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## Global element approach - harmonic functions

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.    Next: References Up: Higher order radiosity approximation Previous: Linear finite element techniques

Szirmay-Kalos Laszlo
Mon Oct 21 14:07:41 METDST 1996