A Matlab Implementation of a Flat Norm Motivated Polygonal Edge Matching Method using a Decomposition of Boundary into Four 1-Dimensional Currents

We describe and provide code and examples for a polygonal edge matching method. Our goal is to find a score to match two polygons P1 and P2 embedded in a rectangle R of the plane, of height I and width J. Using a pixel based representation of the polygon we find pixel based representations of the boundary of each polygon, with four matrices representing top, bottom left and right edges separately. We smooth with a Gaussian kernel, enabling matching of coincident edges and nearby edges. We match top edges to top edges, left edges to left edges and so on. Not allowing cancellation between left and right edges, or between top and bottom edges, gives more sensitivity. Polygons which represent templates and images or occluded images where only part of the original boundary is present can be matched. Polygons which differ by small deformations can be matched. The code here can be combined with registration techniques if required. Following [2], a distance function can be defined using an edge matching score. E(P1,P2) for the two polygons. We can define it as d(P1,P2)=E(P1,P1)+E(P2,P2)-2E(P1,P2). The capability of partial matching of nearby boundary or curves is analogous to properties the flat norm on currents. See [3] for an introduction to currents and the flat norm. Existing implementations of the flat norm [1],[2] however allow for cancellation between top and bottom edges and cancellation between left and right edges within the same polygon. This would prevent the match shown in figure 4 of the slender regions protruding to the right of the two polygons that do not intersect. The four types of edges can also be used to represent unions of oriented curves in the plane. This can include graphs and trees. In this general setting we can consider the four matrices as a decomposition of the curves into four currents. Figure 1: Correspondences between top, bottom, left and right boundary edges and oriented curves With an appropriate sign convention as used in oriented boundary integrals with Stokes theorem, the top edges can be interpreted as horizontal components of oriented curves going to the left, bottom edges as horizontal components of oriented curve going to the right, left edges as vertical components of oriented curves pointing down, and right edges as vertical components of oriented curves pointing up. For unoriented curve matching we only make a distinction between vertical and horizontal components, requiring only two matrices; all vertical components of a curve are represented with a positive number in the vertical component matrix. This unoriented case corresponds to a decomposition into two varifolds[3], one for vertical and one for horizontal. Top