## Lateral light scattering in paper - MTF simulation and measurement |

Optics Express, Vol. 19, Issue 25, pp. 25181-25187 (2011)

http://dx.doi.org/10.1364/OE.19.025181

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### Abstract

The modulation transfer function (MTF) of 22 paper samples is computed using Monte Carlo simulations with isotropic or strongly forward single scattering. The inverse frequency at half maximum of the MTF (*k _{p}*) is found inappropriate as a single metric for the MTF since it is insensitive to the shape of the modeled and simulated MTF. The single scattering phase function has a significant impact on the shape of the MTF, leading to more lateral scattering. However, anisotropic single scattering cannot explain the larger lateral scattering observed in paper. It is argued that the directional inhomogeneity of paper requires a light scattering model with both the phase function and scattering distances being dependent on the absolute direction.

© 2011 OSA

## 1. Introduction

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

4. F. C. Williams and F. R. Clapper, “Multiple internal reflections i photographic color prints,” J. Opt. Soc. Am. **43**, 595–599 (1953). [CrossRef] [PubMed]

5. R. D. Hersch, “Spectral prediction model for color prints on paper with fluorescent additives,” Appl. Opt. **47**, 6710–6722 (2008). [CrossRef] [PubMed]

6. L. Yang, “Probabilistic spectral model of color halftone incorporating substrate fluorescence and interface reflections,” J. Opt. Soc. Am. A **27**, 2115–2122 (2010). [CrossRef]

9. P. Kubelka, “New contributions to the optics of intensely light-scattering materials. part 1,” J. Opt. Soc. Am. **38**, 448–457 (1948). [CrossRef] [PubMed]

*k*, based only on the Kubelka-Munk scattering coefficient

_{p}*S*. The model includes an ad-hoc offset attributed to directional inhomogeneity in paper. This directional inhomogeneity, meaning that lateral scattering is weaker than scattering along the paper thickness direction, was modeled by Mourad [11] who considered lateral fluxes in a KM framework by separating backscattered light and laterally scattered light. This leads to two scattering coefficients that are difficult to determine in practice. Assuming both coefficients to be equal, the model showed good agreement with the measurements of Arney et al. [12

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

13. H. Granberg and M.-C. Béland, “Modelling the angle-dependent light scattering from sheets of pulp fiber fragments,” Nord. Pulp Pap. Res. J. **19**, 354–359 (2004). [CrossRef]

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

15. 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]

16. M. Neuman, L. G. Coppel, and P. Edström, “Point spreading in turbid media with anisotropic single scattering,” Opt. Express **19**, 1915–1920 (2011). [CrossRef] [PubMed]

*k*as a single metric for the MTF, and to investigate whether the larger lateral scattering obtained with forward single scattering can account for the more narrow MTF measured on real paper samples.

_{p}## 2. Method

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

*g*= 0 for isotropic single scattering and

*g*= 0.8 for forward single scattering. The KM model and the two MC simulations give the same reflectance factor and transmittance as the measurements. This leads to different scattering and absorption coefficients for different values of the asymmetry factor. The simulated MTFs allow for a direct comparison of the

*k*obtained from the analytical Arney model, from MC simulations, and from measurements. The impact of anisotropic single scattering on the MTF is assessed by comparing the MTFs obtained with the two different values of the asymmetry factor.

_{p}### 2.1. Material, measurements and parameter estimation

*S*) and absorption (

*K*) coefficients of these samples in a 0°/d geometry. The set of samples includes most uncoloured paper types, from translucent (i.e. low

*S*) papers and newsprint to coated fine papers. The reflectance factor and transmittance of these samples are calculated using KM theory, given the thickness,

*S*, and

*K*of each sample. For the MC simulations, the scattering (

*σ*) and absorption (

_{s}*σ*) coefficients are determined by optimization for the two different values of the asymmetry factor. The optimization uses the non spatially resolved DORT2002 model [19

_{a}19. P. Edström, “A fast and stable solution method for the radiative transfer problem,” SIAM Rev. **47**, 447–468 (2005). [CrossRef]

20. P. Edström, “A two-phase parameter estimation method for radiative transfer problems in paper industry applications,” Inverse Probl. Sci. En. **16**, 927–951 (2008). [CrossRef]

*σ*and

_{s}*σ*are reported in Table (1), together with the paper type, thickness and KM scattering and absorption coefficients.

_{a}### 2.2. Monte Carlo simulation of the edge response

*μ*m. The spatially resolved reflectance (shown in Fig. 1(a)) is then averaged along the illumination edge and normalized to get the edge response (shown in Fig. 1(b)). The edge response is computed for a single layer and for a layer of infinite thickness representing an opaque pad of identical samples. For comparison with the Arney model, the paper samples are modeled as a single layer with refractive index equal to that of the surrounding thus neglecting surface reflection. To reduce noise, 10

^{6}wave packets were used in each simulation.

