Theory: Form Factors

Read CW[10] chapter 4.

We have a system of equations defined as

$$\widehat{L}= (I - \widehat{M}) \widehat{E}$$

where

$$\widehat{M}_{ij} = \left< \mbox{$\widetilde{B_i}$ } \, , \, MB_j \right> .$$

For piecewise constant basis functions and constant reflectivity over the surface, this becomes

$$\widehat{M}_{ij} \int_{\cal E} \mbox{$\widetilde{B_i}$ } (y) \rho(y) \int_{\cal E}... ...\cos(\theta_{y})}{ \left\vert \left\vert x - y \right\vert \right\vert ^2}dx dy$$ = (15)

$$\frac{1}{A_i} \rho_i \int_{{\cal E}_i} \int_{{\cal E}_j} {\rm vis... ...\cos(\theta_{y})}{ \left\vert \left\vert x - y \right\vert \right\vert ^2}dx dy$$ = (16)

where A_i is the area of patch (basis function) i and ${\cal E}_i$ is patch i. We can remove the reflectance term and get an expression that is a function only of geometry,

$$F_{ij} = \frac{1}{A_i} \int_{{\cal E}_i} \int_{{\cal E}_j} {\... ... \left\vert \left\vert x - y \right\vert \right\vert ^2}dx dy.$$

This is almost what is called the form factor (or configuration factor, shape factor, or geometry factor). This expression will sum to $\pi$ over the hemisphere. That's because it is simply the cosine-weighted solid angle of all visible surfaces. The reflectance ($\rho_i$) has a maximum value of $\frac{1}{\pi}$ for a completely reflective surface. The two factors of $\pi$ cancel out, giving what we expect. This is the useful framework for the mathematics, as well as when considering finite element methods in non-diffuse environments. It assumes we are working with radiance and BRDF's. The heat transfer literature and all the early radiosity literature assumes ideal diffuse surfaces. They work with a energy quantity that is directionless ($\frac{W}{m^2}$) and with reflectances, not BRDF's. The directionless quantity is called radiosity. They move the $\pi$ out of the BRDF and into the form factor, resulting in a $\pi$ in the denominator, and form factors that sum to 1 over the hemisphere. They then define the form factor as the fraction of energy leaving surface i that arrives at surface j.

There is a useful identity for form factors (or our version) called reciprocity. This is written

A_i F_ij = A_j F_ji.

Nusselt Analog

The form factor from a differential area to an object is proportional to the projection of the object onto the unit hemisphere, and then the projection from the unit hemisphere down onto the unit circle. Divide by the area of the unit circle to get the form factor, or leave as is to get the cosine-weighted solid angle.

This drove the most popular way of computing form factors back in the late eighties and early nineties, the hemicube[9]. If geometry could be projected onto a sphere and then down onto a circle easily, we could compute form factors easily. It can't be. Computer graphics is good at projecting geometry onto planes, so why not project the geometry onto a planar representation of a hemisphere (a hemicube). Chop the sides of the hemicube up into many elements (pixel) and use it as an item buffer. Compute the (simple) form factors between the pixel and the receiving point and add that to the form factor for the object projected into the pixel. Works fairly quickly, but can miss small objects and can have aliasing artifacts.

A variation on the hemicube is the single-plane method [31]. This replaces the cube by a plane very near to the vertex. It misses things along the edges, but these contribute very little to the result. It's roughly 5 times faster as it only needs to project onto a single plane, not 5.

Another way this is used is the Monte Carlo approach by Malley [25]. Send rays out into the world and the form factor to an object is the number of rays that hit the object divided by the number of rays shot. The rays are uniformly distributed on the circle, then projected up onto the hemisphere. Can easily miss small objects.

Special Situations

First some generally useful ones:

Point to Point

The differential area to differential area term is used as the kernel for numeric integration.

$$F_{dA_i-dA_j} = {\rm vis}(x,y)\frac{\cos(\theta_x)\cos(\theta_{y})}{ \left\vert \left\vert x - y \right\vert \right\vert ^2}$$

Point to Polygon

By doing a contour integration this can be done relatively easily for unoccluded polygons. It's exactness makes it very useful for polygonal environments.

$$F_{dA_i-A_j} = \int_{A_j} {\rm vis}(x,y) \frac{\cos(\theta_x)... ...}{ \left\vert \left\vert x - y \right\vert \right\vert ^2} dy.$$

Now some that are useful only in special situations. These are often used to check results generated in different ways. It's good to know what sorts of geometries are available.

Polygon to Polygon

This was tried by Lambert in 1760 and finally done for arbitrary unoccluded polygons by Schroeder and Hanrahan[30] in 1993. It's complexity makes it completely unusable for all but checking other results.

Special to Special

The heat transfer literature is filled with expressions for the form factor between various simple geometries. These form factors are usually not used because the cases where they work are very constrained (two parallel identically sized aligned rectangles, perpendicular rectangles, differential form factor to a cow.)

Constant Radiosity Assumption and Visibility

As is clear from how we defined them, expressions for form factors assume that both the source and the receiver are constant. This is an assuption that has been made by the heat transfer community and carried over into graphics. Although this is fundamentally flawed, in practice it works fairly well.

All analytic methods assume there is no occlusion between the source and the receiver. In practice this is an unreasonable assumption. Generally, the solution of this is to take the visibility term outside the integral, multiplying the resulting unoccluded form factor by the average visibility between the two objects.

Why do these two approximations work? Because in practice, the source and the receiver are well separated. When they are well separated, the differential form factor is roughly constant between all pairs of points. If the differential form factor were constant, then these approximations would be accurate. The error in the approximation has to do with how far the various terms are from constants.

More Reading

• Hemicube (Cohen, Max) [9,26]
• Hemiplane (Sillion) [31]
• Malley's Method [25]
• Analytic Form Factors Point to disc [36],
• Point to Poly [5]
• Polygon to Polygon [30]
• Shaft culling[17]
• Blocker lists (Teller Hanrahan) [34]