Surface reflection properties of oil paints under various conditions

Surface Reflection Properties of Oil Paints under Various Conditions

Shoji Tominaga∗a and Shogo Nishib aGraduate School of Advanced Integration Science, Chiba University

Inage-ku, Chiba, Japan bDepartment of Engineering Informatics, Osaka Electro-Communication University

Neyagawa, Osaka, Japan

ABSTRACT

This paper describes a method for measurement and analysis of surface reflection properties of oil paints under a variety of conditions. First, the radiance factor of a painting surface is measured at different incidence and viewing angles by using a gonio-spectro photometer. The samples are made from different oil paint materials on supporting boards with different paint thicknesses. Next, typical reflection models are examined for describing 3D reflection of the oil painting surfaces. The models are fitted to the observed radiance factors from the oil paint samples. The Cook-Torrance model describes well the reflection properties. The model parameters are estimated from the least-squared fitting to the genio-photometric measurements. Third, the reflection properties are analyzed on the basis of several material conditions such as pigment, supporting material, oil quantity, paint thickness, and support color.

Keywords: oil paints, surface reflection property, 3D light reflection model, Cook-Torrance type model

1. INTRODUCTION The digital archiving of art paintings was originally based on generating color-calibrated images at high resolution from the paintings [1], where the surface of a painting was assumed to be flat and was photographed under diffuse light sources. This approach, however, achieves a pleasing record only for a single viewpoint and for a limited range of materials with minor surface height variation. It can be called “viewpoint and illumination-dependent digital archiving.” On the other hand, the authors have been discussing the problem from a global point of view that can be called “viewpoint and illumination-independent digital archiving,” (see [2][3]). Viewpoint and illumination-independent digital archiving by rendering art paintings at different viewpoints and illuminations is based on both shape information for describing surface geometries and spectral information for color reproduction. The shape information and the spectral information are estimated from the observed image data of a painting surface. Then, all the estimates of the surface properties are combined for rendering the painting under the arbitrary illumination and viewing condition. A computer graphics technique is used for the realistic image rendering.

A reflection model plays an important role in image analysis and rendering of art paintings. The model describes mathematically three-dimensional light reflection at each point on a painting surface. In previous works, we used the Torrance-Sparrow model [4] and the Cook-Torrance model [5] as a reflection model for oil painting surfaces. These models are effective for describing surface reflection with gross and specular highlight. The models consist of mathematical equations with many parameters, which include surface spectral reflectance, specular intensity, surface roughness, and refractive index. We assumed that a painting had the same model parameters at all points of the painting surface. This assumption can be true if the painting surface is covered with varnish. However, it is clear that light reflection properties depend on painting materials. A painting has not always the same reflection properties on the entire region of its surface. Therefore the authors started to investigate basic reflection properties of painting materials [6][7].

In this paper, we measure oil painting surfaces under a variety of conditions and analyze the surface reflection properties. The spectral reflectance factor of a painting surface is measured at different incidence and viewing angles by using a gonio-spectro photometer. Many paint samples are made by putting different oil paint materials on a two-dimensional supporting board with different thicknesses. Next, typical reflection models are fitted to the observed

∗Present address: Chiba University, Department of Information and Image Sciences, Faculty of Engineering, Chiba 263-8522, Japan

[email protected]; phone/fax; +81-43-290-3486

Color Imaging XIII: Processing, Hardcopy, and Applications, edited by Reiner Eschbach, Gabriel G. Marcu, Shoji Tominaga, Proc. of SPIE-IS&T Electronic Imaging,

SPIE Vol. 6807, 68070M, © 2008 SPIE-IS&T · 0277-786X/08/$18

SPIE-IS&T/ Vol. 6807 68070M-12008 SPIE Digital Library -- Subscriber Archive Copy

Incident light Interface reflection

Body reflection

A

Colorant

spectral reflectance data for the purpose of describing three-dimensional (3D) reflection properties of the painting surface. We show the mathematical expressions with several physical parameters. We estimate the model parameters by fitting to the observed reflectance data. Third, we analyze the reflection behavior in detail on the basis of several material conditions, which are pigment, supporting material, oil ratio, paint thickness, and background color. We investigate relationships between these conditions and the reflection behaviors.

