Cryptography Lecture 7 Stefan Dziembowski [email protected].
The Intrinsic Shape of Point Clouds Stefan Ohrhallinger Ph.D. Defence, July 12, 2012
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Transcript of The Intrinsic Shape of Point Clouds Stefan Ohrhallinger Ph.D. Defence, July 12, 2012
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The Intrinsic Shape of Point Clouds
Stefan Ohrhallinger Ph.D. Defence, July 12, 2012
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Motivation
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Motivation
De-noising
Deforming points
“Fluid” simulation
Visibility culling
Scan reconstruct.
Shape retrieval
Operations on point cloudsrequire an assumed surface
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“[...] to pose a unifying general problem [...]” - [Hoppe et al. 1992]1
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Problem DomainCharacteristics of real-world point data sets
Noise and outliersHoles
Sparse sampling
Local non-uniform Too “random”
Unreliable normals
1 Hoppe, DeRose, Duchamp, McDonald and Stuetzle. Surface reconstruction from unorganized points. Computer Graphics, 1992.
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Related Work
Interpolate Approximate
Non-uniform Uniform
Global Local
Sculpture Filter Optimize
Region grow.Deterministic
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Contributions
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Shape Characteristic Boundary ReconstructionShape Formalization
S. Ohrhallinger, S. Mudur: Interpolating an unorganized 2D point cloud with a single closed shape, Computer-Aided Design, 2011.
S. Ohrhallinger, S. Mudur. An Efficient Algorithm for Determining an Aesthetic Shape Connecting Unorganized 2D Points (under review)
S. Ohrhallinger, S. Mudur. The Intrinsic Shape of Unorganized 3D Point Sets: Computing an Interpolating Orientable Surface (under review)(A concise overview of the 3D method won Best Poster Award at Eurographics 2012, published in the conference proceedings)
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Shape Gestalt
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Proximity Continuity Closure
Derive for piece-wise linear boundary B:
Guided by Gestalt principles of form perception
In R², just minimize boundary length
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R²: Boundary Complex
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NP-hard
Vertex degree
Point set MST
?
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R³: Boundary Complex
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R²: R³:
Mean curvature:
Edge length is 1 factor Longest-edge-in-triangle
R³: R²:
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BC0:Shape Character
BC0 Bmin approximation
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BC0: Varying Density
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36K points 1k points (3%) 0.1k points (0.3%)
Property: Reducing point density does not affect shape approximation
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BC0: Adding Noise
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Catacomb section slightly perturbed extremely perturbed
Property: Adding noise does not impede construction
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Reconstruction in R²
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Inflate Sculpture
Manifold Hull Interpolating Manifold
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R²: Improved Results
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Points
[DW01]1
Ours
1 Dey, Wenger. Detecting undersampling in surface reconstruction. SCG 2001.
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R²: Large Point Sets
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[Dey, Wenger 2001]1 Ours: manifold
10k points
1 Dey, Wenger. Detecting undersampling in surface reconstruction. SCG 2001.
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R²: Extreme details
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Points
[DW01]1
Ours
1 Dey, Wenger. Detecting undersampling in surface reconstruction. SCG 2001.
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R²: Noisy Point Sets
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[Mehra et al. 2010]1 Ours: manifold + interpolating
1 Mehra, Tripathi, Sheffer and Mitra. Visibility of noisy point cloud data, Computers & Graphics, 2010.
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R²: Exhaustive Search
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[Ohrhallinger, Mudur 2011]1Ours: Local Minimum
1 S. Ohrhallinger, S. Mudur: Interpolating an unorganized 2D point cloud with a single closed shape, Computer-Aided Design, 2011.
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Reconstruction in R³
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inflate sculpture
Manifold Hull Interpolating Manifold
thickthin bounds hole
flip edges after, cover holes before
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Covering Hull Holes
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inflate
cover
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Sculpturing
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Sculpturefrom in and outside
Pop Membranes where possible
(Dominantly) interior points may remain (no free lunch)
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Results in R³
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Results in R³ compared
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[DG03]1
Ours
Improves especially for sparse and non-uniform point spacing1 Dey, Goswami. Tight cocone: a water-tight surface reconstructor, SM 2003.
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Varying Density in R³
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Noise Tolerance in R³
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[DG06]1
Ours
1 Dey, Goswami. Provable surface reconstruction from noisy samples, Computational Geometry: Theory and Applications, 2006.
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Limitations
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Very locally non-uniform sampling
Sparse 'overlap':
Saddle-type holes (cover is not a single disk)
Since we require O(n log n) time, local minima may be produced (more extensive searching could avoid that)
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Future Work
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Boundary Operator Minimum Spanning SurfaceSampling Condition
De-noising
Recover topology of deforming point sets
Shape understanding, i.e. retrieval
Invited post-doc in graphics group at Vienna Univ. of Technology
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Conclusion
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Shape Characteristic Boundary ReconstructionShape Formalization
Thesis statement: “Point clouds contain an intrinsic shape, minimizing an objective, which can be efficiently searched.”
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Extension: Sampling
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[ABE98]1 Minimal Umbrella(Boundary Operator)Sampling Condition: empty circumcircle
Boundary neighbors: reciprocal relationReconstruction of Bmin
guaranteed in O(n log n)Dense sampling Sparse sampling
A priori knowledge(shape closedness)permits sub-Nyquist reconstruction
1 Amenta, Bern, Eppstein. The Crust and the β-Skeleton: Combinatorial Curve Reconstruction, Graphical Models and Im. Proc., 1998.
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Extension: MSS
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Minimum Spanning Surface
Boundary ComplexBoundary Complex
Minimum Spanning Tree
R² R³
?
Rn
MSS
BC
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Segmenting Hull Holes
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3D Algorithm Flow
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