Approximations for the distribution of microflake normals

Abstract

Scenes in computer animation can have extreme complexity, especially when high resolution objects are placed in the distance and occupy only a few pixels. A useful technique for level of detail in these cases is to use a sparse voxel octree containing both hard surfaces and a participating medium consisting of microflakes. In this paper, we discuss three different methods for approximating the distribution of normals of the microflakes, which is needed to compute extinction, inscattering of attenuated direct illumination, and multiple scattering in the participating medium. Specifically, we consider (a) k means approximation with k weighted representatives, (b) expansion in spherical harmonics, and (c) the distribution of the normals of a specific ellipsoid. We compare their image quality, data size, and computation time.

Appendix

In this appendix, we give some formulas for the spherical harmonics calculations discussed above. Good references for the mathematics of spherical harmonics as applied to computer graphics are [28] and [5]. Sloan [28] gives simple formulas for the $Y_{lm}(\omega )$ in the form of polynomials of degree l in the x, y, and z coordinates of the unit vector $\omega$, which we used in our implementation.

To compute the extinction integral in Eq. (3) using an $h(\omega )$ of the form of Eq. (1), we expand the nonnegatively clamped cosine of $\theta$, $A(\theta )$, into spherical harmonics. Since this function is independent of $\phi$, all the coefficients with $m \ne 0$ are zero so we get just the “zonal” harmonics expansion

$$\begin{aligned} A(\theta , \phi ) \approx \sum _{l = 0}^L A_{l0}Y_{l0}(\theta , \phi ) \end{aligned}$$

(12)

where, by Eq. (2),

$$\begin{aligned} A_{l0} = 2\pi \int _0^{\pi /2} Y_{l0} (\theta , \phi ) \text { cos}\theta \text { sin}\theta \,\mathrm{d}\theta \,\mathrm{d}\phi . \end{aligned}$$

(13)

Ramamoorthi and Hanrahan [26] (please see the correction of their Eq. (19) at the bottom of the web page http://cseweb.ucsd.edu/ravir/papers/invlamb/) give the formula for the coefficients $A_{l0}$, which are zero for odd $l > 1$, and decrease rapidly for increasing even l. We have used only the first few values $A_{00} = \sqrt{\pi }/2$, $A_{10} = \sqrt{\pi /3}$, $A_{20} = \sqrt{5\pi }/8$, $A_{30} = 0$, and $A_{40} = -\sqrt{\pi }/16$.

This zonal harmonic expansion is only useful for evaluating Eq. (3) if $\omega _o$ is at the north pole of the unit sphere. For other directions of $\omega _o$, one must rotate this zonal harmonic expansion so that its axis of symmetry is along the direction $\omega _o$. This is simpler for zonal harmonics than for general spherical harmonics. According to [26], the result of rotating the expansion in equation (12) to make the axis of symmetry lie in the direction $\omega _o = (\theta _o, \phi _o)$ is

$$\begin{aligned}&R^{\theta _o \phi _o}A(\theta ,\phi ) \nonumber \\&\quad \approx \sum _{l = 0}^L \sum _{m = -l}^l Y_{lm} (\theta _o, \phi _o) \sqrt{\frac{4\pi }{2l+1}}A_{l0} Y_{lm} (\theta ,\phi ). \end{aligned}$$

(14)

Thus, if Eq. (1) is the spherical harmonic expansion of the microflake normal density $h(\omega )$, by equation (3) and the orthonormality of the SH basis functions,

$$\begin{aligned} \sigma _t(x, \theta _o , \phi _o) \approx \sum _{l = 0}^L \sum _{m = -l}^l Y_{lm} (\theta _o, \phi _o) \sqrt{\frac{4\pi }{2l+1}}A_{l0} h_{lm}. \end{aligned}$$

(15)

The inscattering integral in Eq. (4) contains the clamped cosine $\langle \omega _n,\omega _i\rangle$ as well as $\langle \omega _n,\omega _o\rangle$, so if $\langle \omega _n,\omega _i\rangle$ is also expanded in spherical harmonics as in Eq. (14), we get, for the perfectly diffuse reflection of a small light source from direction $\omega _i$ of radiance times solid angle equal to I,

