Self Shadow

Dec 31, 2011 - Rendering

Righting Wrap (Part 1)

A while back, Steve McAuley and I were discussing physically based rendering miscellanea over a quiet pint – hardly a stretch of the imagination, since we’re English 3D programmers after all! Anyway, it turned out that we both had plans to write up a few thoughts in relation to wrap shading, and, following some gentle arm-twisting, Steve has posted his. I suggest that you go and read that first if you haven’t already, then return here for a continuation of the subject.

Bad Wrap

Wrap shading has its uses when more accurate techniques are too expensive, or simply to achieve a certain aesthetic, but common models [1][2] have some deficiencies out of the box. Neither of these is energy conserving and they don’t really play well with shadows either. On top of that, Valve’s Half Lambert model [2] has a fixed amount of wrap, so it can’t be tuned to suit different materials (or, perhaps, to limit shadow oddities). I’ll come back to the point about flexibility in part 2, but first I’d like to discuss another factor that’s easily overlooked: environmental lighting.

If you’re set on using some form of wrap shading, then it’s not just a matter of applying it to your standard direct sources – directional, point and spot lights, for instance – it ought to be carried through to environmental lighting as well! Naturally, the importance of this depends on how strong and directional your secondary lighting is; obviously if you’re only using constant ambient then there’s no problem, but these days it’s fairly common to encode indirect lighting in Spherical Harmonics (SH) [3] and perhaps some additional lights as well. Fortunately, wrap shading in the context of SH lighting is easy, and much like energy conservation it’s a relatively cheap or free addition, so it’s worth considering even if the results prove to be subtle.

Looking After the Environment

So, how do we accomplish this? Well, that’s best explained with a quick recap. If you recall, for diffuse SH lighting, we first project the lighting environment, $f$, into SH:

$$f_{l}^{m} = \int f(s)\,y_{l}^{m}(s)\,\mathrm{d}s.$$

(Of course, in practice, this is commonly performed offline as a numerical integration over a cube map.)

We then convolve this with the SH-projected cosine lobe, $h$, like so:

$$c_{l}^{m} = \sqrt{\frac{4\pi}{2l+1}}h_{l}^{0}f_{l}^{m},\quad \text{where } h_{l}^{0} = \int \mathrm{cos}(s)\,y_{l}^{0}(s)\,\mathrm{d}s.$$

Next, we can evaluate the lighting (more specifically, irradiance) for a given surface direction, $s$:

$$E(s) = \sum_{i=0}^n\sum_{m=-l}^{l}c_l^m y_l^m(s).$$

Finally, a division by $\pi$ gives us outgoing (or exit) radiance. Personally, I find it convenient to roll these extra terms into $h$ itself. The nice thing about this is that the convolution kernel then boils down (via analytical integration) to easy to remember values for the first three SH bands:

$$\begin{array}{lcl} \hat{h}_{l}^{0} &=& \frac{1}{\pi}\sqrt{\frac{4\pi}{2l+1}}h_{l}^{0}, \\ \hat{h} &=& \left[1, \frac{2}{3}, \frac{1}{4}, \cdots \right]. \end{array}$$

For further details, you can find a complete and approachable account in [4].

Now, back to wrap: adjusting things for our shading model of choice is simply a matter of replacing $\hat{h}$. Let’s try this for the simple wrap model from Green [1] that Steve already discussed:

$$\frac{1}{\pi}\int_{0}^{2\pi} \int_{0}^{\alpha} \frac{\mathrm{cos}(\theta)+w}{1+w}\mathrm{sin}(\theta)\,\mathrm{d}\theta\mathrm{d}\phi, \quad \text{where }\alpha=\mathrm{cos}^{-1}(-w).$$

From Steve’s post, we know that we need an additional normalisation factor of $1 + w$ for energy conservation, so the full formula for our new convolution, which I’ll call $\hat{g}$, is:

$$ \hat{g}_{l}^{0} = \frac{1}{1 + w}\sqrt{\frac{4\pi}{2l+1}}\frac{1}{\pi}\int_{0}^{2\pi} \int_{0}^{\alpha} \frac{\mathrm{cos}(\theta)+w}{1+w}y_{l}^{0}(s)\,\mathrm{sin}(\theta)\,\mathrm{d}\theta\mathrm{d}\phi.$$

You can go through a similar process of analytical integration as Steve did, only now with the additional SH basis terms $y_{l}^{0}$, or if you’re lazy like me, you can throw the formula at a package like Mathematica. Either way, once you’re done, you’ll arrive at the following (or something equivalent):

$$\hat{g} = \left[ 1, \frac{1}{3}(2 - w), \frac{1}{4}(1 - w)^2, \cdots \right].$$

We can clearly see that this reduces to the cosine convolution kernel, $\hat{h}$, when $w = 0$. In visual terms, the effect of changing $w$ is evident with a single directional light, as you would expect:

Figure 1: Variable wrap shading (0, 0.5, 1) with a single directional light in SH.

Figure 1: Variable wrap shading (0, 0.5, 1) with a single directional light in SH.

On the other hand, the difference is a lot subtler with a more uniform lighting environment:

Figure 2: Variable wrap shading (0, 0.5, 1) with a general SH environment (Grace Cathedral).

Figure 2: Variable wrap shading (0, 0.5, 1) with a general SH environment (Grace Cathedral).

It’s a Wrap?

Okay, confession time: this post wasn’t just about wrap shading, since it also serves as a foundation for future posts. For instance, part 2 will conveniently segue into shader optimisations, which is a topic I’ve been planning to write – or, perhaps more accurately, rant – about in general, and I’ll also be returning to Spherical Harmonics down the line.


[1] Green, S., “Real-Time Approximations to Subsurface Scattering”, GPU Gems, 2004.
[2] Mitchell, J., McTaggart, G., Green, C., “Shading in Valve’s Source Engine”, Advanced Real-Time Rendering in 3D Graphics and Games, SIGGRAPH Course, 2006.
[3] Green, R., “Spherical Harmonic Lighting: The Gritty Details”, GDC 2003.
[4] Sloan, P.-P, “Efficient Evaluation of Irradiance Environment Maps”, ShaderX^2: Shader Programming Tips and Tricks with DirectX 9.0, 2003.

Tags: physically based shading spherical harmonics wrap shading

Perpendicular Possibilities Righting Wrap (Part 2)