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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
Surface ReconstructionEigenCrust outline
Spectral Theory & PracticePractical (Partial) Diagonalization
References
MotivationSurfacesVoronoi / DelaunayThe Medial AxisVoronoi Poles
Outline
Reconstruction of SurfacesMotivationSurfacesVoronoi / DelaunayThe Medial AxisVoronoi Poles
EigenCrust outline
Spectral Theory & Practice
Practical (Partial) Diagonalization
Nils Erik Flick Spectral Surface Reconstruction
Surface ReconstructionEigenCrust outline
Spectral Theory & PracticePractical (Partial) Diagonalization
References
MotivationSurfacesVoronoi / DelaunayThe Medial AxisVoronoi Poles
Motivation: Reconstruction
I Surface→ cloud of sample points→ watertight approximation
I Robust? Noise, outliers (laser scanner!), holes.
I Geom, top.
Nils Erik Flick Spectral Surface Reconstruction
Surface ReconstructionEigenCrust outline
Spectral Theory & PracticePractical (Partial) Diagonalization
References
MotivationSurfacesVoronoi / DelaunayThe Medial AxisVoronoi Poles
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
Nils Erik Flick Spectral Surface Reconstruction
Surface ReconstructionEigenCrust outline
Spectral Theory & PracticePractical (Partial) Diagonalization
References
MotivationSurfacesVoronoi / DelaunayThe Medial AxisVoronoi Poles
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!
Nils Erik Flick Spectral Surface Reconstruction
Surface ReconstructionEigenCrust outline
Spectral Theory & PracticePractical (Partial) Diagonalization
References
MotivationSurfacesVoronoi / DelaunayThe Medial AxisVoronoi Poles
Voronoi / Delaunay (1)
I Voronoi cell
I Starting point: Spatial closeness
Nils Erik Flick Spectral Surface Reconstruction
Surface ReconstructionEigenCrust outline
Spectral Theory & PracticePractical (Partial) Diagonalization
References
MotivationSurfacesVoronoi / DelaunayThe Medial AxisVoronoi Poles
Voronoi / Delaunay (2)
I Delaunay duals Voronoi.
Nils Erik Flick Spectral Surface Reconstruction
Surface ReconstructionEigenCrust outline
Spectral Theory & PracticePractical (Partial) Diagonalization
References
MotivationSurfacesVoronoi / DelaunayThe Medial AxisVoronoi Poles
Starting Point: Delaunay Contains Surface
I Triangulation contains surface approximation → good startingpoint
Nils Erik Flick Spectral Surface Reconstruction
Surface ReconstructionEigenCrust outline
Spectral Theory & PracticePractical (Partial) Diagonalization
References
MotivationSurfacesVoronoi / DelaunayThe Medial AxisVoronoi Poles
Starting Point: Space partitioning
I Triangulation partitions space
I Label the tetrahedra → inside and outside
I Surface = boundary inside|outside.
Nils Erik Flick Spectral Surface Reconstruction
Surface ReconstructionEigenCrust outline
Spectral Theory & PracticePractical (Partial) Diagonalization
References
MotivationSurfacesVoronoi / DelaunayThe Medial AxisVoronoi Poles
Skeleton
I Medial axis ≈ skeleton
I Deforms to surface’s ambient complement
I (homotopy & homeomorphism!)
Nils Erik Flick Spectral Surface Reconstruction
Surface ReconstructionEigenCrust outline
Spectral Theory & PracticePractical (Partial) Diagonalization
References
MotivationSurfacesVoronoi / DelaunayThe Medial AxisVoronoi Poles
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
Nils Erik Flick Spectral Surface Reconstruction
Surface ReconstructionEigenCrust outline
Spectral Theory & PracticePractical (Partial) Diagonalization
References
MotivationSurfacesVoronoi / DelaunayThe Medial AxisVoronoi Poles
Poles ↔ Skeleton
Nils Erik Flick Spectral Surface Reconstruction
Surface ReconstructionEigenCrust outline
Spectral Theory & PracticePractical (Partial) Diagonalization
References
The Combinatorial ApproachEigenCrust (1)EigenCrust (2)EigenCrust (3)
Outline
Reconstruction of Surfaces
EigenCrust outlineThe Combinatorial ApproachEigenCrust (1)EigenCrust (2)EigenCrust (3)
Spectral Theory & Practice
Practical (Partial) Diagonalization
Nils Erik Flick Spectral Surface Reconstruction
Surface ReconstructionEigenCrust outline
Spectral Theory & PracticePractical (Partial) Diagonalization
References
The Combinatorial ApproachEigenCrust (1)EigenCrust (2)EigenCrust (3)
I Delaunay triangulation is a combinatorial object (graph)
I So is its dual
I Good for algorithms!
