**ken@turtlevax.UUCP (Ken Turkowski)** (07/30/85)

This year's SIGGRAPH impressed on me the profundity of stochastic sampling on the aliasing problem. In a nutshell, distributed ray tracing and stochastic sampling perform sampling on a random (e.g. Poisson disk) rather than regular grid. The result is that the regular artifacts of aliasing are instead converted into white noise (with an appropriate probabilistic distribution of sampling points), which is less apt to be noticed by the eye. I suspect that the aliasing energy is equal to the white noise energy, and that the signal energy is the same. Is there a theory to back this up? Any references to peripheral literature related to the subject? -- Ken Turkowski @ CADLINC, Menlo Park, CA UUCP: {amd,decwrl,hplabs,nsc,seismo,spar}!turtlevax!ken ARPA: turtlevax!ken@DECWRL.ARPA

**wold@ucbvax.ARPA (Erling Wold)** (08/02/85)

In article <843@turtlevax.UUCP> ken@turtlevax.UUCP (Ken Turkowski) writes: >This year's SIGGRAPH impressed on me the profundity of stochastic >sampling on the aliasing problem. > >I suspect that the aliasing energy is equal to the white noise energy, >and that the signal energy is the same. Is there a theory to back this >up? Any references to peripheral literature related to the subject? Yes, there is a large theory on this subject. Stochastic sampling has been used in radar and spectral analysis for at least 20 years. The references given in the papers published in SIGGRAPH are a good place to start, especially those of Lenemann, Shapiro and Silverman, and Masry. Balakrishnan has been referenced by some authors, but his work has been superceded by these others. There is a lot more practical work that needs to be done, however. It is basically true that the error energies for regular and stochastic sampling are the same. However, there are some complications. The reconstruction method used may change the amount of error seen in the two cases. This is because the spectral distributions of the error in the two cases are different. The errors left after reconstruction are (in general) the errors in the spectral band from 0 to 1/2 the display sampling rate. For example, a single low frequency sinusoidal image, sampled regularly at a frequency above 2 times the sinusoidal frequency and reconstructed in a straightforward manner yields an image without error, since no aliasing occurs. However, sampling the same sinusoid stochastically will yield noise in the spectral band of interest and will thus yield an image with errors after reconstruction. There is enough information given in Dippe and Wold, SIGGRAPH `85, to work out the details for any specific case given the image and the reconstruction filter. There is also a paper in the International Computer Music Conference this month which shows some examples and discusses the effect of different ways of calculating the reconstruction integral. ----------------------------------------------------- Erling Wold