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Q.Y. J. Smithwick, A. Weber, A.E. Elsner, M.C. Cheney; Differential Geometric Profiling and Analytic Double Pass Beer’s Law Fitting of Retinal Vessels . Invest. Ophthalmol. Vis. Sci. 2005;46(13):4289.
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© ARVO (1962-2015); The Authors (2016-present)
Purpose:To compare the fits of retinal vessel intensity profiles of a Gaussian model versus a new analytic double pass Beer’s law model. To access the consistency of the Beer’s law model parameters with physical dimensions. Methods:Retinal images of 5 normal human eyes were acquired with scanning laser ophthalmoscopes (SLO) – 4 with a SLO polarimeter (GDx) and 1 with a slit SLO (Laser Scanning Digital Camera). Six vessels with reflexes and four without were selected. For each vessel, 8 adjacent profiles were extracted in the midline maximum curvature directions using differential geometry and then averaged. Gaussian curves were fit to the data by nonlinear regression. The vessel radii were computed as 2.33σ. The Beer’s Law model analytically characterizes the vessel as a cylindrical absorber above a plane reflector and accounts for single/double pass absorption of an incoming ray and outgoing diffuse light. The data was fit using nonlinear regression and the mean, vessel radius, vessel/reflector distance, absorption – k, incoming ray angle – Φ, sensor scale – A and offset – o determined. The RMS errors between data and the two fit types were compared. The vessel radii ratios were compared. To access the consistency of Beer’s law model parameters with corresponding physical dimensions, we fit arteries and veins that pass both the thick nerve fiber layer (NFL) near the optic nerve head and thinner outer NFL. Retinal images of 4 normal human eyes were acquired using a GDx. Profiles for both vessel types in both thick and thin NFL were sampled and fit using the Beer’s Law model. Results:Beer’s Law provided better fits than Gaussians with 10 times smaller RMS errors on average. For vessels with reflexes, Gaussians overestimated peak heights; Beer’s Law fits were flatter. Vessel and Beer’s Law profiles may be asymmetric; Gaussians are symmetric. Radii estimates of the two methods are consistent by a factor of 1.2 ± 0.11. Absorption coefficients and vessel types are mostly consistent, although k, A, and o are not independent. Vessel/reflector distance estimates are consistent with GDx NFL thickness maps. Conclusions:A Beers Law’s model provides more accurate estimation of vessel profiles than Gaussians. The former is a physical model, while the latter is an observation model. Radii estimates are intrinsic to the Beer’s Law model but arbitrary in the Gaussian model. The Beer’s Law model provides asymmetric and flatter profiles, as expected with vessels. The Beer’s Law model provides a better estimate of the vessel profile as it would appear without a reflex, which can further be used to estimate contrasts.
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