Geometrical modeling of light scattering from paper substrates

Transcription

1 Geometrical modeling of light scattering from paper substrates Peter Hansson Department of Engineering ciences The Ångström Laboratory, Uppsala University Box 534, E-75 Uppsala, weden Abstract A light scattering model for opaque objects, such as coated paper, has been developed and verified, with a coefficient of determination between theory and measurement ranging from.84 to.98, for seven different paper samples of selected qualities. The model divides the scattered light into a bulk-scattered and a surface-scattered part. The bulk-scattered part is modeled as Lambertian, with compensation made for the part that is surface reflected. The surface-scattered part is modeled by geometrical optics as a distribution of specular reflections in a rough surface. The reflectance is assumed to be dependent both on the angle of incidence and on an attenuation parameter determined by the standard deviation of the height variations at spatial wavelengths comparable to the wavelength of the illuminating light. Introduction The appearance of a surface is to a very large extent determined by its angular light scattering properties. The angular characteristics can be used directly for visualization and ray tracing purposes and inversely for shape from shading and photometric stereo measurements. Complete characterization of the angular scattering properties of a surface is, however, a very time consuming operation since it generally requires independent variation of four different angles. The scattering properties are also dependent on the wavelength and polarization direction of the illuminating light. The need for physical models of light scattering is therefore obvious. Most existing models apply only to surface scattering from homogenous materials with a surface roughness smaller than the illuminating wavelength 3. Inhomogeneous, bulk scattering materials with rough surfaces, such as paper are often modeled as Lambertian diffusors. This is, however, often a poor description, especially at large angles of incidence. A better model could be used for improved visualization models as well as for more accurate photometric stereo measurements. The aim of this work has been to find a light scattering model that is applicable to most paper types as well as to other opaque materials. In a previous study 4 it was assumed that direct surface reflections from paper surfaces could be eliminated if p- polarized light was used for illumination whereas s-polarized light was detected. A goal here was to verify this assumption. Measurements have been performed on seven different paper qualities ranging from uncoated matte copy paper to very glossy heavy coated paper, selected to be representative of most commercially available paper qualities. ince imaging of surfaces is nearly always made in a fixed geometry the measurements were carried out with a fixed angle between the light source and the detector, while the sample was rotated.

2 Theory Lambert s law 5 of light scattering states that the radiance of an illuminated object should be equal in all directions, meaning that an object will appear equally bright from all viewing directions, regardless of illumination direction, if illuminated with constant irradiance. This is clearly not true for most objects, since almost any surface will show a specular component when illuminated with a large enough angle of incidence. A common way to describe the appearance of an object in more detail is to divide the light scattering into three parts; ) a specular part, due to reflections of the light source in a rough surface where the size of the irregularities are significantly larger than the light wavelength, as illustrated by Fig., ) a diffuse part, due to bulk scattering in the material or diffraction in a rough surface 3) an ambient part, due to diffuse illumination of the surface. This approach is for instance used in the Phong model 6, widely used for visualization purposes. Fig. pecular scattering of a collimated beam illuminating a rough surface. The lateral scale of the surface roughness and of the imaging system used to view the object is very important for the appearance. If for instance the smallest resolved surface area consists of approximately flat surface elements, which are large enough not to cause significant diffraction, but numerous enough to give a smooth distribution of specular reflections, this surface will also have a more or less diffuse appearance. Diffraction of radar waves from a surface with a random height distribution was first described by Davies 7 and then generalized to optical wavelengths by Bennett and Porteus 8. According to their theories the autocorrelation of the surface function is sufficient to determine the polar distribution of light scattering from a surface. Their models require the standard deviation of the surface roughness to be significantly smaller than the wavelength of the illuminating light. Bulk scattering of the material is not considered and the models are therefore mainly applicable to relatively smooth metallic or transmitting materials. For materials with a low index of refraction, such as paper with a refractive index around n =.55, only a small fraction of the light is reflected at the surface. Unless the light is subsequently transmitted through or absorbed in the material it will become bulk scattered. For a white dielectric substrate the dominant part of the reflected light comes from bulk scattering. Nevertheless, the radiance from the surface reflections of a material with low refractive index can be significantly higher than the bulk scattered part within a limited solid angle.

