Hierarchical Face Clustering on Polygonal Surfaces Michael Garland University of Illinois at UrbanaChampaign Andrew Willmott Paul S. Heckbert Carnegie Mellon University Michael Garland University of Illinois at UrbanaChampaign Andrew Willmott Paul S. Heckbert Carnegie Mellon University Overview Surface models are often too complex may exceed processing, storage, capacitymay exceed processing, storage, capacity may represent unnecessary detailmay represent unnecessary detail Must be able to control level of detail has motivated work on surface simplificationhas motivated work on surface simplification Here we describe an alternative approach still produces hierarchical representationstill produces hierarchical representation aggregate, rather than approximate geometryaggregate, rather than approximate geometry Surface models are often too complex may exceed processing, storage, capacitymay exceed processing, storage, capacity may represent unnecessary detailmay represent unnecessary detail Must be able to control level of detail has motivated work on surface simplificationhas motivated work on surface simplification Here we describe an alternative approach still produces hierarchical representationstill produces hierarchical representation aggregate, rather than approximate geometryaggregate, rather than approximate geometry Surface Simplification Iteratively merging adjacent vertices Iterative Face Clustering Repeatedly merge pairs of adjacent clusters Face Clusters A connected set of triangles on the surface use disjoint clusters to partition surfaceuse disjoint clusters to partition surface record certain aggregate propertiesrecord certain aggregate properties representative plane surface area & boundary perimeter Clusters may be non-simple regions individual clusters may have holesindividual clusters may have holes e.g., cluster A has 3 boundary loops A connected set of triangles on the surface use disjoint clusters to partition surfaceuse disjoint clusters to partition surface record certain aggregate propertiesrecord certain aggregate properties representative plane surface area & boundary perimeter Clusters may be non-simple regions individual clusters may have holesindividual clusters may have holes e.g., cluster A has 3 boundary loops A B C Our Sample Application: Multiresolution Radiosity Existing history of hierarchical methods Hierarchical radiosity [Hanrahan et al 91]Hierarchical radiosity [Hanrahan et al 91] Hier. radiosity + volume clustering [Smits et al 94]Hier. radiosity + volume clustering [Smits et al 94] Essential for good performance non-hierarchical methods have O(n ) performancenon-hierarchical methods have O(n ) performance Existing history of hierarchical methods Hierarchical radiosity [Hanrahan et al 91]Hierarchical radiosity [Hanrahan et al 91] Hier. radiosity + volume clustering [Smits et al 94]Hier. radiosity + volume clustering [Smits et al 94] Essential for good performance non-hierarchical methods have O(n ) performancenon-hierarchical methods have O(n ) performance 3.3 million polygon scene For details on radiosity algorithm: Willmott et al. Face Cluster Radiosity. Eurographics Rendering Workshop, 1999. The Basic Problem: Excessively Detailed Input 870,000 input triangles 1,000,000 input triangles Fine detail has little effect on ultimate solution. Dual Graph of Meshes Assume we start with triangulated mesh construct one dual node per faceconstruct one dual node per face connect dual nodes if their faces are adjacentconnect dual nodes if their faces are adjacent Non-manifolds can cause efficiency problems k faces per edge = O(k ) dual edges k faces per edge = O(k ) dual edges Assume we start with triangulated mesh construct one dual node per faceconstruct one dual node per face connect dual nodes if their faces are adjacentconnect dual nodes if their faces are adjacent Non-manifolds can cause efficiency problems k faces per edge = O(k ) dual edges k faces per edge = O(k ) dual edges Clustering = Dual Contraction Consider contracting an edge of dual graph merges two dual nodes into a single nodemerges two dual nodes into a single node Equivalent to merging associated clusters hence, iterative clustering = iterative contractionhence, iterative clustering = iterative contraction Consider contracting an edge of dual graph merges two dual nodes into a single nodemerges two dual nodes into a single node Equivalent to merging associated clusters hence, iterative clustering = iterative contractionhence, iterative clustering = iterative contraction Iterative Clustering Algorithm 1.Construct dual graph for mesh (every face is a singleton cluster) 2.Find cost of contraction of all dual edges (place in heap for efficient queries) 3.Loop until finished a.contract dual edge of least cost b.update costs of neighboring edges 1.Construct dual graph for mesh (every face is a singleton cluster) 2.Find cost of contraction of all dual edges (place in heap for efficient queries) 3.Loop until finished a.contract dual edge of least cost b.update costs of neighboring edges Iterative Clustering Algorithm: Things to Notice Surface is always completely partitioned into disjoint face clusters one per dual nodeinto disjoint face clusters one per dual node does not alter surface geometry at alldoes not alter surface geometry at all Looks very much like surface simplification clustering is the dual of simplificationclustering is the dual of simplification As with simplification, produces