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Radiosity

Brief Description

The radiosity method was one of the first widely used solutions to the global illumination problem. It is normally restricted to diffuse light transfer between surfaces, disregarding all other surface properties, such as specular reflection and refraction to name a few.

Unlike most other global illumination techniques, radiosity is view independent which means that although it can be costly to compute, once a scene's light distribution has been computed using the radiosity method, you can move your camera freely through a scene without having to recompute the illumination of the surfaces, making it particularly suitable for architectural walkthroughs and other scenarios with static environments.

Many people in the computer graphics community tend to use the term radiosity when what they are actually talking about is global illumination or even just diffuse interreflection between objects in a scene. However, in research terms the radiosity method is a particular technique calculating light transfer using finite element methods, which makes it fundamentally different from methods such as path tracing and photon mapping though the results produced by these different methods can sometimes be deceptively similar.

Note the term radiosity in computer graphics differs slightly from the same term used in physics.

The Radiosity Method

The classical radiosity method introduced by Goral et. al in 1984 assumes all surfaces, or patches, in a scene to have a constant radiosity, or radiant exitant power per unit area. This basically means the light energy leaving the surface does not differ across different points on the surface. This basic assumption is normally satisfied by making sure the surfaces are subdivided, or meshed, into elements that are small enough not to cause too big a negative impact on the final result (this will be described further in the meshing section below).

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The Radiosity Equation

The radiosity $B$ of each element $j$ is formulated

$B_j = E_j + \rho_j \sum_{i=1}^n{B_iF_{ji}} \;\;\; \textrm{for} \; j = 1 ... n\, \!$

where:

$E j$ is the emittance of the element.

$ρ j$ is the diffuse reflectance.

$B i$ is the radiosity of another surface element in the scene.

$F j i$ is the form factor describing fraction of the radiosity leaving j that strikes i.

$n$ is the number of surface elements in the scene.

Loose translation "The radiosity of each element in the scene is the sum of its own emittance and the fraction of radiosity arriving at the element from all other elements in the scene that is reflected back into the environment."

The radiosity equation above results in a system of n simultaneous equations, normally solved using iterative methods as described below.

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Form Factor Determination

In radiosity, the form factor $F i j$ describes the fraction of the light leaving surface patch i which eventually strikes surface patch j. The form factor between two patches depends on their sizes, orientations and the distance between them.
Consider the very simple example below. In the left-most example, because the surfaces are parallel, and very close together, a lot of the light leaving i is bound to strike j. In the middle example the surfaces are perpendicular, and so, only a very small amount of the light leaving i will arrive at j. Finally, the right-most image illustrates that when i is much larger than j, only a small amount of the light from i will end up at j, even though they are both close together and parallel.

Note that in the above image, only the light leaving the surface along the surface normal is illustrated. Obviously, as the surfaces are assumed to be diffuse, light is scattered in all directions.

Note: From the radiosity equation in the previous section, we know that n form factors need to be calculated for each element. Given we have a system of n simultaneous equations (one for each element in the scene), we end up with n x n form factors needing to be calculated for a scene with n elements. While this is the case in classical radiosity and research since then has allowed this number to be significanltly reduced, form factor calculation and storage is still an extremely expensive task. Consequently, it is very important to keep form factor calculation costs to a minimum.

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Mathematical Formulation

Formally, the form factor between two surface patches i and j with areas $A i$ and $A j$ as shown in this image (taken from the original radiosity paper by Goral et. al.) is forumlated:

$F_{ij} = F_{A_iA_j} = \frac{1}{A_i} \int_{A_i} \int_{A_j} \frac{\cos{\phi_i} \cos{\phi_j}}{\pi r^2}V_{ij}dA_jdA_i$

where:

$r$ is the distance between the two patches

$φ i$ and $φ j$ are the angles between the line connecting the two patches, and the surface normal at i and j respectively.
$V i j$ is the visibility term which has a value of 0 when the patches are occluded and a value of 1 when unoccluded.

If you are not a mathematician, the above equation can be slightly intimidating. It might be worth mentioning at this point that we never actually solve this equation, but include it simply to illustrate the complexity of the problem at hand. What the equation implies is that to calculate the form factor between two patches precisely, for every single point on the patch i we would have to consider how it relates to every single point on the patch j, in terms of distance between the points, the angle between the line connecting the two points and the surface normals of the patches at these points, and whether or not they are occluded by other surfaces.
Again, if you are not a mathematician, before trying to get your head around how this would be done, rest assured that it is a very complicated problem. In fact, there is no known way of solving it analytically. Instead, we approximate the problem by assuming that i is a patch with a differential area $d A i$ . What this means is that we pretend that its surface area is infinitisemally small, allowing us to represent it as a point in space and reformulate the form factor as:

$F_{ij} \approx F_{dA_iA_j} = \int_{A_j} \frac{\cos{\phi_i}\cos{\phi_j}}{} V_{ij}dA_j$

While this can be transformed into a form that can be solved analytically, it is very expensive and cannot account for occluding surfaces (i.e. the visibility term would be ignored). Instead we normally resort to using the methods described below.

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Numerical Methods for Determining Form Factors

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Stochastically Determined Form Factors

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Meshing

As mentioned above, one of the assumptions of the radiosity method is that each patch is small enough for the radiosity of that patch to be constant without introducing noticable visual artefacts. The process of subdividing the input patches into smaller elements for use with the above equation is known as meshing.
For as accurate and efficient a radiosity solution as possible, we want meshes to be subdivided where there is an abrupt change in lighting (such as along shadow boundaries). Additionally, while we want the elements to be small enough to allow for an accurate solution, we do not want them to be smaller than absolutely necessary, as this will increase $n$ in the above equation. From the form factor discussion above, we know that keeping the number of elements to a minimum is vitally important in terms of both time and space complexity.
In the very early days, meshing was done by the user before the scene was passed to the radiosity engine. However, the way a scene is meshed has far too much of an impact on the radiosity solution (and is, of course, far too tedious) to be done by hand.
There are various meshing strategies for radiosity, but they can normally be divided into a priori or predictive methods and a posteriori or adaptive methods. A priori methods include hierarchical meshing which subdivides a pair of patches when the form factor between them is higher than a certain threshold value, and discontinuity meshing which subdivides the mesh according to predicted light discontinuities in the scene. A posteriori, or adaptive meshing, methods work by calculating a "rough" radiosity solution, and then refining the mesh in areas where the radiosity varies more than a certain threshold.

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Solving for Radiosities

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Gathering

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Shooting

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Limitations

Advances

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Hierarchical Radiosity

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Clustering

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Importance

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Discontinuity Meshing

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Higher Order and Wavelet Radiosity

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Using Finite Element Methods for Non-Diffuse Surfaces

Extensions

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Final Gathering

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Hybrid Methods

Related Literature

Papers

Cindy M. Goral, Kenneth E. Torrance, Donald P. Greenberg, and Bennett Battaile. Modeling the interaction of light between diﬀuse surfaces. In SIGGRAPH ’84: Proceedings of the 11th annual conference on Computer graphics and interactive techniques, pages 213–222, New York, NY, USA, 1984. ACM Press.

Books

Websites

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Categories: Mathematics | Rendering | Technology

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