6.4.2 Interpolating the disambiguation value.

Our disambiguation rule is similar to and generalizes the bilinear interpolation method of Nelson (See [6] ) to disambiguate Marching Cubes (See [7], [8], [9], and [10]). We multilinearly interpolate the disambiguation value c.

Note that the existence of such a value c (so that above c the diagonally opposite Highs are not connected in the face and below c they are) not dependent on linear interpolation, as demonstrated in the discussion above. Regardless of the interpolation method, there will be a value at which they become connected. In higher dimensions the location of the point at which the local components containing the diagonally opposite Highs first meet may be in the cube interior. Still there will be a value at which the two Highs, if initially disconnected, become part of the same level set component within the cubes that share the face. This will only be important if there is no path between the two vertices that passes through any other High vertex within the hypercubes that share the face, at the disambiguation value (see the definition of critical 4-hit faces below and figure 33 through figure 39).

Definition 9  

Disambiguation Rule: In 2 dimensions we do the following: For each 4-hit face we linearly interpolate $\delta$ across each edge. Now interpolate the position of the point p, called the disambiguation point, such that both the vertical and horizontal lines (with respect to the two coordinate directions on the face) that pass through this point intersect identical values on the opposite edges. Interpolate the value $c = \delta(p)$ of this point, called the disambiguation value, and we extend the domain of $\delta$ to include p (we call this the extended $\delta$). For specific $\tau$, if $\tau > c$ regard the Lows as adjacent (knit High), else regard the Highs as adjacent (knit Low).

See figure 10 and figure 11 for clarification. Assume without loss of generality that the face is in the x, y plane and that the origin of the face is x=i y=j.

Let

$$\delta (i, j) = A$$

$$\delta (i+1, j) = B$$

$$\delta (i+1, j+1) = C$$

$$\delta (i, j+1) = D$$

The location of the disambiguation point is given by

i + x₀, j + y₀

where

x₀= (A-B) / ((A+C) - (B+D))

and

y₀= (A-D) / ((A+C) - (B+D))

The disambiguation value

$$c = \delta(i+x_0, j+y_0) = A+ (D-A)x_0$$

In 3 dimensions we use trilinear interpolation to determine the maximum value c at which the diagonally opposite Highs are path connected through the interior of either cube that shares the face F. In higher dimensions we similarly use multi-linear interpolation to choose the maximum value c for which the Highs are connected through the interior of any cube sharing F. Below this value we will regard the Highs as path connected through F. Note that the path may not actually be through the face F, but since we are concerned with topology, this will not change our results. The interior point p so interpolated is called the disambiguation point and is associated with the face F.

For example the disambiguation point of figure 11 is $x = \frac{4}{9}$, $y= \frac{4}{9}$ and the disambiguation value is $92 \frac{2}{9}$.