Nils Erik Flick Spectral Surface Reconstruction
Surface ReconstructionEigenCrust outline
Spectral Theory & PracticePractical (Partial) Diagonalization
References
The Combinatorial ApproachEigenCrust (1)EigenCrust (2)EigenCrust (3)
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.
Nils Erik Flick Spectral Surface Reconstruction
Surface ReconstructionEigenCrust outline
Spectral Theory & PracticePractical (Partial) Diagonalization
References
The Combinatorial ApproachEigenCrust (1)EigenCrust (2)EigenCrust (3)
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
Nils Erik Flick Spectral Surface Reconstruction
Surface ReconstructionEigenCrust outline
Spectral Theory & PracticePractical (Partial) Diagonalization
References
The Combinatorial ApproachEigenCrust (1)EigenCrust (2)EigenCrust (3)
EigenCrust (212)
P
A1
r1
A2
r2
α
delaunay circumcircles
d
α
β
β
Nils Erik Flick Spectral Surface Reconstruction
Surface ReconstructionEigenCrust outline
Spectral Theory & PracticePractical (Partial) Diagonalization
References
The Combinatorial ApproachEigenCrust (1)EigenCrust (2)EigenCrust (3)
EigenCrust (3)
I A priori OUTSIDE / INSIDE supernodes.
I Second step for non-poles / ambiguous.
I Next: a comparison, made by the authors
Nils Erik Flick Spectral Surface Reconstruction
Surface ReconstructionEigenCrust outline
Spectral Theory & PracticePractical (Partial) Diagonalization
References
LaplaciansEigenmodes: Hearing + Seeing = BelievingDiscretizationGraph Vectorspaces for Space Partitioning
Outline
Reconstruction of Surfaces
EigenCrust outline
Spectral Theory & PracticeLaplaciansEigenmodes: Hearing + Seeing = BelievingDiscretizationGraph Vectorspaces for Space Partitioning
Practical (Partial) Diagonalization
Nils Erik Flick Spectral Surface Reconstruction
Surface ReconstructionEigenCrust outline
Spectral Theory & PracticePractical (Partial) Diagonalization
References
LaplaciansEigenmodes: Hearing + Seeing = BelievingDiscretizationGraph Vectorspaces for Space Partitioning
Before we proceed ...
We are going to need some seemingly unrelated stuff.Please bear with me.
Nils Erik Flick Spectral Surface Reconstruction
Surface ReconstructionEigenCrust outline
Spectral Theory & PracticePractical (Partial) Diagonalization
References
LaplaciansEigenmodes: Hearing + Seeing = BelievingDiscretizationGraph Vectorspaces for Space Partitioning
Finite-Dimensional Vector Spaces
I Recall the vector space axioms
I Linear transformation
I Basis
I Matrix
I Square matrix
Nils Erik Flick Spectral Surface Reconstruction
Surface ReconstructionEigenCrust outline
Spectral Theory & PracticePractical (Partial) Diagonalization
References
LaplaciansEigenmodes: Hearing + Seeing = BelievingDiscretizationGraph Vectorspaces for Space Partitioning
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
Nils Erik Flick Spectral Surface Reconstruction
Surface ReconstructionEigenCrust outline
Spectral Theory & PracticePractical (Partial) Diagonalization
References
LaplaciansEigenmodes: Hearing + Seeing = BelievingDiscretizationGraph Vectorspaces for Space Partitioning
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!
Nils Erik Flick Spectral Surface Reconstruction
Surface ReconstructionEigenCrust outline
Spectral Theory & PracticePractical (Partial) Diagonalization
References
LaplaciansEigenmodes: Hearing + Seeing = BelievingDiscretizationGraph Vectorspaces for Space Partitioning
Harmonics
I Eigenfunctions of the Laplacian = Harmonics
I Harmonic Analysis is often a good idea
I Depends on domain
I Demo!
