There are other approaches to computing this prefiltering integration on the fly. One, which was introduced by Frank Crow is called a summed-area table. Basically, a summed-area table is a tabularized two-dimensional cummulative distribution function. Imagine having a 2-D table of numbers the cummulative distribution function could be found as shown below.

To find the sum of region contained in a box bounded by (x₀, y₀) and (x₁, y₁):

T(x₁, y₁) - T(x₀, y₁) - T(x₁, y₀) + T(x₀, y₀)

This approach can be used to compute the integration of pixels that lie under a pixel by divinging the resulting sum by the area of the rectangle,

(y₁ - y₀)(x₁ - x₀). With a little more work you can compute the area under any four-sided polygon (How?).