3.5.1 Convolution of Univariate Gaussian SRBFs
Recall that we discussed the convolution of two univariate SRBFs in Section 3.2.1. For the con-volution of two single-scale univariate Gaussian SRBFs, namely the univariate Gaussian SRBFs with the same bandwidth, Narcowich and Ward [123] derived an analytic solution. However, single-scale univariate SRBFs become inadequate to model univariate spherical functions with scattered SRBFs, whose bandwidths should be adaptive to the given data sets. We thus intro-duce the spherical singular integral of two multi-scale univariate Gaussian SRBFs, and propose Theorem 3.1 for efficient computation.
Proof of Theorem 3.1: Suppose that we intend to compute the convolution of two multi-scale
Algorithm 3.2: Modified soft von Mises-Fisher clustering algorithm for precondition-ing. Interested readers may refer to [6] for the original soft von Mises-Fisher clustering algorithm.
Procedure: UniGaussSRBFPrecondition(F (ω), Ω, J, ˜F )
Input: A univariate spherical function F (ω) on Sm, a set of sampling directions Ω = {ωi ∈ Sm}Ii=1Ω , the number of univariate SRBFs J , and the initial guess
univariate Gaussian SRBFs, which corresponds to Z
without losing generality. Rewrite the integral on the right side of
Equa-tion 3.27 in terms of spherical coordinates in Rm+1, and substitute r with r0 to obtain From Equation 3.71(9) in [199], the order-n modified Bessel function of the first kind can be written in an integral form as
In(x) =
By combining the last line of Equation 3.28 with Equation 3.29 and substituting the result into Equation 3.27, Equation 3.9 can be derived.
3.5.2 Relation between Univariate Gaussian SRBF and Von Mises-Fisher Distribution
In Section 3.2.3, we mentioned about the equivalence of the normalized univariate Gaussian SRBF (with non-negative bandwidth) and the von Mises-Fisher distribution, which serves as a solid base for the preconditioning process in Section 3.4.2. In the following paragraph, we will analytically prove this intriguing relation.
Proof of Remark 3.3: Consider transforming a univariate Gaussian SRBF G(Gau)(ω · ξ|λ), with λ ≥ 0, into a probability distribution function Pr(Gau)(ω · ξ|λ) on Smas
Pr(Gau)(ω · ξ|λ) = G(Gau)(ω · ξ|λ)
g(Gau)(λ) (λ ≥ 0), (3.30)
where the normalizing constant g(Gau)(λ) is obtained by integrating G(Gau)(ω · ξ|λ) over Smas g(Gau)(λ) =
Since the integral on the right side of Equation 3.31 is similar to the integral on the left side of Equation 3.28, Equation 3.16 can be derived by substituting r and λg+ λhwith λ into Equation 3.9. From Equations 3.7 and 3.16, Remark 3.3 is proved by simply comparing Equation 3.30 with the definition of the von Mises-Fisher distribution (Equation 3.14).
Chapter 4
Multivariate Spherical Radial Basis Functions
A compact and efficient representation for large-scale visual data sets remains challenging in practice. The enormous amount of observations frequently becomes the performance bottleneck at run-time and prohibits further analysis in computer graphics and vision applications. In addition to data size, a real-world visual data set is usually a mixed effect of various types of physical factors. This high-dimensional nature is so complicated that simple analytic models often fail to describe the multivariate behavior of a data set.
In this chapter, we propose a novel functional representation, namely multivariate spherical radial basis functions (SRBFs), to solve this issue. The complex behaviors of a visual data set are described as a linear combination of multivariate SRBFs, while each multivariate SRBF is constructed from the product of more than one univariate SRBF. As a result, many popular analytic models and factorization-based approaches would fall within this general weighted sum-of-productsrepresentation.
4.1 Mathematical Formulation
4.1.1 Basic Definitions
In Chapter 3, we introduced a functional model based on univariate SRBFs for a univariate spherical function F (ω) ∈ R. As shown in Equation 3.4, F (ω) can be approximated in uni-variate SRBF expansions. Nevertheless, there are two major problems when applying Equation 3.4 to a model multivariate spherical function, such as the bidirectional reflectance distribution function (BRDF). First, a real-world visual data set may be a heterogenous field that results from various physical factors. Whether these factors are visible or invisible, the observed data distribution is frequently a function of at least two different variables, for example, illumina-tion and view direcillumina-tions for a BRDF. However, Equaillumina-tion 3.4 is a univariate representaillumina-tion that can only account for a single direction on Sm. This suggests that a multivariate representation
may be more favorable to describe the complex behaviors of a multivariate spherical function.
Second, one may suggest that after sampling a multivariate spherical function into multiple univariate spherical functions under different conditions, Equation 3.4 can be applied to sep-arately model the resulting univariate functions. For instance, we can respectively consider the reflectance distribution of each illumination (or view) direction for a BRDF. However, the outcome is a discrete and often non-compact representation. It is also a non-trivial matter to generalize this representation to estimate the distributions under novel conditions, such as novel illumination/view directions.
To represent a multivariate spherical function under different physical conditions, we can construct a multivariate SRBF from the product of several circularly axis-symmetric univariate SRBFs. For a complex or heterogenous multivariate spherical function, multiple multivariate SRBFs can be further linearly mixed to derive a general weighted sum-of-products model. More formally, we introduce the following definition of a multivariate SRBF:
Definition 4.1: Let Ω = {ωn∈ Smn}Nn=1 and Ξ = {ξn ∈ Smn}Nn=1 denote two N -element
The multivariate Abel-Poisson SRBF kernel thus corresponds to
G(Abel) Ω|Ξ, Λ =
and the multivariate Gaussian SRBF kernel is given by
G(Gau) Ω|Ξ, Λ = ePNn=1(λn(ωn·ξn)−λn), (4.3) where Λ = {λn ∈ R}Nn=1 is the set of bandwidth parameters of the involved univariate SRBFs, and Ξ is also known as the SRBF center set. Similar to the univariate SRBF representation, an N -variate spherical function F (Ω) ∈ R, with the n-th variable ωn defined on Smn, thus can be approximated based on a weighted sum-of-products representation as
F (Ω) ≈ ˆF (Ω) = and the bandwidth set of the j-th multivariate SRBF. Moreover, ˆF (Ω), whose dependence on the parameters of multivariate SRBFs is omitted for notational simplicity, denotes the approximate multivariate SRBF representation of F (Ω). As we will present in Section 4.3, Equation 4.4 can
be further extended into a more general model when combined with optimized parameterization.
j=1is the basis coefficient set,ξj,ω(ρ)
l∈ S2 J
j=1andλ(ρ)j,ω
l∈ R J
j=1respectively denote the center set and the bandwidth set for illumination variations, andξ(ρ)j,ωv∈ S2 J
j=1 as well asλ(ρ)j,ω
v∈ R J
j=1 instead specify the center set and the bandwidth set for view variations.
Note that Equation 4.5 is similar to many factorization-based representations for BRDFs, especially principal component analysis [77] and non-negative matrix factorization [91], but the proposed multivariate SRBF representation, like other parametric models, is more compact and intuitive to interpret or edit the derived parameters. This relation to factorization-based methods also suggests the potential of our multivariate SRBF representation to approximate various reflectance functions.