surface reconstruction some figures by turk, curless, amenta, et al
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Surface ReconstructionSurface Reconstruction
Some figures by Turk, Curless, Amenta, et al.Some figures by Turk, Curless, Amenta, et al.
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Two Related ProblemsTwo Related Problems
• Given a point cloud, construct a surfaceGiven a point cloud, construct a surface
• Given several aligned scans (range Given several aligned scans (range images), construct a surfaceimages), construct a surface
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Surface Reconstruction from Surface Reconstruction from Point CloudsPoint Clouds
• Most techniques figure out how to connect up Most techniques figure out how to connect up “nearby” points“nearby” points
• Need sufficiently dense sampling, little noiseNeed sufficiently dense sampling, little noise
• Delaunay triangulation: connect nearest Delaunay triangulation: connect nearest pointspoints– Officially, a triangle is in the Delaunay triangulation Officially, a triangle is in the Delaunay triangulation
iff its circumcircle does not contain any pointsiff its circumcircle does not contain any points
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The “Crust” AlgorithmThe “Crust” Algorithm
• Amenta et al., 1998Amenta et al., 1998
• Medial axis: set of points equidistant Medial axis: set of points equidistant from 2 original pointsfrom 2 original points
• In 2D:In 2D:
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Medial Axes in 3DMedial Axes in 3D
• May contain surfaces as well as edges May contain surfaces as well as edges and verticesand vertices
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Voronoi DiagramsVoronoi Diagrams
• Partitioning of plane according to Partitioning of plane according to closest point (in a discrete point set)closest point (in a discrete point set)
• A subset of Voronoi vertices is an A subset of Voronoi vertices is an approximation to medial axisapproximation to medial axis
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The “Crust” AlgorithmThe “Crust” Algorithm
• Compute Voronoi Compute Voronoi diagramdiagram
• Compute Delaunay Compute Delaunay triangulation of triangulation of original points + original points + Voronoi verticesVoronoi vertices
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Voronoi Cells in 3DVoronoi Cells in 3D
• Some Voronoi vertices lie neither near Some Voronoi vertices lie neither near the surface nor near the medial axisthe surface nor near the medial axis
• Keep the “poles”Keep the “poles”
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Crust ResultsCrust Results
• 36K vertices36K vertices
• 23 minutes (1998)23 minutes (1998)
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Crust ProblemsCrust Problems
• Problems with sharp cornersProblems with sharp corners– Medial axis touches surfaceMedial axis touches surface
– Theoretically need infinitely high samplingTheoretically need infinitely high sampling
– In practice, heuristics to choose polesIn practice, heuristics to choose poles
• Topological problemsTopological problems
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The Ball Pivoting AlgorithmThe Ball Pivoting Algorithm
• Bernardini et al., 1999Bernardini et al., 1999
• Roll ball around surface, connect what it Roll ball around surface, connect what it hitshits
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Alpha ShapesAlpha Shapes
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Problems With Reconstruction Problems With Reconstruction fromfrom
Point CloudsPoint Clouds
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Surface Reconstruction from Surface Reconstruction from Range ImagesRange Images
• Often an easier problem than Often an easier problem than reconstruction from arbitrary point reconstruction from arbitrary point cloudsclouds– Implicit information about adjacency, Implicit information about adjacency,
connectivityconnectivity
– Roughly uniform spacingRoughly uniform spacing
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Surface Reconstruction From Surface Reconstruction From Range ImagesRange Images
• First, construct surface from each range First, construct surface from each range imageimage
• Then, merge resulting surfacesThen, merge resulting surfaces– Obtain average surface in overlapping Obtain average surface in overlapping
regionsregions
– Control point densityControl point density
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Range Image TesselationRange Image Tesselation
• Given a range image, connect up the Given a range image, connect up the neighborsneighbors
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Range Image TesselationRange Image Tesselation
• Caveat #1: can’t be too aggressiveCaveat #1: can’t be too aggressive– Introduce distance threshold for tesselationIntroduce distance threshold for tesselation
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• Caveat #2: Which way to triangulate?Caveat #2: Which way to triangulate?
• Possible heuristics:Possible heuristics:– Shorter diagonalShorter diagonal
– Dihedral angle closer to 180Dihedral angle closer to 180
– Maximize smallest angle in both trianglesMaximize smallest angle in both triangles
– Always the same way (best triangle strips)Always the same way (best triangle strips)
Range Image TesselationRange Image Tesselation
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Scan Merging Using ZipperingScan Merging Using Zippering
• Turk & Levoy, 1994Turk & Levoy, 1994
• Erode geometry in overlapping areasErode geometry in overlapping areas
• Stitch scans together along seamStitch scans together along seam
• Re-introduce all dataRe-introduce all data– Weighted averageWeighted average
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ZipperingZippering
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Point WeightingPoint Weighting
• Higher weights to points facing the Higher weights to points facing the cameracamera– Favor higher sampling ratesFavor higher sampling rates
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Point WeightingPoint Weighting
• Lower weightsLower weights(tapering to 0)(tapering to 0)near boundariesnear boundaries– Smooth blendsSmooth blends
between viewsbetween views
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Point WeightingPoint Weighting
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Consensus GeometryConsensus Geometry
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Zippering ExampleZippering Example
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Zippering ExampleZippering Example