Topic : Radiosity in English
Author : Paul Nettle
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Radiosity in English
Paul Nettle


I've found many people shy away from radiosity simply because of the mystique surrounding it. I’ll admit that I was a bit intimidated by the topic when I decided to tackle it. But the truth be known, it boils down to some pretty simple techniques, most of which are very common. As a matter of fact, if you’ve got access to a rendering architecture that supports z-buffering, you’ve got 90% of a basic radiosity processor that can produce some excellent results.

I understand that "Aggravated Nosebleed" (the source of this Q&A entry) has some basic knowledge of radiosity, but I’d like to take this opportunity to cover the basics from the very beginning. Hopefully he/she as well as some other readers will gain enough of a fundamental understanding to help ease the learning curve from some of the more notable references (which I will list later.)

In the Beginning

The introduction of Radiosity came in 1984 from Cornell University in a paper titled "Modelling the Interaction of Light Between Diffuse Surfaces" written by Goral, Torrance & Greenberg. The idea was to simulate energy (light) transference from diffuse surfaces. Diffuse surfaces are surfaces that reflect light equally in all directions - the opposite of a shiny surface.

This result was considered "view independent." This simply meant that the illumination on a surface looked the same no matter what angle you were viewing it from. For the sake of clarity, an example of the opposite ("view dependent") would be a reflective surface. Reflective surfaces are view dependent because the specular highlights would appear at a different position on the surface based on the angle at which the surface was viewed.

This view independence was nice, especially considering the cost (in processor power & running time) of radiosity processing. The illumination could be calculated once and the scene could then be rendered very quickly from any angle. This translates directly into many of today’s modern "first-person shooter" games.

The First Approach

Consider a simple room with only four walls, a ceiling and a floor. Can you see it in your mind’s eye? You better not be able to; I haven’t specified a light source yet. :-) In radiosity, light sources aren’t your typical point or spot light sources. In radiosity, light is emitted from surfaces. So, rather than adding a surface for a light source, lets just make the entire ceiling an "area light source." In the real world, this would translate to a cubic room where the ceiling was a huge panel of fluorescent lights behind a huge diffuse reflector (those smoked-glass looking things that spread light out.)

This example is a simple one since every surface can see every other surface. In other words, there’s nothing to block the light from reaching any surface (i.e. no shadows.)

Each surface has two values associated with it. An amount of how brightly it is illuminated (its illumination) and how much of a surplus of energy it has (its radiative energy.) To start with, only the ceiling will have any radiative energy and all other surfaces will have no radiative energy or illumination.

What we need to do now is calculate the interaction of energy from every surface to every other surface. This is an n^2 problem since we need to calculate this interaction from each surface to every other surface in the scene. This can be calculated based on their geometrical relationships (distance between surfaces, relative orientation, relative area, etc.) The math that calculates this relationship results in a single value. This value is called a "form factor."

The attentive readers (the one’s that are still awake) might have already guessed that there are only (n^2)/2 individual form factors since the relationship between surfaces 5&6 is the same as the relationship between 6&5. However, this is not true since relative area is taken into consideration.

We can calculate all the form factors in a scene and store them in a grid that is n elements wide by n elements tall. This grid is referred to as the "radiosity matrix" and it works just like a 2-dimensional table. Each element in this matrix contains a form factor for the interaction from the surface indexed by the column and the surface indexed by the row.

Remember how I said that there are n^2 interactions and not (n^2)/2? This is because each form factor is like a diode in that it only handles energy going in one direction: from a "source surface" to a "destination surface." In this case, we’ll say that the source surfaces are index by columns and destination surfaces are indexed by rows. Source surfaces will emit their energy to the destination surfaces.

Now lets solve the matrix. To do this, we simply visit each column (source) in the matrix and emit energy to each row (destination) in that column. When we do this, we’ll be placing some of that radiated energy (from the source) in the illumination value for the destination. But these surfaces are reflectors, which means they’re going to reflect SOME energy back into the scene. Based on the surface’s reflectivity, we’ll add a little bit of energy to the destination’s radiative energy. This radiative energy will eventually make its way back into the scene (i.e. to the other surfaces) as we progress through the matrix.

If the destination is a perfect reflector (i.e. it reflects every single bit of energy it receives - a mirror) then there will be no energy stored in the destination’s illumination, it would all go to its radiative energy. The inverse is also true: a perfectly black surface might not reflect any energy back into the scene, absorbing it all, so every bit of energy it receives is stored in its illumination value. If you’re starting to think that we’re making a black surface white, we’re not. Remember, we’re dealing with light, so the color of a surface is ultimately multiplied by its illumination. In the case of the perfectly black surface, the surface remains visually black.

Once we’ve gone through the matrix once, we do it all over again. This is necessary because we we’re storing some energy as illumination, and some as radiative energy. Now it’s time to go through the matrix again and start distributing that reflected radiative energy.

We’ll go through this matrix over and over again until the total amount of radiative energy for all surfaces is relatively small.

The Next Step

If you made it this far without getting lost, you’re in the home stretch. There’s still a lot we haven’t covered yet, so let’s move on. I’ll start with a few shortcomings of the basic radiosity matrix as I’ve described it thus far and common solutions to these issues.

Our surfaces have only one illumination value for the entire surface, so there is no change in illumination across a single surface. To solve this problem, we can simply subdivide each surface into a series smaller polygons called "patches." If you do this, you simply treat each patch as its own surface as a replacement for the original surface. Your matrix will grow to the number of patches in the scene squared.

This brings us to our next issue: the matrix can be quite large (especially if you subdivide into a number of patches) If the scene is very simple (say, a meager 1,000 polygons) then our illustrious matrix will be pretty big (1,000,000 elements.) If you've subdivided each of those surfaces to a meager 8x8 grid of patches per surface, then we're talking about 4,096,000,000 total elements in our matrix (8*8 = 64 patches per surface, 64*1000 = 64000 patches per scene, 64000*64000 is = 4,096,000,000 total matrix elements.) This is pretty tough for any computer to swallow.

Before I discuss the solutions to this ever-increasing matrix, let’s talk about a related issue: a matrix of this magnitude would take a long time to solve. Especially considering the fact that we’ll have to solve it multiple times. If a mistake was made in the modeled scene, wouldn’t it be nice to know this sooner rather than later?

Progressive Refinement

In 1988, Cohen, Chen, Wallace & Greenberg published a paper called "A Progressive Refinement Approach to Fast Radiosity Image Generation." This paper described a new way of solving radiosity. It was quite clever in that it reordered the way things were done.

In the matrix method, illumination was gathered by each destination element from its source element. Ironically, this is called "gathering." The progressive refinement approach reversed this and defined (the other incredibly ironic term) "shooting."

The basic idea behind progressive refinement starts by finding the surface with the most energy to contribute to the scene (i.e. has the highest amount of radiative energy.) This surface would then iterate through all other surfaces, distributing its energy along the way. After this process was completed, the image was then rendered for the user, and the process began again, finding the surface

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