662/ project a Scientific grant from Ministry of Education of Slovak Republic and Slovak Academy of Science, and Comenius University Grant UK/34/21. References [1] ISO Paints and varnishes -Determination of specular gloss of non-metallic paint films at 2 degrees, 6 degrees and 85 degrees. Internation Organization for Standardization, [2] ASTM D Standard Test Method for Specular Gloss. ASTM, [3] Shruthi Achutha. Brdf acquisition with basis illumination. Master s thesis, Department of Computer Science, University of British Columbia, Vancouver, Canada, 26. [4] M. Bodart, R. de Pearanda, A. Deneyer, and G. Flamant. Photometry and colorimetry characterisation of materials in daylighting evaluation tools. Building and Environment, 43(12): , 28. [5] R. L. Cook and K. E. Torrance. A reflectance model for computer graphics. ACM Trans. Graph., 1(1):7 24, [6] Julie Dorsey, Holly Rushmeier, and Francois Sillion. Digital Modeling of Material Appearance. Morgan Kaufmann, San Francisco, USA, 28. [7] R. Ďurikovič and I. Šeďo. Real-time friendly representation of arbitrary brdf with appearance industry measurements. Journal of the Applied Mathematics, Statistics and Informatics, 3(1):17 26, 27. [8] Roman Ďurikovič and Tomáš Ágošton. Prediction of optical properties of paints. Central European Journal of Physics, 5(3): , 27. [9] V. Goossens, N. Gotzen, S. Van Gils, E. Stijns, G. Van Assche, R. Finsy, and H. Terryn. Predicting reflections of thin coatings. Surface and Coatings Technology, 24(5): ,

14 Virtual Gonio-spectrophotometer for Validation of BRDF Designs [1] Richard S. Hunter and Richard W. Harold. The Measurement of Appearance. John Wiley & Sons, Inc., New York, NY, USA, [11] Fabio Pellacini James A. Ferwerda and Donald P. Greenberg. A psychophysically-based model of surface gloss perception. In Proceedings SPIE Human Vision and Electronic Imaging 1, pages , 21. [12] Mario Noël, Joanne Zwinkels, and Jian Liu. Optical characterization of a reference instrument for gloss measurements in both a collimated and a converging beam geometry. Appl. Opt., 45(16): , Jun 26. [13] K. E. Torrance and E. M. Sparrow. Theory for off-specular reflection from roughened surfaces. JOSA, 57(9): , [14] Gregory J. Ward. Measuring and modeling anisotropic reflection. SIGGRAPH Comput. Graph., 26(2): , [15] Harold B. Westlund and Gary W. Meyer. Applying appearance standards to light reflection models. In SIGGRAPH 1: Proceedings of the 28th annual conference on Computer graphics and interactive techniques, pages , New York, NY, USA, 21. ACM. [16] Josh Wills, Sameer Agarwal, David J. Kriegman, and Serge J. Belongie. Toward a perceptual space for gloss. ACM Trans. Graph., 28(4),