1. visualization and graphics research group CIPIC January 21, 2003Multiresolution (ECS 289L) - Winter 20031 Surface Simplification Using Quadric Error Metrics By Michael Garland and Paul S. Heckbert Carnegie Mellon University Presented by Lok Hwa and Taylor Holliday

2. visualization and graphics research group CIPIC January 21, 2003Multiresolution (ECS 289L) - Winter 20032 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 Background / Related Work

3. visualization and graphics research group CIPIC January 21, 2003Multiresolution (ECS 289L) - Winter 20033 Introduction Iterative Pair Contraction

4. visualization and graphics research group CIPIC January 21, 2003Multiresolution (ECS 289L) - Winter 20034 Cow Simplification in under 1 second

5. visualization and graphics research group CIPIC January 21, 2003Multiresolution (ECS 289L) - Winter 20035 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. Advantages

6. visualization and graphics research group CIPIC January 21, 2003Multiresolution (ECS 289L) - Winter 20036 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 Algorithm Setup

7. visualization and graphics research group CIPIC January 21, 2003Multiresolution (ECS 289L) - Winter 20037 Algorithm: Vertex Pair Contraction (v 1, v 2 )  v –Move v 1 and v 2 to v –Replace all occurances of v 2 with v 1 –Remove v 2 and degenerate triangles Valid Pairs –Edges –|| v 1 - v 2 || < t, where t is a threshhold parameter

8. visualization and graphics research group CIPIC January 21, 2003Multiresolution (ECS 289L) - Winter 20038 Algorithm: Aggregation

9. visualization and graphics research group CIPIC January 21, 2003Multiresolution (ECS 289L) - Winter 20039 Q is a 4x4 symmetric matrix representing the error at each vertex Evaluated at vertex v = [v x v y v z 1]: (V) = v T Q v Need a new Q for contraction (v 1, v 2 )  v Q = Q 1 + Q 2 Want to minimize (v) Using Quadrics to Approximate Error

10. visualization and graphics research group CIPIC January 21, 2003Multiresolution (ECS 289L) - Winter 200310 Using Quadrics to Approximate Error If matrix is not invertible, then try to find optimal vertex along segment v 1 v 2, else choose from v 1 v 2 or midpoint

11. visualization and graphics research group CIPIC January 21, 2003Multiresolution (ECS 289L) - Winter 200311 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. Algorithm Summary

12. visualization and graphics research group CIPIC January 21, 2003Multiresolution (ECS 289L) - Winter 200312 Algorithm Summary 1.Compute Q matrices for all the initial vertices. 2.Select all the valid pairs 3.Compute the optimal contraction target for each valid pair. The error becomes the cost of contracting that pair. 4.Place all the pairs in a priority queue keyed on cost with the minimum cost pair at the front

13. visualization and graphics research group CIPIC January 21, 2003Multiresolution (ECS 289L) - Winter 200313 Algorithm Summary (cont..) 5.Iteratively remove the least cost pair (v 1, v 2 ), contract it, and updated the costs of all valid pairs involving v 1 until the simplification goals are satisfied

14. visualization and graphics research group CIPIC January 21, 2003Multiresolution (ECS 289L) - Winter 200314 Derivation of QEMs

15. visualization and graphics research group CIPIC January 21, 2003Multiresolution (ECS 289L) - Winter 200315 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

16. visualization and graphics research group CIPIC January 21, 2003Multiresolution (ECS 289L) - Winter 200316 Derivation of QEMs

17. visualization and graphics research group CIPIC January 21, 2003Multiresolution (ECS 289L) - Winter 200317 Derivation of QEMs

18. visualization and graphics research group CIPIC January 21, 2003Multiresolution (ECS 289L) - Winter 200318 Derivation of QEMs

19. visualization and graphics research group CIPIC January 21, 2003Multiresolution (ECS 289L) - Winter 200319 Derivation of QEMs

20. visualization and graphics research group CIPIC January 21, 2003Multiresolution (ECS 289L) - Winter 200320 Derivation of QEMs Now we combine these quadrics for a set of planes

21. visualization and graphics research group CIPIC January 21, 2003Multiresolution (ECS 289L) - Winter 200321 Derivation of QEMs Now we combine these quadrics for a set of planes Note Q is positive definite

22. visualization and graphics research group CIPIC January 21, 2003Multiresolution (ECS 289L) - Winter 200322 Geometric Interpretation  Vertex errors have either planar, cylindrical, or ellipsoid level surfaces

23. visualization and graphics research group CIPIC January 21, 2003Multiresolution (ECS 289L) - Winter 200323 Geometric Interpretation  Why ellipsoids? If Q is diagonalizable,

24. visualization and graphics research group CIPIC January 21, 2003Multiresolution (ECS 289L) - Winter 200324 Geometric Interpretation  Why ellipsoids? If Q is diagonalizable,  Let

25. visualization and graphics research group CIPIC January 21, 2003Multiresolution (ECS 289L) - Winter 200325 Geometric Interpretation  Why ellipsoids? If Q is diagonalizable,  Let  Apply the rotation y = S T v

26. visualization and graphics research group CIPIC January 21, 2003Multiresolution (ECS 289L) - Winter 200326 Details  Recall: minimize  Do not double-count planes on an original vertex  Planes can still be double-counted if they coincide between vertices

27. visualization and graphics research group CIPIC January 21, 2003Multiresolution (ECS 289L) - Winter 200327 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

28. visualization and graphics research group CIPIC January 21, 2003Multiresolution (ECS 289L) - Winter 200328 Preventing Mesh Inversion  prevent normals from changing direction by more than (“flipping”)  Contractions resulting in flipped normals are rejected

29. visualization and graphics research group CIPIC January 21, 2003Multiresolution (ECS 289L) - Winter 200329 Results

30. visualization and graphics research group CIPIC January 21, 2003Multiresolution (ECS 289L) - Winter 200330 Evaluation Metric  Need a metric for approximation error

31. visualization and graphics research group CIPIC January 21, 2003Multiresolution (ECS 289L) - Winter 200331 Effect of vertex placement

32. visualization and graphics research group CIPIC January 21, 2003Multiresolution (ECS 289L) - Winter 200332 Bunny Model

33. visualization and graphics research group CIPIC January 21, 2003Multiresolution (ECS 289L) - Winter 200333 Crater Lake Model

34. visualization and graphics research group CIPIC January 21, 2003Multiresolution (ECS 289L) - Winter 200334 Timings  t is non-edge contraction threshold  Crater Lake was a large dataset in 1997

35. visualization and graphics research group CIPIC January 21, 2003Multiresolution (ECS 289L) - Winter 200335 Foot Model

36. visualization and graphics research group CIPIC January 21, 2003Multiresolution (ECS 289L) - Winter 200336 Aggregation Recall: aggregation is the joining of disconnected components Aggregation effects approximation error

37. visualization and graphics research group CIPIC January 21, 2003Multiresolution (ECS 289L) - Winter 200337 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

38. visualization and graphics research group CIPIC January 21, 2003Multiresolution (ECS 289L) - Winter 200338 Conclusion  Quality  Only other algorithm supporting aggregation (vertex clustering) has low quality  Efficiency  Compact error information (10 floats/vertex)  Generality  Aggregation  Non-manifold topology