2. MEASURING SYSTEM We use genio-spectrophotometric measurement for determining the basic reflection properties of oil painting materials. In a gonio-spectrophotometer used in this study, a target sample has a flat surface. The sensor position is fixed, and the light source can rotate around the sample. The sample table also can rotate on its axis. Figure 1 depicts the geometry in reflectance measurement, where N, L and V are normal vector, light vector and viewing vector respectively. Moreover θi and θr are incident angle and viewing angle. The spectral distribution of the reflected light is observed by the sensor of a concave diffraction grating. For obtaining spectral reflectance, a perfect diffuser is illuminated in the same way as the given surface, and the radiance is observed in the same geometries. The ratio of the radiance from the sample to the one from the reference diffuser is output as reflectance. This quantity is called the radiance factor.

The spectral radiance of the reference white diffuser is expressed as

( ) ( )cosR iY Eλ α θ λ= , (1)

where E(λ) is the illuminant spectrum, and α is a coefficient that is proportional to incidence light strength. The spectral radiance factor r(λ) measured by the gonio-spectrophotometer is then calculated as a ratio of the spectral radiance from the object Y(λ) and YR(λ),

r(λ) =( )( )

( )( )cosR i

Y YY E

λ λλ α θ λ

= . (2)

3. 3D REFLECTION MODELS There have been a lot of works for modeling light reflection, such as the Torrance-Sparrow model [4], the Phong model [8], and the Cook-Torrance model [5]. Oil paints are regarded as inhomogeneous dielectric substances. Light reflected from a paint surface is composed of light reflected from two different physical paths as shown in Figure 2. Some light is reflected at the interface between the paint’s surface and the air. This type of reflection is called specular (interface) reflection; the reflected light can be seen only over a narrow range of viewing geometries. The second reflection path occurs for light crossing the paint’s interface. Scattering of the light among the pigment particles occurs. This type of light reflection is said to be diffuse (body) reflection. The characterization of the reflected light by the two additive components can be described using the Phong model and the Cook and Torrance model.

3.1 Phong model

The Phong model [8] is one of the well-known 3D light reflection models used for computer graphics. This model is available for describing the dichromatic reflection property of inhomogeneous dielectric materials like plastics. The mathematical expression of the model is relatively simple, and the number of model parameters are small.

Figure 1 Surface reflectance measurement.

Figure 2 Reflection geometry for oil paint.

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Let us N, L, and V be normal vector, light vector and viewing vector respectively, as shown in Figure1. Let Rl and Rv be the mirrored reflection vector of L and V, respectively. All the vectors are unit vectors. Then the spectral radiance of light reflected from a painting surface is described as follows:

( ) ( ) ( ) ( ) ( )cos cosmiY S E Eλ α θ λ λ β ρ λ= + , (3)

where S(λ) represents the surface-spectral reflectance. The scalars α and β are the weighting coefficients for representing the relative intensity of the diffuse and specular reflection components. The symbol ρ indicates the angle between Rv and L. The specular reflection is observed only within a restricted range of the viewing angle. This reflection falls off sharply as the angle ρ increases, and the rapid falloff is described by cosm(ρ). The index value m is smaller for rougher surfaces and larger for smoother surfaces. Therefore, the unknown parameters to be estimated are α , β, and m.

3.2 Cook-Torrance type model

The characterization of the reflected light by the two additive components was summarized well by Cook and Torrance. The specular term of the Cook-Torrance model [5] can be derived from the Torrance-Sparrow model [4].

The spectral radiance distribution Y(λ) from a painting surface is modeled as a function of the wavelength λ as follows:

( ) ( ) ( )( ) ( ) ( )( ) ( )Q, , ,

coscosi

r

D F n kY S E E

ϕ γ θ λ λλ α θ λ λ β λ

θ= + , (4)

where the first and second terms represent, respectively, the diffuse and specular reflection components. α and β are respectively the diffuse and specular reflection coefficients. The restriction α + β = 1 shown in the original Cook-Torrance model is neglected here. These coefficients vary independently. A specular surface is assumed to be an isotropic collection of planar microscopic facets by Torrance and Sparrow [4]. The area of each microfacet is much smaller than the pixel size of an image, and so the surface normal vector N represents the normal vector of a macroscopic surface. Let Q be the vector bisector of an L and V vector pair, that is, the normal vector of a microfacet. The symbol θi is the incidence angle, θr is the viewing angle, ϕ is the angle between N and Q, and θQ is the angle between L and Q.

