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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 19, Iss. 3 — Jan. 31, 2011
  • pp: 1915–1920
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Point spreading in turbid media with anisotropic single scattering

Magnus Neuman, Ludovic G. Coppel, and Per Edström  »View Author Affiliations


Optics Express, Vol. 19, Issue 3, pp. 1915-1920 (2011)
http://dx.doi.org/10.1364/OE.19.001915


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Abstract

Point spreading is investigated using general radiative transfer theory. We find that the single scattering anisotropy plays a significant role for point spreading together with the medium mean free path, single scattering albedo and thickness. When forward scattering dominates, the light will on average undergo more scattering events to give a specific optical response in reflectance measurements. This will significantly increase point spreading if the medium is low absorbing with large mean free path. Any fundamental and generic model of point spreading must capture the dependence on all of these medium characteristics.

© 2011 Optical Society of America

1. Introduction

The point spread function (PSF) describes the response of a medium to a point light source. The lateral distribution D(x,y) of the reflected light is obtained by integrating the PSF and the illumination I over the reflecting medium, that is
D(x,y)=ΔvΔuPSF(x,y;u,v)I(u,v)dudv
(1)
where Δu and Δv delimit the medium. In the present work we deal with laterally homogeneous turbid media with point source illumination. The PSF is then isotropic, i.e., it depends on radial distance only, and the reflected light is given by D(r) = PSF(r)I if the illumination is incident at radial coordinate r = 0.

The PSF is closely connected to the phenomenon known as optical dot gain, when printed dots are perceived larger than their actual physical size due to lateral spreading of light. This is important in graphic arts where continuous tones are reproduced through halftoning using dots varying in size and spacing. Optical dot gain then makes the printed image appear darker than expected, which has to be compensated for. Point spreading is also important in computer rendering for a realistic appearance of rendered images [8

8. A. S. Glassner, Principles of Digital Image Synthesis, Volume Two, (Morgan Kauffman, 1995).

], and in optical tomography where light scattering is used for medical diagnosis [9

9. J. M. Schmitt, “Optical coherence tomography (OCT): A review,” IEEE J. Sel. Top. Quantum Electron. 5, 1205–1215 (1999). [CrossRef]

].

General radiative transfer (RT) theory can also be used to study the PSF and optical dot gain of a medium. While the KM model approximates the light intensity within a medium with two diffuse fluxes, general RT theory is angle resolved. A basic reference for RT is Chandrasekhar [10

10. S. Chandrasekhar, Radiative Transfer, (Dover, 1960).

]. Considering light of a single wavelength, the familiar RT equation can be stated as
dI(x,y,z;θ,φ)ds=σe[I(x,y,z;θ,φ)+S].
(2)
The symbols have their usual meaning; I(x,y,z,θ,φ) is intensity at position (x,y,z) at polar angle θ and azimuthal angle φ, s = (x2 + y2 + z2)1/2 is distance, σe is the extinction coefficient and S is a source function. The extinction coefficient is the sum of the scattering and absorption coefficients σs and σa and also the inverse of the mean free path e. The source function accounts for light scattered to θ, φ at position (x,y,z) from all other directions. It can be written
S=a4π4πp(cosΘ)I(x,y,z;θ,φ)dω,
(3)
where a is the single scattering albedo defined as a = σs/(σs + σa), ω is solid angle and p(cos Θ) is the phase function. Here Θ is the angle between the directions of the incident and scattered light. The phase function describes the angular distribution of the single scattering process. A commonly used phase function is the Henyey–Greenstein (HG) phase function [11

11. L. G. Henyey and J. L. Greenstein, “Diffuse Radiation in the Galaxy,” Astrophys. J. 93, 70–83 (1941). [CrossRef]

] which contains a single parameter, the asymmetry factor g. It ranges from −1 to 1 with g = −1 meaning complete back scattering, g = 0 isotropic scattering and g = 1 complete forward scattering. Also, g is the average of the cosine of the scattering angle and the first moment in an expansion of any phase function. For organic materials such as paper it has been shown that g has values in the interval 0.6–0.9 approximately [12

12. W.-F. Cheong, S. A. Prahl, and A. J. Welch, “A review of the optical properties of biological tissues,” IEEE J. Quantum Electron. 26, 2166–2185 (1990). [CrossRef]

14

14. N. Joshi, C. Donner, and H. W. Jensen, “Noninvasive measurement of scattering anisotropy in turbid materials by nonnormal incident illumination,” Opt. Lett. 31, 936–938 (2006). [CrossRef] [PubMed]

].

