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”

delaunay circumcircles

References

EigenCrust (212)

delaunay circumcircles

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

{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(√

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)

References

Questions for you

I Proof of properties (reconstruction quality)?

I Incremental computations?

I Sharp corners → hybrid

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.

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

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