Θ x Θ x Θ. y Detector. Detector y

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1 The Potential Equation and Importance in Illumination Computations S.N.Pattanaik and S.P.Mudur Graphics & CAD Division, National Centre for Software Technolog Juhu, Bomba , INDIA Abstract An equation adoint tothe luminance equation for describing the global illumination can be formulated using the notion of a surface potential to illuminate the region of interest. This adoint equation which we shall call as the potential equation, is fundamental to the adoint radiosit equation used to devise the importance driven radiosit algorithm. In this paper we rst brie derive the adoint sstem of integral equations and then show that the adoint linear equations used in the above algorithm are basicall discrete formulations of the same. We also show that the importance entit of the linear equations is basicall the potential function integrated over a patch. Further we prove that the linear operators in the two equations are indeed transposes of each other. 1. Introduction In a recent paper [1] Smits et al present a new importance driven radiosit algorithm for ecientl computing global illumination solutions with respect to a constrained set of views. The basis of this new algorithm is a pair of adoint light transport equations. One of these is the standard radiosit equation [2] which uses the form-factor matri to relate the radiosit of an surface patch to the emitting sources in the environment. The other is the adoint equation, which uses the transpose of the form-factor matri to relate so called importance to the receivers. Importance and receivers are deemed as the duals of radiosit and emitting sources respectivel. Smits et al visualise importance to be owing out from the ee or camera and distributing itself amongst the patches of the environment, ver much like light. In our earlier work [3] we have similarl formulated a pair of integral equations as an adoint set for describing the light transport. The rst of these equations is basicall Kaia's rendering equation [4] dened in terms of luminance, emittance and brdfs, while the second is its adoint dened in terms of potential, hpothetical detector(s) and brdfs. The potential equation is fundamental to the adoint radiosit equation ust as as much as the rendering equation is to the standard radiosit equation. In this paper we show how the adoint radiosit equation can be derived from the potential equation. We show that importance is basicall 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 derive the adoint equations below. For a detailed discussion on the subect the reader is referred to [3]. 2. Potential Equation Because of the optical properties of surfaces, such as reection, transmission, etc the light emitted from an surface in an direction can illuminate man other surfaces of an environment. Alternativel we can sa that a surface can be illuminated b lights placed anwhere in

2 Θ Θ Θ Detector Detector g(; ) = 1 and W (; )=1. g(; )=0,g(; R )=1and W (; )=0+ ( ; )W (; )cos d! Figure 1. Direct and Indirect Components of the Potential Function. the environment. Suppose we place a light detector in the environment. Let the detector be hpothetical in the sense, it does not in an wa aect the ow of light in the environment. Emission from a given point,, along a given direction,, ma result in some light being detected b the detector. The light can be detected directl from the emission point and/or indirectl from other points of the environment due to reection, transmission etc. (see Figure 1). The quantit of light detected will var depending on the point of emission and at a given point depending on the direction of emission. This quantit, ma be termed as potential, W, of a point and direction to contribute light into the detector. Further more as W is a function of the surface position, and the direction, around, wehave in the process dened a potential function W (; ). We shall now derive an epression for such a function. Light on being emitted from (; ) can enter the detector directl if the path (; ) leads to the detector. So to represent the direct component we will use a function g(; ) which has a value 1 if the light path starting from along reaches the detector unhidered, 0 otherwise. The quantit of light entering the detector due to one or more reections ma be epressed recursivel as follows. The emission from an (; ) will reach the nearest surface point and then possibl be reected. If we take the probabilit of the whole amount of u getting reected in an one of the hemispherical directions around as ( ; )cos d!, where the smbols used are as in Figure 2, then the quantit of light entering the detector due to this reection will be this probabilit times the potential of the point along, i.e. ( ; )cos d! W (; ). Then the cumulative result of the reection in the hemispherical direction around will be ( ; )W (; )cos d! The complete epression for the potential function is therefore given b: W (; )=g(; )+ ( ; )W (; )cos d! (1)

3 Θ θ Θ Figure 2. Hemispherical Directions for the Outgoing illumination. Θin Θ out n θin δωin Figure 3. Hemispherical Directions for the Incoming illumination. 3. Luminance Equation From the denition of surface bidirectional reectance function [5;page64], the outgoing luminance (L) at an point of a surface in the environment in an direction out, due to the luminance incident at from direction in can be given b L out (; out )= ( out ; in )L in (; in )cos in d! in where in and d! in are as shown in the Figure 3. Taking into account incoming luminance from all the directions in the incoming hemisphere around the point, the outgoing luminance can be epressed as L out (; out )= ( out ; in )L in (; in )cos in d! in where the integration range represents the hemisphere around. If we include emitting surfaces also in the general epression for the outgoing luminance then it takes the form: L out (; out )= out (; out )+ ( out ; in )L in (; in )cos in d! in

