Abstract
This work presents a measurement uncertainty analysis for a system designed to simultaneously capture specular in-plane and out-of-plane bidirectional reflectance distribution function (BRDF) data with high spatial resolution by augmenting the Complete Angle Scatter Instrument (CASI®) with a charge-coupled device (CCD) camera. Various scatter flux, incident flux, scatter angle, and detector solid angle uncertainty contributions are considered and evaluated based on imperfectly known system parameters. In particular, incident flux temporal fluctuation, detector noise and non-linearity, and out-of-plane aperture misalignment considerations each require significant adjustment from original CASI® uncertainty analysis, and expressions for neutral density (ND) filter, scatter angle, and solid angle uncertainties each require new formulations. Ultimately, ND filter uncertainty produces the largest contribution for the augmented system—at least when using unrefined worst-case tolerances—followed by solid angle uncertainty and pixel non-linearity. Total BRDF uncertainty and its contributing terms are compiled for several measurement scenarios, and compared with those from original analyses for single-pixel detectors. In particular, when ND filter uncertainty can be ignored or mitigated, total BRDF uncertainty values are comparable to those for the original system.
Highlights
The bi-directional reflectance distribution function (BRDF) defines the spatial distribution of light reflected from a material surface as the ratio of scattered radiance to incident irradiance.[1]This ratio can be written purely in terms of measurable quantities as f r ðω^ i ; EQ-TARGET;temp:intralink-;e001;116;293 ω^ sÞ 1⁄4δΦi δΦs cos θs δΩd ; (1)which is a useful formulation for computing BRDF values from measurement data.[2,3]In this expression, ω^ i and ω^ s represent the incident and scattered directions, respectively, which are often written in spherical coordinates ðθi; φiÞ and ðθs; φsÞ
If a lens was placed in front of the charge-coupled device (CCD), each pixel would have an independent field of view, meaning each pixel would instead measure the flux reflected from a different location on the material sample, essentially imaging the material instead of measuring its BRDF
When BRDF values are shown on a log scale over the full dynamic range, as in Fig. 2(a), the CCD uncertainty bounds are indistinguishable from the measurement data; there is no optical density (OD) uncertainty for the beam signature and so the average total uncertainty remains below 4% (See Table 2)
Summary
The bi-directional reflectance distribution function (BRDF) defines the spatial distribution of light reflected from a material surface as the ratio of scattered radiance to incident irradiance.[1] This ratio can be written purely in terms of measurable quantities as f r ðω^ i ; EQ-TARGET;temp:intralink-;e001;116;293 ω^ sÞ. Which is a useful formulation for computing BRDF values from measurement data.[2,3] In this expression, ω^ i and ω^ s represent the incident and scattered directions, respectively, which are often written in spherical coordinates ðθi; φiÞ and ðθs; φsÞ. Butler, and Marciniak: Uncertainty analysis for CCD-augmented CASI® BRDF measurement system plane of incidence (out-of-plane) with high spatial resolution, surrounding specular peaks.[4,6]. Polarizing elements are not incorporated, so polarization misalignment terms are ignored
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