3 . pecular light scattering z θ y φ x Fig. Definition of the scattering direction. It is assumed, in this model, that the inclination of the surface elements follows a Gaussian distribution: p ( ( f + f )/ σ ) () ( f x, f k ) = exp x y πσ where f x and f y are the slopes in the x- and y-directions respectively and σ is the standard deviation of the slope variation. If the elevation, π/-θ, and azimuth, φ, defined by Fig., are introduced and specular scattering is assumed we can write: α θ α θ f x = tan () where α is the average angle of incidence and the error of approximation is small when α is close to θ. The slope in the y-direction is approximately: f y φ / (3) From geometrical optics it then follows that the probability for a light ray, hitting a surface element from the distribution specified by Eq. to be scattered in the direction (θ, φ) is: 3

4 p ( ( θ α ) φ )/ σ ) = (4) ( θ, φ) exp + 8 8πσ It follows that the intensity from a surface of area A, illuminated with an angle of incidence, α, can be expressed as: I ( θ, φ) = E Acos( α) p( θ, φ) R( α) (5) where E is the irradiance of a surface perpendicular to the light direction and R(α) is the reflectance in the direction (θ, φ). For moderate σ we can use the approximations: (( θ α ) ) cos( α ) cos + (6) and (( θ α ) ) R ( α ) R + (7) Hence we have: ( ( θ α ) + φ )/ 8σ ) (( θ + α ) ) cos( ( θ α ) ) E A I ( θ, φ) = exp R + (8) 8πσ 4

5 .. Attenuation of specular light As derived by Davies 6, the specular reflection from a surface with normally distributed heights, with standard deviation σ H, will be attenuated by a factor: R( α) (4πσ = = H cosα) D ( α ) exp () R ( α) λ where λ is the wavelength of the illuminating light, α is the angle of incidence and R (α) is the Fresnel reflectance of a perfectly smooth surface. For a rough surface the resulting intensity of the attenuated specular surface reflections can therefore be written as: I = D (( θ + α ) / ) E AR (( θ + α ) / ) cos( ( θ + α ) / ) 8πσ exp ( ( θ α ) + φ )/ 8σ ) (). Diffuse light scattering According to Lambert s law the intensity at a point with elevation π/-θ, relative to a surface of area A, illuminated with an incident angle α, can be expressed as: E Acosα cosθ I D = () π where E is the irradiance of a surface perpendicular to the light direction. Energy conservation, however, requires that light that is specularily reflected at the surface cannot contribute to the diffuse (bulk) scattering. A more realistic model is therefore: I DD E Acosα cosθ = ( R( α ))( R ( θ )) (3) π D where D D is an attenuation factor for the diffuse light, introduced to account for absorption in the material. R(α ) is the reflectance for light entering the material and R (θ) is the reflectance for light leaving the material, after a number of scattering events. The reflectance values are determined by Fresnels law of reflection. The above compensation was introduced by aundersen 9 in his studies of pigmented materials. 5

6 .3 Polarization It is assumed that the bulk-scattered light has changed direction so many times that the original polarization direction has been forgotten. Hence, the diffusely scattered light is depolarized. The surface reflected light, however, is assumed to have been reflected only once. If the polarization of the illuminating light is in one direction, so is therefore also the polarization direction of the scattered light..4 Composite light scattering If (θ+α )/ is approximated by α, addition of Eq. and Eq. 3 gives a total intensity of: I Tot E Acosα ( ( θ α ) φ )/ 8σ ) D ( α ) R ( α )exp + D ( )( D cosθ R( α ) R ( θ )) + 8σ (4) = π where the first term represents bulk scattering and the second represents surface scattering. 3 Experimental 3. amples For verification of the light-scattering model, seven of the most common commercially available paper qualities, listed in Table, were selected. Three of the qualities were uncoated, namely the copy, newsprint and super-calendered paper samples. Two of the coated samples, the cast coated and the heavy-coated-glossy, were glossy and the remaining two, heavy-coated-matte and light-weight-coated, were less glossy. 3. Measurement of the slope distribution The slope distributions of the samples were determined using a stylus instrument, a Perthometer, with a µm tip radius. On each sample, five mm long height profiles with a sampling distance of 5 µm were recorded. The slope distributions were obtained as histograms of the differentiated height profiles. Normal distributions could be fitted to the actual distributions with coefficients of determination ranging from r >.97, for the newsprint and copy paper, to r >.99 for the other qualities. The corresponding standard deviations, σ, are given in the second column of Table. 6

