Spectral Surface Reconstruction Nils Erik Flick January 13, 2009 - PowerPoint PPT Presentation

Spectral Surface Reconstruction Nils Erik Flick January 13, 2009

Surface Reconstruction EigenCrust outline Spectral Theory & Practice Practical (Partial) Diagonalization References Reconstruction of Surfaces EigenCrust outline Spectral Theory & Practice Practical (Partial) Diagonalization Nils Erik Flick Spectral Surface Reconstruction

Surface Reconstruction Motivation EigenCrust outline Surfaces Spectral Theory & Practice Voronoi / Delaunay Practical (Partial) Diagonalization The Medial Axis References Voronoi Poles Outline Reconstruction of Surfaces Motivation Surfaces Voronoi / Delaunay The Medial Axis Voronoi Poles EigenCrust outline Spectral Theory & Practice Practical (Partial) Diagonalization Nils Erik Flick Spectral Surface Reconstruction

Surface Reconstruction Motivation EigenCrust outline Surfaces Spectral Theory & Practice Voronoi / Delaunay Practical (Partial) Diagonalization The Medial Axis References Voronoi Poles Motivation: Reconstruction ◮ Surface → cloud of sample points → watertight approximation ◮ Robust? Noise, outliers (laser scanner!), holes. ◮ Geom, top. Nils Erik Flick Spectral Surface Reconstruction

Surface Reconstruction Motivation EigenCrust outline Surfaces Spectral Theory & Practice Voronoi / Delaunay Practical (Partial) Diagonalization The Medial Axis References Voronoi Poles The EigenCrust Algorithm ◮ Geometric heuristics ◮ Transcends local problems by taking a global view ◮ No holes even in the presence of noise and unsampled patches Nils Erik Flick Spectral Surface Reconstruction

Surface Reconstruction Motivation EigenCrust outline Surfaces Spectral Theory & Practice Voronoi / Delaunay Practical (Partial) Diagonalization The Medial Axis References Voronoi Poles What is a Surface? ◮ Codimension 1 submanifold of ambient space ◮ No intersections, no boundary, manifold ◮ Surface = boundary of a volume . ◮ Search for manifold → automatically watertight! Nils Erik Flick Spectral Surface Reconstruction

Surface Reconstruction Motivation EigenCrust outline Surfaces Spectral Theory & Practice Voronoi / Delaunay Practical (Partial) Diagonalization The Medial Axis References Voronoi Poles Voronoi / Delaunay (1) ◮ Voronoi cell ◮ Starting point: Spatial closeness Nils Erik Flick Spectral Surface Reconstruction

Surface Reconstruction Motivation EigenCrust outline Surfaces Spectral Theory & Practice Voronoi / Delaunay Practical (Partial) Diagonalization The Medial Axis References Voronoi Poles Voronoi / Delaunay (2) ◮ Delaunay duals Voronoi. Nils Erik Flick Spectral Surface Reconstruction

Surface Reconstruction Motivation EigenCrust outline Surfaces Spectral Theory & Practice Voronoi / Delaunay Practical (Partial) Diagonalization The Medial Axis References Voronoi Poles Starting Point: Delaunay Contains Surface ◮ Triangulation contains surface approximation → good starting point Nils Erik Flick Spectral Surface Reconstruction

Surface Reconstruction Motivation EigenCrust outline Surfaces Spectral Theory & Practice Voronoi / Delaunay Practical (Partial) Diagonalization The Medial Axis References Voronoi Poles Starting Point: Space partitioning ◮ Triangulation partitions space ◮ Label the tetrahedra → inside and outside ◮ Surface = boundary i nside | o utside . Nils Erik Flick Spectral Surface Reconstruction

Surface Reconstruction Motivation EigenCrust outline Surfaces Spectral Theory & Practice Voronoi / Delaunay Practical (Partial) Diagonalization The Medial Axis References Voronoi Poles Skeleton ◮ Medial axis ≈ skeleton ◮ Deforms to surface’s ambient complement ◮ (homotopy & homeomorphism!) Nils Erik Flick Spectral Surface Reconstruction

Surface Reconstruction Motivation EigenCrust outline Surfaces Spectral Theory & Practice Voronoi / Delaunay Practical (Partial) Diagonalization The Medial Axis References Voronoi Poles Voronoi Poles ◮ Denser sampling → elongated cells ◮ Pole p + = furthest vertex of cell ◮ Pole p − : only if a ngle > π 2 ◮ Convergence to Medial Axis in 2D ◮ In 3D, “Surface” tetrahedra occur Nils Erik Flick Spectral Surface Reconstruction

Surface Reconstruction Motivation EigenCrust outline Surfaces Spectral Theory & Practice Voronoi / Delaunay Practical (Partial) Diagonalization The Medial Axis References Voronoi Poles Poles ↔ Skeleton Nils Erik Flick Spectral Surface Reconstruction

