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RADIOSITY METHOD

   

The radiosity method is based on the numerical solution of the shading equation by the  finite element method. It subdivides the surfaces into small elemental surface patches. Supposing these patches are small, their intensity distribution over the surface can be approximated by a constant value which depends on the surface and the direction of the emission. We can get rid of this directional dependency if only diffuse surfaces are allowed, since diffuse surfaces generate the same intensity in all directions. This is exactly the initial assumption of the simplest radiosity model, so we are also going to consider this limited case first. Let the energy leaving a unit area of surface i in a unit time in all directions be tex2html_wrap_inline1772 , and assume that the light density is homogeneous over the surface. This light density plays a crucial role in this model and is also called the radiosity of surface i.

The dependence of the intensity on tex2html_wrap_inline1772 can be expressed by the following argument:

  1. Consider a differential dA element of surface A. The total energy leaving the surface dA in unit time is tex2html_wrap_inline1784 , while the flux in the solid angle tex2html_wrap_inline1786 is tex2html_wrap_inline1788 if tex2html_wrap_inline1790 is the angle between the surface normal and the direction concerned.
  2. Expressing the total energy as the integration of the energy contributions over the surface in all directions and assuming diffuse reflection only, we get:

      equation93

    since tex2html_wrap_inline1792 .

Consider the energy transfer of a single surface on a given wavelength. The total energy leaving the surface ( tex2html_wrap_inline1794 ) can be divided into its own emission and the diffuse reflection of the radiance coming from other surfaces (figure 1.1).

  
Figure: Calculation of the radiosity

The emission term is tex2html_wrap_inline1796 if tex2html_wrap_inline1798 is the emission density which is also assumed to be constant on the surface.

The diffuse reflection is the multiplication of the diffuse coefficient tex2html_wrap_inline1800 and that part of the energy of other surfaces which actually reaches surface i. Let tex2html_wrap_inline1804 be a factor, called the  form factor, which determines that fraction of the total energy leaving surface j which actually reaches surface i.

Considering all the surfaces, their contributions should be integrated, which leads to the following formula of the radiosity of surface i:

equation113

Before analyzing this formula any further, some time will be devoted to the meaning and the properties of the form factors.

The fundamental law of photometry (equation gif) expresses the energy transfer between two differential surfaces if they are visible from one another. Replacing the intensity by the radiosity using equation 1.1, we get:

equation118

If tex2html_wrap_inline1812 is not visible from tex2html_wrap_inline1814 , that is, another surface is obscuring it from tex2html_wrap_inline1814 or it is visible only from the ``inner side'' of the surface, the energy flux is obviously zero. These two cases can be handled similarly if an indicator variable tex2html_wrap_inline1818 is introduced:

equation125

Since our goal is to calculate the energy transferred from one finite surface ( tex2html_wrap_inline1824 ) to another ( tex2html_wrap_inline1826 ) in unit time, both surfaces are divided into infinitesimal elements and their energy transfer is summed or integrated, thus:

equation130

By definition, the form factor tex2html_wrap_inline1804 is a fraction of this energy and the total energy leaving surface j ( tex2html_wrap_inline1832 ):

  equation139

It is important to note that the expression of tex2html_wrap_inline1834 is symmetrical with the exchange of i and j, which is known as the  reciprocity relationship:

equation151

We can now return to the basic radiosity equation.  Taking advantage of the homogeneous property of the surface patches, the integral can be replaced by a finite sum:

equation156

Applying the reciprocity relationship, the term tex2html_wrap_inline1834 can be replaced by tex2html_wrap_inline1842 :

equation161

Dividing by the area of surface i, we get:

  equation164

This equation can be written for all surfaces, yielding a linear equation where the unknown components are the surface radiosities ( tex2html_wrap_inline1772 ):

equation168

or in matrix form, having introduced matrix tex2html_wrap_inline1848 :

equation200

( tex2html_wrap_inline1850 stands for the unit matrix).

The meaning of tex2html_wrap_inline1852 is the fraction of the energy reaching the very same surface. Since in practical applications the elemental surface patches are planar polygons, tex2html_wrap_inline1852 is 0.

Both the number of unknown variables and the number of equations are equal to the number of surfaces (N). The solution of this linear equation is, at least theoretically, straightforward (we shall consider its numerical aspects and difficulties later). The calculated tex2html_wrap_inline1772 radiosities represent the light density of the surface on a given wavelength. Recalling Grassman's laws, to generate color pictures at least three independent wavelengths should be selected (say red, green and blue), and the color information will come from the results of the three different calculations.

Thus, to sum up, the basic steps of the radiosity method   are these:

  1. tex2html_wrap_inline1862 form factor calculation.
  2. Describe the light emission ( tex2html_wrap_inline1798 ) on the representative wavelengths, or in the simplified case on the wavelength of red, green and blue colors. Solve the linear equation for each representative wavelength, yielding tex2html_wrap_inline1866 , tex2html_wrap_inline1868 ... tex2html_wrap_inline 1870 .
  3. Generate the picture taking into account the camera parameters by any known hidden surface algorithm. If it turns out that surface i is visible in a pixel, the color of the pixel will be proportional to the visible in a pixel, the color of the pixel will be pro proportional to its radiosity (equation 1.1) and is independent of the direction of the camera.

Constant color of surfaces results in the annoying effect of faceted objects, since the eye psychologically accentuates the discontinuities of the color distribution. To create the appearance of smooth surfaces, the tricks of Gouraud shading can be applied to replace the jumps of color by linear changes. In contrast to Gouraud shading as used in incremental methods, in this case vertex colors are not available to form a set of knot points for interpolation. These vertex colors, however, can be approximated by averaging the colors of adjacent polygons (see figure  1.2).  

  
Figure: Color interpolation for images created by the radiosity method

Note that the first two steps of the radiosity method are independent of the actual view, and the form factor calculation depends only on the geometry of the surface elements. In camera animation, or when the scene is viewed from different perspectives, only the third step has to be repeated; the computationally expensive form factor calculation and the solution of the linear equation should be carried out only once for a whole sequence. In addition to this, the same form factor matrix can be used for sequences, when the lightsources have time varying characteristics.




next up previous index
Next: Form factor calculation Up: Radiosity Method - Contents Previous: Radiosity Method - Contents

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