Finding near neighbors

Question

I need to find "near" neighbors among a set of points.

There are 10 points in the above image. Red lines are edges from the Delaunay Triangulation, black stars mark the mid-lines of the edges, blue lines are the Voronoi tesselation. Point 1 has three "near" neighbors, i.e. 4, 6, and 7, but not 2 and 3, who are almost in line with the edge 1-7, but much further away.

What is a good way to identify the near neighbors (or "good" edges)? Looking at the figure, it seems to me that either selecting edges whose mid-point falls onto the intersection with the Voronoi lines, or considering as "near" neighbors those with touching Voronoi cells could be a good solution (the classification of 3-5 can go either way). Is there an efficient way of implementing either of the solutions in Matlab (I'd be happy to get a good general algorithm that I can then translate to Matlab, btw)?

Answer 1

You can implement your first idea of selecting edges whose mid-points fall on the intersection with the Voronoi lines by making use of the DelaunayTri class and its edges and nearestNeighbor methods. Here's an example with 10 random pairs of x and y values:

x = rand(10,1);                     %# Random x data
y = rand(10,1);                     %# Random y data
dt = DelaunayTri(x,y);              %# Compute the Delaunay triangulation
edgeIndex = edges(dt);              %# Triangulation edge indices
midpts = [mean(x(edgeIndex),2) ...  %# Triangulation edge midpoints
          mean(y(edgeIndex),2)];
nearIndex = nearestNeighbor(dt,midpts);  %# Find the vertex nearest the midpoints
keepIndex = (nearIndex == edgeIndex(:,1)) | ...  %# Find the edges where the
            (nearIndex == edgeIndex(:,2));       %#   midpoint is not closer to
                                                 %#   another vertex than it is
                                                 %#   to one of its end vertices
edgeIndex = edgeIndex(keepIndex,:);      %# The "good" edges

And now edgeIndex is an N-by-2 matrix where each row contains the indices into x and y for one edge that defines a "near" connection. The following plot illustrates the Delaunay triangulation (red lines), Voronoi diagram (blue lines), midpoints of the triangulation edges (black asterisks), and the "good" edges that remain in edgeIndex (thick red lines):

triplot(dt,'r');  %# Plot the Delaunay triangulation
hold on;          %# Add to the plot
plot(x(edgeIndex).',y(edgeIndex).','r-','LineWidth',3);  %# Plot the "good" edges
voronoi(dt,'b');  %# Plot the Voronoi diagram
plot(midpts(:,1),midpts(:,2),'k*');  %# Plot the triangulation edge midpoints

How it works...

The Voronoi diagram is comprised of a series of Voronoi polygons, or cells. In the above image, each cell represents the region around a given triangulation vertex which encloses all the points in space that are closer to that vertex than any other vertex. As a result of this, when you have 2 vertices that aren't close to any other vertices (like vertices 6 and 8 in your image) then the midpoint of the line joining those vertices falls on the separating line between the Voronoi cells for the vertices.

However, when there is a third vertex that is close to the line joining 2 given vertices then the Voronoi cell for the third vertex may extend between the 2 given vertices, crossing the line joining them and enclosing that lines midpoint. This third vertex can therefore be considered a "nearer" neighbor to the 2 given vertices than the 2 vertices are to each other. In your image, the Voronoi cell for vertex 7 extends into the region between vertices 1 and 2 (and 1 and 3), so vertex 7 is considered a nearer neighbor to vertex 1 than vertex 2 (or 3) is.

In some cases, this algorithm may not consider two vertices as "near" neighbors even though their Voronoi cells touch. Vertices 3 and 5 in your image are an example of this, where vertex 2 is considered a nearer neighbor to vertices 3 or 5 than vertices 3 or 5 are to each other.