4 For a nonparticipating environment consisting of mainl opaque solids the luminance in an incoming direction at must be due to the outgoing luminance from some surface point in an outgoing direction where is dened b the vector oining the point to. If we now wish to rewrite the luminance equation in terms of outgoing luminance and outgoing directions onl, then b representing the outgoing directions at and as and we get: L(; )=(; )+ ( ; )L(; )cos d! (2) If we look back at the potential equation (Equation 1), we nd a striking similarit in the forms of these equations. In both the cases there is an implicit assumption that represents a surface point visible to. However it must be noted that in Equation 2 the integration is over the incoming hemisphere around whereas in Equation 1 the integration is over the outgoing hemisphere around. 4. Discrete Equations for a Diuse Environment Because of their inherentl comple nature it is dicult to solve Equations 1 and 2. However, simplied forms of these have been made amenable to analtical solutions. A simplied discrete formulation of the Equation 2, widel known as radiosit equation [2],is i = E i + k i N X F i where i is the radiosit, i.e. u per unit area, E i is the emission u densit, k i is the hemispherical diuse reectance of patch i and F i is the form factor between patch i and patch. This discrete formulation is derived from the continuous Equation 2 b making the following assumptions: 1. The environment is a collection of a nite number, sa N, of small diusel reecting patches with uniform radiosit. 2. As the luminance from an point of an such uniforml diuse patch is 1= times the u per unit area we shall compute this total u from an patch leaving that patch in all the hemispherical directions. 3. The solution is carried out in an enclosure, i.e. the hemispherical direction around an point,, in the environment is assumed to be covered b one or more of the patches of that environment and ever patch,, ma be assumed to occup a solid angle,! (which ma be zero) in the hemisphere over a surface point. Using this equation we arrive at a sstem of equations for the whole environment of the form: L = E where L is , k 1 F 11 :::,k 1 F 1i :::,k 1 F 1N,k i F i1 ::: 1, k i F ii :::,k i F in,k N F N 1 :::,k N F Ni ::: 1, k N F NN 3 7 5

5 which ma be solved to arrive at the equilibrium state radiosit for each patch. Similarl we shall derive the discrete formulation using the potential equation. For this we shall dene a quantit called importance, sa i, as the potential of the i-th patch over the full hemispherical direction. We can arrive at an epression for i bintegrating W (; ) for ever point of the i-th patch and over the hemisphere around each point of the patch. Thus i = = 2Ai 2Ai W (; )cos d! d g(; )cos d! d + 2Ai " ( ; )W (; )cos d! # Using the earlier mentioned assumptions the equation for i ma be simplied to i = 2Ai g(; )cos d! d + A i " ( ; )W (; )cos d! # = R i + A i N X R i + A i N X = R i + A i N X R i + = R i + NX NX! "! A 2A k A i A F i k F i W (; )cos d! # A cos d! Vis(; ) cos cos da D 2 cos d! cos d! d cos d! where k i = i, Vis(; ) is the visibilit between two surface points and, D is the distance between and, and R i = R 2A i R g(; )cos d! d. As g(; ) is simpl a visibilit function between detector and the point along direction, R i is related to the detector-to-patch form-factor. Thus the sstem of equations for the whole environment ma be written as L = R where L is , k 1 F 11 :::,k i F i1 :::,k N F N 1,k 1 F 1i ::: 1, k i F ii :::,k N F Ni,k 1 F 1N :::,k i F in ::: 1, k N F NN This set of equations ma be solved to arrive at the equilibrium importance for the whole environment. It is eas to see that the discrete transport operators L and L as described above are transpose of each other.

