Hierarchical Face Clustering on Polygonal Surfaces

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Hierarchical Face Clustering on Polygonal Surfaces

• Michael Garland
• University of Illinois at UrbanaChampaign
• Andrew Willmott
• Paul S. Heckbert
• Carnegie Mellon University

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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

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Surface Simplification
Iteratively mergingadjacent vertices
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Iterative Face Clustering
Repeatedlymerge pairs ofadjacent clusters
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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

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Our Sample ApplicationMultiresolution 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 algorithmWillmott et
al. Face Cluster Radiosity.Eurographics
Rendering Workshop, 1999.
3.3 million polygon scene
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The Basic ProblemExcessively Detailed Input
1,000,000 input triangles
870,000 input triangles
Fine detail has little effecton ultimate
solution.
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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

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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

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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

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Iterative Clustering AlgorithmThings 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

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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

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Measuring Contraction Cost

• This is the big outstanding question
• choice of metric has great effect on results
• Our ChoiceWant mostly planar clusters
• Why? Consider radiosity application
• clusters approximated with planar elements
• non-planarity leads to imprecise solution

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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

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Planarity Metric

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

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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

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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

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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

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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

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Example Results
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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)
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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

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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

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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,

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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

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The Final Cost Metric
Without Shape Bias
With Shape Bias
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Example Results Isis Statue
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Running Times
Model InputTriangles Init Time(sec) Cluster Time(sec)
5804 0.21 0.51
11,036 0.4 1.43
375,736 13.34 48.31
on 450 MHz Intel Pentium III system
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Sample Radiosity Results

• Large scene complexity
• (3,350,000 triangles)
• Solution time 450 sec
• (on 195 MHz MIPS R10000)

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

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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 wouldnt work
• transport links established between node pairs
• desired LOD varies with transport partner
• nearby patches need finer detail than far away
  ones

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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

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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

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Future DirectionsOther 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

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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

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More Details Available Online
Radiosity software
Clustering software
Related papers
http//graphics.cs.uiuc.edu/garland/research/clus
ter.html