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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, tex2html_wrap_inline2740 , where parameters u and v are in the range of [0,1].

Let a representative of the basis functions be:

equation929

( tex2html_wrap_inline2748 substitutes tex2html_wrap_inline2750 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

displaymath2752

for each k,l is discussed. Since tex2html_wrap_inline2756 , it can be regarded as a function defined over the square tex2html_wrap_inline2758 . Using the equalities of surface integrals, and introducing the notation tex2html_wrap_inline2760 for surface element magnification, we get:

  equation941

Let us mirror the function tex2html_wrap_inline2762 onto coordinate system axes u and v, and repeat the resulting function having its domain in tex2html_wrap_inline2768 infinitely in both directions with period 2. Due to mirroring and periodic repetition, the final function tex2html_wrap_inline2770 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:

equation949

All the Fourier coefficients tex2html_wrap_inline2772 can be calculated by a single, two-dimensional FFT. (A D-dimensional FFT of N samples can be computed by taking tex2html_wrap_inline2778 number of one-dimensional FFTs [Nus82] [PFTV88].)

Since tex2html_wrap_inline2780 if tex2html_wrap_inline2782 , this Fourier series and the definition of the basis functions can be applied to equation 1.98, resulting in:

displaymath2784

equation970

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

displaymath2802

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

To calculate

displaymath2806

the Fourier transform of

displaymath2808

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 up previous index
Next: References Up: Higher order radiosity approximation Previous: Linear finite element techniques

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