Hierarchical Face Clustering on Polygonal Surfaces

1. Hierarchical Face Clustering on Polygonal Surfaces Michael Garland University of Illinois at Urbana–Champaign Andrew Willmott Paul S. Heckbert Carnegie Mellon University

2. Overview • Surface models are often too complex • may exceed processing, storage, … capacity • may represent unnecessary detail • Must be able to control level of detail • has motivated work on surface simplification • Here we describe an alternative approach • still produces hierarchical representation • aggregate, rather than approximate geometry

3. Surface Simplification Iteratively mergingadjacent vertices

4. Iterative Face Clustering Repeatedlymerge pairs ofadjacent clusters

5. B C A Face Clusters • A connected set of triangles on the surface • use disjoint clusters to partition surface • record certain aggregate properties • representative plane • surface area & boundary perimeter • Clusters may be non-simple regions • individual clusters may have holes e.g., cluster A has 3 boundary loops

6. Our Sample Application:Multiresolution Radiosity • Existing history of hierarchical methods • Hierarchical radiosity [Hanrahan et al 91] • Hier. radiosity + volume clustering [Smits et al 94] • Essential for good performance • non-hierarchical methods haveO(n2) performance For details on radiosity algorithm:Willmott et al. Face Cluster Radiosity.Eurographics Rendering Workshop, 1999. 3.3 million polygon scene

7. The Basic Problem:Excessively Detailed Input 1,000,000 input triangles 870,000 input triangles Fine detail has little effecton ultimate solution.

8. Dual Graph of Meshes • Assume we start with triangulated mesh • construct one dual node per face • connect dual nodes if their faces are adjacent • Non-manifolds can cause efficiency problems • k faces per edge = O(k2) dual edges

9. Clustering = Dual Contraction • Consider contracting an edge of dual graph • merges two dual nodes into a single node • Equivalent to merging associated clusters • hence, iterative clustering = iterative contraction

10. Iterative Clustering Algorithm • Construct dual graph for mesh (every face is a singleton cluster) • Find cost of contraction of all dual edges (place in heap for efficient queries) • Loop until finished • contract dual edge of least cost • update costs of neighboring edges

11. Iterative Clustering Algorithm:Things to Notice • Surface is always completely partitioned • into disjoint face clusters — one per dual node • does not alter surface geometry at all • Looks very much like surface simplification • clustering is the dual of simplification • As with simplification, produces hierarchy • simplification — tree of vertex neighborhoods • clustering — tree of face clusters

12. Face Cluster Hierarchies • Contraction merges 2 clusters • producing new larger cluster • Iteration forms binary tree • original faces at leaves • children merge & form parent • 1 root per connected component • Similar to vertex hierarchies • result from simplification • used in applications such asview-dependent refinement

13. Measuring Contraction Cost • This is the big outstanding question • choice of metric has great effect on results • Our Choice:Want “mostly planar” clusters • Why? Consider radiosity application • clusters approximated with planar elements • non-planarity leads to imprecise solution

14. Measuring Planarity • Consider the set of all vertices in a cluster • can fit some plane to this set of points • quality of the fit will measure planarity • We choose the least squares best plane • find a plane that minimizes the mean squared distance of points to plane • the mean itself reflects the degree of planarity

15. Planarity Metric • Formally, this measure of planarity is • Using the dual quadric error metric Pi = the squared distance of point i to given plane

16. Using Quadric Metrics • Each node has an associated quadric • initially constructed from 3 corners of each face • sum quadrics when merging nodes • To compute cost of contracting an edge • add quadrics of endpoints • find representative plane, and evaluate:

17. Finding Planes for Clusters • Fit least squares best plane to set of points • we use principal component analysis (PCA) • a very common approach to the problem • Construct the sample covariance matrix • smallest eigenvector is normal of optimal plane • assume plane passes through mean of points

18. Finding Planes for Clusters • Can derive this directly from quadrics • this ignores the averaging factor • because only relative eigenvalue order matters • smallest eigenvector provides normal • other 2 eigenvectors provide axesfor oriented bounding box

