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By N.E. Norlund

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3 We consider a two-dimensional discrete Morse function (a) and its discrete gradient field (b). The ascending manifold of the bottom left minimum is identical to the descending manifold of the maximum. These cells are highlighted in (c). The intersection of these manifolds is every cell except for the middle vertex. However, as illustrated in (d) and (e), there are two distinct quads of the two-dimensional MS complex in this intersection Fig. 4 Gradient vectors are shown for cells contained in V-paths originating at the 2-saddle (red quad) in a volumetric discrete gradient field.

Testing this theory hinges on extracting valid cells; as we will show, previous techniques fall short of computing all cells of three-dimensional MS complexes. The following contributions are made in this paper: • We identify the challenges in extracting topologically valid cells from a discrete gradient vector field. • We present an algorithm that computes the distinct cells of the MS complex connecting two critical points. • We propose data structures to enable an efficient implementation of the algorithm.

Each of the stages of the algorithm maintains both of the invariants stated above. The initial condition satisfies both invariants trivially: the oriented boundary of a 2-cell is an alternating list of 0- and 1-cells; the 2-cell and its faces form a disk. Extending the disk replaces a 1-cell on the boundary with a sequence of 1-, 0-, 1-, 0-, and 1-cells, maintaining the first invariant. Each zip step removes a contiguous even length sequence from the boundary, and each corner extension step in finalizing the boundary adds a 1-, 0-, 1-, 0-cell sequence right after a 0-cell; both these steps maintain the first invariant.

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