Nils Erik Flick Spectral Surface Reconstruction
Surface ReconstructionEigenCrust outline
Spectral Theory & PracticePractical (Partial) Diagonalization
References
LaplaciansEigenmodes: Hearing + Seeing = BelievingDiscretizationGraph Vectorspaces for Space Partitioning
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
Nils Erik Flick Spectral Surface Reconstruction
Surface ReconstructionEigenCrust outline
Spectral Theory & PracticePractical (Partial) Diagonalization
References
LaplaciansEigenmodes: Hearing + Seeing = BelievingDiscretizationGraph Vectorspaces for Space Partitioning
Harmonics & Eigenmodes
I Vibrations: Boundary value problem; but also ...I finite modelI Resonances
Nils Erik Flick Spectral Surface Reconstruction
Surface ReconstructionEigenCrust outline
Spectral Theory & PracticePractical (Partial) Diagonalization
References
LaplaciansEigenmodes: Hearing + Seeing = BelievingDiscretizationGraph Vectorspaces for Space Partitioning
Rn doesn’t fit in my computer!
I Basic finite difference approximation
I Square grid
I Convergence
I generalizes to arbitrary grids & graphs
Nils Erik Flick Spectral Surface Reconstruction
Surface ReconstructionEigenCrust outline
Spectral Theory & PracticePractical (Partial) Diagonalization
References
LaplaciansEigenmodes: Hearing + Seeing = BelievingDiscretizationGraph Vectorspaces for Space Partitioning
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!
Nils Erik Flick Spectral Surface Reconstruction
Surface ReconstructionEigenCrust outline
Spectral Theory & PracticePractical (Partial) Diagonalization
References
LaplaciansEigenmodes: Hearing + Seeing = BelievingDiscretizationGraph Vectorspaces for Space Partitioning
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.
Nils Erik Flick Spectral Surface Reconstruction
Surface ReconstructionEigenCrust outline
Spectral Theory & PracticePractical (Partial) Diagonalization
References
LaplaciansEigenmodes: Hearing + Seeing = BelievingDiscretizationGraph Vectorspaces for Space Partitioning
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.
Nils Erik Flick Spectral Surface Reconstruction
Surface ReconstructionEigenCrust outline
Spectral Theory & PracticePractical (Partial) Diagonalization
References
LaplaciansEigenmodes: Hearing + Seeing = BelievingDiscretizationGraph Vectorspaces for Space Partitioning
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
Nils Erik Flick Spectral Surface Reconstruction
Surface ReconstructionEigenCrust outline
Spectral Theory & PracticePractical (Partial) Diagonalization
References
Lanczos IterationThe Wider PerspectiveOpen Questions
Outline
Reconstruction of Surfaces
EigenCrust outline
Spectral Theory & Practice
Practical (Partial) DiagonalizationLanczos IterationThe Wider Perspective: Other Interesting Uses of RelatedTechniquesOpen Questions
Nils Erik Flick Spectral Surface Reconstruction
Surface ReconstructionEigenCrust outline
Spectral Theory & PracticePractical (Partial) Diagonalization
References
Lanczos IterationThe Wider PerspectiveOpen Questions
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.
Nils Erik Flick Spectral Surface Reconstruction
Surface ReconstructionEigenCrust outline
Spectral Theory & PracticePractical (Partial) Diagonalization
References
Lanczos IterationThe Wider PerspectiveOpen Questions
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
Nils Erik Flick Spectral Surface Reconstruction
Surface ReconstructionEigenCrust outline
Spectral Theory & PracticePractical (Partial) Diagonalization
References
Lanczos IterationThe Wider PerspectiveOpen Questions
More Applications
I Simulation (physics)
I Data Clustering: importance-weighted criterion
I / Text Mining
I Transductive Learning (global view)
I ...
Nils Erik Flick Spectral Surface Reconstruction
Surface ReconstructionEigenCrust outline
Spectral Theory & PracticePractical (Partial) Diagonalization
References
Lanczos IterationThe Wider PerspectiveOpen Questions
Questions for you
I Proof of properties (reconstruction quality)?
I Incremental computations?
I Sharp corners → hybrid
I ...
Nils Erik Flick Spectral Surface Reconstruction
Surface ReconstructionEigenCrust outline
Spectral Theory & PracticePractical (Partial) Diagonalization
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.
Nils Erik Flick Spectral Surface Reconstruction
Surface ReconstructionEigenCrust outline
Spectral Theory & PracticePractical (Partial) Diagonalization
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)
Nils Erik Flick Spectral Surface Reconstruction
Surface ReconstructionEigenCrust outline
Spectral Theory & PracticePractical (Partial) Diagonalization
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 Spectral Surface Reconstruction