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# The Complete Rendering Equation

The generation of a synthetic image was originally characterized as the solution of a rendering equation by Kajiya . A complete rendering equation gives the values to be set on an image display device as a function of the radiometric properties of the synthetic scene and the display device. For convenience in examining the source of image noise we will examine the rendering equation in three parts:

- - - Equation 1 is the equation of transport for visible light in the synthetic scene. Equation 1 gives the radiance at visible surface point (x,y) in the direction that would reach the observer of a physical realization of the scene. is the emitted radiance of the point (non-zero only for light sources) and is the bidirectional reflection distribution function (BRDF) for the point. The integral on the right hand side accounts for all reflection of incident radiance from solid angles dw. Radiance has dimensions of energy per unit time, area and solid angle.

Equation 2 expresses the radiance Lp of a discrete pixel from the function . The function g(x,y) is the 2-D filter used to eliminate spatial aliasing errors. The function g(x,y) has dimensions of 1/area and is normalized so that Lp has the same units and range as .

Equation 3 expresses the conversion of pixel radiance, which can take on any physically realizable value (i.e. from starlit to sunlit scenes), and converts it to a dimensionless setting for the display device Np - usually to an integer between 0 and 255. The function T() is the tone operator, which is constructed using properties of human perception and characteristics of the display device. Various forms for T() are discussed in , , and . Even simple tone operators are inherently non-linear because of quantization effects and clipping of out of range values.

Typical Monte Carlo renderers compute an estimate L^p by forming and averaging many sample values L'p. As stated by Purgathofer , the number of samples Mt is determined by the number required to reduce the estimated deviation Sp of the average to less than a specified tolerance d with a specified confidence alpha using the percentage point of the t distribution t(1-alpha/2),(Mt-1). That is:

- - - The calculation of each value L'p begins by selecting a random value of (x,y) in the right hand side of Eq. 2. At this location Eq. 1 is estimated by choosing a random direction for evaluating the integral on the right hand side. Since the value of is unknown, and also governed by Eq. 1 , this estimation is done recursively, and a path of rays is generated . After values of L^p are computed the tone operator in Eq. 3 is applied to display the image.

Using this naive approach, there is often high variance in the estimate of the integral on the right hand side of Eq. 1. This is because the integrand sample values can vary from the light source radiance to the radiance of dark, shaded objects in the room - with a dynamic range of 100,000 being common . To make Monte Carlo solutions practical, the integral on the right hand side is rewritten as a sum of integrals .

- In Eq. 7, fr,d is the diffuse-like component of the BRDF, and fr,s is the specular-like. The integral for the diffuse-like component is divided into parts. The first is the integral for direct illumination. The direct illumination is an integral over all light sources s in terms of the area of the sources As, visibility of the source vis(s), emission from the source surface Le(xs,ys), angle from the source and distance to the source, rs. The second is an integral for indirect illumination, i.e. over all incident light that does not come directly from the light source. The final integral on the right of Eq. 7 gives the specularly reflected light. Each of the integrals on the right of Eq. 7 generally has a lower variance than the integral on the left of Eq. 1, so the number of trials required is greatly reduced .

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