Docstoc

spectral_surface_reconstruction

Document Sample
spectral_surface_reconstruction Powered By Docstoc
					Spectral Surface
Reconstruction from
Noisy Point Clouds
Paper by: R. Kolluri, J. Shewchuck, J.
    O’brien
Presented by: Gautam Kumar

                                                 Gautam Kumar
          Spectral Surface Reconstruction from Noisy Point Clouds
Outline
• Problem & Applications
• Definitions
• Previous Work
• Algorithm
• Results
• Limitations


                                                  Gautam Kumar
           Spectral Surface Reconstruction from Noisy Point Clouds
The Problem
• Laser Range Finders =
  VERY imperfect
  – Measurement Errors
  – Outliers
  – Under-sampled areas
• Surface Reconstruction
  – Point Cloud -> Watertight
    mesh

                                                   Gautam Kumar
            Spectral Surface Reconstruction from Noisy Point Clouds
Applications
• Computer Graphics
• Medical Imaging
• Computer-aided Design
• Solid Modeling




                                                 Gautam Kumar
          Spectral Surface Reconstruction from Noisy Point Clouds
Definitions
• Voronoi Cells
• Delaunay
  Triangulation
• Delaunay
  Tetrahedralization




                                                   Gautam Kumar
            Spectral Surface Reconstruction from Noisy Point Clouds
Surface Reconstruction
Approaches
• Zero-Set Approach
  – Define a function over the space where
    the zero-set is the surface.
• Level-Set
• Mesh-Zippering
• Ball-Pivoting



                                                   Gautam Kumar
            Spectral Surface Reconstruction from Noisy Point Clouds
“Crusty” Algorithms
• CRUST (Amenta, Bern, Kamvysselis)
• COCONE (Amenta, Choi, Dey, Leekha)
• POWER CRUST (Amenta, Choi, Kolluri)
• TIGHT COCONE (Dey, Goswami)
• ROBUST COCONE (Dey, Goswami)




                                                 Gautam Kumar
          Spectral Surface Reconstruction from Noisy Point Clouds
The Eigencrust Algorithm
• Partition the tetrahedra
  of a Delaunay
  tetrahedralization into
  inside/outside
• Identify the triangular
  faces that interface
  between the subgraphs
• And voila… the
  eigencrust

                                                    Gautam Kumar
             Spectral Surface Reconstruction from Noisy Point Clouds
Algorithm Overview
• Compute Voronoi vertices & poles
• Weight edges between Delaunay
  tetrahedra
• Divide the tetrahedra (inside/outside)
• Label remaining tetrahedra and find
  the “eigencrust”



                                                   Gautam Kumar
            Spectral Surface Reconstruction from Noisy Point Clouds
Algorithm Overview
• Compute Voronoi vertices & poles
• Weight edges between Delaunay
  tetrahedra
• Divide the tetrahedra (inside/outside)
• Label remaining tetrahedra and find
  the “eigencrust”



                                                   Gautam Kumar
            Spectral Surface Reconstruction from Noisy Point Clouds
Without Poles
• For 3D Voronoi diagrams, many Voronoi
  vertices lie near the surface
• As a result, the corresponding Delaunay
  tetrahedra are often flat along the surface
• This leads to ambiguity when labeling
  tetrahedra as inside/outside




                                                    Gautam Kumar
             Spectral Surface Reconstruction from Noisy Point Clouds
Poles to the Rescue
• Amenta, Bern,
  Kamvysselis ’98
• Voronoi vertices
  that are farthest
  away from a
  sample point on
  opposite sides of
  the surface

                                                   Gautam Kumar
            Spectral Surface Reconstruction from Noisy Point Clouds
Delaunay Tetrahedralization
• Now the Delaunay tetrahedra are far
  from the surface
• This makes labeling them
  inside/outside much easier!