The specular reflection component consists of several terms: D is the distribution function of the microfacet orientation, and F represents the Fresnel spectral reflectance [9] of the microfacets. D is assumed as a Gaussian distribution function with rotational symmetry about the surface normal N as

( )2

2, exp log(2)D ϕϕ γ

γ⎧ ⎫

= −⎨ ⎬⎩ ⎭

, (5)

where the parameter γ is a constant that represents surface roughness. The Fresnel reflectance function is originally a function of the incidence angle, the refractive index, and the absorption coefficient. The absorption coefficient can be assumed to be 0 for inhomogeneous dielectric. Moreover, the refractive index is approximated with a constant value. Then the Fresnel reflectance can be described with the parameter of the refractive index n as

2

2Q QQ

Q 22Q Q Q

cos( )( cos( )) 1( cos( ))1( , ) 12 ( cos( )) cos( )( cos( )) 1

ggF n

g g

θ θθθ

θ θ θ

⎧ ⎫⎡ ⎤+ −− ⎪ ⎪⎣ ⎦= +⎨ ⎬+ ⎡ ⎤− +⎪ ⎪⎣ ⎦⎩ ⎭

, (6)

where 2 2Qcos 1g n θ= + − .

4. FITTING OF REFLECTION MODELS 4.1 Principle

We have investigated that the Phong model and the Cook-Torrance model are available for describing light reflection of oil painting surfaces. Figure 3 shows an example of the measured spectral radiance factors from a flat oil paint surface of Vermilion Hue by using a gonio-spectrophotometer, where the incidence angle is 40 degrees and the spectral radiance is depicted as a function of viewing angle. The specular reflection occurs around the viewing angle of

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6

C

0 ia

0-80 -40

0 40Viewing angle (deg]

700

500 6OO80 400 e\1'

Diffuse reflection component+ Specular reflection component

Diffuse reflecction cont

40 degrees. The spectral reflectance data from the gonio-spectrophotometer are useful for investigating the validity of the reflection models. The radiance factor is defined as the ratio of Y(λ) from the given surface to YR(λ).

The Phong model and the Cook-Torrance model can be written, respectively, in terms of the radiance factor as,

( ) ( ) ( )cos'

cos

m

i

r Sρ

λ λ βθ

= + , (7)

( ) ( ) ( ) Q, ( , )'

cos cosi r

D F nr S

ϕ γ θλ λ β

θ θ= + , (8)

where β ’ = β / α. Here the Phong model has the unknown parameters of the weighting coefficient β ’ and the index value m. The Cook-Torrance model has the unknown parameters of the refractive index n, the weighting coefficient β ’ and the surface roughness γ.

The reflectance models are fitted to the observed spectral radiance factors by the method of least squares. In the fitting computation, we used the average radiance factors on wavelength in the visible range. We can use a priori knowledge about physical parameters. First, the refractive index n can be limited to the range [1.3, 2.0] because the assumption of an inhomogeneous dielectric material. Moreover, the surface roughness is limited to [0.01, 1.0] empirically. The weighting coefficient β ’ is determined in an arbitrary range. Then, minimize the squared sum of the fitting error

( ) ( ){ }2ˆminj

e r j r j= −∑ , (9)

where r(j) is j-th measurement of the radiance factors acquired from the gonio-photometer and ˆ r (j) is the corresponding model estimate. The optimal parameters are determined to minimize the error in the given ranges of the parameters.

4.2 Sample measurement

We have made samples of flat painting surfaces for analyzing the reflection properties. Figure 4 shows a set of samples of oil paints where color paints called Permanent Green and Permanent Yellow Light were painted on black acrylic boards (upper) and on white canvas (lower). These were painted using a film applicator to keep a smooth surface and a fixed thickness 100 µm. We should note in Figure 4 that surface appearance looks different in the supporting materials of plastic board and canvas. We cannot neglect the influence of texture and reflection of the supporting materials.

The spectral radiance factors of the samples were measured by using the gonio-spectrophotometer in the range of incidence angle from 10 to 60 degrees at 10 degree interval. For each incidence angle, the viewing angle was changed from -80 to 80 degrees at 2 degree interval. The viewing angle was sampled in finer intervals within the specular highlight region for improving the measurement accuracy. In fact, we sampled the viewing angle at 0.5 degree interval within the range from -10 to 10 degrees of the highlight peak.

Figure 3 Spectral radiance factors measured

from a flat oil paint Vermilion Hue

Figure 4 Oil paint samples on black acrylic

boards (upper) and on white canvas (lower).

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Measured —Phong

— —

Cook-Torrance — -

Permanent Greenon acrylic board

-80 -60 -40 -20 0 20 40 60 80Viewing angle [deg]

50C

(540

30

Incident angle 0: 40

-80 -60 -40 -20 0 20 40 60 80Viewing angle [deg]

30

25

20

15

10

C0

a

4.3 Fitting results

The reflection models, the Phong model (7) and the Cook-Torrance model (8) were fitted to the measured radiance factors on the basis of the performance index (9). In the fitting computation, we used the average radiance factors on wavelength in the visible range. First, we analyzed the reflection properties of oil paints on an acrylic board. Figure 5 shows the fitting results for Permanent Green and Permanent Yellow Light on the acrylic board shown in Figure 4. Both models were used for curve fitting at incident angles 10, 20, …, 60 degrees. In Figure 5, the measurements by the gonio-photometer are indicated in solid curves. The fitting results by Phong and Cook-Torrance are indicated in broken curves and dot-dashed curves, respectively. The Phong model parameters were estimated as β ’ = 14 and m = 540 and the Cook-Torrance model parameters were estimated as n = 1.92, β ’ = 103 and γ = 0.024 for Permanent Green in Figure 5(a), Moreover the Phong model parameters are β ’ = 4 and m = 540 , and the Cook-Torrance model parameters are n = 1.64, β ’ = 57 and γ = 0.04 for Permanent Yellow Light in Figure 5(b). These figures suggest clearly large fitting errors in the Phong model. The fitting error increases as the viewing angle becomes large. We should note that the specular highlight region of reflectance curves cannot be described using the Phong model at all. On the other hand, in the Cook-Torrance model, although there is a little error, good fitting results are obtained even in the specular highlight region. Note that the dot-dashed curves of the Cook-Torrance model are almost coincident with the solit curves of the measurements.

Next, we investigated the reflection properties of oil paints on a canvas shown in the lower of Figure 4. Figure 6 shows the fitting results for Permanent Green and Permanent Yellow Light by the two models in the same manner as in Figure 5. The Phong model parameters were estimated as β ’ = 3 and m = 96 and the Cook-Torrance model parameters were estimated as n = 1.51, β ’ = 39.5 and γ = 0.06 for Permanent Green in Figure 6(a). Moreover, the Phong model parameters are β ’ = 1 and m = 45 , and the Cook-Torrance model parameters are n = 1.41, β ’ = 22 and γ = 0.08 for Permanent Yellow Light in Figure 6(b). Again we see large fitting errors in the Phong model. This model has serious problems not only in insufficient height of specular highlight but also in its position. Note the viewing angle position of specular highlight. The Phong model makes the specular highlight peak at the viewing angle corresponding to the mirrored reflection vector Rl of the incident light vector L. However this is not true because the real specular peak occurs at an off-specular angle. We can see a difference in peak positions between the real measurements and the Phong model in Figure 6 (b). On the other hand, the Cook-Torrance model describes well the entire region of the measured reflectance curves.

We should note that there is a discrepancy between the specular reflections of oil paints on the acrylic board and the canvas. The specular reflection on the acrylic board exhibits much stronger and sharper highlight than the reflection on the canvas. In fact, the specular coefficient β ’is much large and the roughness γ is smaller the acrylic than the parameter values in the canvas. These differences are based on the influence by the supporting material of white canvas.

