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.