nils erik flick january 13, 2009 - uni-hamburg.detchernia/slides/eigencr… · i codimension 1...

Spectral Surface Reconstruction

Nils Erik Flick

January 13, 2009

Surface ReconstructionEigenCrust outline

Spectral Theory & PracticePractical (Partial) Diagonalization

References

Reconstruction of Surfaces

EigenCrust outline

Spectral Theory & Practice

Practical (Partial) Diagonalization

Nils Erik Flick Spectral Surface Reconstruction



References

MotivationSurfacesVoronoi / DelaunayThe Medial AxisVoronoi Poles

Outline

Reconstruction of SurfacesMotivationSurfacesVoronoi / DelaunayThe Medial AxisVoronoi Poles

EigenCrust outline






References


Motivation: Reconstruction

I Surface→ cloud of sample points→ watertight approximation

I Robust? Noise, outliers (laser scanner!), holes.

I Geom, top.




References


The EigenCrust Algorithm

I Geometric heuristics

I Transcends local problems by taking a global view

I No holes even in the presence of noise and unsampled patches




References


What is a Surface?

I Codimension 1 submanifold of ambient space

I No intersections, no boundary, manifold

I Surface = boundary of a volume.

I Search for manifold → automatically watertight!




References


Voronoi / Delaunay (1)

I Voronoi cell

I Starting point: Spatial closeness




References


Voronoi / Delaunay (2)

I Delaunay duals Voronoi.




References


Starting Point: Delaunay Contains Surface

I Triangulation contains surface approximation → good startingpoint




References


Starting Point: Space partitioning

I Triangulation partitions space

I Label the tetrahedra → inside and outside

I Surface = boundary inside|outside.




References


Skeleton

I Medial axis ≈ skeleton

I Deforms to surface’s ambient complement

I (homotopy & homeomorphism!)




References


Voronoi Poles

I Denser sampling → elongated cells

I Pole p+ = furthest vertex of cell

I Pole p−: only if angle > π2

I Convergence to Medial Axis in 2D

I In 3D, “Surface” tetrahedra occur




References


Poles ↔ Skeleton




References

The Combinatorial ApproachEigenCrust (1)EigenCrust (2)EigenCrust (3)

Outline


EigenCrust outlineThe Combinatorial ApproachEigenCrust (1)EigenCrust (2)EigenCrust (3)






References


I Delaunay triangulation is a combinatorial object (graph)

I So is its dual

I Good for algorithms!




References


EigenCrust proper

I Augment point cloud with bounding box

I Form pole graph (V ,E ,w):

I Poles belonging to a single vertex

I Poles of delaunay-neighboring vertices

I Edge weights: Geometrical Heuristic (sorry).

I Partition the pole graph

I Unlabel suspicious tetrahedra and re-partition.




References


EigenCrust (2)

I True MAT goes off into infinity → bounding box

I Authors use negative weights to great effect

I Weight: −e4+4 cos φ – e4−4 cos φ

I (unproven) justification: “Angle between circumspheres”

P

A1

r1

A2

r2

α

delaunay circumcircles

d

α

β

β




References


EigenCrust (212)

P

A1

r1

A2

r2

α

delaunay circumcircles

d

α

β

β




References


EigenCrust (3)

I A priori OUTSIDE / INSIDE supernodes.

I Second step for non-poles / ambiguous.

I Next: a comparison, made by the authors




References

LaplaciansEigenmodes: Hearing + Seeing = BelievingDiscretizationGraph Vectorspaces for Space Partitioning

Outline


EigenCrust outline

Spectral Theory & PracticeLaplaciansEigenmodes: Hearing + Seeing = BelievingDiscretizationGraph Vectorspaces for Space Partitioning





References


Before we proceed ...

We are going to need some seemingly unrelated stuff.Please bear with me.




References


Finite-Dimensional Vector Spaces

I Recall the vector space axioms

I Linear transformation

I Basis

I Matrix

I Square matrix




References


We are talking ... Hilbert Spaces!

I Inner product: distances, angles

I f · g =∫ 10 f (x)g(x)dx

I Importance of linear operators

I Importance of hermitean operators




References


Operators

I Laplacian on Rn as second derivative vector

I It frequently appears in physics

I It is a linear operator.

I You already know its eigenvectors!




References


Harmonics

I Eigenfunctions of the Laplacian = Harmonics

I Harmonic Analysis is often a good idea

I Depends on domain

I Demo!




References


Natural Modes of Vibration

I Consider some solid object

I Tap it, it sounds

I You are hearing its spectrum!

I Normal Mode ↔ Eigenvalue ↔ Frequency ↔ Energy (Why?)