(a) Permanent Green (b) Permanent Yellow Light

Figure 5 Fitting results for oil paint samples on acrylic board.

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Measured —PhongCook-Torrance — —

12Permanent Green

10 on canvas

Incident angle 0,:

-80 -60 -40 -20 0 20 40 60 80Viewing angle [deg]

Measured 60PhongCook-Torrance

I Permanent Yellow Light

LA1

on canvas

tncident angle 0:4O/tx\102030-._tcv, %;s.4

-80 -60 -40 -20 0 20 40 60 80Viewing angle [deg]

Viewing angle [deg]

10

9

8

I)

0

60

5040

(a) Permanent Green (b) Permanent Yellow Light

Figure 6 Fitting results for oil paint sample on canvas.

5. ANALYSIS OF REFLECTION PROPERTIES We have analyzed the reflection properties of oil paints in connection with the conditions of support material, paint thickness and medium quantity.

5.1 Supporting material

Light reflection on oil painting surfaces depends on the supporting material as shown in the previous section. Figure 7 demonstrates a good example for large reflection difference between two supporting materials. The solid curves and the broken curves in Figure 7 indicate, respectively, the reflectance measurements from Burnt Sienna on the black acrylic plate and on the white canvas. The shape of specular highlight reflection is quite different between the two sets. The acrylic board produces strong and sharp peaks of the specular reflection. On the other hand the canvas produces lower and duller specular peaks. The peak positions shift greatly from the mirror reflection angle.

Next, we investigate the influence of color of the supporting material. Figure 8 shows a paper used for supporting oil paints. The paper is segmented into white area (upper) and black area (lower). There are two color areas as shown in Figure 8. The oil paint Cobalt Blue Hue was painted on the two areas with different pain thicknesses. Figure 9 (a) and (b) show the reflectance curves in the specular reflection area in relatively large viewing angles and in the diffuse reflection area in small viewing angles, respectively. The thicknesses are 100 µm and 225 µm. These curves show that the white paper produces a little higher reflectance in both specular and diffuse regions than the back paper. As a result, the reflectance of an oil painting surface depends on the reflectance of the supporting material.

Figure 7 Reflections of Burnt Sienna on two supports. Figure 8 Paper used for supporting oil paints.

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White support

50Thickness 100 pmThickness 225 pm

Black support

40 Thickness 100 pm — —

Thickness 225 pm

30 Cobalt Blue Hue

50

20

Incident angle 0:

-80 -60 -40 -20 0 20 40 60 80

Viewing angle [deg]

White supportThickness 100 pmThickness 225 pm

Black support:Thickness 100 pm — —

0.6 Thickness 225 pm

Cobalt Blue Hue

-20 -10 0-80 -70 -60 -50 -40 -30Viewing angle [deg]

0.1

00 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

x0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

x

(a) Difference in the specular reflection (b) Difference in the diffuse reflection.

Figure 9 Reflections of Cobalt Blue Hue on two supporting colors with two paint thicknesses.

5.2 Paint thickness

Figure 9 also shows how the reflection of oil surfaces depends on the thickness of painted used. A comparison between the sets of curves for two thicknesses 100 µm and 225 µm suggests that the reflectance of the thick paint is much higher in the specular reflection region than the one of the thin paint. The paint thickness affects the specular reflectance larger than the supporting color. However the reflectance in the diffuse reflection region is almost independent of the paint thickness.

Next we investigated how the object color of oil painting surface depends on the paint thickness and the support color. Figure 10 shows Chromaticity changes of eight oil paints with different support colors, where the chromaticity coordinates of the oil paint surfaces under D65 are plotted in the CIE-xy chromaticity diagram. We assume that the incident angle is fixed to 10 degrees and the viewing angle changes from -80 to +80 degrees. We should note that the chromaticity for the specular reflectance is located in the center and, as the specularity decreases, the chromaticity moves to the peripheral region of the diagram. It is important to note that a linear line of each chromaticity locus combines the two extremes of the chromaticity without specularity and the chromaticity with maximal specularity. Moreover, all the chromaticity loci are coincident even though the thickness changes and the support color changes.

