1. Progressive Meshes with Controlled Topology Modification University of Bonn Institute II. for Computer Science Computer Graphics Group Pavcl Borodin Rchinhard Klcin

2. Introduction ● Progressive Meshes Which was introduced by Hoppe(1996) ● Retain the topology of original model Vertex Contraction Edge Contraction ● To sew the small cracks,gaps,holes Utilizing Vertex-Edge Contraction(description in this paper)

3. Background Edge Contraction Unifying two vertices lying on a common edge

4. Background Vertex Contraction Which is essentially a generalization of the edge contraction,the difference is that vertices not necessarily connected by an edge are contracted

5. Background Vertex-Edge Contraction

6. The Operator of Vertex-edge Contraction 1. Project the vertex v orthogonally onto the edge e. 2. Insert a vertex v ’ into the edge at the geometric position of the projection. 3. Split the triangle t1 into two triangles t1’ = ( v0, v ’, v2 ) and t2 = ( v1, v2, v ’). 4. Perform a vertex contraction to v and v ’. The new position of the vertex will be a convex combination of v and v ’, we move the new vertex Vnew into position l v +( 1 −l) v ’.

7. Error Measure The most appropriate error measure is simply the Euclidean distance Euclidean distance: The straight line distance between two points. In a plane with p 1 at (x 1, y 1 ) and p 2 at (x 2, y 2 ), it is √((x 1 - x 2 ) ² + (y 1 - y 2 ) ²).

8. Algorithm 1. degenerate faces without finite area 2. unwanted gaps and cracks between regions of the mesh resulting from erroneous scan data reconstruction or modeling and/or tessellation of analytical surfaces 3. holes in the model due to missing polygons 4. T-vertices lying on interior of an edge of a face The mesh to be processed by our method will possibly include the following artifacts:

9. Algorithm Preprocessing Phase 1. Reading the mesh 2. Identification of boundaries 3. Identification of corresponding vertex-vertex and vertex-edge pairs 4. Computation of error 5. Ordering according to error

10. Boundary Edges: The edges having only one incident triangle Boundary Vertices: The vertices are simply incident to boundary edge

11. Algorithm Decimation 1. Pop the boundary vertex or inner edge with minimal error from the priority queue. If the error e of the popped feature is larger than a prescribed error threshold e then STOP. 2. Perform the according operation, i.e. vertex-edge contraction for a boundary vertex, or an edge contraction for an inner edge.

12. Algorithm Decimation 3. Encode the modification and store in the progressive data structure. 4. Recompute the error for vertices and edges influenced by the operation, accordingly maintain the priority queue. GOTO 1.

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