The Potential Equation and Importance in Illumination Computations

An equation adjoint to the luminance equation for describing the global illumination can be formulated using the notion of a surface potential to illuminate the region of interest This adjoint equation which we shall call as the potential equation is fundamental to the adjoint radiosity equation used to devise the importance driven radiosity algorithm In this paper we rst brie y derive the adjoint system of integral equations and then show that the adjoint linear equations used in the above algorithm are basically discrete formulations of the same We also show that the importance entity of the linear equations is basically the potential function integrated over a patch Further we prove that the linear operators in the two equations are indeed transposes of each other Introduction In a recent paper Smits et al present a new importance driven radiosity algorithm for e ciently computing global illumination solutions with respect to a constrained set of views The basis of this new algorithm is a pair of adjoint light transport equations One of these is the standard radiosity equation which uses the form factor matrix to relate the radiosity of any surface patch to the emitting sources in the environment The other is the adjoint equation which uses the transpose of the form factor matrix to relate so called importance to the receivers Importance and receivers are deemed as the duals of radiosity and emitting sources respectively Smits et al visualise importance to be owing out from the eye or camera and distributing itself amongst the patches of the environment very much like light In our earlier work we have similarly formulated a pair of integral equations as an adjoint set for describing the light transport The rst of these equations is basically Kajiya s rendering equation de ned in terms of luminance emittance and brdfs while the second is its adjoint de ned in terms of potential hypothetical detector s and brdfs The potential equation is fundamental to the adjoint radiosity equation just as as much as the rendering equation is to the standard radiosity equation In this paper we show how the adjoint radiosity equation can be derived from the potential equation We show that importance is basically the potential integrated over a patch we derive the detector values and prove that the linear operators in the two equations are indeed transposes of each other In order to make the contents of this paper self contained we brie y derive the adjoint equations below For a detailed discussion on the subject the reader is referred to Potential Equation Because of the optical properties of surfaces such as re ection transmission etc the light emitted from any surface in any direction can illuminate many other surfaces of an environ ment Alternatively we can say that a surface can be illuminated by lights placed anywhere in