We can characterize a turbid medium with the albedo a, the mean free path e and the asymmetry factor g. These medium characteristics are independent of position in a homogeneous medium but normally vary between wavelengths. Equation (2) can then be solved for each wavelength if the wavelengths are independent. This is not the case in fluorescing media, but in the present work we do not consider fluorescence.

Considering only one spatial coordinate Eq. (2) becomes a 3-dimensional problem and can be solved numerically using e.g. the method of discrete ordinates (DORT) and a Fourier expansion of the azimuthal dependence [15

15. P. Edström, “A fast and stable solution method for the radiative transfer problem,” SIAM Rev. 47, 447–468 (2005). [CrossRef]

]. In the present work we are interested in laterally resolved (5-dimensional) radiative transfer and we choose to use Monte Carlo simulations since this problem has not been solved using DORT-methods. Coppel et al. [16

16. L. G. Coppel, P. Edström, and M. Lindquister, “Open source Monte Carlo simulation platform for particle level simulation of light scattering from generated paper structures,” in Proc. Papermaking Res. Symp., E. Madetoja, H. Niskanen, and J. Hämäläinen, eds. (Kuopio, 2009).

] presented an open source Monte Carlo model called Open PaperOpt (available at http://openpaperopt.sourceforge.net), adapted to simulations of light scattering in paper and print that is suitable for the problem at hand.

2. Method

2.1. Material

Relevant numerical values for the medium parameters e and a are assessed using a set of paper samples. We choose two lightly dyed and two non-dyed 30 g/m2 paper samples with and without fillers, giving four samples in total. Samples varying in filler content differ in mean free path le since the addition of fillers significantly increases scattering, thus decreasing the mean free path. The addition of a blue dye increases absorption, thus decreasing the albedo, in the wavelength interval 550–700 nm approximately, and the effect of increased absorption on point spreading can be investigated by studying wavelengths in this interval. No samples contain fluorescent whitening agents. We denote the samples M1–M4, where M1 contains no dye or filler, M2 contains dye but no filler, M3 contains no dye but filler and M4 contains both dye and filler.

2.2. Estimation of medium parameters using DORT simulations

The paper industry uses standardized reflectance measurements to assess the reflectance factor in d/0 geometry (diffuse illumination and detection in the normal direction) [21

21. ISO 2469: Paper, Board and Pulps - Measurement of Diffuse Reflectance Factor, (International Organization for Standardization, 1994).

]. We use these measurements to determine e and a by measuring the reflectance factor from a single sheet and an opaque pad of paper sheets. We then get a well-posed optimization problem that can be solved for e and a, e.g., by using the RT based DORT2002 model [15

15. P. Edström, “A fast and stable solution method for the radiative transfer problem,” SIAM Rev. 47, 447–468 (2005). [CrossRef]

, 22

22. P. Edström, “A Two-Phase Parameter Estimation Method for Radiative Transfer Problems in Paper Industry Applications,” J. Comput. Appl. Math. 16, 927–951 (2008).

] (freely available). This model can accurately describe the illumination and detection conditions of the d/0 instrument. The asymmetry factor g is varied from 0 to 0.8 in steps of 0.2, and the inverse RT problem is solved for each of these g values. We include g = 0 since this is an assumption in the KM model. Each parameter setup [a, ℓe, g] will then give the same optical response in the d/0 instrument for the particular medium studied, despite the variations in g, and the medium thus has the same scattering power irrespective of the g value.

2.3. Monte Carlo simulations of the PSF

When we have the parameter setup [a, ℓe, g] we can estimate the PSF of a medium with a given thickness t through Monte Carlo simulations. We do simulations using both the thickness t corresponding to 30 g/m2 paper and an opaque medium where t → ∞ in order to study the influence of transmittance on the PSF. The thickness of the paper samples is measured with a micrometer and found to be 65 μm. We simulate illumination incident normally on a point and to minimize noise 108 wave packets are simulated in each run. We choose the wavelength of light that is most heavily absorbed by the blue dye (620 nm) to represent the four media M1–4.

