The generation of a synthetic image was originally characterized as the solution of a rendering equation by Kajiya [7]. 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 [4], [12],
and [13]. 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 [11], 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 [7].
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 [4]. To make Monte Carlo solutions practical, the integral on the right hand side is rewritten as a sum of integrals [3].

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 [1].