miun.sePublications
Change search
CiteExportLink to record
Permanent link

Direct link
Cite
Citation style
  • apa
  • harvard1
  • ieee
  • modern-language-association-8th-edition
  • vancouver
  • Other style
More styles
Language
  • de-DE
  • en-GB
  • en-US
  • fi-FI
  • nn-NO
  • nn-NB
  • sv-SE
  • Other locale
More languages
Output format
  • html
  • text
  • asciidoc
  • rtf
Mathematical modeling and numerical tools for simulation and design of light scattering in paper and print
Mid Sweden University, Faculty of Science, Technology and Media, Department of Engineering, Physics and Mathematics. (FSCN - Fibre Science and Communication Network)ORCID iD: 0000-0002-0529-1009
Responsible organisation
2007 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

This work starts with a real industrial problem - the perceived need for a moredetailed and more accurate model for light scattering in paper and print than theKubelka‐Munk model of today. A careful analysis transfers this problem into aphysical description of the phenomena involved. This is then given a mathematicalformulation, and a detailed analysis leads to numerical solution procedures forspecific sub problems. Methods from scientific computing make it possible to meetindustrial demands made on speed and stability, and implementation in computercode is then followed by analysis of accuracy and stability.A problem formulation and a solution method are outlined for the forwardradiative transfer problem. First, all necessary steps to arrive at a numericallystable solution procedure are treated, and then methods are introduced to increasethe speed by a factor of several thousands or millions compared to a naiveapproach. The method is shown to be unconditionally stable, though the problemwas previously considered numerically intractable, and systematic studies ofnumerical performance are presented.The inverse radiative transfer problem is given a least‐squares formulation, anddifferent solution methods are analyzed and compared. Specifically, a two‐phasemethod for estimation of the scattering and absorption coefficients and theasymmetry factor (σs, σa and g) is presented. A sensitivity analysis is given, and it isshown how it can be used for designing measurements with minimal impact frommeasurement noise.It is shown how the standardized use of Kubelka‐Munk and the d/0°instrument leads to errors, and that the errors arising from an over‐idealized viewof the instrument - due to the fact that instrument readings are incorrectlyinterpreted - can be larger than any errors inherent in the Kubelka‐Munk modelitself. It is argued that the measurement device and the simulation model cannot beviewed as separate instances, which is a widespread implicit practice in appliedreflectance measurements. Rather, given a measurement device, measurement datashould be interpreted through a model that takes into consideration the actualgeometry, function and calibration of the instrument.The resulting tool, DORT2002, is in all aspects the Next Generation Kubelka‐Munk, and provides a greater range of applicability, higher accuracy and increasedunderstanding. It offers better interpretation of measurement data, and facilitatesthe exchange of data between the paper and graphical arts industries. It opens forunderstanding of anisotropic reflectance and for the utilization of the asymmetryfactor to design anisotropy, and thereby for the design of different visualappearance or optical performance in new printed or paper products.

Place, publisher, year, edition, pages
Sundsvall: Mittuniversitetet , 2007. , 32 p.
Series
Mid Sweden University doctoral thesis, ISSN 1652-893X ; 22
Keyword [en]
mathematical modeling, radiative transfer, integro-differential equations, inverse problems, parameter estimation, solution method, numerical performance, light scattering, paper industry applications, Kubelka-Munk
National Category
Mathematics
Identifiers
URN: urn:nbn:se:miun:diva-5908Local ID: 5026ISBN: 978-91-85317-50-9 (print)OAI: oai:DiVA.org:miun-5908DiVA: diva2:30941
Public defence
(English)
Available from: 2008-09-30 Created: 2009-05-06 Last updated: 2009-07-13Bibliographically approved
List of papers
1. A Fast and Stable Solution Method for the Radiative Transfer Problem
Open this publication in new window or tab >>A Fast and Stable Solution Method for the Radiative Transfer Problem
2005 (English)In: SIAM Review, ISSN 0036-1445, Vol. 47, no 3, 447-468 p.Article in journal (Refereed) Published
Abstract [en]

