Branch and bound algorithm for accurate estimation of analytical isotropic bidirectional reflectance distribution function models

Chanki Yu; Sang Wook Lee

doi:10.1364/AO.55.004193

Applied Optics
Vol. 55,
Issue 15,
pp. 4193-4200
(2016)
•https://doi.org/10.1364/AO.55.004193

Branch and bound algorithm for accurate estimation of analytical isotropic bidirectional reflectance distribution function models

Chanki Yu and Sang Wook Lee

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Chanki Yu and Sang Wook Lee, "Branch and bound algorithm for accurate estimation of analytical isotropic bidirectional reflectance distribution function models," Appl. Opt. 55, 4193-4200 (2016)

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Table of Contents Category
- Vision, Color, and Visual Optics
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About this Article
History
- Original Manuscript: January 29, 2016
- Revised Manuscript: April 19, 2016
- Manuscript Accepted: April 23, 2016
- Published: May 19, 2016

Abstract

We present a reliable and accurate global optimization framework for estimating parameters of isotropic analytical bidirectional reflectance distribution function (BRDF) models. This approach is based on a branch and bound strategy with linear programming and interval analysis. Conventional local optimization is often very inefficient for BRDF estimation since its fitting quality is highly dependent on initial guesses due to the nonlinearity of analytical BRDF models. The algorithm presented in this paper employs $L_{1}$ -norm error minimization to estimate BRDF parameters in a globally optimal way and interval arithmetic to derive our feasibility problem and lower bounding function. Our method is developed for the Cook–Torrance model but with several normal distribution functions such as the Beckmann, Berry, and GGX functions. Experiments have been carried out to validate the presented method using 100 isotropic materials from the MERL BRDF database, and our experimental results demonstrate that the $L_{1}$ -norm minimization provides a more accurate and reliable solution than the $L_{2}$ -norm minimization.

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Input: an initial surface roughness parameter interval $S_{0}$ for $σ$ .
Output: the globally optimal solution of $ρ_{d}$ , $ρ_{s}$ and $σ$ .
1.	Initialize $j = 0$ , $Ω_{0} = {S_{0}}$ , $L_{0} = - \infty$ , $U_{0} = \infty$ .
2.	While $(U_{j} - L_{j}) / U_{j} > ε$ , then ${$
3.	Choose $S \in Ω_{j}$ for which $Φ_{l b} (S) = L_{j}$ .
4.	Bisect $S$ into ${S_{1}, S_{2}}$ .
5.	$Ω_{j + 1} \leftarrow {Ω_{j} ∖ S} ⋃ {S_{1}, S_{2}}$ ; $j \leftarrow j + 1$ .
6.	Compute lower bounds ${Φ_{l b} (S_{1}), Φ_{l b} (S_{2})}$ and upper bounds ${Φ_{u b} (S_{1}), Φ_{u b} (S_{2})}$ .
7.	Update the global lower and upper bounds: $L_{j}$ and $U_{j}$ .
8.	Prune $Ω_{j}$ by discarding $\forall S \in Ω_{j}$ if $Φ_{l b} (S) > U_{j} .}$

	$L_{1}$ -Norm	$L_{2}$ -Norm
Beckmann	1.05	6.18
GGX	3.11	13.89
Berry	2.5	15.93

	$θ_{d} = 10 °$	$θ_{d} = 20 °$	$θ_{d} = 30 °$	$θ_{d} = 40 °$	$θ_{d} = 50 °$
Black-obsidian	0.6107	0.6107	0.6107	0.6107	0.6107
Black-oxidized-steel	0.0035	0.0035	0.0034	0.0038	0.0066
Gold-paint	0.0100	0.0099	0.0092	0.0098	0.0167
Delrin	0.0113	0.0105	0.0195	0.0101	0.0153
Cherry-235	0.0059	0.0056	0.0052	0.0054	0.0081
Blue-rubber	0.0041	0.0040	0.0038	0.0042	0.0067
Nickel	0.1754	0.1583	0.1557	0.1905	0.2992
Red-phenolic	0.1200	0.1129	0.1147	0.1366	0.1966

Input: an initial surface roughness parameter interval $S_{0}$ for $σ$ .
Output: the globally optimal solution of $ρ_{d}$ , $ρ_{s}$ and $σ$ .
1.	Initialize $j = 0$ , $Ω_{0} = {S_{0}}$ , $L_{0} = - \infty$ , $U_{0} = \infty$ .
2.	While $(U_{j} - L_{j}) / U_{j} > ε$ , then ${$
3.	Choose $S \in Ω_{j}$ for which $Φ_{l b} (S) = L_{j}$ .
4.	Bisect $S$ into ${S_{1}, S_{2}}$ .
5.	$Ω_{j + 1} \leftarrow {Ω_{j} ∖ S} ⋃ {S_{1}, S_{2}}$ ; $j \leftarrow j + 1$ .
6.	Compute lower bounds ${Φ_{l b} (S_{1}), Φ_{l b} (S_{2})}$ and upper bounds ${Φ_{u b} (S_{1}), Φ_{u b} (S_{2})}$ .
7.	Update the global lower and upper bounds: $L_{j}$ and $U_{j}$ .
8.	Prune $Ω_{j}$ by discarding $\forall S \in Ω_{j}$ if $Φ_{l b} (S) > U_{j} .}$

	$L_{1}$ -Norm	$L_{2}$ -Norm
Beckmann	1.05	6.18
GGX	3.11	13.89
Berry	2.5	15.93

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Applied Optics