1. Mesh Sekeltonization פרויקט סוף תואר איל רז מנחה – ג ' יהאד אל - סנה

2. Context  Intro  Definitions  The Algorithm  Model Loading and 3D triangulation  Distance Calculation  Tree Building  Mesh & Tree Simplification

3. Intro  Mesh Skeletonization – The aim of the skeletonization is to extract a region-based shape feature representing the general form of an object

4. 3D triangulation (Tetrahedralization)  3D equivalent of triangulation  decompose a given polytope (geometric object with flat sides) into non- overlapping tetrahedra, where the vertices of the tetrahedra must be vertices of the original polytope

5. Mesh Representation  In CGAL, a 3D triangulation data structure is a container of cells and vertices  Each cell gives access to its four incident vertices and to its four adjacent cells. Each vertex gives direct access to one of its incident cells, which is sufficient to retrieve all the incident cells when needed.  The four vertices of a cell are indexed with 0, 1, 2 and 3. The neighbors of a cell are also indexed with 0, 1, 2, 3 in such a way that the neighbor indexed by i is opposite to the vertex with the same index

6. Distance Calculation

7. Scan Vertices & Build Tree  From most inner vertex do:  For each u  Adj(v)  If d(v) >= d(u) and u has no inbound edges add edge (v,u)

8. Problems  More than one tree may appear, due to the fact that the next most inner vertex is not a neighbor to the current one  Solved by checking the number of roots (a vertex with no parent)  If more than one exists – find the nearest leaf of the closest tree and add an edge from the leaf to the root

9. Mesh Simplification  Performe Mesh surface simplification (reduces the number of vertices and faces on the surface)  For each vertex v that was removed by the simplification, remove the edge (u,v)

11. Implementation  Implemented in Visual Studio C++ with CGAL and OpenGL libraries  2 Classes  I/O – used to load an off file and save the Tetrahedralized mesh  Analyzer – performing all the operations on the mesh