hierarchy simplification tree of vertex neighborhoodssimplification tree of vertex neighborhoods clustering tree of face clustersclustering tree of face clusters Surface is always completely partitioned into disjoint face clusters one per dual nodeinto disjoint face clusters one per dual node does not alter surface geometry at alldoes not alter surface geometry at all Looks very much like surface simplification clustering is the dual of simplificationclustering is the dual of simplification As with simplification, produces hierarchy simplification tree of vertex neighborhoodssimplification tree of vertex neighborhoods clustering tree of face clustersclustering tree of face clusters Face Cluster Hierarchies Contraction merges 2 clusters producing new larger clusterproducing new larger cluster Iteration forms binary tree original faces at leavesoriginal faces at leaves children merge & form parentchildren merge & form parent 1 root per connected component1 root per connected component Similar to vertex hierarchies result from simplificationresult from simplification used in applications such as view-dependent refinementused in applications such as view-dependent refinement Measuring Contraction Cost This is the big outstanding question choice of metric has great effect on resultschoice of metric has great effect on results Our Choice: Want mostly planar clusters Why? Consider radiosity application clusters approximated with planar elementsclusters approximated with planar elements non-planarity leads to imprecise solutionnon-planarity leads to imprecise solution This is the big outstanding question choice of metric has great effect on resultschoice of metric has great effect on results Our Choice: Want mostly planar clusters Why? Consider radiosity application clusters approximated with planar elementsclusters approximated with planar elements non-planarity leads to imprecise solutionnon-planarity leads to imprecise solution Measuring Planarity Consider the set of all vertices in a cluster can fit some plane to this set of pointscan fit some plane to this set of points quality of the fit will measure planarityquality of the fit will measure planarity We choose the least squares best plane find a plane that minimizes thefind a plane that minimizes the mean squared distance of points to plane the mean itself reflects the degree of planaritythe mean itself reflects the degree of planarity Consider the set of all vertices in a cluster can fit some plane to this set of pointscan fit some plane to this set of points quality of the fit will measure planarityquality of the fit will measure planarity We choose the least squares best plane find a plane that minimizes thefind a plane that minimizes the mean squared distance of points to plane the mean itself reflects the degree of planaritythe mean itself reflects the degree of planarity Planarity Metric Formally, this measure of planarity is Using the dual quadric error metric P i the squared distance of point i to given plane Formally, this measure of planarity is Using the dual quadric error metric P i the squared distance of point i to given plane Using Quadric Metrics Each node has an associated quadric initially constructed from 3 corners of each faceinitially constructed from 3 corners of each face sum quadrics when merging nodessum quadrics when merging nodes To compute cost of contracting an edge add quadrics of endpointsadd quadrics of endpoints find representative plane, and evaluate:find representative plane, and evaluate: Each node has an associated quadric initially constructed from 3 corners of each faceinitially constructed from 3 corners of each face sum quadrics when merging nodessum quadrics when merging nodes To compute cost of contracting an edge add quadrics of endpointsadd quadrics of endpoints find representative plane, and evaluate:find representative plane, and evaluate: Finding Planes for Clusters Fit least squares best plane to set of points we use principal component analysis (PCA)we use principal component analysis (PCA) a very common approach to the problema very common approach to the problem Construct the sample covariance matrix smallest eigenvector is normal of optimal planesmallest eigenvector is normal of optimal plane assume plane passes through mean of pointsassume plane passes through mean of points Fit least squares best plane to set of points we use principal component analysis (PCA)we use principal component analysis (PCA) a very common approach to the problema very common approach to the problem Construct the sample covariance matrix smallest eigenvector is normal of optimal planesmallest eigenvector is normal of optimal plane assume plane passes through mean of pointsassume plane passes through mean of points Finding Planes for Clusters Can derive this directly from quadrics this ignores the averaging factorthis ignores the averaging factor because only relative eigenvalue order matters smallest eigenvector provides normalsmallest eigenvector provides normal other 2 eigenvectors provide axes for oriented bounding boxother 2 eigenvectors provide axes for oriented bounding box Can derive this directly from quadrics this ignores the averaging factorthis ignores the averaging factor because only relative eigenvalue order matters smallest eigenvector provides normalsmallest eigenvector provides normal other 2 eigenvectors provide axes for oriented bounding boxother 2 eigenvectors provide axes for oriented bounding