computing stable and compact representation of medial axis wenping wang the university of hong kong
TRANSCRIPT
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Computing Stable and Compact Representation of Medial Axis
Wenping Wang
The University of Hong Kong
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Properties of Medial Axis Transform
• Medial representation of a shape1. First proposed by Blum (1967) – the set of
centers and radii of inscribed maximal circles
2. Encodes symmetry, thickness and structural components
3. A complete shape representation of both object interior and boundary
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“A transformation for extracting new descriptorsof shape”, Harry Blum (1967).
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“A transformation for extracting new descriptorsof shape”, Harry Blum (1967).
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“A transformation for extracting new descriptorsof shape”, Harry Blum (1967).
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Applications
• Object recognition
• Shape matching
• Path planning and collision detection
• Skeleton-controlled animation
• Geometric processing
• Mesh generation
• Network communication
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Voronoi-based Computation of MAT
• Voronoi-based method (e.g. Amenta and Bern 1998)
Every Voronoi vertex is the circum-center of a triangle/tet in Delaunay triangulation.
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Instability of MAT
• Small variations of the object boundary may cause large changes to the medial axis
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Instability in Computation of MAT
• Medial axis of a shape with noisy boundary typically has numerous unstable branches (spikes), making it highly non-manifold
Smooth boundary Noisy boundary
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Structural Redundancy
Causes for spikes in 3D: (1) Boundary noise; and (2) Slivers in Delaunay triangulation of boundary sample points.
When four sample points are co-circular, its circumscribing sphere is not unique.
# of MA vertices = 54,241
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Instability of MAT
• Small variations of the object boundary may cause large changes to the medial axis
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Principle of Approximating MAT
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Analogies
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Different Methods for Medial Axis Simplification
• Angle-based filtering (Attali and Montanvert 1996; Amenta et al. 2001; Dey and Zhao 2002; Foskey et al. 2003) • Scale-invariant. Does not ensure approximation accuracy
• The λ-medial axis (Chazal and Lieutier 2005; Chaussard et al. 2009)• Incapable of preserving fine feature of the original shape
• Scale axis transform - SAT (Giesen et al. 2009; Miklos et al. 2010).
Removes spikes effectively. May change topology
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Different Approaches to Pruning Spikes
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3D Medial Axis Simplification
Several methods exist for pruning unstable spikes on the medial axis
• Issues• Efficiency: Inefficient representation—MAT represented
as the union of a large number of circles/spheres.• Accuracy: Inaccurate representation—the simplified
medial axis may have large approximation error to the original shape
• Our goal• To efficiently compute a clean, compact and
accurate medial axis approximation
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Data Redundancywith too many mesh vertices
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Compact Representation by Medial Meshes
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Medial Meshes-- Approximation of MAT in 3D
• The medial mesh is 2D simplicial complex approximating the medial axis of a 3D object.
• Medial vertex: v = (p, r) where p is a 3D point, r the medial radius
• Medial edge: (1−t) v1 + t v2, t [0,1] .
• Medial face: a1v1+a2v2+a3v3, where ai ≥ 0 and a1+a2+a3=1.
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Medial Meshes
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Instability of MAT of 3D Objects
• Voronoi-based method generates unstable initial medial axis for 3D objects, due to noisy boundary sampling or slivers
Noise-free mesh approximating an ellipsoid
Medial axis computed byVoronoi-based method
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Understanding Unstable BranchesStability Ratio
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Two Extreme Cases Stability Ratio = 0 or 1
ratio = 0 ratio = 1
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Understanding Unstable Branches
Visualization of stability ratio
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Simplification by Edge Contraction Based on QEM by Garland and Heckbert (1997)
• Least squares errors are minimized with quadratic error minimization (QEM). (v1 and v2 are merged to v0)
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QEM for Mesh Decimation in 3DGarland and Heckbert (1997)
#v = 6,938 #v = 500
#v = 250
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Metric for MAT Simplification
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Geometric Interpretations
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Quadratic Error for MAT Simplification
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Which part to simplify first?
Spikes vs. Dense Smooth Region
• Mesh decimation
• Spike pruning
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Remove Spikes First
• The merge cost is defined by
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Experiments
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Plane (#v= 20 in 2 sec)
#v = 100
#v = 20
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Dolphin (#v=100 in 12 sec)
#v = 54,241
#v = 100
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Bear (#v =50 in 7 sec)
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Initial MAT from Voronoi Diagram
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Compared with Angle Filtering
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Compared with lambda-medial axis
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Comparison with SAT
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Comparison with SAT
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Comparison with SAT
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Comparison with SAT
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Medial Axis of Sphere(Degeneracy Test)
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Noise Test
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Results
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Results
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More Results
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Further Issues to Address
• Topology preservation
• Sharp feature preservation, e.g. for CAD models
• Converting medial meshes to boundary surfaces
• MAT for point clouds, noisy and incomplete data
• MAT used for shape modeling and deformation
• MAT as shape descriptor for matching and retrieval
• ….
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Thank you!
Acknowledgements:
Pan Li, Bin Wang, Feng Sun, Xiaohu Guo
Caiming Zhang