Path: utzoo!attcan!uunet!brunix!jfh
From: jfh@brunix (John Forbes Hughes)
Newsgroups: comp.graphics
Subject: Re: iso-surfaces in SciVi
Message-ID: <16722@brunix.UUCP>
Date: 30 Sep 89 16:16:58 GMT
References: <19889@mimsy.UUCP> <5430@portia.Stanford.EDU>
Sender: news@brunix.UUCP
Reply-To: jfh@euclid.UUCP (John Forbes Hughes)
Organization: Brown University Department of Computer Science
Lines: 46

In article <5430@portia.Stanford.EDU> rick@hanauma.UUCP (Richard Ottolini) writes:
>In article <19889@mimsy.UUCP> rmr@mimsy.umd.edu (Randy M. Rohrer) writes:
>>
>>what is an iso-surface?
>In a multi-dimensional (2,3,+) sampling of data points, an iso-surface
>is a surface passes through data of the same value.
>This implies a certain degree of smoothness in the data samples.
>For example, the temperature surface of 200-degrees surrounding a flame.

I don't want to nit-pick (actually, of course, I do, which is why I'm writing),
but it might be better to say the following:

If  
     f : A --> R

is a function, an *iso-set* of f is a set of the form

   J = { x in A :  f(x) = C }

where C is some constant in R. (R is the set of real numbers) (footnote 1). In 
the event that A = n-space, and f is smooth, and its derivative has maximal rank
at each point of J, the iso-set is a codimension 1 submanifold of n-space.
For example, if A = 3-space, and f is nice, then J will be a (possibly empty)
surface.  If f is merely continuous, then J may contain various degeneracies.

   If L is a lattice in n-space, and f is defined only on L, then there
are infinitely many functions F such that F|L = f (the restriction of F
to L is f). An iso-set of any such function F *can* be called an
iso-set of f (footnote 2). If L is a finite lattice, and f is known to come 
from a function with reasonable behavior (e.g., it has no high-frequency 
components when fourier analyzed on the lattice), then the "correct" F can 
sometimes be synthesized from the values of f. The iso-sets of F deserve to be 
called the iso-sets of f in this case. In the cases where the data samples
are taken from data whose exact continuous nature is unknown, typically
some "reasonable" inference is made--for example people will do cubic
interpolations between nearby sets of data values and use this as F. The
results are, of course, only as good as the assumptions about the nature
of the original signal.
 -John Hughes, Dept.'s of Math and CS, Brown University, Providence, RI
  jfh@cs.brown.edu

(1) Actually, the codomain, which I have written as R, can actually be any
set, and an "iso-set" is seen to be a specific case of the notion of
"inverse image" or "preimage" of a point. 

(2) This is what is typically done is scientific visualization.