### 2.3. MTF calculation and characterization

*w*is the spatial frequency. Equation (1) leads to since MTF(1/

*k*) = 0.5. The

_{p}*k*values from the MC simulations are given by the the inverse frequency at which MTF = 0.5 (Fig. 1(c)).

_{p}## 3. Results

*g*= 0 than for the case with

*g*= 0.8. The shape of the MTF in Fig. 1(c) also differs but the

*k*values are close. For these samples, the Arney MTF model (Eq. (1)) gives MTFs similar to the MC simulated MTF with

_{p}*g*= 0. For non translucent samples, Eq. (1) and MC simulations differ at larger frequencies, but the

*k*values are close. Hence the

_{p}*k*metric does not reflect the different MTFs obtained with the different models, nor does it reflect the effect of the asymmetry factor on the MTF.

_{p}*k*. This is in line with the broadening of the point spread function with increasing forward scattering observed by Neuman et al. [16

_{p}16. M. Neuman, L. G. Coppel, and P. Edström, “Point spreading in turbid media with anisotropic single scattering,” Opt. Express **19**, 1915–1920 (2011). [CrossRef] [PubMed]

*k*versus the

_{p}*k*predicted by the Arney model for all 22 samples. Figure 2(a) shows a single sample over a black background and Fig. 2(b) shows an opaque pad of identical samples. The Arney value is close to MC value for single sheets, and for opaque pads of sheets when

_{p}*k*is low. The asymmetry factor has a negligible impact, except for the single sheet samples with low

_{p}*S*(and thus larger

*k*). Figure 3 shows the MC simulated

_{p}*k*versus the measured

_{p}*k*. The two models give similar predictions of

_{p}*k*for non translucent samples (lower

_{p}*S*and lower

*k*). For the opaque pads of translucent samples, the MC simulations predict larger

_{p}*k*, which better correlate with the measured

_{p}*k*(Fig. 3).

_{p}*k*and measured

_{p}*k*is high. However, the MC simulations underestimate

_{p}*k*, i.e. the width of the MTF at half maximum value, meaning that the models predict less lateral scattering than what is actually measured. Anisotropic scattering does not significantly increase

_{p}*k*, and thus cannot explain the narrower MTF measured from paper.

_{p}## 4. Discussion and conclusions

*k*is close for MC and Arney models. This means that

_{p}*k*as single metric of the MTF is inappropriate since it does not reflect the different MTF obtained with different asymmetry factors and with the Arney MTF model. Using

_{p}*k*may lead to the wrong conclusion that the asymmetry factor does not affect the MTF.

_{p}*σ*will then lead to less lateral scattering within the layer. Although single forward scattering leads to more lateral scattering than isotropic single scattering, this effect does not explain the narrower MTF measured. Directional inhomogeneity in uncoated papers and other fibrous materials such as tissue calls for a radiative transfer model with both the single scattering phase function and scattering distances being dependent on the absolute direction within the material. MC modeling of the lateral light scattering in fiber networks is ongoing to address this.

_{s}## Acknowledgments

## References and links

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

2. | J. M. Schmitt, “Optical coherence tomography (OCT): A review,” IEEE J. Sel. Top. Quant. |

3. | J. Yule and W. Neilsen, “The penetration of light into paper and its effect on halftone reproduction,” in Proceedings of TAGA ,vol. |

4. | F. C. Williams and F. R. Clapper, “Multiple internal reflections i photographic color prints,” J. Opt. Soc. Am. |

5. | R. D. Hersch, “Spectral prediction model for color prints on paper with fluorescent additives,” Appl. Opt. |

6. | L. Yang, “Probabilistic spectral model of color halftone incorporating substrate fluorescence and interface reflections,” J. Opt. Soc. Am. A |

7. | S. Gustavson, “Dot Gain in Colour Halftones,” Ph. D. thesis, Linköping university (1997). |

8. | P. Oittinen, “Limits of microscopic print quality,” in |

9. | P. Kubelka, “New contributions to the optics of intensely light-scattering materials. part 1,” J. Opt. Soc. Am. |

10. | J.S. Arney, J. Chauvin, J. Nauman, and P.G. Anderson, “Kubelka-Munk theory and the MTF of paper,” |

11. | S. Mourad, P. Emmel, K. Simon, and R. D. Hersch, “Extending Kubelka-Munk’s theory with lateral light scattering,” in IS&T’s NIP17: International Conference on Digital Printing Technologies, Lauderdale, Florida, USA , (2001), pp. 469–473. |

12. | S. Mourad, “Improved Calibration of Optical Characteristics of Paper by an Adapted Paper-MTF Model,” J. Imaging Sci. Techn. |

13. | H. Granberg and M.-C. Béland, “Modelling the angle-dependent light scattering from sheets of pulp fiber fragments,” Nord. Pulp Pap. Res. J. |

14. | M. Neuman and P. Edström, “Anisotropic reflectance from turbid media. II. Measurements,” J. Opt. Soc. Am. A |

15. | 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 |

16. | M. Neuman, L. G. Coppel, and P. Edström, “Point spreading in turbid media with anisotropic single scattering,” Opt. Express |