The PSF is obtained by averaging the simulation data over azimuthal angle. We then introduce a single number metric to represent point spreading. This metric is the mean radial distance a wave packet travels before exiting the medium, i.e.,
r¯=[iPSF(ri)]1iriPSF(ri).
(4)
The metric we introduce here is in the distance domain, as opposed to the frequency domain metric kp used by other authors and introduced by Arney [4

4. J.S. Arney, J. Chauvin, J. Nauman, and P.G. Anderson, “Kubelka-Munk theory and the MTF of paper,” J. Imaging Sci. Technol. 47, 339–345 (2003).

], which is the frequency where MTF = 0.5. A distance domain metric is more useful when dealing with point spreading since the frequency resolution of the reflected light then is of minor interest.

Furthermore, we calculate using the Monte Carlo simulations the mean number of scattering events that a wave packet undergoes before leaving a medium. This allows for important conclusions to be drawn about the dependence of point spreading on the medium characteristics.

3. Results

3.1. Medium parameters

Table 1 shows the values of a and e obtained from d/0 measurements when g is varied. As expected M2 and M4 have the lowest albedos and M3 and M4 have the shortest mean free path. We can see that the mean free path decreases as g increases. This can be understood intuitively since if the light is scattered more in the forward direction when it impinges on the medium surface, the medium must be highly scattering in order to reflect the measured amount of light towards the detector. In this way the scattering power is the same irrespective of the g value.

Table 1. Albedo (a) and mean free path (e) of the media at 620 nm obtained from d/0 measurements when the asymmetry factor g is varied. Parameters here are used in the Monte Carlo simulations

table-icon
View This Table

3.2. Point spread simulations

The data in Table 1 is used in Monte Carlo simulations with the Open PaperOpt model. Fig. 1 shows how our PSF metric varies with g for the different media and for the two different thicknesses. It can be seen that increases with g for all media. The high albedo medium with large mean free path (M1) has the largest mean radial distance . The low albedo medium with short mean free path (M4) has the smallest . We can also see that is larger for the opaque media, with the most noticeable difference for M1. Hence, transmittance can obviously have a significant effect on point spreading. We can thus conclude that point spreading, as represented by , depends on asymmetry factor, albedo, mean free path and medium thickness.

Fig. 1 Point spreading as represented by for two thicknesses of media M1–4 when g is varied. Point spreading increases with g for all media. M1 (high albedo and large mean free path) has the largest point spreading and the most noticeable difference when varying the thickness. M4 (low albedo and short mean free path) has the smallest point spreading.

Figure 2 shows the average number of scattering events that the wave packets undergo before exiting the medium. It can be seen that increasing g increases the number of scattering events. This holds for both thin and opaque media, but when the mean free path e is large (as for M1 and M2) the light is scattered fewer times in thin media. When the effect of transmittance is eliminated (Fig. 2(b)) there is only a small difference in the number of scattering events when varying the mean free path (M1 vs. M3 and M2 vs. M4).

Fig. 2 The mean number of scattering events that the wave packets undergo before exiting the medium. We see that the asymmetry factor g affects the number of scattering events, and that increasing the albedo leads to more scattering events in opaque media.

Based on these observations we can relate point spreading to the scattering properties of a medium. Media with high albedo (M1 and M3) have a similar contribution to the reflectance from different scattering orders. Media with low albedo (M2 and M4) are also similar in this respect. This means that the point spreading is determined by the distance that the multiply scattered light can travel. This distance is larger if the mean free path is large and absorption is low. A medium with high albedo and large mean free path will thus give the largest point spreading, and it will increase further if light is scattered more in the forward direction.

4. Discussion and conclusions

This is of importance for example when predicting the optical dot gain to adjust halftone prints. By using a model based on fundamental knowledge it can be generic in the sense that the dot gain of any paper substrate can be estimated. Furthermore, a thorough understanding of lateral diffusion of light is important for fields such as image rendering and optical tomography.

Acknowledgments

This work was financially supported by VINNOVA (The Swedish Governmental Agency for Innovation Systems) which is gratefully acknowledged.

References and links

1.

P. Oittinen, “Limits of microscopic print quality,” in Advances in Printing Science and Technology, W. H. Banks, ed. (Pentech, London, 1982), Vol. 16, pp 121–128.

2.

P. Kubelka and F. Munk, “Ein beitrag zur optik der farbanstriche,” Z. Tech. Phys. (Leipzig) 11a, 593–601 (1931).

3.

P. G. Engeldrum and B. Pridham, “Application of turbid medium theory to paper spread function measurements,” Tech. Assoc. Graphic Arts Proc. 47, 339–352 (1995).

