Hybrid Method for Solving the Radiative Transport Equation
The spherical harmonics method (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>P</mi><mi>N</mi></msub></semantics></math></inline-formula> method) is of...
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MDPI AG
2025-04-01
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| Series: | Photonics |
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| Online Access: | https://www.mdpi.com/2304-6732/12/5/409 |
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| author | André Liemert Dominik Reitzle Alwin Kienle |
| author_facet | André Liemert Dominik Reitzle Alwin Kienle |
| author_sort | André Liemert |
| collection | DOAJ |
| description | The spherical harmonics method (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>P</mi><mi>N</mi></msub></semantics></math></inline-formula> method) is often used for solving the radiative transport equation in terms of analytical functions. A severe and unsolved problem in this context was the evaluation of the angle-resolved radiance near sources and boundaries, which is a serious limitation of this method in view of concrete applications, e.g., in biomedical optics for investigating the different types of optical microscopy, within NIR spectroscopy, such as for the determination of ingredients in foods or in pharmaceuticals, and within physics-based rendering. In this article, we report on a hybrid method that enables accurate evaluation of the angle-resolved radiance directly at the boundary of an anisotropically scattering medium, avoiding the problems of the traditional <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>P</mi><mi>N</mi></msub></semantics></math></inline-formula> methods. The derived integral equation needed for the realization of the hybrid <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>P</mi><mi>N</mi></msub></semantics></math></inline-formula> method is formally valid for an arbitrary convex bounded medium. The proposed approach can be evaluated with practically the same computational effort as the traditional <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>P</mi><mi>N</mi></msub></semantics></math></inline-formula> method while being far more accurate. |
| format | Article |
| id | doaj-art-44b21d59d4df4ce39a484792dd10e6b8 |
| institution | Kabale University |
| issn | 2304-6732 |
| language | English |
| publishDate | 2025-04-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Photonics |
| spelling | doaj-art-44b21d59d4df4ce39a484792dd10e6b82025-08-20T03:47:57ZengMDPI AGPhotonics2304-67322025-04-0112540910.3390/photonics12050409Hybrid Method for Solving the Radiative Transport EquationAndré Liemert0Dominik Reitzle1Alwin Kienle2Institut für Lasertechnologien in der Medizin und Meßtechnik an der Universität Ulm, Helmholtzstr. 12, D-89081 Ulm, GermanyInstitut für Lasertechnologien in der Medizin und Meßtechnik an der Universität Ulm, Helmholtzstr. 12, D-89081 Ulm, GermanyInstitut für Lasertechnologien in der Medizin und Meßtechnik an der Universität Ulm, Helmholtzstr. 12, D-89081 Ulm, GermanyThe spherical harmonics method (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>P</mi><mi>N</mi></msub></semantics></math></inline-formula> method) is often used for solving the radiative transport equation in terms of analytical functions. A severe and unsolved problem in this context was the evaluation of the angle-resolved radiance near sources and boundaries, which is a serious limitation of this method in view of concrete applications, e.g., in biomedical optics for investigating the different types of optical microscopy, within NIR spectroscopy, such as for the determination of ingredients in foods or in pharmaceuticals, and within physics-based rendering. In this article, we report on a hybrid method that enables accurate evaluation of the angle-resolved radiance directly at the boundary of an anisotropically scattering medium, avoiding the problems of the traditional <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>P</mi><mi>N</mi></msub></semantics></math></inline-formula> methods. The derived integral equation needed for the realization of the hybrid <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>P</mi><mi>N</mi></msub></semantics></math></inline-formula> method is formally valid for an arbitrary convex bounded medium. The proposed approach can be evaluated with practically the same computational effort as the traditional <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>P</mi><mi>N</mi></msub></semantics></math></inline-formula> method while being far more accurate.https://www.mdpi.com/2304-6732/12/5/409radiative transportspherical harmonics methodphoton migrationturbid medialight propagation in tissues |
| spellingShingle | André Liemert Dominik Reitzle Alwin Kienle Hybrid Method for Solving the Radiative Transport Equation Photonics radiative transport spherical harmonics method photon migration turbid media light propagation in tissues |
| title | Hybrid Method for Solving the Radiative Transport Equation |
| title_full | Hybrid Method for Solving the Radiative Transport Equation |
| title_fullStr | Hybrid Method for Solving the Radiative Transport Equation |
| title_full_unstemmed | Hybrid Method for Solving the Radiative Transport Equation |
| title_short | Hybrid Method for Solving the Radiative Transport Equation |
| title_sort | hybrid method for solving the radiative transport equation |
| topic | radiative transport spherical harmonics method photon migration turbid media light propagation in tissues |
| url | https://www.mdpi.com/2304-6732/12/5/409 |
| work_keys_str_mv | AT andreliemert hybridmethodforsolvingtheradiativetransportequation AT dominikreitzle hybridmethodforsolvingtheradiativetransportequation AT alwinkienle hybridmethodforsolvingtheradiativetransportequation |