17. | L. G. Henyey and J. L. Greenstein, “Diffuse Radiation in the Galaxy,” Astrophys. J. |

18. | M. Ukishima, “Prediction and evaluation of color halftone print quality based on microscopic measurement,” Ph.D. thesis, University of Eastern Finland (2010). |

19. | P. Edström, “A fast and stable solution method for the radiative transfer problem,” SIAM Rev. |

20. | P. Edström, “A two-phase parameter estimation method for radiative transfer problems in paper industry applications,” Inverse Probl. Sci. En. |

21. | 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 |

**OCIS Codes**

(030.5620) Coherence and statistical optics : Radiative transfer

(100.2810) Image processing : Halftone image reproduction

(290.4210) Scattering : Multiple scattering

(290.7050) Scattering : Turbid media

(290.2558) Scattering : Forward scattering

**ToC Category:**

Scattering

**History**

Original Manuscript: September 26, 2011

Revised Manuscript: November 9, 2011

Manuscript Accepted: November 10, 2011

Published: November 23, 2011

**Virtual Issues**

Vol. 7, Iss. 2 *Virtual Journal for Biomedical Optics*

**Citation**

Ludovic G. Coppel, Magnus Neuman, and Per Edström, "Lateral light scattering in paper - MTF simulation and measurement," Opt. Express **19**, 25181-25187 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-25-25181

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### References

- A. S. Glassner, Principles of Digital Image Synthesis, Volume Two, (Morgan Kauffman, 1995).
- J. M. Schmitt, “Optical coherence tomography (OCT): A review,” IEEE J. Sel. Top. Quant.5, 1205–1215 (1999). [CrossRef]
- J. Yule and W. Neilsen, “The penetration of light into paper and its effect on halftone reproduction,” in Proceedings of TAGA ,vol. 3 (1951), pp. 65–67.
- F. C. Williams and F. R. Clapper, “Multiple internal reflections i photographic color prints,” J. Opt. Soc. Am.43, 595–599 (1953). [CrossRef] [PubMed]
- R. D. Hersch, “Spectral prediction model for color prints on paper with fluorescent additives,” Appl. Opt.47, 6710–6722 (2008). [CrossRef] [PubMed]
- L. Yang, “Probabilistic spectral model of color halftone incorporating substrate fluorescence and interface reflections,” J. Opt. Soc. Am. A27, 2115–2122 (2010). [CrossRef]
- S. Gustavson, “Dot Gain in Colour Halftones,” Ph. D. thesis, Linköping university (1997).
- P. Oittinen, “Limits of microscopic print quality,” in Advances in Printing Science and Technology, L. . W. H. Banks, ed. (Pentech, London, 1982), Vol. 16, pp. 121–128.
- P. Kubelka, “New contributions to the optics of intensely light-scattering materials. part 1,” J. Opt. Soc. Am.38, 448–457 (1948). [CrossRef] [PubMed]
- J.S. Arney, J. Chauvin, J. Nauman, and P.G. Anderson, “Kubelka-Munk theory and the MTF of paper,” J. Imaging Sci. Techn.47, 339–345 (2003).
- S. Mourad, P. Emmel, K. Simon, and R. D. Hersch, “Extending Kubelka-Munk’s theory with lateral light scattering,” in IS&T’s NIP17: International Conference on Digital Printing Technologies, Lauderdale, Florida, USA, (2001), pp. 469–473.
- S. Mourad, “Improved Calibration of Optical Characteristics of Paper by an Adapted Paper-MTF Model,” J. Imaging Sci. Techn.51, 283–292 (2007). [CrossRef]
- H. Granberg and M.-C. Béland, “Modelling the angle-dependent light scattering from sheets of pulp fiber fragments,” Nord. Pulp Pap. Res. J.19, 354–359 (2004). [CrossRef]
- M. Neuman and P. Edström, “Anisotropic reflectance from turbid media. II. Measurements,” J. Opt. Soc. Am. A27, 1040–1045 (2010). [CrossRef]
- 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. A26, 1403–1413 (2009). [CrossRef]
- M. Neuman, L. G. Coppel, and P. Edström, “Point spreading in turbid media with anisotropic single scattering,” Opt. Express19, 1915–1920 (2011). [CrossRef] [PubMed]
- L. G. Henyey and J. L. Greenstein, “Diffuse Radiation in the Galaxy,” Astrophys. J.93, 70–83 (1941). [CrossRef]
- M. Ukishima, “Prediction and evaluation of color halftone print quality based on microscopic measurement,” Ph.D. thesis, University of Eastern Finland (2010).
- P. Edström, “A fast and stable solution method for the radiative transfer problem,” SIAM Rev.47, 447–468 (2005). [CrossRef]
- P. Edström, “A two-phase parameter estimation method for radiative transfer problems in paper industry applications,” Inverse Probl. Sci. En.16, 927–951 (2008). [CrossRef]
- 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).

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