Strict unimodality of the exact time-domain diffuse-light response under a Robin boundary condition
Abstract
Time-domain boundary measurements of diffuse light are temporal point-spread functions whose peak position is used as a robust scalar datum in diffuse optical and fluorescence imaging. We consider the exact response generated by a single fluorescent point target in a homogeneous half-space with a Robin boundary condition. The response is a temporal convolution of two diffusion Green-function factors containing a complementary-error-function boundary correction. For every positive diffusion, absorption, depth, and source--target and detector--target distance parameter, and for every nonnegative Robin parameter, we prove that the normalized exact response is a generalized gamma convolution density. Its extension by zero to the negative half-line is bell-shaped. A differentiable small-time expansion, established by sectorial analyticity and Cauchy estimates, together with the sign-change structure of the first two derivatives, excludes tangential critical points and yields a unique peak with strict increase before it and strict decrease after it. The argument is exact and requires no large-distance, source--detector balance, symmetry, restricted-time, or small-parameter assumption. The same conclusion holds after convolution with an exponential fluorescence-lifetime density for every positive lifetime. The key ingredients are an explicit Stieltjes representation of the Robin factor, hyperbolic complete monotonicity, closure of the generalized gamma convolution class, and bell-shape theory. A derivation of the exact half-space model and the differentiated endpoint asymptotics are included in two appendices.
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BibTeX
@misc{eom2026strict,
title = {{Strict unimodality of the exact time-domain diffuse-light response under a Robin boundary condition}},
author = {Junyong Eom and Jinwan Park},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.21431},
eprint = {2610.21431},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.21431},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Strict unimodality of the exact time-domain diffuse-light response under a Robin boundary condition
AU - Junyong Eom
AU - Jinwan Park
PY - 2026
DA - 2026/10/15
PB - arXiv
DO - 10.48550/arxiv.2610.21431
UR - https://arxiv.org/abs/2610.21431
AB - Time-domain boundary measurements of diffuse light are temporal point-spread functions whose peak position is used as a robust scalar datum in diffuse optical and fluorescence imaging. We consider the exact response generated by a single fluorescent point target in a homogeneous half-space with a Robin boundary condition. The response is a temporal convolution of two diffusion Green-function factors containing a complementary-error-function boundary correction. For every positive diffusion, absorption, depth, and source--target and detector--target distance parameter, and for every nonnegative Robin parameter, we prove that the normalized exact response is a generalized gamma convolution density. Its extension by zero to the negative half-line is bell-shaped. A differentiable small-time expansion, established by sectorial analyticity and Cauchy estimates, together with the sign-change structure of the first two derivatives, excludes tangential critical points and yields a unique peak with strict increase before it and strict decrease after it. The argument is exact and requires no large-distance, source--detector balance, symmetry, restricted-time, or small-parameter assumption. The same conclusion holds after convolution with an exponential fluorescence-lifetime density for every positive lifetime. The key ingredients are an explicit Stieltjes representation of the Robin factor, hyperbolic complete monotonicity, closure of the generalized gamma convolution class, and bell-shape theory. A derivation of the exact half-space model and the differentiated endpoint asymptotics are included in two appendices.
ER -