In contrast to previous cases, the application of harmonic functions does not require the subdivision of surfaces into planar polygons, but deals with the original geometry. This property makes it especially useful when the view-dependent rendering phase uses ray-tracing.

Suppose surface *A* is defined parametrically by a
position vector function, , where parameters *u* and *v* are in
the range of [0,1].

Let a representative of the basis functions be:

( substitutes for notational simplicity). Note that the basis functions have two indices, hence the sums should also be replaced by double summation in equation 1.79. Examining the basis functions carefully, we can see that the goal is the calculation of the Fourier series of the radiosity distribution.

In contrast to the finite element method, the basis functions are now
non-zero almost everywhere in the domain, so they can approximate the
radiosity distribution in a wider range. For that reason, approaches
applying this kind of basis function are called **global element
methods**.

In the radiosity method the most time consuming step is the evaluation of the integrals appearing as coefficients of the linear equation system (equation 1.79). By the application of cosine functions, however, the computational time can be reduced significantly, because of the orthogonal properties of the trigonometric functions, and also by taking advantage of effective algorithms, such as Fast Fourier Transform (FFT).

In order to illustrate the idea, the calculation of

for each *k*,*l* is discussed.
Since , it can be regarded as a function defined
over the square . Using the equalities of surface integrals, and
introducing the notation
for surface element magnification, we get:

Let us mirror the function
onto coordinate system axes *u* and *v*, and repeat the resulting
function having its domain in
infinitely in both directions with period 2. Due to mirroring and periodic
repetition, the final function will be even and periodic with
period 2 in both
directions. According to the theory of the Fourier series, the function can be
approximated by the following sum:

All the Fourier coefficients can be calculated by a
single, two-dimensional FFT. (A *D*-dimensional FFT of *N* samples
can be computed by taking
number of one-dimensional FFTs [Nus82] [PFTV88].)

Since if , this Fourier series and the definition of the basis functions can be applied to equation 1.98, resulting in:

Consequently, the integral can be calculated in closed form, having replaced the original function by Fourier series. Similar methods can be used to evaluate the other integrals. In order to compute

*J*(*u*,*v*) must be Fast Fourier Transformed.

To calculate

the Fourier transform of

is needed. Unfortunately the latter requires a 4D FFT which involves many
operations. Nevertheless, this transform can be realized by two
two-dimensional FFTs if *g*(*p*,*p*') can be assumed to be
nearly independent of either *p* or *p*', or it can be
approximated by a product form of *p* and *p*' independent
functions.

Finally, it should be mentioned that other **global function bases** can also be useful. For
example, Chebyshev polynomials are effective in approximation, and similar
techniques to FFT can be developed for their computation.

Mon Oct 21 14:07:41 METDST 1996