Recently at work I was chatting with a colleague, and the topic of energy conservation for specular reflections came up. This reminded me that I’ve been sitting on a blog post for a while about just this subject, so I thought it was time to finish it.
First of all, I’d like to start by looking at the standard diffuse reflection model. In games, the typical formula for calculating diffuse reflection from a particular light is:
Where Cd is the diffuse material color, Li is the light color, N is the normal, and L is the normalized direction to the light. What’s the problem with this? Well, it’s not energy conserving. In itself, this isn’t really a problem since we don’t calculate multiple bounces of light in games, so we’re not adding energy to the scene as light bounces around like would happen in a ray tracer. It’s a good starting point for discussion though.
As the name suggests, energy conservation is a restriction on the reflection model that requires that the total amount of reflected light cannot be more than the incoming light. It sounds sensible, but it’s often not practiced. A more formal way of stating this restriction is:
The function ρ represents the bidirectional reflection distribution function (BRDF), and could be anything from a simple Lambertian diffuse model, to a complicated microfacet model. Either way, the energy conservation restriction still stands.
Diffuse Energy Conservation
I’m going to show why the diffuse lighting equation above isn’t energy conserving, and how to make it so. Let’s start by replacing the BRDF with the constant diffuse color:
The incoming light direction is fixed here, and we are integrating over outgoing directions. Because of this, both Cd and Li are constant over the integral, and can be pulled outside. Also, the incoming light appears on both sides of the inequality, so we can divide by Li leaving:
This integral can be solved analytically. First of all, we need to rewrite it as a double integral of the two polar coordinates φ and θ:
The extra sin θ may seem a little bit confusing at first, but it’s necessary to take into account the smaller area towards the polar region. Using the double angle trigonometric identity, this is the same as:
Now we can start using some trigonometric integrals to integrate, firstly over θ:
This integral completely disappears down to nothing:
So now, integrate over φ:
This gives us the final inequality:
Assuming that we want to keep our diffuse material color in the range [0,1], all this says is that we need to divide it by π in order to remain energy conserving. Since this is just a constant scale, it might not be worth doing in a game if the only lighting model it uses is diffuse. Most games use a more sophisticated reflection model though, at least including specular reflections.
Specular Energy Conservation
The standard Blinn-Phong specular model is also not energy conserving. In fact, in some ways it is even worse than the diffuse model, because as you increase the specular power, you lose more and more energy. A manifestation of this problem can be that artists find it hard to get a really tight specular highlight.
Here’s an example of a sphere rendered at three different specular powers. Notice how much light there appears to be on the left image, and how it seems to have disappeared on the right, even though it’s supposed to be more focused. It’s at this point that artists start to ask to ramp the specular reflection color over one to compensate. This isn’t a good idea!
Instead of boosting the specular reflection color, you can switch to an energy conserving specular model. If you do this, the same spheres with the same specular powers now look like this:
The specular reflection on sphere on the left is actually dimmer than the non-conserving model. This is because the non-conserving specular model reflected too much light in this case. As the specular power increases, this time we compensate for the energy loss, and you get a really nice and tight hotspot on the right.
The typical Blinn-Phong reflection model is:
Somebody who knows more than me has worked out that to ensure energy conservation, the normalization factor is:
[Edit: Thanks to Fabian “ryg” Giesen for showing that this is actually the normalization factor for the regular Phong specular model, not Blinn-Phong. A commonly used normalization factor (according to Real-Time Rendering) for Blinn-Phong is shown below.]
So this makes the energy conserving function for Blinn-Phong specular reflection:
I’d love to know how this was derived, but I haven’t found anything so far that explains it. The integral for raising a cosine function to a power gets pretty hairy very quickly. All I know is that it appears to work. If anyone reading this knows why, then please let me know!
[Edit: Fabian “ryg” Giesen very kindly posted up a derivation of the normalization factor for both the regular Phong, and Blinn-Phong specular reflection models here. Interestingly he doesn’t come up with the exact same answer for the Blinn-Phong normalization factor as shown in Real-Time Rendering. Check out the comments below to find out more about this.]
Combined Diffuse and Specular
Once you have energy-conserving models for diffuse and specular, it’s easy to make sure that the combined model is also energy conserving. You just need to make sure that your diffuse and specular material colors don’t sum to more than one:
This means that if you want your material to have more specular, you may have to reduce the diffuse. Here’s a range of variations of the same material, going from 100% diffuse to 100% specular:
Is It Worth It For Games?
Clearly, the specular model benefits significantly from being energy conserving so I think most people would say that it’s worth it. Switching to a model where the diffuse and specular have to compete for energy might not be though, since it’s can be harder to tweak. I personally use this kind of model for my projects at home, but that’s because I don’t have artists to please.
One thing I do like about an energy conserving reflection model is that it enforces some kind of reasonable limits to the material reflections. This might help to make materials created by different people sit better together.