How to calculate a Delaunay mesh nerve?

Question

How to calculate a Delaunay mesh nerve? A Delaunay mesh nerve is a collection of triangles that surround a single triangle called the nerve nucleus. The triangles in a Delaunay mesh nerve either have a common edge or a common vertex with the nerve nucleus. Here, nerve nucleus means that triangle which has maximum area in a Delaunay mesh.

Here is the code I developed to find the Delaunay mesh. But, I can't calculate the mesh nerve. Please kindly let me know. Thanks.

c = ImageCorners[img, MaxFeatures -> 200];
dm = DelaunayMesh[c];

Answer 1

The approach I would take is the following:

1. Clear definitions, load a sample image, find corners in the image using the method you indicated in the OP:

   Clear[img, points, dm, cells, nerve, nervenucleus]
   img = ExampleData[{"TestImage", "Aerial2"}];
   points = ImageCorners[img, MaxFeatures -> 200];
   

2. Generate a Delaunay mesh from those points:

   dm = DelaunayMesh[
          points, PlotTheme -> "Lines",
          MeshCellStyle -> Directive[EdgeForm[GrayLevel[0.3]], Opacity[0]]
        ];
   
   Show[img, dm]
   

   

3. Extract the coordinates for the two-dimensional cells (triangles) that make up the mesh:

   cells = MeshPrimitives[dm, 2];
   

4. Find the "nucleus", which you defined as the triangle with the largest area in the mesh:

   nucleus = First@MaximalBy[cells, Area];
   Show[img, dm, Graphics[{Opacity[0.4], Red, nucleus}]]
   

   

5. Find the "nerve", that you defined as the set of cells that share a vertex or an edge with the nucleus (note that the nucleus itself is not included in this list thanks to the second part of the condition):

   nerve = Cases[
      cells,
      Polygon[{p1_, p2_, p3_}] /;
       (
        ContainsAny[{p1, p2, p3}, nucleus[[1]] ] && {p1, p2, p3} != nucleus[[1]]
       )
     ];
   

6. Show all results:

   Show[img, dm, Graphics[{Opacity[0.4], Red, nucleus, Darker@Green, nerve}]]