                                                  Gautam Kumar
           Spectral Surface Reconstruction from Noisy Point Clouds
Algorithm Overview
• Compute Voronoi vertices & poles
• Weight edges between Delaunay
  tetrahedra
• Divide the tetrahedra (inside/outside)
• Label remaining tetrahedra and find
  the “eigencrust”



                                                   Gautam Kumar
            Spectral Surface Reconstruction from Noisy Point Clouds
Weighting Edges
• Partitioning (inside/outside) is similar to
  a system of masses & springs
• Positive weight between two
  tetrahedra means attractive force
• Negative weight between two
  tetrahedra means repulsive force



                                                    Gautam Kumar
             Spectral Surface Reconstruction from Noisy Point Clouds
Weight Values
• There are edge
  weights between
  pairs of Voronoi poles
  – Greater weight
  – Greater angle
    between tetrahedra
  – Greater likelihood to
    be on same side of the
    surface


                                                     Gautam Kumar
              Spectral Surface Reconstruction from Noisy Point Clouds
Algorithm Overview
• Compute Voronoi vertices & poles
• Weight edges between Delaunay
  tetrahedra
• Divide the tetrahedra (inside/outside)
• Label remaining tetrahedra and find
  the “eigencrust”



                                                   Gautam Kumar
            Spectral Surface Reconstruction from Noisy Point Clouds
Spectral Partitioning
• Used for image
  segmentation, circuit
  layout, document
  clustering, etc…
• Cut a graph into two
  sub-graphs, each
  approximately half
  the size, minimizing
  weight of cut edges.

                                                  Gautam Kumar
           Spectral Surface Reconstruction from Noisy Point Clouds
Computing Eigenvectors
• Create Laplacian matrix for the poles
  where L(i,j) = -w(i,j)
• Eigensystem of L represents modes of
  vibration in our mass-spring system
• Use Lanczos algorithm to compute the
  eigenvector with smallest eigenvalue
  (lowest frequency)
• Runs in O(n√n) time
                                                  Gautam Kumar
           Spectral Surface Reconstruction from Noisy Point Clouds
The Poles Separate
• Eigenvector with
  lowest eigenvalue
  tells us where to cut
  the graph most
  effectively
• Now, we have poles
  labeled inside &
  outside

                                                   Gautam Kumar
            Spectral Surface Reconstruction from Noisy Point Clouds
Algorithm Overview
• Compute Voronoi vertices & poles
• Weight edges between Delaunay
  tetrahedra
• Divide the tetrahedra (inside/outside)
• Label remaining tetrahedra and find
  the “eigencrust”



                                                   Gautam Kumar
            Spectral Surface Reconstruction from Noisy Point Clouds
Label Remaining Tetrahedra
• Non-pole edges
  weighted with
  “aspect ratio”
• We can use
  Powercrust
  algorithm to create
  power cell for each
  pole to eliminate
  non-pole labeling
  – Captures sharp
    corners better but
    produces many
    more vertices
                                                     Gautam Kumar
              Spectral Surface Reconstruction from Noisy Point Clouds
The Final Mesh
• The final mesh is the “eigencrust”
  – The triangles where the inside and outside
    tetrahedra meet




                                                    Gautam Kumar
             Spectral Surface Reconstruction from Noisy Point Clouds
Optional Features
• Removing labels from small tetrahedra
  in Stage 1 reduces noise and gives
  greater accuracy
• Irregularities can be fixed to guarantee
  manifoldness of surface
• Laplacian smoothing as a last step
  produces a great quality mesh


                                                   Gautam Kumar
            Spectral Surface Reconstruction from Noisy Point Clouds
Results
• Problems solved
  – Outliers solved easily: if all adjacent tetrahedra
    are labeled the same, the point is an outlier
  – Undersampled regions are handled easily without
    holes
  – Measurement errors are minimized using global
    view when partitioning




                                                     Gautam Kumar
              Spectral Surface Reconstruction from Noisy Point Clouds
Comparisons
• Eigencrust: Genus-0 manifold but
  disintegrates with high noise
• Tight Cocone: Can only handle very
  small amount of noise
• Powercrust: Handles errors but is holey




                                                   Gautam Kumar
            Spectral Surface Reconstruction from Noisy Point Clouds
Limitations
• Occasionally creates
  unwanted handles
• Eigen computation is
  VERY slow
• Sharp corners are not
  handled well
  – Fixed with power cells at
    the cost of model
    complexity

                                                    Gautam Kumar
             Spectral Surface Reconstruction from Noisy Point Clouds
Questions?
• Please raise your hands… just kidding




                                                  Gautam Kumar
           Spectral Surface Reconstruction from Noisy Point Clouds

				
DOCUMENT INFO
Shared By:
Categories:
Tags:
Stats:
views:2
posted:10/17/2012
language:Unknown
pages:28