7 3.3 Angle resolved light scattering measurement The scattering distributions were measured using the setup in Fig 3. A HeNe laser with λ = 63.8 nm was used for illumination. The light goes through a chopper and hits the sample with a spot size A of about mm at normal incidence. A detector connected to a lock-in amplifier detects the scattered light. The angle between the laser and the detector direction was fixed to γ = 6 deg. The sample was rotated around an axis perpendicular to the plane of incidence. The scattering angle is given by: θ = γ α. (5) Detector P CH θ v α n LAER ample P Fig. 3 etup for light scattering measurement. CH, chopper; P and P, optional polarizers; n, surface normal; α, angle of incidence; θ, scattering angle. In a first series of measurements p-polarized light was used for illumination and s- polarized light was detected. The angle of incidence, α was varied from to 9 deg. ince the surface reflected light is assumed to keep its polarization direction whereas the bulk scattered light is assumed to be depolarized, only the latter part should be detected. The detected intensity is therefore given by Eq. 3, which can be rewritten as: ( R ( α ))( R ( )) I = I cos( θ ) θ (6) Tot p s 7

8 where I = E Acosα/π is constant, since the spot size increases as A = A cosα. R p (α ) denotes the reflectance for p-polarized light entering the material: tan ( α α ) R p ( α ) = (7) tan ( α + α ) and R s(θ) is the reflectance for the light leaving the material: sin ( θ θ ) R s ( θ ) = (8) sin ( θ + θ ) The angles of refraction, α and θ, were obtained using nell s law, with an index of refraction n =.55. In a second measurement series unpolarized light was used both for illumination and detection. The detected intensity is then obtained from Eq. 4 and Eq. 5. I Tot D ( γ ) R ( γ )exp = I ( )( DD cos( γ α ) R( α ) R ( γ α )) + σ ( ( γ / α ) )/ σ ) (9) Here the reflectance values are obtained as the average of the two polarization directions R = R p / + R s /. 4 Results and discussion The measurement results for the seven paper samples are shown in Fig. 4 to Fig., where the upper graphs show the polarized results and the lower graphs show the unpolarized results, for each sample. The agreement between theory and measurement is good, with a coefficient of determination, listed in Table,.84 r.98 in the polarized case and.93 r.97 in the unpolarized case. For the theoretical curves the slope standard deviation, σ, and the ratio D D /D in Table are used. D D /D was chosen to give good correlation between theory and measurement. This gives a theoretical surface height standard deviation, σ H, also listed in Table. Note that σ H corresponds to a much smaller scale than σ. The measurement results for the cast-coated sample, is seen in Fig. 4. The correspondence between measurement and theory is excellent in the polarized case. The disagreement around the specular peak in the unpolarized case can probably be explained by a slight misalignment of the detector in combination with insufficient angular resolution, with the consequence that the specular peak can have missed the detector. 8

9 The measurements on the copy paper show excellent agreement with theory in both cases, as seen in Fig. 5. A noisier signal in the polarized case seems to be the explanation for the slightly lower correlation there. For the heavy-coated-glossy sample the agreement is excellent in both cases, see Fig. 6, although the measured peak in the unpolarized case is slightly narrower than predicted by theory. This could be a result of varying surface slope statistics across the surface. The measurement results on the heavy-coated-matte sample, seen in Fig. 7, shows excellent agreement with theory in the unpolarized case and a slightly lower correlation in the polarized case, seemingly due to a noisier signal. Note the large damping of the specular part in the unpolarized case. As seen in Fig. 8, the light-weight-coated sample shows excellent agreement in both cases, with an oddity for large angles of incidence in the unpolarized case. A possible explanation for this behavior could be self-shadowing at large angles of incidence. For the newsprint paper, excellent agreement is seen in both cases in Fig. 9. However, the measured curve in the polarized case seems to be slightly shifted to the left compared to the theoretical curve. This could indicate that specular reflections are not completely eliminated by the polarizers, perhaps due to multiple reflections in the surface. Measurements on the super-calendered sample show excellent agreement in both cases, with the exception of an unexplained deviation from theory at small angles of incidence in the unpolarized case. This is seen in Fig.. The noisier results in the polarized cases can probably be explained by the fact that each polarizer reduced the amount of light by 5 %, meaning that only one fourth of the light was available for detection. peckle problems can probably also be more severe when polarizers are used. ample σ [deg] D D /D σ H [nm] r, pspolarized r, unpolarized Cast-coated Copy paper Heavy-coated-Glossy Heavy-coated-Matte Light-weight-coated Newsprint uper-calendered Table tandard deviation for the slope and height distributions, inverse attenuation factor and correlation between theory and measurements for the 7 paper samples. 9

