Chapter 10 RADIOSITY METHOD The radiosity metho d is based on the numerical solution of the shading equation by the nite element metho d. It sub divides the surfaces into small elemental surface patches. Supp osing these patches are small, their intensity distribution over the surface can b e approximated by a constant value which dep ends on the surface and the direction of the emission. We can get rid of this directional dep endency if only di use surfaces are allowed, since di use surfaces generate the same intensity in all directions. This is exactly the initial assumption of the simplest radiosity mo del, so we are also going to consider this limited case rst. Let the energy leaving a unit area of surface i in a unit time in all directions b e B , and assume that the light i density is homogeneous over the surface. This light density plays a crucial role in this mo del and is also called the radiosity of surface i. The dep endence of the intensityon B can b e expressed by the following i argument: 1. Consider a di erential dA element of surface A. The total energy leaving the surface dA in unit time is B dA, while the ux in the solid angle d! is d=IdA cos d! if is the angle b etween the surface normal and the direction concerned. 2. Expressing the total energy as the integration of the energy contribu- tions over the surface in all directions and assuming di use re ection 265 266 10. RADIOSITY METHOD only,we get: =2 2 2 2 Z Z Z Z d 1 cos sin dd=I I cos d! = I B = d! = dA d! =0 =0 (10:1) since d! = sin dd . Consider the energy transfer of a single surface on a given wavelength. The total energy leaving the surface (B dA ) can b e divided into its own i i emission and the di use re ection of the radiance coming from other surfaces ( gure 10.1). Ej dA j Φ ji B i B j dA i E i Figure 10.1: Calculation of the radiosity The emission term is E dA if E is the emission density which is also i i i assumed to b e constant on the surface. The di use re ection is the multiplication of the di use co ecient % and i that part of the energy of other surfaces which actually reaches surface i. Let F b e a factor, called the form factor, which determines that fraction ji of the total energy leaving surface j which actually reaches surface i. Considering all the surfaces, their contributions should b e integrated, which leads to the following formula of the radiosity of surface i: Z B dA = E dA + % B F dA : (10:2) i i i i i j ji j 10. RADIOSITY METHOD 267 Before analyzing this formula any further, some time will b e devoted to the meaning and the prop erties of the form factors. The fundamental law of photometry (equation 3.15) expresses the en- ergy transfer b etween two di erential surfaces if they are visible from one another. Replacing the intensityby the radiosity using equation 10.1, we get: dA cos dA cos dA cos dA cos i i j j i i j j d=I = B : (10:3) j 2 2 r r If dA is not visible from dA , that is, another surface is obscuring it from i j dA or it is visible only from the \inner side" of the surface, the energy ux j is obviously zero. These two cases can b e handled similarly if an indicator variable H is intro duced: ij 8 1ifdA is visible from dA < i j (10:4) H = ij : 0 otherwise Since our goal is to calculate the energy transferred from one nite sur- face (A ) to another (A ) in unit time, b oth surfaces are divided into j i in nitesimal elements and their energy transfer is summed or integrated, thus: Z Z dA cos dA cos i i j j = B H : (10:5) ji j ij 2 r A A i j By de nition, the form factor F is a fraction of this energy and the total ji energy leaving surface j ( B A ): j j Z Z dA cos dA cos 1 i i j j H F = : (10:6) ij ji 2 A r j A A j i It is imp ortant to note that the expression of F A is symmetrical with ji j the exchange of i and j , which is known as the recipro city relationship: F A = F A : (10:7) ji j ij i We can now return to the basic radiosity equation. Taking advantage of the homogeneous prop erty of the surface patches, the integral can b e replaced by a nite sum: X B A = E A + % B F A : (10:8) i i i i i j ji j j 268 10. RADIOSITY METHOD Applying the recipro city relationship, the term F A can b e replaced ji j by F A : ij i X B A = E A + % B F A : (10:9) i i i i i j ij i j Dividing by the area of surface i,we get: X B = E + % B F : (10:10) i i i j ij j This equation can b e written for all surfaces, yielding a linear equation where the unknown comp onents are the surface radiosities (B ): i 3 3 2 3 2 2 E B 1 % F % F ::: % F 1 1 1 11 1 12 1 1N 7 7 6 7 6 6 E B % F 1 % F ::: % F 7 7 6 7 6 6 2 2 2 21 2 22 2 2N 7 7 6 7 6 6 7 7 6 7 6 6 : : : 7 7 6 7 6 6 (10:11) = 7 7 6 7 6 6 : : : 7 7 6 7 6 6 7 7 6 7 6 6 5 5 4 5 4 4 : : : E B % F % F ::: 1 % F N N N N 1 N N 2 N NN or in matrix form, having intro duced matrix R = % F : ij i ij (1 R) B = E (10:12) (1 stands for the unit matrix). The meaning of F is the fraction of the energy reaching the very same ii surface. Since in practical applications the elemental surface patches are planar p olygons, F is 0. ii Both the numb er of unknown variables and the numb er of equations are equal to the numb er of surfaces (N ). The solution of this linear equation is, at least theoretically, straightforward (we shall consider its numerical as- p ects and diculties later). The calculated B radiosities represent the light i density of the surface on a given wavelength. Recalling Grassman's laws, to generate color pictures at least three indep endentwavelengths should b e selected (say red, green and blue), and the color information will come from the results of the three di erent calculations. 10. RADIOSITY METHOD 269 Thus, to sum up, the basic steps of the radiosity metho d are these: 1. F form factor calculation. ij 2. Describ e the light emission (E ) on the representativewavelengths, or i in the simpli ed case on the wavelength of red, green and blue colors. Solve the linear equation for each representativewavelength, yielding 1 2 n B , B ... B . i i i 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 b e prop ortional to the cal- culated radiosity, since the intensity of a di use surface is prop ortional to its radiosity (equation 10.1) and is indep endent of the direction of the camera. Constant color of surfaces results in the annoying e ect of faceted ob jects, since the eye psychologically accentuates the discontinuities of the color distribution. To create the app earance of smo oth surfaces, the tricks of Gouraud shading can b e applied to replace the jumps of color by linear changes. In contrast to Gouraud shading as used in incremental metho ds, in this case vertex colors are not available to form a set of knot p oints for interp olation. These vertex colors, however, can b e approximated by averaging the colors of adjacent p olygons (see gure 10.2). B2 B1 B3 B4 BB+++ BB B=1234 v 4 Figure 10.2: Color interpolation for images created by the radiosity method Note that the rst two steps of the radiosity metho d are indep endent of the actual view, and the form factor calculation dep ends only on the 270 10. RADIOSITY METHOD geometry of the surface elements. In camera animation, or when the scene is viewed from di erent p ersp ectives, only the third step has to b e rep eated; the computationally exp ensive form factor calculation and the solution of the linear equation should b e carried out only once for a whole sequence. In addition to this, the same form factor matrix can b e used for sequences, when the lightsources have time varying characteristics. 10.1 Form factor calculation The most critical issue in the radiosity metho d is ecient form factor cal- culation, and thus it is not surprising that considerable research e ort has gone into various algorithms to evaluate or approximate the formula which de nes the form factors: Z Z 1 dA cos dA cos i i j j F = : (10:13) H ij ij 2 A r i A A i j As in the solution of the shading problem, the di erent solutions represent di erent compromises b etween the con icting ob jectives of high calculation sp eed, accuracy and algorithmic simplicity.