4.

J.S. Arney, J. Chauvin, J. Nauman, and P.G. Anderson, “Kubelka-Munk theory and the MTF of paper,” J. Imaging Sci. Technol. 47, 339–345 (2003).

5.

S. Gustavson, “Dot Gain in Colour Halftones,” Doctoral thesis, Linköping university (1997).

6.

P. Emmel, “Modèles de Prédiction Couleur Appliqués à l’Impression Jet d’Encre,” Doctoral thesis, Ecole Polytechnique Fédérale de Lausanne (1998).

7.

S. Mourad, “Improved Calibration of Optical Characteristics of Paper by an Adapted Paper-MTF Model,” J. Imaging Sci. Technol. 51, 283–292 (2007). [CrossRef]

8.

A. S. Glassner, Principles of Digital Image Synthesis, Volume Two, (Morgan Kauffman, 1995).

9.

J. M. Schmitt, “Optical coherence tomography (OCT): A review,” IEEE J. Sel. Top. Quantum Electron. 5, 1205–1215 (1999). [CrossRef]

10.

S. Chandrasekhar, Radiative Transfer, (Dover, 1960).

11.

L. G. Henyey and J. L. Greenstein, “Diffuse Radiation in the Galaxy,” Astrophys. J. 93, 70–83 (1941). [CrossRef]

12.

W.-F. Cheong, S. A. Prahl, and A. J. Welch, “A review of the optical properties of biological tissues,” IEEE J. Quantum Electron. 26, 2166–2185 (1990). [CrossRef]

13.

S. A. Prahl, M. J. C. van Gemert, and A. J. Welch, “Determining the optical properties of turbid media by using the adding-doubling method,” Appl. Opt. 32, 559–568 (1993). [CrossRef] [PubMed]

14.

N. Joshi, C. Donner, and H. W. Jensen, “Noninvasive measurement of scattering anisotropy in turbid materials by nonnormal incident illumination,” Opt. Lett. 31, 936–938 (2006). [CrossRef] [PubMed]

15.

P. Edström, “A fast and stable solution method for the radiative transfer problem,” SIAM Rev. 47, 447–468 (2005). [CrossRef]

16.

L. G. Coppel, P. Edström, and M. Lindquister, “Open source Monte Carlo simulation platform for particle level simulation of light scattering from generated paper structures,” in Proc. Papermaking Res. Symp., E. Madetoja, H. Niskanen, and J. Hämäläinen, eds. (Kuopio, 2009).

17.

M. Sormaz, T. Stamm, S. Mourad, and P. Jenny, “Stochastic modeling of light scattering with fluorescence using a Monte Carlo-based multiscale approach,” J. Opt. Soc. Am. A 26, 1403–1413 (2009). [CrossRef]

18.

T. F. Chen, G. V. G. Baranoski, and K. F. Lin, “Bulk scattering approximations for HeNe laser transmitted through paper,” Opt. Express 16, 21762–21771 (2008). [CrossRef] [PubMed]

19.

M. Neuman and P. Edström, “Anisotropic reflectance from turbid media. I. Theory,” J. Opt. Soc. Am. A 27, 1032–1039 (2010). [CrossRef]

20.

M. Neuman and P. Edström, “Anisotropic reflectance from turbid media. II. Measurements,” J. Opt. Soc. Am. A 27, 1040–1045 (2010). [CrossRef]

21.

ISO 2469: Paper, Board and Pulps - Measurement of Diffuse Reflectance Factor, (International Organization for Standardization, 1994).

22.

P. Edström, “A Two-Phase Parameter Estimation Method for Radiative Transfer Problems in Paper Industry Applications,” J. Comput. Appl. Math. 16, 927–951 (2008).

OCIS Codes
(030.5620) Coherence and statistical optics : Radiative transfer
(100.2810) Image processing : Halftone image reproduction
(170.3660) Medical optics and biotechnology : Light propagation in tissues
(290.4210) Scattering : Multiple scattering
(290.7050) Scattering : Turbid media
(290.2558) Scattering : Forward scattering

ToC Category:
Scattering

History
Original Manuscript: November 18, 2010
Revised Manuscript: January 12, 2011
Manuscript Accepted: January 12, 2011
Published: January 18, 2011

