It was pointed out to me that in a ray-tracer it would be good to pre-filter an image with a sinc function before sampling it. It seemed obvious to me that we could evaluate the required convolution by using Monte Carlo integration. So we picked out a function that would map a uniform distribution on [0, 1] to a non-uniform distribution on [-∞, ∞], did the appropriate tweaking based on our pdf, and plugged it into RISE.

Problem was, the image was very noisy. We added more samples, and still noise. So, I used Maple to calculate the variance of our method. The computed variance was ∞.

There doesn’t seem to be any way around this problem. I think the infinite variance results from the fact that the integral of the sinc function is not absolutely convergent (I believe the integral of |sinc| diverges).

We are multiplying the sinc function by the image data, which presumably goes to black the pixels go off to infinity, so in theory we should get convergence of our Monte Carlo integration; however, in practice the attempt was a failure.