(a) Black support and 100µm thickness (b) White support and 100µm thickness

Figure 10 Chromaticity changes of eight oil paints with different support colors.

5.3 Oil quantity

We investigate how the surface reflectance depends on the quantity of oil. We made samples of Cadmium Red on a black acrylic board with different oil quantities of 5, 10, 20, and 30 %. The notation x% indicates that the quantity of added oil is equal to x percent of the weight of the original paint. Figure 11 shows the observed radiance factors from

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Oil quantity5% —

10% ——20%

30%

Cadmium Redon acrylic board

-80 -60 -40 -20 0 20 40 60 80Viewing angle degI

6040

35

30C

25

I 5(

Incident angle e: 403(

1020

Oil quantity5% —

10%

2OO/30%0.4

0.3Cadmium Red

0.2 on acrylic board

0.6

0.5

400 450 500 550 600 650 700Wavelength [rim]

the oil painted surfaces with different oil quantities. As the quantity of oil increases, the highlight peak of specular reflection is getting larger and shaper. However we have found that the diffuse reflection component is stable for the change in oil quantity. Figure 12 shows the surface-spectral reflectances of Cadmium Red with the oil quantities of 5, 10, 20, and 30 %, which were obtained the spectral radiance factors without the specular reflection component. Note that the surface-spectral reflectance is very little affected by the oil quantities.

Figure 11 Radiance factors with different oil quantities. Figure 12 Surface-spectral reflectances with different oil quantities.

6. CONCLUSION We have measured oil painting surfaces under a variety of conditions and analyzed the surface reflection properties in detail. First, the radiance factor of a painting surface was measured at different incidence and viewing angles by using a gonio-spectro photometer. Next, the Phong model and the Cook-Torrance model were examined for describing 3D reflection of the oil painting surfaces. These models were fitted to the observed radiance factors for oil paints on a black acrylic board and on a white canvas. We confirmed that the Cook- Torrance model could describe the reflection properties of oil paints with sufficient accuracy. The model parameters were estimated from the least-squared fitting to the measurements. Third, we have analyzed the reflection properties on the basis of several conditions such as pigment, supporting material, oil quantity, paint thickness, and support color. The analysis results are summarized as follows: Reflections of oil paints are influenced by the supporting material. The specular reflection on the acrylic board exhibited much stronger and sharper highlight than the reflection on the canvas. If the support is the same material, the reflectance of an oil painting surface slightly depends on the reflectance of the support material. The thickness of oil paint affected greatly the specular reflection, rather than the diffuse reflection. The quantity of oil also affects the specular reflection.

REFERENCES

1 K. Martinez et al., "Ten Years of Art Imaging Research," Proc. IEEE, 90(1), 28-41 (2002) . 2 S. Tominaga and N. Tanaka, "3D Recording and Rendering of Art Paintings, " Proc. Ninth CIC, 337-341 (2001). 3 S. Tominaga and N. Tanaka, "Measuring and Rendering Art Paintings Using a RGB Camera, " Proc. of EUROGRAPHICS, 299-306 (2002). 4 K. E. Torrance and E. M. Sparrow, "Theory for Off-Specular Reflection From Roughened Surfaces, " J. Opt. Soc. Am., 57(9), 1105-1114 (1967). 5 R. L. Cook and K. E. Torrance, "A Reflectance Model for Computer Graphics, " ACM Trans. Gr., 1(1), 7-24 (1982). 6 S. Nishi and S. Tominaga, "Estimation of Reflection Properties of Paintings and its Application to Image Rendering, " Proc. 30th ICIS 292-293 (2006). 7 S. Nishi and S. Tominaga, "Measurement and Analysis of Reflection Properties of Art Paints," Proc. of AIC Midterm Meeting, 170-173 (2007). 8 B. T. Phong, "Illumination for Computer-generated Pictures," Comm. ACM, 18(6), 311-317 (1975). 9 M. Born and E. Wolf, Principles of Optics, Pergamon Press, Oxford, pp.36-51, 1987.

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