Radiative transfer theory considers radiation in turbid media, and is used in a wide range of applications. This paper outlines a problem formulation and a solution method for the radiative transfer problem in multilayer scattering and absorbing media, using discrete ordinate model geometry. A selection of different steps is brought together. The main contribution here is the synthesis of these steps, all of which have been used in different areas, but never all together in one method. First all necessary steps to get a numerically stable solution procedure are treated, and then methods are introduced to increase the speed by a factor of several thousand. This includes methods for handling strongly forward-scattering media. The method is shown to be unconditionally stable, whilst the problem was previously considered numerically intractable.

Keyword
radiative transfer, discrete ordinates, solution method, numerical stability, speed
National Category
Mathematics
Identifiers
urn:nbn:se:miun:diva-2798 (URN)10.1137/S0036144503438718 (DOI)000231301400004 ()2321 (Local ID)2321 (Archive number)2321 (OAI)
Projects
DPC - Digital printing center
Available from: 2008-09-30 Created: 2009-07-10 Last updated: 2009-07-30Bibliographically approved
2. Numerical Performance of Stability Enhancing and Speed Increasing Steps in Radiative Transfer Solution Methods
Open this publication in new window or tab >>Numerical Performance of Stability Enhancing and Speed Increasing Steps in Radiative Transfer Solution Methods
2009 (English)In: Journal of Computational and Applied Mathematics, ISSN 0377-0427, E-ISSN 1879-1778, Vol. 228, no 1, 104-114 p.Article in journal (Refereed) Published
Abstract [en]

Methods for solving the radiative transfer problem, which are crucial for a number of sectors of industry, involve several numerical challenges. DORT2002 is a fast and numerically stable solution method for this problem, and has been implemented in MATLAB. This paper studies the numerical performance of DORT2002, in terms of stability, speed and accuracy. Focus is on the effects of the steps that are needed to make the method numerically efficient, and that differ from a naive implementation. Performance tests show that the steps that are included to improve stability and speed of DORT2002 are very successful. Together they give an unconditionally stable solution method to a problem previously considered numerically intractable, and decrease computation time compared to a naive implementation with a factor 1 000-10 000 in typical cases and with a factor up to and beyond 10 000 000 in extreme cases. It is also shown that the speed increasing steps are not introduced at the cost of reduced accuracy. Further studies and developments, that can have a positive impact on computation time, are suggested.

Keyword
radiative transfer, solution method, numerical performance, stability, speed, accuracy
National Category
Mathematics
Identifiers
urn:nbn:se:miun:diva-2799 (URN)10.1016/j.cam.2008.08.045 (DOI)000265892700011 ()2-s2.0-63349094792 (Scopus ID)2322 (Local ID)2322 (Archive number)2322 (OAI)
Available from: 2008-09-30 Created: 2009-01-19 Last updated: 2016-09-26Bibliographically approved
3. Comparison of the DORT2002 Radiative Transfer Solution Method and the Kubelka-Munk Model
Open this publication in new window or tab >>Comparison of the DORT2002 Radiative Transfer Solution Method and the Kubelka-Munk Model
2004 (English)In: Nordic Pulp & Paper Research Journal, ISSN 0283-2631, E-ISSN 2000-0669, Vol. 19, no 3, 397-403 p.Article in journal (Refereed) Published
Abstract [en]

The need for optical modeling of paper is obvious to provide connections between its optical response and the actual properties of the paper. It is argued that modern solution methods from radiative transfer theory could be considered instead of the Kubelka-Munk model, and a specific example, DORT2002, is tested. It is shown that Kubelka-Munk is a simple special case of DORT2002, and the two models and their coefficients are compared. A comprehensive list of advantages for the applied user of a model with higher dimensionality is supplied. It is shown, by the use of DORT2002, that when the medium has finite thickness, the light distribution deviates from the perfectly diffuse even under the theoretically ideal conditions for which Kubelka-Munk was created. This effect causes errors in Kubelka-Munk reflectance calculations that may be up to 20% and more, even for a grammage of 80 g/m2. The magnitude of the error shows a strong dependence on the degree of light absorption, with higher absorption giving greater error. DORT2002 can well be considered for increased understanding in cases where the level of accuracy of Kubelka-Munk reflectance calculations is not sufficient.

