Simplified spherical harmonics approximation of the time-dependent equation of radiative transfer for the forward problem in time-domain diffuse optical tomography
Bibliographic record
Abstract
The equation of radiative transfer (ERT) is generally accepted as the most accurate model for light propagation in biological tissues. The ERT is notoriously expensive to solve numerically. Recently, Klose and Larsen have approximated the time-independent ERT using the simplified spherical harmonics equations ( <i>SP<sub>N</sub></i> approximation). In this work, we outline how to derive the <i>SP<sub>N</sub></i> approximation of the time-dependent ERT and obtain the associated integro- partial differential equations involving temporal convolution integrals. No approximation is made as regards the time variable in our derivation. To simplify the numerical solution of these equations, we introduce a "memory function". We discuss the numerical solution for <i>N</i> = 1 in the 2D and homogeneous case. We provide time evolution maps of the solution and compare it with the diffusion approximation of the ERT. The findings presented here straightforwardly extend to 3D inhomogeneous media and for higher values of <i>N</i>. These more complicated cases along with further details will be reported elsewhere.
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How this classification was reachedexpand
Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one teacher head, not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".