19. Why Use Quadrics? • Expresses the error we want to measure • namely planarity (in the least squares sense) • Fairly compact, efficient representation • storage: 10 doubles per quadric • time: easy formula to evaluate • Very easy updating rules • 2 nodes are merged … added their quadrics • so only 10 additions per contraction

20. 11,036 clusters 6000 clusters 1000 clusters 200 clusters Example Results

21. Adding Orientation Bias • Both of the following are equally “planar” • least squares plane is roughly the same • Leftmost region shape would be preferable • we want regions with consistent normals • so we add an additional error term (a) (b)

22. Orientation Bias Metric • Each cluster has a representative normal • we measure deviation of all normals from it • As with planarity, can be written as quadric

23. Resulting Cluster Shape • Result of using planarity + shape bias • very jagged boundaries • long, irregular shapes “gerrymandering” • May be undesirable (application dependent) • yields poor radiosity solutions due to shadows

24. Measuring Cluster Irregularity • We define the irregularity of a cluster as • minimum value is 1 (achieved by a circle) • higher irregularity values mean less circular • This is a fairly common definition • image processing, surface clustering, politics, …

25. Cluster Shape Bias • And we can introduce additional shape bias • penalizes contractions that increase irregularity • Why bias and not hard limit? • guarantees that progress can always be made • will always produce a complete cluster hierarchy

26. The Final Cost Metric Without Shape Bias With Shape Bias

27. Example Results: Isis Statue

28. Running Times on 450 MHz Intel Pentium III system

29. Sample Radiosity Results • Large scene complexity (3,350,000 triangles) • Solution time: 450 sec (on 195 MHz MIPS R10000)

30. Radiosity Solution Time 2000 Running Time (seconds) Progressive Vol. clustering Face clustering 120,000 Input Triangles

31. Why Nearly Flat Growth? • Solutions are run with fixed error threshold • settles on an “appropriate” level in the hierarchy • sufficient detail for requested precision • far above the leaves of the hierarchy • once found, never descends below this level • Works because clusters fit surface well • poor clusters = descend further for accuracy • this is a problem for volume clustering • often little coherence of elements in a single cell

32. Why Not Use Simplification? • Vertex hierarchies are less effective • empirically tested against face hierarchies • intuitively, they optimize the wrong thing • a planar vertex neighborhood is a useless one • Static levels of detail wouldn’t work • transport links established between node pairs • desired LOD varies with transport partner • nearby patches need finer detail than far away ones

33. Related Work • General idea of clustering is an old one • most commonly on (multi-dimensional) point sets • Duality is often used in geometric algorithms • “simplex meshes” representation [Delingette 94] • Some region clustering work on … • subdivision surfaces [DeRose et al 98] • (range) images [Faugeras et al 87, Willersinn et al 94] • polygon meshes [Kalvin & Taylor 96]

34. Future Directions • Alternative clustering heuristics • planarity is quite well suited to radiosity • but is certainly not ideal for all applications • Balancing competing goals • summing error terms causes some problems • might want other terms (e.g., balanced tree) • Non-greedy algorithm framework • might produce noticeably better partitions

35. Future Directions:Other Possible Applications • Distance & intersection queries • oriented bounding boxes for all clusters • similar to STRIP tree [Ballard 81] curve representation • could be used for ray tracing, for example • Collision detection • very similar to OBBTrees [Gottschalk et al 96] • bottom up merging vs. top down partitioning • partitions surface rather than partitioning space

36. Summary • Face clustering is the dual of simplification • same LOD problem — same framework works • greedy, iterative edge contraction on dual graph • dual quadric error metric for guiding iteration • aggregate properties vs. approximate geometry • Important features of our method • hierarchical representation of surface partitions • practical & efficient construction algorithm • an effective (dual) quadric error metric

37. More Details Available Online Radiosity software Clustering software Related papers http://graphics.cs.uiuc.edu/~garland/research/cluster.html