Keyword
light scattering, mathematical models, radiative transfer, Kubelka-Munk, light absorption, errors, reflectance calculations
National Category
Mathematics
Identifiers
urn:nbn:se:miun:diva-2686 (URN)10.3183/NPPRJ-2004-19-03-p397-403 (DOI)000224636500019 ()2-s2.0-6344280895 (Scopus ID)2118 (Local ID)2118 (Archive number)2118 (OAI)
Projects
DPC - Digital printing center
Available from: 2008-09-30 Created: 2009-07-10 Last updated: 2016-09-26Bibliographically approved
4. Quantification of the intrinsic error of the kubelka–munk model caused by strong light absorption
Open this publication in new window or tab >>Quantification of the intrinsic error of the kubelka–munk model caused by strong light absorption
2003 (English)In: Journal of Pulp and Paper Science (JPPS), ISSN 0826-6220, Vol. 29, no 11, 386-390 p.Article in journal (Refereed) Published
Abstract [en]

The Kubelka-Munk (KM) model is widely used within the paper industry to interpret diffuse reflectance factor measurements of paper and its components. It has been found in the literature that the addition of a dye colorant to a paper sheet not only increases its KM light absorption coefficient, but for strong absorption also decreases its KM light scattering coefficient. This effect has previously been attributed to the intrinsic error of the KM model induced by light absorption that tends to orient of the light fluxes perpendicular to the sheet. In the present work we have mapped the intrinsic error of the KM model by comparing light scattering calculations from the KM model with the more accurate Discrete Ordinate Radiative Transfer model DORT2002. We found that the models agree within 2.3% in reflectance, and that the intrinsic error in the KM model explains about 1/5 of the previously observed interdependence of the KM coefficients for heavily dyed sheets.

Keyword
light scattering, light absorption, reflectance, Kubelka-Munk, DORT2002, optical properties, errors
National Category
Mathematics
Identifiers
urn:nbn:se:miun:diva-1799 (URN)000187624200006 ()507 (Local ID)507 (Archive number)507 (OAI)
Available from: 2008-09-30 Created: 2008-09-30Bibliographically approved
5. Levenberg-Marquardt Methods for Parameter Estimation Problems in the Radiative Transfer Equation
Open this publication in new window or tab >>Levenberg-Marquardt Methods for Parameter Estimation Problems in the Radiative Transfer Equation
2007 (English)In: Inverse Problems, ISSN 0266-5611, E-ISSN 1361-6420, Vol. 23, no 3, 879-891 p., 002Article in journal (Refereed) Published
Abstract [en]

A discrete ordinate method is developed for solving the radiative transfer equation, and the corresponding parameter estimation problem is given a least-squares formulation. Two Levenberg-Marquardt methods, a feasible-path approach and an SQP type method, are analyzed and compared. A sensitivity analysis is given, and it is shown how it can be used for designing measurements with minimal impact of measurement noise. Numerical experiments are performed to exemplify the usefulness of the theory.

Keyword
Radiative transfer, parameter estimation
National Category
Mathematics
Identifiers
urn:nbn:se:miun:diva-4111 (URN)10.1088/0266-5611/23/3/002 (DOI)000246789100002 ()2-s2.0-34249701208 (Scopus ID)4718 (Local ID)4718 (Archive number)4718 (OAI)
Available from: 2008-09-30 Created: 2008-09-30 Last updated: 2016-10-05Bibliographically approved
6. A Two-Phase Parameter Estimation Method for Radiative Transfer Problems in Paper Industry Applications
Open this publication in new window or tab >>A Two-Phase Parameter Estimation Method for Radiative Transfer Problems in Paper Industry Applications
2008 (English)In: Inverse Problems in Science and Engineering, ISSN 1741-5977, Vol. 16, no 7, 927-951 p.Article in journal (Refereed) Published
Abstract [en]

