4.1 Axiomatic treatment of level set boundary interpolation problem.

First let us summarize the previous discussion by giving the definitions of connected sets of lattice points, or hypercube vertices, of $\bf {X}_{\delta}^{\geq \tau}$ and its complement. These definitions are reminiscent of both graph connectivity and digital topology (See [11],[12], [13]).

In the following, we assume that f interpolates the extended $\delta$.

Definition 11  

For a given threshold $\tau$, two High (respectively Low) vertices are adjacent iff they are edge adjacent or diagonally adjacent according to the disambiguation rule at $\tau$. Path connectivity is the transitive closure of the adjacency relation. A set $S \subset \bf {Z}^n$ (of lattice vertices) is connected iff there is path, completely contained in the set, between each pair of vertices. A connected set of High (resp. Low) vertices is defined to be a path-connected set containing only High (resp. Low) vertices. A connected component of Highs (resp. Lows) is just a maximal path connected set of Highs (resp. Lows).

Definition 12   We say that two sets of points overlap if they have nonempty intersection. Given a finite set S of oriented manifolds, we define the relative interior of a manifold M to be the intersection of the interior of M with all interiors of manifolds in S that overlap the interior of M. We define the relative exterior of M similarly.

Definition 13  

By a d-dimensional hypercube we mean a hypercube defined by 2^d adjacent integer lattice points, so that the boundary of a d-dimensional hypercube will consist of d-1-dimensional hypercubes.

Definition 14  

Let C be a d-dimensional hypercube and M a component of $B (\bf {X}_{f}^{\geq \tau})$. A (hyper)surface patch of M, with respect to C, is a d-1-dimensional connected component of $M \cap C$.

Definition 15  

Toward developing a Digital Morse Theory, let us make the following assumptions (or axioms) on the components (iso-surfaces) of $B (\bf {X}_{f}^{\geq \tau})$, for each $\tau$ not in the range of the extended $\delta$. If an interpolation function satisfies the axioms it is called a Digital Morse interpolant.