Surface Simplification Using Quadric Error Metrics

1. Surface Simplification Using Quadric Error Metrics By Michael Garland and Paul S. Heckbert Carnegie Mellon University Presented by Lok Hwa and Taylor Holliday Multiresolution (ECS 289L) - Winter 2003

2. Background / Related Work • Vertex Decimation(Schroeder et al 92; Soucy-Laurendeau 96)- usually limited to non-manifold surfaces- carefully maintains the model topology • Vertex Clustering(Rossignac-Borrel 93; Low-Tan 96)- fast and general, but quality is usually comparably lower- cannot input a specific face count; needs more control- can alter topology drastically • Iterative Edge Contraction(Hoppe 96; Gueziec 95; Ronfard-Rossignac 96)- mainly designed for manifold surfaces- can close holes but not join unconnected regions Multiresolution (ECS 289L) - Winter 2003

3. Introduction Iterative Pair Contraction Multiresolution (ECS 289L) - Winter 2003

4. Cow Simplification in under 1 second Multiresolution (ECS 289L) - Winter 2003

5. Advantages Efficiency: fast running time and error approximation is compact (10 floating point numbers per vertex) Quality: Main features are preserved even in highly simplified models Generality: Joins unconnected regions (aggregation) Better approximations of many disconnected parts. Works on non-manifold surface and actually creates non-manifold surfaces. Multiresolution (ECS 289L) - Winter 2003

6. Algorithm Setup • Polygon models consisting of triangles only • Better results achieved if intersecting triangle corners are defined as a shared vertex • Applications are for rendering systems, not visualization • Input is typically either a desired face count or maximum tolerable error Multiresolution (ECS 289L) - Winter 2003

7. Algorithm: Vertex Pair Contraction • (v1, v2)  v • Movev1 and v2to v • Replace all occurances of v2withv1 • Removev2 and degenerate triangles • Valid Pairs • Edges • ||v1 - v2 || < t , where t is a threshhold parameter Multiresolution (ECS 289L) - Winter 2003

8. Algorithm: Aggregation Multiresolution (ECS 289L) - Winter 2003

9. Using Quadrics to Approximate Error • Qis a 4x4 symmetric matrix representing the error at each vertex • Evaluated at vertex v = [vx vy vz 1]:(V) =vTQv • Need a new Q for contraction(v1, v2)  v Q = Q1 + Q2 • Want to minimize (v) Multiresolution (ECS 289L) - Winter 2003

10. Using Quadrics to Approximate Error If matrix is not invertible, then try to find optimal vertex along segment v1v2, else choose from v1 v2 or midpoint Multiresolution (ECS 289L) - Winter 2003

11. Algorithm Summary • Start by representing the models in an adjacency graph structure:vertices, edges, and faces all explicitly linked together. - Keep in mind the mesh must handle arbitrary topology • Each vertex maintains a list of the pairs of which it is a member. Multiresolution (ECS 289L) - Winter 2003

12. Algorithm Summary • ComputeQmatrices for all the initial vertices. • Select all the valid pairs • Compute the optimal contraction target for each valid pair. The error becomes the cost of contracting that pair. • Place all the pairs in a priority queue keyed on cost with the minimum cost pair at the front Multiresolution (ECS 289L) - Winter 2003

13. Algorithm Summary (cont..) • Iteratively remove the least cost pair (v1, v2), contract it, and updated the costs of all valid pairs involving v1 until the simplification goals are satisfied Multiresolution (ECS 289L) - Winter 2003

14. Derivation of QEMs Multiresolution (ECS 289L) - Winter 2003

15. Derivation of QEMs • Intuitively: sum of squared distances from a set of planes • First, we show that squared distance to a plane can be represented as a quadratic form Multiresolution (ECS 289L) - Winter 2003

16. Derivation of QEMs Multiresolution (ECS 289L) - Winter 2003

17. Derivation of QEMs Multiresolution (ECS 289L) - Winter 2003

18. Derivation of QEMs Multiresolution (ECS 289L) - Winter 2003

19. Derivation of QEMs Multiresolution (ECS 289L) - Winter 2003

20. Derivation of QEMs • Now we combine these quadrics for a set of planes Multiresolution (ECS 289L) - Winter 2003

21. Derivation of QEMs • Now we combine these quadrics for a set of planes • Note Q is positive definite Multiresolution (ECS 289L) - Winter 2003

22. Geometric Interpretation • Vertex errors have either planar, cylindrical, or ellipsoid level surfaces Multiresolution (ECS 289L) - Winter 2003

23. Geometric Interpretation • Why ellipsoids? If Q is diagonalizable, Multiresolution (ECS 289L) - Winter 2003

24. Geometric Interpretation • Why ellipsoids? If Q is diagonalizable, • Let Multiresolution (ECS 289L) - Winter 2003

25. Geometric Interpretation • Why ellipsoids? If Q is diagonalizable, • Let • Apply the rotation y = ST v Multiresolution (ECS 289L) - Winter 2003

26. Details • Recall: minimize • Do not double-count planes on an original vertex • Planes can still be double-counted if they coincide between vertices Multiresolution (ECS 289L) - Winter 2003

27. Preserving Boundaries • Error quadrics do not make allowance for boundary edges - such as those found in terrains • We want to simplify the shape, but preserve the boundary • Generate heavily weighted planes perpendicular to boundary triangles Multiresolution (ECS 289L) - Winter 2003

28. Preventing Mesh Inversion • prevent normals from changing direction by more than (“flipping”) • Contractions resulting in flipped normals are rejected Multiresolution (ECS 289L) - Winter 2003

29. Results Multiresolution (ECS 289L) - Winter 2003

30. Evaluation Metric • Need a metric for approximation error Multiresolution (ECS 289L) - Winter 2003

31. Effect of vertex placement Multiresolution (ECS 289L) - Winter 2003

32. Bunny Model Multiresolution (ECS 289L) - Winter 2003

33. Crater Lake Model Multiresolution (ECS 289L) - Winter 2003

34. Timings • t is non-edge contraction threshold • Crater Lake was a large dataset in 1997 Multiresolution (ECS 289L) - Winter 2003

35. Foot Model Multiresolution (ECS 289L) - Winter 2003

36. Aggregation • Recall: aggregation is the joining of disconnected components • Aggregation effects approximation error Multiresolution (ECS 289L) - Winter 2003

37. Discussion • Primary weakness of QEMs: measuring the distance to a set of planes only works well in local neighborhoods • Note that there is usually no longer a zero-distance point after we add quadrics • Removed planes are still counted during simplification • No easy way to subtract them from quadrics Multiresolution (ECS 289L) - Winter 2003

38. Conclusion • Quality • Only other algorithm supporting aggregation (vertex clustering) has low quality • Efficiency • Compact error information (10 floats/vertex) • Generality • Aggregation • Non-manifold topology Multiresolution (ECS 289L) - Winter 2003