Graphics Reference
In-Depth Information
Application to diffuse integral.
We would like to apply the Monte Carlo method
to our diffuse integral. In order to simplify that process, we assume that the
light shape has uniform surface normal
n
. Another assumption is that
dA
does
not change for integrated points, meaning the whole shape is always visible.
Therefore, we can conceptually move
dAcosθ
o
out of our integral and compute it
once for the entire domain:
dAcosθ
o
A
cosθ
i
r
2
.
(1.13)
The integrated function is dependent only on the distance and cosine between
the surface normal and the directions subtended by the light shape. Those are
continuous and well-defined functions.
Immediately we can see that the global minimum and maximum of the in-
tegrated function are inside the sampling domain defined by the light shape.
Therefore, there is a single point, on the light shape, that is able to represent the
integral, thus minimizing the error estimator.
If we could find this point, we could essentially solve the integral by approx-
imation using importance sampling with one point of highest importance. We
would just need to evaluate the function once for that specific point. However, we
are only interested in points that are easy to find at GPU program runtime. The
ideal candidate would be given by a function of light parameters, the position of
point
p
,andthenormal
n
. We need to find a function that returns a point min-
imizing the error estimator. We also need to minimize the function complexity,
to maintain runtime performance. To simplify the problem's domain, we prefer
to work in the 2D space of a light quadrilateral.
Our function of interest is bounded by the light shape and defined as
cosθ
i
r
2
.
(1.14)
We know that the function integral over area is bounded by local minima and
maxima in limits of given area. The function from equation (1.14) has one global
maximum and one global minimum at the shape bounds. Therefore, we know that
a single point best representing the integral can be found on a segment between
the maximum and minimum. In order to find it, we would have to calculate
the global minimum and maximum, which we deemed to be too computationally
expensive.
Instead, we decided to find an approximate important point in the proximity
of the global maximum, accepting the risk of overestimation. In order to do so, we
need to set boundary conditions for our search. The function from equation (1.14)
is a component-wise multiplication of two functions. The maximum of their
products can be found along a segment connecting their local maximums.