I Can even be made visible

I Degeneracies




References


Harmonics & Eigenmodes

I Vibrations: Boundary value problem; but also ...I finite modelI Resonances




References


Rn doesn’t fit in my computer!

I Basic finite difference approximation

I Square grid

I Convergence

I generalizes to arbitrary grids & graphs




References


Self-Adjointness or Why the Spectrum is Real

I A desirable property: 〈Ax , y〉 = 〈x ,Ay〉I Corresponds to symmetric matrix

I Real spectrum and orthogonal set of eigenvectors.

I Find A = UΛU∗ (U rotation). Often miraculous!




References


Introducing Normalized Cuts

I Flexible formalism

I Segmentation by graph cuts

I Good, globally consistent solution

I Graph Theory ↔ Linear Algebra

I Combinatorics ↔ Numeric Methods

I Weighted, undirected graphs.




References


Combinatorial Laplacian / Graph Matrices

I Combinatorial Laplacian

I Affinity (generalized adjacency) matrix

Aij =

{wij if (i , j) ∈ E0 else

I Degree matrix D, diagonal Dii = degree of vertex i

I Graph Laplacian L = D − A

I Degree-normalized: W = D− 12 LD− 1

2 (transform vectors asneeded)

I W remains sparse.




References


Outline of the NCuts Algorithm

I Construct a graph with weighted edges.

I Connectivity and Weights = adapt to model

I Partition along nodal sets

I Corresponding to λ2

I (lowest is trivial)

I Object splits naturally – tightly connected parts vibratetogether




References

Lanczos IterationThe Wider PerspectiveOpen Questions

Outline


EigenCrust outline


Practical (Partial) DiagonalizationLanczos IterationThe Wider Perspective: Other Interesting Uses of RelatedTechniquesOpen Questions




References


Sparsely connected graphs → sparse matrices

I Small eigenvalue problems can be solved by direct methods(matrix factorizations)

I Prohibitive for large problems.

I Sparse matrices are made of zeroes ... mainly

I Matrix-Vector multiplication tends to be inexpensive

I Iterative methods very welcome.




References


Selective calculation of eigenvectors

I Calculating eigenpairs in O(n) time per iteration

I Typical number of iterations O(√

n)

I But varies according to eigenstructure

I Positive definite; xT AxxT x

I Get lowest first




References


More Applications

I Simulation (physics)

I Data Clustering: importance-weighted criterion

I / Text Mining

I Transductive Learning (global view)

I ...




References


Questions for you

I Proof of properties (reconstruction quality)?

I Incremental computations?

I Sharp corners → hybrid

I ...




References

References I

Amenta, N., Choi, S., Dey, T. K., & Leekha, N. (2000). A simplealgorithm for homeomorphic surface reconstruction. InInternational journal of computational geometry andapplications (pp. 213–222).

Amenta, N., Choi, S., & Kolluri, R. K. (2001). The power crust,unions of balls, and the medial axis transform.Computational Geometry: Theory and Applications, 19,127–153.

Choi, H. I., Choi, S. W., & Moon, H. P. (1997). Mathematicaltheory of medial axis transform. Pacific Journal ofMathematics, 181, 57–88.




References

References II

Fowlkes, C., Belongie, S., Chung, F., & Malik, J. (2004). Spectralgrouping using the nystrom method. IEEE Transactions onPattern Analysis and Machine Intelligence, 26, 214–225.

Kac, M. (1966). Can you hear the shape of a drum. Amer. Math.Monthly, 73(1).

Rahimi, A., & Recht, B. (2004). Clustering with normalized cuts isclustering with a hyperplane. Statistical Learning inComputer Vision.

Shi, J., Belongie, S., Leung, T. K., & Malik, J. (1998). Image andvideo segmentation: The normalized cut framework. In Icip(1) (p. 943-947).

Cgal, Computational Geometry Algorithms Library. (n.d.).(http://www.cgal.org)




References

References III

Ietl, the Iterative Eigensolver Template Library. (n.d.).(http://www.comp-phys.org/software/ietl/)

Wardetzky, M., Mathur, S., Kalberer, F., & Grinspun, E. (2007).Discrete laplace operators: no free lunch.

Zhou, D., & Scholkopf, B. (2005). Regularization on DiscreteSpaces. LECTURE NOTES IN COMPUTER SCIENCE,3663, 361.


seppke

Textfeld

Ravikrishna Kolluri, Jonathan Richard Shewchuk, and James F. O'Brien, Spectral Surface Reconstruction from Noisy Point Clouds, Symposium on Geometry Processing 2004 (Nice, France), pages 11-21,Eurographics Association, July 2004.

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Accepted festgelegt von seppke

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Completed festgelegt von seppke

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nils erik flick january 13, 2009 - uni-hamburg.detchernia/slides/eigencr… · i codimension 1...

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