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

“[...] 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

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)

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

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:

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

BC0: Adding Noise

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Catacomb section slightly perturbed extremely perturbed

Property: Adding noise does not impede construction

Reconstruction in R²

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Inflate Sculpture

Manifold Hull Interpolating Manifold

R²: Improved Results

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Points

[DW01]1

Ours

1 Dey, Wenger. Detecting undersampling in surface reconstruction. SCG 2001.

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.

R²: Extreme details

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Points

[DW01]1

Ours

1 Dey, Wenger. Detecting undersampling in surface reconstruction. SCG 2001.

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.

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.

Reconstruction in R³

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inflate sculpture

Manifold Hull Interpolating Manifold

thickthin bounds hole

flip edges after, cover holes before

Covering Hull Holes

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inflate

cover

Sculpturing

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Sculpturefrom in and outside

Pop Membranes where possible

(Dominantly) interior points may remain (no free lunch)

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.

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.

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)

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

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.”

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.

Extension: MSS

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Minimum Spanning Surface

Boundary ComplexBoundary Complex

Minimum Spanning Tree

R² R³

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Rn

MSS

BC

Segmenting Hull Holes

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3D Algorithm Flow

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