A two-phase method for estimation of the scattering and absorption coefficients and the asymmetry factor (sigma_s, sigma_a and g) in the radiative transfer problem is presented. The first phase parameterizes sigma_s and sigma_a through g via a simplified model and performs - at a relatively low cost - a scalaroptimization over g. It is shown that this gives such a good starting point that the second phase can be accurately performed by a simple Gauss-Newton method. It is also shown that a part of the first phase can be used on its own when only sigma_s and sigma_a are wanted, and it is noted that this gives higheraccuracy than the commonly used Kubelka-Munk method when using standardized paper industry reflectance factor measurements.The parameter estimation problem is shown to be non-trivial and ill-conditioned, and its character is analyzed. It is discussed that standard optimization methods are so sensitive to the choice ofstarting point for this problem that it is hard to find a starting point that gives convergence at all. The new two-phase method is illustrated by application to relevant paper industry problems, and efficiency and sensitivity measures are given.

Keyword
Radiative transfer, Integro-ordinary differential equations, Inverse problems, Paper industry applications
National Category
Mathematics Other Engineering and Technologies
Identifiers
urn:nbn:se:miun:diva-6447 (URN)10.1080/17415970802080066 (DOI)000259645600007 ()2-s2.0-52949137438 (Scopus ID)4930 (Local ID)4930 (Archive number)4930 (OAI)
Available from: 2008-11-30 Created: 2008-11-30 Last updated: 2009-03-11Bibliographically approved
7. Examination of the Revised Kubelka-Munk Theory: Considerations of Modeling Strategies
Open this publication in new window or tab >>Examination of the Revised Kubelka-Munk Theory: Considerations of Modeling Strategies
2007 (English)In: Journal of the Optical Society of America A, ISSN 0740-3232, Vol. 24, no 2, 548-556 p.Article in journal (Refereed) Published
Abstract [en]

The revised Kubelka-Munk theory is examined theoretically and experimentally. Systems of dyed paper sheets are simulated, and the results are compared with other models. The results show that the revised Kubelka-Munk model yields significant errors in predicted dye-paper mixture reflectances, and is not self-consistent. The absorption is noticeably overestimated. Theoretical arguments show that properties in the revised Kubelka-Munk theory are inadequately derived. The main conclusion is that the revised Kubelka-Munk theory is wrong in the inclusion of the so-called scattering-induced-path-variation factor. Consequently, the theory should not be used for light scattering calculations. Instead, the original Kubelka-Munk theory should be used where its accuracy is sufficient, and a radiative transfer tool of higher resolution should be used where higher accuracy is needed.

Keyword
mathematical methods in physics, multiple scattering, turbid media
National Category
Mathematics Other Engineering and Technologies
Identifiers
urn:nbn:se:miun:diva-918 (URN)10.1364/JOSAA.24.000548 (DOI)000243517600029 ()2-s2.0-33847278408 (Scopus ID)4894 (Local ID)4894 (Archive number)4894 (OAI)
Available from: 2008-09-30 Created: 2009-07-30 Last updated: 2016-10-04Bibliographically approved

Open Access in DiVA

No full text

Search in DiVA

By author/editor
Edström, Per
By organisation
Department of Engineering, Physics and Mathematics
Mathematics

Search outside of DiVA

GoogleGoogle Scholar

Total: 3150 hits
CiteExportLink to record
Permanent link

Direct link
Cite
Citation style
  • apa
  • harvard1
  • ieee
  • modern-language-association-8th-edition
  • vancouver
  • Other style
More styles
Language
  • de-DE
  • en-GB
  • en-US
  • fi-FI
  • nn-NO
  • nn-NB
  • sv-SE
  • Other locale
More languages
Output format
  • html
  • text
  • asciidoc
  • rtf