1 Dr. Scott Schaefer Surface Simplification

2/32 Surface Simplification Given a closed polygon model, reduce the number of polygons and maintain appearance of the shape 5804 tris2500 tris1000 tris500 tris

3/32 Reducing Polygons Perform local, topological operations to reduce number of polygons Vertex removal

4/32 Reducing Polygons Perform local, topological operations to reduce number of polygons Vertex removal Edge Collapse

5/32 Reducing Polygons Perform local, topological operations to reduce number of polygons Vertex removal Edge Collapse Face Collapse, …

6/32 Reducing Polygons Perform local, topological operations to reduce number of polygons Vertex removal Edge Collapse Face Collapse, …

7/32 Surface Simplification How do we determine the order of edge collapse operations? Where do we place new vertex after collapse?

8/32 Error Metrics For Simplification QEF: Quadratic Error Function Measures distance to infinite planes

9/32 Error Metrics For Simplification QEF: Quadratic Error Function Measures distance to infinite planes

10/32 Error Metrics For Simplification QEF: Quadratic Error Function Measures distance to infinite planes symmetric 3x33x11x1

11/32 Error Metrics For Simplification QEF: Quadratic Error Function Measures distance to infinite planes symmetric 3x33x11x1 Requires 10 floats independent of number of polygons!!!

12/32 Combining QEFS

13/32 Combining QEFS Add 10 numbers to combine QEFs!!!

14/32 Placement of Vertices Using QEFs Place new vertex at minimum of error function

15/32 Placement of Vertices Using QEFs Place new vertex at minimum of error function

16/32 Placement of Vertices Using QEFs Place new vertex at minimum of error function

17/32 Placement of Vertices Using QEFs Place new vertex at minimum of error function Not invertible in flat areas or straight edges!!!

18/32 Placement of Vertices Using QEFs Place new vertex at minimum of error function Pseudoinverse minimizes |v|

19/32 Placement of Vertices Using QEFs Let where c is a point we want to minimize the distance to

20/32 Placement of Vertices Using QEFs Let where c is a point we want to minimize the distance to

21/32 Placement of Vertices Using QEFs Let where c is a point we want to minimize the distance to

22/32 Placement of Vertices Using QEFs Let where c is a point we want to minimize the distance to

23/32 Plane-Based Quadratic Error Function Compact representation (10 numbers) Fast to combine multiple functions (addition) Relatively easy to minimize (pseudoinverse) Suffers from numerical instabilities

24/32 Surface Simplification Algorithm Build QEFs for each vertex For each edge  Compute combined QEF and error  Insert edge into priority queue sorted by error While poly # > target #  Collapse edge

25/32 Surface Simplification: Edge Collapse Place new vertex at minimizer of QEF

26/32 Surface Simplification: Edge Collapse Place new vertex at minimizer of QEF QEF of new vertex is combined QEF

27/32 Surface Simplification: Edge Collapse Place new vertex at minimizer of QEF QEF of new vertex is combined QEF Remove all edges touching collapsed edge from priority queue

28/32 Surface Simplification: Edge Collapse Place new vertex at minimizer of QEF QEF of new vertex is combined QEF Remove all edges touching collapsed edge from priority queue Recompute QEF/error for all edges touching new vertex and insert into priority queue

29/32 Surface Simplification: Edge Collapse Edge collapse may alter topology of surface Test for topology change and exclude unsafe edge collapses Unsafe edge may become safe after another collapse Alternatively, perform two edge collapses

30/32 Surface Simplification: Edge Collapse Edge collapse may alter topology of surface Test for topology change and exclude unsafe edge collapses Unsafe edge may become safe after another collapse Alternatively, perform two edge collapses unsafe edge collapses

31/32 Surface Simplification: Edge Collapse Edge collapse may alter topology of surface Test for topology change and exclude unsafe edge collapses Unsafe edge may become safe after another collapse Alternatively, perform two edge collapses

32/32 Surface Simplification Extremely fast Somewhat memory intensive  Limits maximum surface size Greedy algorithm  Does not guarantee optimal sequence of edge collapses!!!