1. Bigyan Ankur Mukherjee University of Utah

2. Given a set of Points P sampled from a surface Σ,  Find a Surface Σ * that “approximates” Σ  Σ * is generally given as a polygon mesh

5.  P+ : Farthest Vertex in Voronoi Cell V p (adjusted for unbounded case)  P- : Farthest vertex in V x ∃ < 0  The vector P + P - (Pole Vector) approximates the normal at p P+P+ P-P- P+ P-

6.  Draw two cones at p with angle 3 π/8 around the pole vector  The region complementary to these two cones and clipped by the Voronoi Cell is the co-cone at point p

7. 1. Compute Delaunay Triangulation D 2. Consider only those Triangles in D that intersects all the three co-cones at the three vertices [FILTER] 3. Delete all Triangles incident on “Sharp Edges”[PRUNE] 4. Extract a manifold out of the remaining Triangles[WALK]

8.  Compute Voronoi Diagram  For each point p  Store the pole vector (p)= P + P - with p  For each triangle t in D  For each vertex v of t  If (v) or - (v) makes an angle greater than π/8 with the normal to t,  Remove t from D

9.  Pending = Ф  For each edge e in D  Pending.push(e)  While Pending ≠Ф  e = Pending.pop()  If e is sharp  For each triangle t incident on e  Remove t from D An edge is sharp if there is only one triangle incident on it

10.  For each triangle t in D  Orient the edges of t w.r.t. a global orientation  Surface = Ф  Choose one triangle t from D that is on the convex hull  Surface.insert (t)  Grow the surface starting from t by walking along the edges and choosing the neighboring triangle that best fits the orientation of current triangle at each step

12. 1. Voronoi-based:  For each point p, the pole vector (v) gives the estimated normal 2. Principal Component Analysis  Fit a linear least square plane to all points inside a ball of predefined radius and return the normal The normals obtained in both cases are not oriented  For orienting normals, we use the algorithm presented in Hoppe et.al. ‘92

13.  From a point set with oriented normals, computes a scalar field which is zero inside the surface and >0 outside (by solving a PDE)  Extracts the zero level set of the surface