box Why Use Quadrics? Expresses the error we want to measure namely planarity (in the least squares sense)namely planarity (in the least squares sense) Fairly compact, efficient representation storage: 10 doubles per quadricstorage: 10 doubles per quadric time: easy formula to evaluatetime: easy formula to evaluate Very easy updating rules 2 nodes are merged added their quadrics2 nodes are merged added their quadrics so only 10 additions per contractionso only 10 additions per contraction Expresses the error we want to measure namely planarity (in the least squares sense)namely planarity (in the least squares sense) Fairly compact, efficient representation storage: 10 doubles per quadricstorage: 10 doubles per quadric time: easy formula to evaluatetime: easy formula to evaluate Very easy updating rules 2 nodes are merged added their quadrics2 nodes are merged added their quadrics so only 10 additions per contractionso only 10 additions per contraction Example Results 1000 clusters 6000 clusters 200 clusters 11,036 clusters Adding Orientation Bias Both of the following are equally planar least squares plane is roughly the sameleast squares plane is roughly the same Leftmost region shape would be preferable we want regions with consistent normalswe want regions with consistent normals so we add an additional error termso we add an additional error term Both of the following are equally planar least squares plane is roughly the sameleast squares plane is roughly the same Leftmost region shape would be preferable we want regions with consistent normalswe want regions with consistent normals so we add an additional error termso we add an additional error term (a)(b) Orientation Bias Metric Each cluster has a representative normal we measure deviation of all normals from itwe measure deviation of all normals from it As with planarity, can be written as quadric Each cluster has a representative normal we measure deviation of all normals from itwe measure deviation of all normals from it As with planarity, can be written as quadric 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 shadowsyields poor radiosity solutions due to shadows Result of using planarity + shape bias very jagged boundaries long, irregular shapes gerrymandering May be undesirable (application dependent) yields poor radiosity solutions due to shadowsyields poor radiosity solutions due to shadows Measuring Cluster Irregularity We define the irregularity of a cluster as minimum value is 1 (achieved by a circle)minimum value is 1 (achieved by a circle) higher irregularity values mean less circularhigher irregularity values mean less circular This is a fairly common definition image processing, surface clustering, politics, image processing, surface clustering, politics, We define the irregularity of a cluster as minimum value is 1 (achieved by a circle)minimum value is 1 (achieved by a circle) higher irregularity values mean less circularhigher irregularity values mean less circular This is a fairly common definition image processing, surface clustering, politics, image processing, surface clustering, politics, Cluster Shape Bias And we can introduce additional shape bias penalizes contractions that increase irregularitypenalizes contractions that increase irregularity Why bias and not hard limit? guarantees that progress can always be madeguarantees that progress can always be made will always produce a complete cluster hierarchywill always produce a complete cluster hierarchy And we can introduce additional shape bias penalizes contractions that increase irregularitypenalizes contractions that increase irregularity Why bias and not hard limit? guarantees that progress can always be madeguarantees that progress can always be made will always produce a complete cluster hierarchywill always produce a complete cluster hierarchy The Final Cost Metric Without Shape Bias With Shape Bias Example Results: Isis Statue Running Times Model Input Triangles Init Time (sec) Cluster Time (sec) , , on 450 MHz Intel Pentium III system Sample Radiosity Results Large scene complexity (3,350,000 triangles) Solution time: 450 sec (on 195 MHz MIPS R10000) Radiosity Solution Time Progressive Vol. clustering Face clustering Input Triangles 120, Running Time (seconds) Why Nearly Flat Growth? Solutions are run with fixed error threshold settles on an appropriate level in the hierarchysettles 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 levelonce found, never descends below this level Works because clusters fit surface well poor clusters = descend further for accuracypoor clusters = descend further for accuracy this is a problem for volume clusteringthis is a problem for volume clustering often little coherence of elements in a single cell Solutions are run with fixed error threshold settles on an appropriate level in the hierarchysettles 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 levelonce found, never descends below this level Works because clusters fit surface well poor clusters = descend further for accuracypoor clusters = descend further for accuracy this is a problem for volume clusteringthis is a problem for volume clustering often little coherence of elements in a single cell Why Not Use Simplification? Vertex hierarchies are less effective empirically tested against face hierarchiesempirically tested against face hierarchies intuitively, they optimize the wrong thingintuitively, they optimize the wrong thing a planar vertex