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Efficient Polygon Reduction Techniques for Shape Simplification

Learn about advanced methods for reducing polygon count while maintaining shape appearance. Uncover edge collapse, vertex removal, and face collapse operations to streamline your models efficiently. Discover error metrics such as the Quadratic Error Function (QEF) and optimal vertex placement strategies. Dive into the Surface Simplification Algorithm, QEF usage, and the Lazy Method for seamless shape simplification. Explore how to ensure topological consistency while collapsing edges and leveraging memory usage for speed.

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Efficient Polygon Reduction Techniques for Shape Simplification

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  1. Surface Simplification Dr. Scott Schaefer

  2. Surface Simplification • Given a closed polygon model, reduce the number of polygons and maintain appearance of the shape 5804 tris 2500 tris 1000 tris 500 tris

  3. Reducing Polygons • Perform local, topological operations to reduce number of polygons • Vertex removal

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

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

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

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

  8. Error Metrics For Simplification • QEF: Quadratic Error Function • Measures distance to infinite planes

  9. Error Metrics For Simplification • QEF: Quadratic Error Function • Measures distance to infinite planes

  10. Error Metrics For Simplification • QEF: Quadratic Error Function • Measures distance to infinite planes symmetric 3x3 3x1 1x1

  11. Error Metrics For Simplification • QEF: Quadratic Error Function • Measures distance to infinite planes symmetric 3x3 3x1 1x1 Requires 10 floats independent of number of polygons!!!

  12. Combining QEFS

  13. Combining QEFS Add 10 numbers to combine QEFs!!!

  14. Placement of Vertices Using QEFs • Place new vertex at minimum of error function

  15. Placement of Vertices Using QEFs • Place new vertex at minimum of error function

  16. Placement of Vertices Using QEFs • Place new vertex at minimum of error function

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

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

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

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

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

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

  23. 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. 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. Surface Simplification: Edge Collapse • Place new vertex at minimizer of QEF

  26. Surface Simplification: Edge Collapse • Place new vertex at minimizer of QEF • QEF of new vertex is combined QEF

  27. 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. 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. Surface Simplification: Lazy Method • Place new vertex at minimizer of QEF • QEF of new vertex is combined QEF • Mark all edges touching collapsed edge as dirty

  30. Surface Simplification: Lazy Method • Place new vertex at minimizer of QEF • QEF of new vertex is combined QEF • Mark all edges touching collapsed edge as dirty • Mark edges to be removed as invalid

  31. Surface Simplification: Lazy Method • Place new vertex at minimizer of QEF • QEF of new vertex is combined QEF • Mark all edges touching collapsed edge as dirty • Mark edges to be removed as invalid • When removing an edge from the priority queue, throw away if invalid. If dirty, recompute QEF and insert into priority queue

  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

  33. 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

  34. 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

  35. Surface Simplification • Extremely fast • Somewhat memory intensive • Limits maximum surface size • Greedy algorithm • Does not guarantee optimal sequence of edge collapses!!!

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