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Next: Piecewise constant radiosity approximation Up: RADIOSITY METHOD Previous: Combination of radiosity and

Higher order radiosity approximation

 

The original radiosity method is based on finite element techniques. In other words, the radiosity distribution is searched in a piecewise constant function form, reducing the original problem to the calculation of the values of the steps.

The idea of piecewise constant approximation is theoretically simple and easy to accomplish, but an accurate solution would require a large number of steps, making the solution of the linear equation difficult. Besides, the constant approximation can introduce unexpected artifacts in the picture even if it is softened by Gouraud shading.  

This section addresses this problem by applying a variational method for the solution of the integral equation [SK93].

The variational solution consists of the following steps [Mih70]:  

  1. It establishes a functional which is extreme for a function (radiosity distribution) if and only if the function satisfies the original integral equation (the basic radiosity equation).
  2. It generates the extreme solution of the functional by  Ritz's method, that is, it approximates the function to be found by a function series, where the coefficients are unknown parameters, and the extremum is calculated by making the partial derivatives of the functional (which is a function of the unknown coefficients) equal to zero. This results in a linear equation which is solved for the coefficients defining the radiosity distribution function.

Note the similarities between the second step and the original radiosity method. The proposed variational method can, in fact, be regarded as a generalization of the finite element method, and, as we shall see, it contains that method if the basis functions of the function series are selected as piecewise constant functions being equal to zero except for a small portion of the surfaces. Nevertheless, we are not restricted to these basis functions, and can select other function bases, which can approximate the radiosity distribution more accurately and by fewer basis functions, resulting in a better solution and requiring the calculation of a significantly smaller linear equation.

  
Figure: Geometry of the radiosity calculation

Let the diffuse coefficient be tex2html_wrap_inline2514 at point p and the visibility indicator between points p and p' be H(p,p'). Using the notations of figure 1.9, and denoting the radiosity and emission at point p by B(p) and E(p) respectively, the basic radiosity equation is:

equation695

where f(p,p') is the point-to-point form factor:

  equation698

Dividing both sides by dA, the radiosity equation is then:

  equation704

Let us define a linear operator tex2html_wrap_inline2534 :

  equation707

Then the radiosity equation can also be written as follows:

  equation711

The solution of the radiosity problem means to find a function B satisfying this equation. The domain of possible functions can obviously be restricted to functions whose square has finite integration over surface A. This function space is usually called tex2html_wrap_inline2540 space where the scalar product is defined as:

equation715

If tex2html_wrap_inline2534 were a symmetric and positive operator, that is, for any u,v in tex2html_wrap_inline2540 ,

equation717

were an identity and

equation721

then according to the minimal theorem of quadratic functionals [Ode76] the solution of equation 1.62 could also be found as the stationary point of the following functional:

  equation729

Note that tex2html_wrap_inline2548 makes no difference in the stationary point, since it does not depend on B, but it simplifies the resulting formula.

To prove that if and only if some tex2html_wrap_inline2552 satisfies

equation733

for a symmetric and positive operator tex2html_wrap_inline2534 , then tex2html_wrap_inline2552 is extreme for the assumption that tex2html_wrap_inline2534 is positive and symmetric can be used:

displaymath2560

displaymath2562

displaymath2564

equation750

Since only the term tex2html_wrap_inline2566 depends on B and this term is minimal if and only if tex2html_wrap_inline2570 is zero due to the assumption that tex2html_wrap_inline2534 is positive, therefore the functional is really extreme for that tex2html_wrap_inline2552 which satisfies equation 1.62.

Unfortunately tex2html_wrap_inline2534 is not symmetric in its original form (equation 1.61) due to the asymmetry of the radiosity equation which depends on tex2html_wrap_inline2514 but not on tex2html_wrap_inline2580 . One possible approach to this problem is the subdivision of surfaces into finite patches having constant diffuse coefficients, and working with multi-variate functionals, but this results in a significant computational overhead.

Now another solution is proposed that eliminates the asymmetry by calculating B(p) indirectly through the generation of tex2html_wrap_inline2584 . In order to do this, both sides of the radiosity equation are divided by tex2html_wrap_inline2586 :

equation761

Let us define tex2html_wrap_inline2588 , tex2html_wrap_inline2590 and g(p,p') by the following formulae

  equation771

Using these definitions, we get the following form of the original radiosity equation:

  equation780

Since g(p,p') = g(p',p), this integral equation is defined by a symmetric linear operator tex2html_wrap_inline2596 :

equation784

As can easily be proven, operator tex2html_wrap_inline2596 is not only symmetric but also positive taking into account that for physically correct models:

  equation788

This means that the solution of the modified radiosity equation is equivalent to finding the stationary point of the following functional:

equation792

This extreme property of functional I can also be proven by generating the functional's first variation and making it equal to zero:

equation795

Using elementary derivation rules and taking into account the following symmetry relation:

equation801

the formula of the first variation is transformed to:

equation803

The term closed in brackets should be zero to make the expression zero for any tex2html_wrap_inline2604 variation. That is exactly the original radiosity equation, hence finding the stationary point of functional I is really equivalent to solving integral equation 1.71.

In order to find the extremum of functional tex2html_wrap_inline2608 , Ritz's method is used. Assume that the unknown function tex2html_wrap_inline2610 is approximated by a function series:

equation806

where tex2html_wrap_inline2612 form a complete function system (that is, any piecewise continuous function can be approximated by their linear combination), and tex2html_wrap_inline2614 are unknown coefficients. This assumption makes functional tex2html_wrap_inline2608 an n-variate function tex2html_wrap_inline2620 , which is extreme if all the partial derivatives are zero. Having made every tex2html_wrap_inline2622 equal to zero, a linear equation system can be derived for the unknown tex2html_wrap_inline2624 -s ( tex2html_wrap_inline2626 ):

  equation809

This general formula provides a linear equation for any kind of complete function system tex2html_wrap_inline2628 , thus it can be regarded as a basis of many different radiosity approximation techniques, because the different selection of basis functions, tex2html_wrap_inline2630 , results in different methods of determining the radiosity distribution.

Three types of function bases are discussed:

 
Figure: One-dimensional analogy of proposed basis functions




next up previous index
Next: Piecewise constant radiosity approximation Up: RADIOSITY METHOD Previous: Combination of radiosity and

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