The collocation method for solving the radiosity equation for unoccluded surfaces
The radiosity equation occurs in computer graphics, and its solution leads to more realistic illumination for the display of surface \(S.\) The equation is \[ (I-\mathcal K)u:=u(P) - \frac{\rho(P)}{\pi}\int_S u(Q)G(P,Q)V(P,Q)dS_Q = E(P),\quad P\in S, \tag{1} \] with \(u(P)\) the ``brightness'' or radiosity and \(E(P)\) the emissivity at \(P\in S.\) The function \(\rho(P)\) gives the reflectivity with \(0\leq\rho(P) < 1.\) The function \(G\) is given by \[ G(P,Q)=\left[(Q-P)\cdot\mathbf n_P\right]\left[(P-Q)\cdot\mathbf n_Q\right]/| P-Q|^4. \] In this, \(\mathbf n_p\) is the inner unit normal to \(S\) at \(P,\) and \(\mathbf n_Q\) is defined analogously. The function \(V(P,Q)\) is a ``line of sight'' function. An unoccluded surface is one for which \(V \equiv 1\) on \(S,\) and it is the simplest case investigated here in order to make clearer the behavior of the approximation methods being used. In the numerical solution of (1), the Galerkin method has been the predominant form of numerical solution. In this paper, the collocation method with approximations of all possible orders is investigated. The case of piecewise smooth \(S,\) which is a more practical situation, is considered, and the effect of using linear interpolation to approximate \(S\) is also analyzed. For one particular choice of such interpolation, superconvergence results are obtained when solving on a smooth surface. Numerical results conclude the paper.
scientific article; zbMATH DE number 1240965