Persistence-sensitive simplification of a terrain

We wrote a preliminary implementation using the python language and specialized to the topological simplification of terrains. The terrain is a triangulated 2D grid whose simplices are assigned a height value in [0, 1]. The terrain is made manifold by gluing dummy triangles from the boundary of the terrain to a dummy vertex. The dummy simplices are assigned height greater that one. In practice, this is not ideal, as the many dummy triangles and edges tend to interfere with the pairing of the actual terrain’s simplices; it would be preferable to use a single dummy 2-dimensional face whose boundary spans the whole terrain boundary edges.

• The left image is the initial terrain.
• The middle image is simplified with epsilon = 0.2.
• The right image is fully simplified.
• Note: When simplifying a subtree of the spanning tree, we did not redistribute the height values evenly among the subtree's vertices. This results in the many small flats around each vertex. They appear when the barycentric subdivision is computed.
• The name of the game is to spot the new (reddish) ridges and (blueish) valleys that appear when the simplification threshold is raised. Many of them are difficult to see. Can you spot them all ?
• Top view, constant shading:
  
• Top view, smooth shading:
  
• Front view, constant shading:
  
• Front view, smooth shading:
  
• Initial terrain:
  
• Simplified with epsilon=0.2:
  
• Fully simplified: