Ambiguity Problem Certain Marching Cubes cases have more

Ambiguity Problem Certain Marching Cubes cases have more than one possible triangulation Mismatch!!! Hole! + + Case 6 + + Case 3

The Problem Ambiguous Face : a face that has two diagonally oppsed points with the same sign + + Connecting either way is possible

To fix it … Match!!! + + Case 6 + Case 3 B + The goal is to come up with a consistent triangulation

Solutions There are many solutions available – we present a method called: Asymptotic Decider by Nielson and Hamann (IEEE Vis’ 91)

Asymptotic Decider Based on bilinear interpolation over faces B 11 B 01 (s, t) B 00 B(s, t) = (1 -s, s) B 10 B 01 B 10 B 11 1 -t t The contour curves of B: {(s, t) | B(s, t) = a } are hyperbolas

Asymptotic Decider (2) (1, 1) (0, 0) Where the hyperbolas go through the cell depends on the values at the corners, I. e. , B 00, B 01, B 10, B 11

Asymptotic Decider (3) (1, 1) (Sa, Ta) If (0, 0) Asymptote a < B(Sa, Ta)

Asymptotic Decider (4) (1, 1) (Sa, Ta) If (0, 0) Asymptote a> B(Sa, Ta)

Asymptotic Decider (5) (1, 1) (Sa, Ta) Sa = B 00 - B 01 B 00 + B 11 – B 01 – B 10 Ta= B 00 – B 10 B 00 + B 11 – B 01 – B 10 (0, 0) B(Sa, Ta) = B 00 B 11 + B 10 B 01 B 00 + B 11 – B 01 – B 10

Asymptotic Decider (6) Based on the result of asymptotic decider, we expand the marching cube case 3, 6, 12, 10, 7, 13 (These are the cases with at least one ambiguious faces). Let’s look at Nielon and Hamann’s paper…