10 Angle of incidence [deg] Fig. 4 Measured (points) and theoretical (solid line) intensities as a function of incident angle for the Cast Coated sample in the polarized case (top) and the unpolarized case (bottom) Angle of incidence [deg] Fig. 5 Measured (points) and theoretical (solid line) intensities as a function of incident angle for the Copy paper sample in the polarized case (top) and the unpolarized case (bottom).

11 Angle of incidence [deg] Fig. 6 Measured (points) and theoretical (solid line) intensities as a function of incident angle for the Heavy Coated-Glossy sample in the polarized case (top) and the unpolarized case (bottom) Angle of incidence [deg] Fig. 7 Measured (points) and theoretical (solid line) intensities as a function of incident angle for the Heavy Coated-Matte sample in the polarized case (top) and the unpolarized case (bottom).

12 Angle of incidence [deg] Fig. 8 Measured (points) and theoretical (solid line) intensities as a function of incident angle for the LWC sample in the polarized case (top) and the unpolarized case (bottom) Angle of incidence [deg] Fig. 9 Measured (points) and theoretical (solid line) intensities as a function of incident angle for the Newsprint paper sample in the polarized case (top) and the unpolarized case (bottom).

13 Angle of incidence [deg] Fig. Measured (points) and theoretical (solid line) intensities as a function of incident angle for the uper Calendered sample in the polarized case (top) and the unpolarized case (bottom). 5 Conclusions The measurements show that the developed light scattering model correlates very well to the results in nearly all cases. Most of the deviations are probably a result of instrumental noise. It has also been shown that surface reflections can be effectively eliminated using orthogonal polarizers for the illuminating and detected light. In the model, four material parameters are sufficient for a complete description of the light scattering properties of an opaque surface. These parameters are refractive index (n) diffuse reflectance (D D ), the standard deviation of slope variations (σ ) corresponding to spatial wavelengths significantly longer than the wavelength of the illuminating light, λ, and the standard deviation of surface height variations (σ H ) corresponding to spatial wavelengths significantly longer than λ. One possible application of the method could be to determine these parameters from light scattering measurements. Other applications could be for improved visualization models or to achieve more accurate photometric stereo measurements. 3

14 References B. K. P. Horn, Understanding image intensities, Artificial Intelligence 8, -3 (977). R. J. Woodham, Photometric method for determining surface orientation from multiple images, Opt. Eng. 9(), (98). 3 T. R. Lettieri, E. Marx, J.-F. ong and T. D. Vorburger, Light scattering from glossy coatings on paper, Applied Optics 3(3), (99). 4 P. Hansson and G. Manneberg, Topography and Reflectance Analysis of Paper urfaces Using a Photometric tereo Method, Opt. Eng. ep. 5 M. V. Klein and T. E. Furtak, Optics, Wiley, New York, B. T. Phong, Illumination for computer generated pictures, Communications of the Association for Computing Machinery (ACM) 8(6), 3-37 (975). 7 H. Davies, The reflection of electromagnetic waves from a rough surface, Inst. of Electrical Eng., Vol, pp 9-4, H. E. Bennett and J. O. Porteus, Relation between surface roughness and specular reflectance at normal incidence, J. Opt. oc. Am. 5(), 3-9 (96). 9 J. L. aunderson, Calculation of the color of pigmented plastics, JOA 3, (94). G. Bryntse, A method for the analysis of ink mottle using polarized light reflection, Ph.D. dissertation, KTH, tockholm, (98). 4