2.3 Data-Driven Rendering
2.3.2 Radiance Transfer
Precomputed Radiance Transfer
Recently, precomputed radiance transfer (PRT) has received a growing interest owing to its ability of rendering complex illumination and shadowing effects, such as self-inter-reflections, sub-surface scattering, caustics, and self-shadows, in dynamic lighting environments at real-time rates. The key concept is to precompute and model light scattering between an object and its surroundings by representing both incident radiance and light transport functions in the spherical harmonic basis. Thus, run-time rendering of exitant radiance can be reduced to a simple dot product for a diffuse object, or a matrix-vector multiplication for a glossy one. Since the spherical harmonic basis is inadequate to approximate high-frequency signals, PRT methods based on spherical harmonics are also called low-frequency PRT [99, 161, 162].
Beyond low-frequency radiance transfer, the all-frequency PRT methods [105, 125, 126, 195, 196] pre-sampled high-resolution light transport data to accurately capture hard and soft shadows in all-frequency lighting environments. The densely-sampled PRT data were then com-pressed using sophisticated compression techniques, for instance, non-linear wavelet
approxi-mation [125, 126, 195, 196] and matrix factorization [105, 195, 196]. Green et al. [50] further introduced a hybrid PRT method for static scenes. While view-independent effects, including direct and indirect diffuse terms, were modeled with spherical harmonics or wavelets, high-frequency view-dependent signals, such as direct and indirect glossy terms, were approximated with Gaussian functions using non-linear optimization. The success of PRT has stimulated the development of sophisticated approximation algorithms for large-scale light transport data sets [49, 108, 109, 206] and practical graphics applications such as editing systems [10, 11, 171].
Interested readers may refer to [98] for a comprehensive survey of recent progress of PRT.
Apart from static scenes and distant illumination, PRT has been extended to deformable ob-jects [67, 149, 164], dynamic scenes [76, 149, 170, 211], and local lights [85, 149, 211]. Sloan et al. [164] adopted rotation-invariant zonal harmonics to model radiance transfer functions us-ing non-linear optimization, so that rotatus-ing the transfer functions becomes simple and trivial.
Although the zonal harmonic basis may yield a more compact representation than spherical harmonics, it is still restricted to low-frequency signals and lighting environments. Moreover, Kristensen et al. [85] introduced unstructured light clouds by precomputing radiance transfer functions with respect to densely-sampled local lights and then clustering the results based on heuristic metrics. Zhou et al. [211] also presented a shadowing approach, namely precomputed shadow fields, for dynamic scenes and local lights. The shadow field was built from concentric shells that surround a light source or an object. At run-time, the shadow fields of scene entities were combined to obtain incident radiance distributions for rendering.
In this dissertation, we focus on static PRT since previous representations and approxima-tion algorithms may be inadequate for harnessing the power of PRT. Available methods either only handle low-frequency information, or suffer from unwieldy amount of data even after com-pression. For dynamic scenes, the amount of PRT data sets further expands to an impractical degree for real-time applications. The enormous data sets often prohibit high-quality rendering, and subsequently restrict the practical use of PRT. Therefore, we employ the proposed univari-ate SRBFs and tensor approximation algorithms to permit real-time performance and compact storage space for view-dependent all-frequency PRT in a unified framework. Extending the proposed PRT approach to cope with dynamic scenes remains as the future work.
Bi-Scale Radiance Transfer
While most PRT algorithms tackled spatially-uniform surface appearance, bi-scale radiance transfer(BRT) [163] effectively extended PRT with spatially-varying materials in low-frequency lighting environments. By generalizing light transport into macro- and meso-scale radiance transfer, BRT effectively combined coarsely-sampled global illumination data with detailed surface appearance. Thus, different sampling rate, precomputation technique, and data repre-sentation can be utilized at each scale to efficiently approximate the full global illumination solution.
Based on the same concept, radiance transfer volume [197] and shell radiance texture
func-tions [166] were proposed to respectively model the meso-scale radiance transfer with gener-alized displacement maps [197] and shell texture functions [21]. In this way, rendered image quality can be greatly improved, while considerably reducing precomputation costs. Neverthe-less, previous articles on BRT are restricted to low-frequency light transport since the spherical harmonic basis is employed to approximate the precomputed data. Thus, hard self-shadows and specular highlights at both scales due to the geometric details of meso-structures can not be accurately rendered. By contrast, we focus on the issue of combining all-frequency radiance transfer data at both scales, and propose an all-frequency BRT framework based on the pro-posed univariate SRBFs and tensor approximation algorithms to achieve real-time rendering on GPUs.