neighborhood is a useless one Static levels of detail wouldnt work transport links established between node pairstransport links established between node pairs desired LOD varies with transport partnerdesired LOD varies with transport partner nearby patches need finer detail than far away ones Vertex hierarchies are less effective empirically tested against face hierarchiesempirically tested against face hierarchies intuitively, they optimize the wrong thingintuitively, they optimize the wrong thing a planar vertex neighborhood is a useless one Static levels of detail wouldnt work transport links established between node pairstransport links established between node pairs desired LOD varies with transport partnerdesired LOD varies with transport partner nearby patches need finer detail than far away ones Related Work General idea of clustering is an old one most commonly on (multi-dimensional) point setsmost commonly on (multi-dimensional) point sets Duality is often used in geometric algorithms simplex meshes representation [Delingette 94]simplex meshes representation [Delingette 94] Some region clustering work on subdivision surfaces [DeRose et al 98]subdivision surfaces [DeRose et al 98] (range) images [Faugeras et al 87, Willersinn et al 94](range) images [Faugeras et al 87, Willersinn et al 94] polygon meshes [Kalvin & Taylor 96]polygon meshes [Kalvin & Taylor 96] General idea of clustering is an old one most commonly on (multi-dimensional) point setsmost commonly on (multi-dimensional) point sets Duality is often used in geometric algorithms simplex meshes representation [Delingette 94]simplex meshes representation [Delingette 94] Some region clustering work on subdivision surfaces [DeRose et al 98]subdivision surfaces [DeRose et al 98] (range) images [Faugeras et al 87, Willersinn et al 94](range) images [Faugeras et al 87, Willersinn et al 94] polygon meshes [Kalvin & Taylor 96]polygon meshes [Kalvin & Taylor 96] Future Directions Alternative clustering heuristics planarity is quite well suited to radiosityplanarity is quite well suited to radiosity but is certainly not ideal for all applicationsbut is certainly not ideal for all applications Balancing competing goals summing error terms causes some problemssumming error terms causes some problems might want other terms (e.g., balanced tree)might want other terms (e.g., balanced tree) Non-greedy algorithm framework might produce noticeably better partitionsmight produce noticeably better partitions Alternative clustering heuristics planarity is quite well suited to radiosityplanarity is quite well suited to radiosity but is certainly not ideal for all applicationsbut is certainly not ideal for all applications Balancing competing goals summing error terms causes some problemssumming error terms causes some problems might want other terms (e.g., balanced tree)might want other terms (e.g., balanced tree) Non-greedy algorithm framework might produce noticeably better partitionsmight produce noticeably better partitions Future Directions: Other Possible Applications Distance & intersection queries oriented bounding boxes for all clustersoriented bounding boxes for all clusters similar to STRIP tree [Ballard 81] curve representation could be used for ray tracing, for examplecould be used for ray tracing, for example Collision detection very similar to OBBTrees [Gottschalk et al 96]very similar to OBBTrees [Gottschalk et al 96] bottom up merging vs. top down partitioningbottom up merging vs. top down partitioning partitions surface rather than partitioning spacepartitions surface rather than partitioning space Distance & intersection queries oriented bounding boxes for all clustersoriented bounding boxes for all clusters similar to STRIP tree [Ballard 81] curve representation could be used for ray tracing, for examplecould be used for ray tracing, for example Collision detection very similar to OBBTrees [Gottschalk et al 96]very similar to OBBTrees [Gottschalk et al 96] bottom up merging vs. top down partitioningbottom up merging vs. top down partitioning partitions surface rather than partitioning spacepartitions surface rather than partitioning space Summary Face clustering is the dual of simplification same LOD problem same framework workssame LOD problem same framework works greedy, iterative edge contraction on dual graph dual quadric error metric for guiding iteration aggregate properties vs. approximate geometryaggregate properties vs. approximate geometry Important features of our method hierarchical representation of surface partitionshierarchical representation of surface partitions practical & efficient construction algorithmpractical & efficient construction algorithm an effective (dual) quadric error metrican effective (dual) quadric error metric Face clustering is the dual of simplification same LOD problem same framework workssame LOD problem same framework works greedy, iterative edge contraction on dual graph dual quadric error metric for guiding iteration aggregate properties vs. approximate geometryaggregate properties vs. approximate geometry Important features of our method hierarchical representation of surface partitionshierarchical representation of surface partitions practical & efficient construction algorithmpractical & efficient construction algorithm an effective (dual) quadric error metrican effective (dual) quadric error metric More Details Available OnlineRadiosity software Clustering software Related papers