Virtual Issues
Vol. 6, Iss. 2 Virtual Journal for Biomedical Optics

Citation
Magnus Neuman, Ludovic G. Coppel, and Per Edström, "Point spreading in turbid media with anisotropic single scattering," Opt. Express 19, 1915-1920 (2011)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-3-1915


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References

  1. P. Oittinen, "Limits of microscopic print quality," in Advances in Printing Science and Technology, W. H. Banks, ed. (Pentech, London, 1982), Vol. 16, pp 121-128.
  2. P. Kubelka, and F. Munk, "Ein beitrag zur optik der farbanstriche," Z. Tech. Phys. (Leipzig) 11a, 593-601 (1931).
  3. P. G. Engeldrum, and B. Pridham, "Application of turbid medium theory to paper spread function measurements," Tech. Assoc. Graphic Arts Proc. 47, 339-352 (1995).
  4. J. S. Arney, J. Chauvin, J. Nauman, and P. G. Anderson, "Kubelka-Munk theory and the MTF of paper," J. Imaging Sci. Technol. 47, 339-345 (2003).
  5. S. Gustavson, "Dot Gain in Colour Halftones," Doctoral thesis, Linköping university (1997).
  6. P. Emmel, "Modèles de Prédiction Couleur Appliqués à l’Impression Jet d’Encre," Doctoral thesis, Ecole Polytechnique Fédérale de Lausanne (1998).
  7. S. Mourad, "Improved Calibration of Optical Characteristics of Paper by an Adapted Paper-MTF Model," J. Imaging Sci. Technol. 51, 283-292 (2007). [CrossRef]
  8. A. S. Glassner, Principles of Digital Image Synthesis, Vol. 2, (Morgan Kauffman, 1995).
  9. J. M. Schmitt, "Optical coherence tomography (OCT): A review," IEEE J. Sel. Top. Quantum Electron. 5, 1205-1215 (1999). [CrossRef]
  10. S. Chandrasekhar, Radiative Transfer, (Dover, 1960).
  11. L. G. Henyey, and J. L. Greenstein, "Diffuse Radiation in the Galaxy," Astrophys. J. 93, 70-83 (1941). [CrossRef]
  12. W.-F. Cheong, S. A. Prahl, and A. J. Welch, "A review of the optical properties of biological tissues," IEEE J. Quantum Electron. 26, 2166-2185 (1990). [CrossRef]
  13. S. A. Prahl, M. J. C. van Gemert, and A. J. Welch, "Determining the optical properties of turbid media by using the adding-doubling method," Appl. Opt. 32, 559-568 (1993). [CrossRef] [PubMed]
  14. N. Joshi, C. Donner, and H. W. Jensen, "Noninvasive measurement of scattering anisotropy in turbid materials by nonnormal incident illumination," Opt. Lett. 31, 936-938 (2006). [CrossRef] [PubMed]
  15. P. Edström, "A fast and stable solution method for the radiative transfer problem," SIAM Rev. 47, 447-468 (2005). [CrossRef]
  16. L. G. Coppel, P. Edström, and M. Lindquister, "Open source Monte Carlo simulation platform for particle level simulation of light scattering from generated paper structures," in Proc. Papermaking Res. Symp., E. Madetoja, H. Niskanen and J. Hämäläinen, eds. (Kuopio, 2009).
  17. M. Sormaz, T. Stamm, S. Mourad, and P. Jenny, "Stochastic modeling of light scattering with fluorescence using a Monte Carlo-based multiscale approach," J. Opt. Soc. Am. A 26, 1403-1413 (2009). [CrossRef]
  18. T. F. Chen, G. V. G. Baranoski, and K. F. Lin, "Bulk scattering approximations for HeNe laser transmitted through paper," Opt. Express 16, 21762-21771 (2008). [CrossRef] [PubMed]
  19. M. Neuman, and P. Edström, "Anisotropic reflectance from turbid media. I. Theory," J. Opt. Soc. Am. A 27, 1032-1039 (2010). [CrossRef]
  20. M. Neuman, and P. Edström, "Anisotropic reflectance from turbid media. II. Measurements," J. Opt. Soc. Am. A 27, 1040-1045 (2010). [CrossRef]
  21. ISO 2469: Paper, Board and Pulps - Measurement of Diffuse Reflectance Factor, (International Organization for Standardization, 1994).
  22. P. Edström, "A Two-Phase Parameter Estimation Method for Radiative Transfer Problems in Paper Industry Applications," J. Comput. Appl. Math. 16, 927-951 (2008).

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