Chapter 3
Univariate Spherical Radial Basis Functions
In computer graphics and vision, previous functional models for radiance functions seem to be insufficient for solving related problems. Based on harmonic functions, it may take tens of thou-sands of terms to represent regional illumination and shadowing effects. As for wavelet-based methods, there is usually no analytic solution for efficient rotation of wavelet coefficients of a univariate spherical function. In this chapter, we therefore introduce a functional representation, namely univariate spherical radial basis functions1(SRBFs), to overcome the above-mentioned issues.
This chapter is a more detailed version of Section 4.1 and Appendix A in our published paper [179]. Unlike the published paper that only briefly presented the background of univariate SRBFs, here we additionally discuss many important characteristics of univariate SRBFs, and describe more implementation details of different univariate SRBF representations.
3.1 Mathematical Formulation
Univariate SRBFs are circularly axis-symmetric functions defined on Sm, where Sm is the unit hyper-sphere embedded in the (m+1)-dimensional Euclidean space Rm+1. Let ω and ξ denote two points on Sm, ω · ξ represent the dot product of ω and ξ, and φ ∈ R be the geodesic distance between ω and ξ, namely φ = arccos(ω · ξ). A univariate SRBF is defined as a function that depends on φ, and can be expressed in terms of expansions in Legendre polynomials as
G(cos φ) = G(ω · ξ) =
∞
X
n=0
`(G)n Pn(ω · ξ), (3.1)
1
Throughout this dissertation, the univariate SRBF is referred to as the original SRBF that was proposed by
Freeden et al. [44].
λ = 2 λ = 4 λ = 16
0 π
−π
φ (radians) 0.0
0.2 0.4 0.6 0.8 1.0
(a) (b)
(a) (b) (c)
Figure 3.1: (a) A two-dimensional plot of univariate Gaussian SRBFs with different band-widths. Three-dimensional plots of (b) a univariate Gaussian SRBF and the results by changing its (c) coefficient, (d) center, and (e) bandwidth.
where Pn(ω · ξ) ∈ R is the normalized Legendre polynomial of degree n so that Pn(1) = 1, the Legendre coefficientsof G(ω · ξ), namely`(G)n ∈ R ∞
n=0, satisfy `(G)n ≥ 0 for each n as well as
∞
P
n=0
`(G)n < ∞, and ξ is also known as the center of a univariate SRBF.
Two common examples of univariate SRBFs are the univariate Abel-Poisson SRBF kernel (Equation 3.2) and the univariate Gaussian SRBF kernel (Equation 3.3):
G(Abel)(ω · ξ|λ) = 1 − λ2
1 − 2λ(ω · ξ) + λ232 (0 < λ < 1), (3.2)
G(Gau)(ω · ξ|λ) = e−λeλ(ω·ξ), (3.3)
where λ ∈ R denotes the bandwidth parameter that controls the coverage of a univariate SRBF.
By choosing an appropriate value for λ, a univariate SRBF can be adaptive to the spatial varia-tion of local region. Therefore, univariate SRBFs not only overcome one of the major disadvan-tages of spherical harmonics, but also possess more degrees of freedom than zonal harmonics.
Different types of univariate SRBFs define distinctive distributions on Sm, and may exhibit diverse behaviors in various applications. The two-dimensional and three-dimensional plots of univariate Gaussian SRBFs are illustrated in Figure 3.1, and more examples of univariate SRBFs can be found in [44].
Similar to RBFs in the Euclidean space, given a set of distinct points Ξ = {ξj ∈ Sm}Jj=1,
Figure 3.2: A univariate spherical function in univariate SRBF expansions. From left to right, top to bottom: the result after incorporating the first univariate SRBF, the second one, and so forth. Right-most bottom: the final summed result with six univariate SRBFs.
which is called the SRBF center set, and another set of real scalars Λ = {λj ∈ R}Jj=1, which is named the SRBF bandwidth set, a univariate spherical function F (ω) ∈ R can be approximated with a linear combination of J univariate SRBFs as
F (ω) ≈ ˆF (ω) =
J
X
j=1
βjG(ω · ξj|λj), (3.4)
where βj ∈ R denotes the basis coefficient of the j-th univariate SRBF. Furthermore, ˆF (ω), whose dependence on the parameters of univariate SRBFs is omitted for notational simplicity, specifies the approximate univariate SRBF representation of F (ω). Univariate SRBFs thus be-have as reproducing kernels for interpolating F (ω) on Sm. An example of a univariate spherical function in univariate SRBF expansions is illustrated in Figure 3.2.
To learn the parameters in Equation 3.4, we can formulate an unconstrained least-squares optimization problem as follows:
min
{βj,ξj,λj}Jj=1
1 2
Z
Sm
F (ω) − ˆF (ω)2
dω. (3.5)
In Sections 3.3 and 3.4, we will further discuss this subject and present practical algorithms for solving Equation 3.5.