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From: wold@ucbvax.ARPA (Erling Wold)
Newsgroups: net.graphics,net.math
Subject: Re: Stochastic Sampling and Distributed Ray Tracing
Message-ID: <9568@ucbvax.ARPA>
Date: Thu, 1-Aug-85 19:52:40 EDT
Article-I.D.: ucbvax.9568
Posted: Thu Aug  1 19:52:40 1985
Date-Received: Sat, 3-Aug-85 02:46:28 EDT
References: <843@turtlevax.UUCP>
Reply-To: wold@ucbvax.UUCP (Erling Wold)
Organization: University of California at Berkeley
Lines: 38
Keywords: aliasing, sampling, Shannon/Nyquist Sampling Theorem
Xref: watmath net.graphics:978 net.math:2153


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authors, but his work has been superceded by these others.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