$$\begin{aligned}&S(\omega _o) \nonumber \\&\quad = \int _\Omega \int _\Omega h(\omega _n) \frac{\alpha }{\pi } \langle \omega _n,\omega _i\rangle \langle \omega _n,\omega _o\rangle L(\omega _i)\mathrm{d}\omega _n \mathrm{d}\omega _i \nonumber \\&\quad = \frac{I\alpha }{\pi }\int _0^{2\pi }\int _0^{\pi } h(\theta ,\phi )R^{\theta _i \phi _i}A(\theta ,\phi )R^{\theta _o \phi _o}A(\theta ,\phi ) \nonumber \\&\quad \text { sin}\theta \mathrm{d}\theta \mathrm{d}\phi \nonumber \\&\quad \approx \frac{I\alpha }{\pi }\int _0^{2\pi }\int _0^{\pi } \sum _{l = 0}^L \sum _{m = -l}^l h_{lm}Y_{lm}(\theta , \phi ) \nonumber \\&\quad \sum _{l' = 0}^L \sum _{m' = -l'}^{l'} Y_{l'm'} (\theta _o, \phi _o) \sqrt{\frac{4\pi }{2l'+1}}A_{l'0} Y_{l'm'} (\theta ,\phi )\sum _{l'' = 0}^L \nonumber \\&\quad \sum _{m'' = -l''}^{l''} Y_{l''m''} (\theta _i, \phi _i) \sqrt{\frac{4\pi }{2l''+1}}A_{l''0} Y_{l''m''} (\theta ,\phi )\text { sin}\theta \mathrm{d}\theta \mathrm{d}\phi \nonumber \\&\quad = \frac{I\alpha }{\pi }\sum _{l = 0}^L \sum _{m = -l}^l h_{lm} \sum _{l' = 0}^L \sum _{m' = -l'}^{l'} Y_{l'm'}(\theta _o, \phi _o) \sqrt{\frac{4\pi }{2l'+1}} \nonumber \\&\quad A_{l'0} \sum _{l'' = 0}^L \sum _{m' = -l''}^{l''} Y_{l''m''} (\theta _i, \phi _i) \sqrt{\frac{4\pi }{2l''+1}}A_{l''0}T_{lml'm'l''m''}\nonumber \\ \end{aligned}$$

(16)

where $T_{lml'm'l''m''}$ is the “triple product integral”

$$\begin{aligned}&T_{lml'm'l''m''} \nonumber \\&\quad = \int _0^{2\pi }\int _0^{\pi }Y_{lm}(\theta ,\phi )Y_{l'm'} (\theta ,\phi )Y_{l''m''}(\theta ,\phi ) \ \text { sin}\theta \,d\theta \,d\phi .\nonumber \\ \end{aligned}$$

(17)

For order $L = 4$, the potential number of triple product integrals is $25^3$ = 15,625, not accounting for symmetries in the indices, but only 1,158 of them are nonzero, and these are precomputed and stored in a sparse table. In fact, since $A_{30} = 0$, only 605 terms are actually included in the sum in Eq. (16), and if the microflakes are double-sided, and we approximate $|\mathrm {cos}(\theta )|$ by $A(\theta , \phi ) + A(-\theta , \phi )$, all terms with $l'$ = 1 or $l''$ = 1 also disappear, and only 585 terms remain.

Note that if $\omega _o = (\theta _o, \phi _o)$ is fixed, grouping the terms and factors from $h(\omega _n) \langle \omega _n,\omega _o\rangle$ in the last form in Eq. (16) into the large parentheses below gives a spherical harmonic expansion in the direction variable $\omega _i = (\theta _i, \phi _i)$.

$$\begin{aligned}&S(\omega _o) \nonumber \\&\quad = \sum _{l'' = 0}^L \sum _{m'' = -l''}^{l''} \bigg (\frac{I\alpha }{\pi }\sum _{l = 0}^L \sum _{m = -l}^l h_{lm} \sum _{l' = 0}^L \sum _{m' = -l'}^{l'} Y_{l'm'}(\theta _o, \phi _o) \nonumber \\&\qquad \times \sqrt{\frac{4\pi }{2l'+1}}A_{l'0} \sqrt{\frac{4\pi }{2l''+1}}A_{l''0} T_{lml'm'l''m''}\bigg )\; Y_{l''m''} (\theta _i, \phi _i)\nonumber \\ \end{aligned}$$

(18)

Thus, it can be importance-sampled for path tracing using the techniques of [14], or by rejection sampling. Note also from this form that once the basis function coefficients in the large parentheses have been computed, the inscattering of illumination from multiple light sources can be computed with only $(L+1)^2$ multiplications and adds for each. For a light source “at infinity” with constant $\omega _i$, a different grouping of the terms involving instead the factors $\langle \omega _n,\omega _i\rangle \langle \omega _n,\omega _o\rangle$ can be computed once per viewing ray that hits microflakes, allowing only $(L+1)^2$ multiplications per octree cell (9.2 s for the image in Fig. 9 instead of 21.1). For finite distance light sources, using more storage, the inscattering and the extinction coefficient can be computed once per voxel as first needed and saved for subsequent viewing rays crossing the same voxel (8.4 s for the image in Fig. 9).