# Merging Discrete Morse Vector Fields: A Case of Stubborn Geometric Parallelization
## 2 Discrete Gradient Vector Fields  
All complexes in this paper are regular CW complexes in the sense that characteristic maps of all cells are homeomorphisms onto the closures of cells in the complex. We refer, for the general background in discrete Morse theory on regular CW complexes, to [3]. **Notation 1**. : _Suppose \(K\) is a regular complex, and suppose \(f\) is a discrete Morse function on \(K\). Then there are well-defined sets of critical cells \(C(K,f)\) and the discrete vector field that can be thought of as a set of pairs of regular cells \(V(K,f)\) or \(V(K)\). When we use the notation \((\sigma\leq\tau)\) for a pair in \(V(K)\), we mean that \(\sigma\) is a codimension \(1\) face of \(\tau\)._  
**Definition 1**. : _We introduce the following relation among cells in \(K\). Two cells \(\sigma_{1}\) and \(\sigma_{2}\) are adjacent if the closures of the two cells in \(K\) have a point in common. The star of a cell \(\sigma\) is the smallest cellular subcomplex of \(K\) that contains all cells adjacent to \(\sigma\). Given a collection of cells \(\mathcal{C}\) of \(K\), let the star of \(\mathcal{C}\) be the smallest subcomplex that contains the stars of all of the cells in \(\mathcal{C}\). If \(K^{\prime}\) is a subcomplex, we also call its star the \(1\)-neighborhood of \(K^{\prime}\) denoted \(K^{\prime}[1]\). The star of a star is the \(2\)-neighborhood, etc. We use the notation \(K^{\prime}[n]\) for the \(n\)-neighborhood of \(K^{\prime}\). The \(k\)-border of a subcomplex \(L\) of \(K\) consists of those cells in \(L\) whose \(k\)-neighborhoods are not contained entirely in \(L\)._  
We assume that the lists \(C(K,f)\) and \(V(K,f)\) are results of applying an algorithm \(\alpha:K\mapsto V\) of the following nature. **Definition 2**. : _Given a face-cofree pair of cells \((\sigma\leq\tau)\), the algorithm decides whether the pair \((\sigma\leq\tau)\) is placed in the list \(V(K)\) based on data specific to \(\alpha\) coming from some \(k\)-neighborhood \(\sigma[k]\), for some uniformly fixed number \(k>0\). The result of the algorithm is a discrete vector field \(V_{\alpha}(K)\) which can be therefore realized as \(V(K,f)\) for some discrete Morse function \(f\). We say that \(\alpha\) with this property is uniformly local or, more precisely, uniformly \(k\)-local._  
In fact, in all applications that generate \(V(K)\) that the authors see in the literature, it is always done using a uniformly \(1\)-local algorithm. This is not surprising, of course, as traditional smooth vector fields are generated through differentiation procedures, which are inherently local. **Example 1**. : _An example of such an algorithm is ProcessLowerStars, which produces \(V(K)\), starting with an arbitrary positive bounded function (also known as a grayscale function) on cells (pixels or voxels) of a \(2D\) or \(3D\) pixel grid \(K\). Since the algorithm qualifies \((\sigma\leq\tau)\) based on the values of a function on vertices in \(\sigma[1]\), this algorithm is uniformly \(k\)-local for \(k=1\)._  
_To be more specific, to make these images into a regular CW complex, we translate them to cubical complexes by making the pixels or voxels correspond to \(0\)-cells of the cubical complex, \(K\). This gives us a positive bounded function, called \(g\), on a cubical complex: more specifically, a cubulated plane or space. To apply the algorithm, it is ideal to have the values of \(g\) on all of the \(0\)-cells be unique. If they are not, one can make small changes (i.e., a linear ramp) to \(g\) to ensure they are unique. Once there are unique values of \(g\), the algorithm inspects all of the cells in the lower star of a \(0\)-cell, \(x\). The lower star of \(x\), \(L(x)\), contains all cells \(\sigma\in K\) in the cubical complex such that \(g(x)=\max\limits_{y\in\sigma}g(y)\).