The collocation method for solving the radiosity equation for unoccluded surfaces.

*(English)*Zbl 0914.65137The 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.

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.

Reviewer: Nikolay Yakovlevich Tikhonenko (Odessa)

##### MSC:

65R20 | Numerical methods for integral equations |

45E10 | Integral equations of the convolution type (Abel, Picard, Toeplitz and Wiener-Hopf type) |

##### Keywords:

numerical results; computer graphics; unoccluded surfaces; reflectivity; emissivity; smooth and piecewise smooth surfaces; Lambertian diffuse reflector; radiosity equation; piecewise linear collocation method; superconvergence##### Software:

BIEPACK##### References:

[1] | K. Atkinson, A survey of numerical methods for the solution of Fredholm integral equations of the second kind , SIAM Publications, Philadelphia, 1976. · Zbl 0353.65069 |

[2] | ——–, A survey of boundary integral equation methods for the numerical solution of Laplace’s equation in three dimensions , in Numerical solution of integral equations (M. Golberg, ed.), Plenum, New York, 1990. · Zbl 0737.65085 |

[3] | ——–, User’s guide to a boundary element package for solving integral equations on piecewise smooth surfaces , Reports on Computational Mathematics #43, Dept of Mathematics, University of Iowa, Iowa City. (This is available by anonymous ftp from math .uiowa.edu, under pub /atkinson/bie.package. Version#2 (1998 release) contains programs for the unoccluded radiosity equation.) |

[4] | ——–, The numerical solution of Fredholm integral equations of the second kind , Cambridge University Press, Cambridge, 1997. · Zbl 0882.65133 |

[5] | K. Atkinson and D. Chien, Piecewise polynomial collocation for boundary integral equations , SIAM J. Sci. Statist. Comput. 16 (1995), 651-681. · Zbl 0826.65095 |

[6] | K. Atkinson, I. Graham, and I. Sloan, Piecewise continuous collocation for integral equations , SIAM J. Num. Anal. 20 (1983), 172-186. JSTOR: · Zbl 0514.65094 |

[7] | S. Brenner and R. Scott, The mathematical theory of finite element methods , Springer-Verlag, New York, 1994. · Zbl 0804.65101 |

[8] | M. Cohen and J. Wallace, Radiosity and realistic image synthesis , Academic Press, New York, 1993. · Zbl 0814.68138 |

[9] | N. Günter, Potential theory , Ungar Pub. Co., New York, 1967. |

[10] | S. Mikhlin, Mathematical physics : An advanced course , North-Holland, Amsterdam, 1970. · Zbl 0202.36901 |

[11] | W. Pogorzelski, Integral equations and their applications , Vol. I, Pergamon Press, New York, 1966. · Zbl 0137.30502 |

[12] | W. Wendland, Boundary element methods for elliptic problems , in Mathematical theory of finite and boundary element methods (A. Schatz, V. Thomée, and W. Wendland, eds.), Birkhäuser, Boston, 1990, 219-276. · Zbl 0712.65099 |

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