I want to do a reduced-affine-arithmetic raytracer.
The idea is that an “image” is a function from pixel coordinates x and y to color component intensities r, g, and b, and we merely want to compute an adequate approximation to that function. We recursively subdivide the image into rectangular regions, and restrict ourselves to a linear approximation within each region, so that the overall approximation is piecewise linear (though not necessarily continuous between the pieces).
In this way, we can avoid spending much computation time on smooth gradient regions, concentrating on the regions where aliasing is possible.
Extending this, a “video” is a similar function, but has three independent variables: x, y, and t. This allows us to avoid spending computation time on parts of the scene that don’t change much from frame to frame.
You can derive such an approximation by applying a self-validated arithmetic model from a mathematical description of the ray-traced scene. Most self-validated arithmetic models only give you zeroth-order approximations in any given region; interval arithmetic and the use of Lipschitz constants are examples. Affine arithmetic gives you a first-order approximation, but it is crushingly computationally expensive; reduced affine arithmetic, though it doesn’t provide such tight bounds, is more efficient, and has been successfully used for raytracing.