Surface Reconstruction The Combinatorial Approach EigenCrust outline EigenCrust (1) Spectral Theory & Practice EigenCrust (2) Practical (Partial) Diagonalization EigenCrust (3) References Outline Reconstruction of Surfaces EigenCrust outline The Combinatorial Approach EigenCrust (1) EigenCrust (2) EigenCrust (3) Spectral Theory & Practice Practical (Partial) Diagonalization Nils Erik Flick Spectral Surface Reconstruction

Surface Reconstruction The Combinatorial Approach EigenCrust outline EigenCrust (1) Spectral Theory & Practice EigenCrust (2) Practical (Partial) Diagonalization EigenCrust (3) References ◮ Delaunay triangulation is a combinatorial object (graph) ◮ So is its dual ◮ Good for algorithms! Nils Erik Flick Spectral Surface Reconstruction

Surface Reconstruction The Combinatorial Approach EigenCrust outline EigenCrust (1) Spectral Theory & Practice EigenCrust (2) Practical (Partial) Diagonalization EigenCrust (3) References EigenCrust proper ◮ Augment point cloud with bounding box ◮ Form pole graph ( V , E , w ): ◮ Poles belonging to a single vertex ◮ Poles of delaunay-neighboring vertices ◮ Edge weights: Geometrical Heuristic (sorry). ◮ Partition the pole graph ◮ Unlabel suspicious tetrahedra and re-partition. Nils Erik Flick Spectral Surface Reconstruction

Surface Reconstruction The Combinatorial Approach EigenCrust outline EigenCrust (1) Spectral Theory & Practice EigenCrust (2) Practical (Partial) Diagonalization EigenCrust (3) References EigenCrust (2) ◮ True MAT goes off into infinity → bounding box ◮ Authors use negative weights to great effect ◮ Weight: − e 4+4 cos φ – e 4 − 4 cos φ ◮ (unproven) justification: “Angle between circumspheres” α β P r r 1 2 β α A 1 A 2 d delaunay circumcircles Nils Erik Flick Spectral Surface Reconstruction

Surface Reconstruction The Combinatorial Approach EigenCrust outline EigenCrust (1) Spectral Theory & Practice EigenCrust (2) Practical (Partial) Diagonalization EigenCrust (3) References EigenCrust (2 1 2 ) α β P r r 2 1 β α A 1 A 2 d delaunay circumcircles Nils Erik Flick Spectral Surface Reconstruction

Surface Reconstruction The Combinatorial Approach EigenCrust outline EigenCrust (1) Spectral Theory & Practice EigenCrust (2) Practical (Partial) Diagonalization EigenCrust (3) References EigenCrust (3) ◮ A priori OUTSIDE / INSIDE supernodes. ◮ Second step for non-poles / ambiguous. ◮ Next: a comparison, made by the authors Nils Erik Flick Spectral Surface Reconstruction

Surface Reconstruction Laplacians EigenCrust outline Eigenmodes: Hearing + Seeing = Believing Spectral Theory & Practice Discretization Practical (Partial) Diagonalization Graph Vectorspaces for Space Partitioning References Outline Reconstruction of Surfaces EigenCrust outline Spectral Theory & Practice Laplacians Eigenmodes: Hearing + Seeing = Believing Discretization Graph Vectorspaces for Space Partitioning Practical (Partial) Diagonalization Nils Erik Flick Spectral Surface Reconstruction

Surface Reconstruction Laplacians EigenCrust outline Eigenmodes: Hearing + Seeing = Believing Spectral Theory & Practice Discretization Practical (Partial) Diagonalization Graph Vectorspaces for Space Partitioning References Before we proceed ... We are going to need some seemingly unrelated stuff. Please bear with me. Nils Erik Flick Spectral Surface Reconstruction

Surface Reconstruction Laplacians EigenCrust outline Eigenmodes: Hearing + Seeing = Believing Spectral Theory & Practice Discretization Practical (Partial) Diagonalization Graph Vectorspaces for Space Partitioning References Finite-Dimensional Vector Spaces ◮ Recall the vector space axioms ◮ Linear transformation ◮ Basis ◮ Matrix ◮ Square matrix Nils Erik Flick Spectral Surface Reconstruction

Surface Reconstruction Laplacians EigenCrust outline Eigenmodes: Hearing + Seeing = Believing Spectral Theory & Practice Discretization Practical (Partial) Diagonalization Graph Vectorspaces for Space Partitioning References We are talking ... Hilbert Spaces! ◮ Inner product: distances, angles � 1 ◮ f · g = 0 f ( x ) g ( x ) dx ◮ Importance of linear operators ◮ Importance of hermitean operators Nils Erik Flick Spectral Surface Reconstruction

Surface Reconstruction Laplacians EigenCrust outline Eigenmodes: Hearing + Seeing = Believing Spectral Theory & Practice Discretization Practical (Partial) Diagonalization Graph Vectorspaces for Space Partitioning References Operators ◮ Laplacian on R n as second derivative vector ◮ It frequently appears in physics ◮ It is a linear operator. ◮ You already know its eigenvectors! Nils Erik Flick Spectral Surface Reconstruction