Variational Shape Approximation
Variational Shape Approximation
Variational Shape Approximation
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IntroductionThe Problem:
¡IntroductionOther approaches:PartitioningMesh decimation: Garland, ...Greedy Suboptimal meshes
¡IntroductionOther approaches:PartitioningMesh decimation: Garland, ...Greedy Suboptimal meshesGlobal OptimizationEnergy Functional: Hoppe et al.Irregular mesh, geometric efficiencyAnisotropyRemeshing: Turk, Lee, Kobelt, ...Quality of the mesh?
<strong>Shape</strong> <strong>Approximation</strong>
<strong>Shape</strong> <strong>Approximation</strong><strong>Approximation</strong> Theory:Functional SettingStrong Results in : Splines, ...Relies on Parameterization :-(
<strong>Shape</strong> <strong>Approximation</strong><strong>Approximation</strong> Theory:Height Field <strong>Approximation</strong>Obvious Parameterization.Few results about optimality in .Few results on approximation of error.Results are fairly narrow in scoperestricted to height fields,triangulations are assumed to beinterpolating the height field at vertices,asymptotical case does not help for asmall number of triangles.
¡¢£¡¤¢ £¡¤¦¦¥¡§¨©¨ ¡¢£ ¡¤¡ <strong>Shape</strong> <strong>Approximation</strong><strong>Approximation</strong> Theory:Arbitrary GeometryLack of knowledge on optimally efficientpiecewise linear approximation.Greedy Algorithms: (- triangles, ? error) or(- error, ? triangles)Metric commonly used:dinf
¥¥¡¡¡¢¡¢¤<strong>Shape</strong> <strong>Approximation</strong><strong>Variational</strong> Partitioning and ProxiesRemove topology from the searchPartitioningiteratively seek a partition that minimizes agiven error metric.Partition and proxiesGiven a mesh partitionA shape proxy is a pair"average" Point"average" Normal£ ¡¤ ¡£ ¡ £¤ ¡, where
d¤¡¡ £¡¢¤¡<strong>Shape</strong> <strong>Approximation</strong>Metrics on Proxies ¡¢¢¤£§¨
d¤¤¢¤¡¡ £¡¢¤ ¡¢<strong>Shape</strong> <strong>Approximation</strong>Metrics on Proxies¡ ¡¢ £§¨
¢£¤¡¡¡¢¢¡ £¡¤<strong>Shape</strong> <strong>Approximation</strong>Optimal shape proxies
Optimizing <strong>Shape</strong> Proxies
Optimizing <strong>Shape</strong> ProxiesThe algorithmIt is inspired in Lloyd’s Clustering AlgorithmGeometry Partitioning.It is used an error-minimizing region-growingalgorithm. The object is segmented innon-overlapping regions.Proxy Fitting.For each partition it is computed an optimallocal representative.
¡¡¡¡Optimizing <strong>Shape</strong> ProxiesGeometry Partitioning: Initial seedGiven a partition , a set of proxies , and anerror metric :For each region , select the trianglethat is most similar to .In the first step, select arbitrary triangles.For each one, assign as proxy its barycenterand normal.of
¢£¡¤¡¡Optimizing <strong>Shape</strong> ProxiesGeometry Partitioning: Region growingUse a priority queue to "propagate" the regions:For each seed triangle , insert its 3neighbors in the queuePriority =Tag them with the same label ofFor each popped triangle:IF it has been assigned to a proxy, ENDELSE Assign it to the region with its tag andpush its (untagged) neighbors in the queue
¡£ ¡¤ ¡¡¡¡¡Optimizing <strong>Shape</strong> ProxiesProxy FittingMetric:= Barycenter of the region .= Eigenvector of the smallest eigenvalueof the covariance matrix of .¡ ¤ ¡ £¢Metric:= Barycenter of the region .= area-weighted average of normals oftriangles in .
Optimizing <strong>Shape</strong> ProxiesImprovementsChoosing the number of proxiesRegion TeleportationFarthest-point initializationTailoring RefinementsSmoothing the normal field
Optimizing <strong>Shape</strong> Proxies
Final remarks and results
Final remarks and resultsApplication to meshingAnchor VerticesEdge ExtractionTriangulationPolygons
Final remarks and resultsLimitationsCannot compete (in time) with greedymethodsNot Real-Time. I mean, just for offlinecomputationsMeshing does not handle non-manifoldmeshes.
Final remarks and results
Final remarks and results
Final remarks and results
Final remarks and results
Final remarks and results
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