6 5. Concluding Remarks The primar use of the adoint equations has been to increase computational ecienc. In the area of neutron transport [6] adoint equations have been used for the estimation of u and to provide a good choice for importance function biasing. In [3] we have presented algorithms for ecientl computing the global illumination in the Monte Carlo simulation of the particle model of light. In a Monte Carlo simulation of the particle model of light [7; 8], particles, packets of energ, are shot out in dierent directions from dierent positions of the surface of the light source. The interaction of each particle in the environment is traced. Global illumination is then computed from the histor of these interactions. The potential equation forms the basis for this algorithm. Furthermore, it is used to derive biasing functions that are used to carr out importance sampling of the emission and reection functions. These biasing functions ensure that more and more particles are directed towards those parts of the environment which are of greater interest and need more accurate illumination computation. Similarl in [1] the importance of patches is used to increase the ecienc of the hierarchic radiosit algorithm [9] b subdividing patches depending on their importance. So far the application of the adoint equations has been restricted to global illumination computations in diuse environments. Techniques for dealing with global illumination in general environments with diuse, specular and other more comple optical behaviour [10,,13] are still ecessivel demanding in terms of computational resources. We believe that the benets of using the adoint equations to devise new algorithms in such cases will be much more. Similarl a number of multi-pass algorithms [14,,16] have been devised for dealing with the problem of accuratel reconstructing the luminance function with eects such as shadows, highlights, caustics, etc.. Once again these algorithms tend to be resource intensive and more optimal solutions could be sought b the use of these equations. 6. Acknowledgements We are grateful to Dr S Ramani, Director, NCST for his support and encouragement. We also wish to thank Dr David Salesin, Universit of Washington, for lucidl eplaining their importance-driven radiosit algorithm to one of the authors. REFERENCES 1. Brian E. Smits, James R. Arvo, and David H. Salesin. An importance driven radiosit algorithm. Computer Graphics (SIGGRAPH '92 Proceedings), 26(2):273{282, Jul Cind M. Goral, Kenneth E. Torrance, Donald P. Greenberg, and Bennett Battaile. Modelling the interaction of light between diuse surfaces. Computer Graphics (SIGGRAPH '84 Proceedings), 18(3):212{222, Jul S. N. Pattanaik and S. P. Mudur. Adoint equations and random walks for illumination computation. Submitted for Publication, Ma James T. Kaia. The rendering equation. Computer Graphics (SIGGRAPH '86 Proceedings), 20(4):143{150, Aug Robert Siegel and John R. Howell. Thermal Radiation Heat Transfer. Hemisphere Publishing Corporation, R. R. Coveou, V. R. Cain, and K. J. Yost. Adoint and importance in monte carlo application. Nuclear Science and Engineering, 27:219{234, S. N. Pattanaik and S. P. Mudur. Computation of global illumination b monte carlo simulation of the particle model of light. In Proceedings of the Third Eurographics Workshop

7 on Rendering, pages 71{83, Bristol, UK, Ma S. N. Pattanaik and S. P. Mudur. Computation of global illumination in a participating medium b monte carlo simulation. The Journal of Visualisation and Computer Animation, To appear, 4, Pat Hanrahan, David Salzman, and Larr Aupperle. A rapid hierarchical radiosit algorithm. Computer Graphics (SIGGRAPH '91 Proceedings), 25(4):197{206, Jul Dave S. Immel, Michael Cohen, and Donald P. Greenberg. A radiosit method for nondiuse environments. Computer Graphics (SIGGRAPH '86 Proceedings), 20(4):133{142, Aug Holl E. Rushmeier and Kenneth E. Torrance. The zonal method for calculating light intensities in the presence of a participating medium. Computer Graphics (SIGGRAPH '87 Proceedings), 21(4):293{302, Jul S.P. Mudur and S.N. Pattanaik. Multidimensional illumination functions for visualisation of comple 3d environments. The Journal of Visualisation and Computer Animation, 1(2):49{ 58, Xiao D. He, Kenneth E. Torrance, Francois X. Sillion, and Donald P. Greenberg. A comprehensive phsical model for light reection. Computer Graphics (SIGGRAPH '91 Proceedings), 25(4):175{186, Jul John R. Wallace, Michael F. Cohen, and Donald P. Greenberg. A two-pass solution to the rendering equation: A snthesis of ra tracing and radiosit methods. Computer Graphics (SIGGRAPH '87 Proceedings), 21(4):311{320, Jul Paul Heckbert. Adaptive radiosit tetures for bidirectional ra tracing. Computer Graphics (SIGGRAPH '90 Proceedings), 24(4):145{154, Aug Shenchang Eric Chen, Holl E. Rushmeier, Gavin Miller, and Douglass Turner. A progressive multi-pass method for global illumination. Computer Graphics (SIGGRAPH '91 Proceedings), 25(4):164{174, Jul 1991.

8 List of Figures 1 Direct and Indirect Components of the Potential Function Hemispherical Directions for the Outgoing illumination Hemispherical Directions for the Incoming illumination