4.3.1 Overview
Previous articles have reported that heuristic parameterizations of a reflectance function, such as the half-way and difference vectors [151], can make a great impact on the efficiency of approxi-mation algorithms. Nevertheless, since a pre-defined parameterization method frequently relies on certain assumptions of data characteristics, it may be inadequate to handle various real-world visual data sets. For example, the half-way parameterization tends to align the specular peak of a reflectance function, but the shadowing and masking effects of micro-facets are ignored. This situation will become even worse for a real-world multivariate visual data set, since it is usually measured under diverse and complex physical conditions.
To overcome the disadvantages of fixed parameterization, we propose to learn a set of opti-mized transformation functions for a given visual data set. Since our goal is to obtain a compact representation, we choose to model the transformation functions using parametric equations.
This particularly allows the parameterization process to be tightly integrated into the proposed multivariate SRBF representation. Although the derived optimal solution is constrained to a certain functional form, experimental results show that even a linear mixture of the parameters of a multivariate reflectance function, followed by projection onto the unit hyper-sphere, can be more effective than previous heuristic approaches. Finding the truly optimal parameterizations using non-parametric models thus is left as the future work.
Let ψ(Ω|Θ) ∈ Sm be a transformation function that depends on a given set of parameter-ization coefficients Θ = {θi ∈ R}Ii=1Θ , where IΘ denotes the total number of parameterization coefficients in Θ and is specified by users. We would like to find an optimal solution to Θ so that a multivariate spherical function F (Ω) can be efficiently approximated by transforming it into a univariate spherical function F0 ψ(Ω|Θ) ∈ R that is more suitable for univariate SRBF
Algorithm 4.2: Initial guess for the scattered multivariate SRBF representation.
Procedure: MultiInitial(F (Ω), J, {Ωn}Nn=1)
Input: An N -variate spherical function F (Ω) on Sm1× Sm2× · · · × SmN, the number of multivariate SRBFs J , and N sets of sampling directions {Ωn}Nn=1.
Output: The initial guess of parametersβj, Ξj, Λj
J
j=1in Equation 4.6.
begin
for j ← 1 to J do βj,0(Ω) ← F (Ω) for n ← 1 to N do
// Initialization Ω ← Ω \ {ωn}
Initialize {ξj,n, λj,n}Jj=1with respect to β1,n−1 ωn, {ω1,k}Nk=n+1 (Section 3.4.2) for j ← 1 to J do βj,n(Ω) ← βj,n−1(ξj,n, Ω)
// Model parameter estimation repeat
for in← 1 to Iωn do
for in+1← 1 to Iωn+1 do ...
for iN ← 1 to IωN do for j ← 1 to J do
Update basis coefficients βj,n {ωik,k}Nk=n+1 with respect to βj,n−1 ωn, {ωik,k}Nk=n+1
end end end end
Update center set {ξj,n}Jj=1 with respect toβj,n−1(ωn, Ω) J j=1
Update bandwidth set {λj,n}Jj=1with respect toβj,n−1(ωn, Ω) J j=1
until convergence
Update all parametersβj,n(Ω), ξj,n, λj,n
J
j=1to obtain a locally optimal solution with respect toβj,n−1(ωn, Ω) J
j=1
end
for j ← 1 to J do βj ← βj,N end
expansions:
From Equation 4.7, it is also intuitive to extend the same concept to transform F (Ω) into an N0-variate spherical function F0(Ψ) ∈ R as specifies the parameterization coefficient set with IΘnelements for the n-th transformation func-tion, and ˆF0(Ψ) denotes the approximate multivariate SRBF representation of F0(Ψ). Note that the number of variables of F0(Ψ), namely N0, is not necessarily identical to that of F (Ω), but rather can be specified by users. This flexibility particularly allows our paramerterized mul-tivariate SRBF representation to accurately model various complex behaviors of a real-world multivariate spherical function.
In summary, we combine the scattered multivariate SRBF representation and optimized parameterization to derive a parametric model (Equation 4.8), and obtain its parameters by solving the following unconstrained least-squares optimization problem:
minΥ
, and Eaddl is the additional energy term similar to Equation 4.6. In Section 4.3.3, we will further present a general algorithm for solving Equation 4.9.
4.3.2 Example
We again take the BRDF ρ(ωl, ωv) for instance. Based on Equation 4.8, Equation 4.5 can be extended into an N0-variate SRBF representation as
ρ(ωl, ωv) ≈ ˆρ0 Ψ(ρ) = Θ(ρ)n respectively represent the center set, the bandwidth set, and the parameterization coefficient
set for the n-th transformed variable. Each transformation function in Equation 4.10 can be modeled with the normalization of a linear combination of ωl and ωv as follows:
∀n, ψn ωl, ωv|Θ(ρ)n = θ(ρ)1,nωl+ θ2,n(ρ)ωv θ(ρ)1,nωl+ θ(ρ)2,nωv
2
, (4.11)
where θ(ρ)1,n∈ R and θ(ρ)2,n∈ R are the parameterization coefficients in Θ(ρ)n for the n-th transfor-mation function.
When N0 = 3 and J = 1, Equation 4.10 is similar to the mathematical formulation of homo-morphic factorization [116], but the parameterized multivariate SRBF representation allows a linear combination of multiple multivariate functions to model heterogeneous materials, which is particularly important for representing spatially-varying reflectance data sets. Moreover, Equation 4.11 was inspired by the fact that many common parameterizations for reflectance functions, such as the half-way, illumination, and view vectors, are its special cases. It is also remarkable that these three heuristic parameterizations were employed in the implementation of homomorphic factorization, which further implies the practical effectiveness of our methods.
4.3.3 Fitting Algorithm
Algorithm 4.3 presents the pseudo-code of a practical algorithm for solving Equation 4.9. At a time, we optimize only one out of four types of parameters, including basis coefficients, center sets, bandwidth sets, and parameterization coefficient sets, while the other types of parameters are fixed. Similar to Algorithm 4.1, the optimization of different parameterization coefficient sets is not decoupled for performance issues. Furthermore, it is also straightforward to develop a GPU-based implementation and an incremental variant of Algorithm 4.3.
As for the initial guess of parameters in Equation 4.9, since parameterization coefficient sets strongly depend on the functional form of transformation functions, we currently do not have a common solution to this issue. Nevertheless, previous heuristic parameterizations generally provide an appropriate starting point if they are special cases of the adopted transformation functions. Take Equation 4.11 for example, the initial guess of the first three parameterization coefficient sets can be explicitly set to the half-way, illumination, and view parameterizations, while the remainders are randomly generated. Once the initial guess of parameterization coef-ficient sets is determined, that of basis coefcoef-ficients, center sets, and bandwidth sets then can be estimated using Algorithm 4.2.