Meshless geometric subdivision

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Meshless geometric subdivision

• By
• Carsten Moenning
• Facundo Mémoli
• Guillermo Sapiro
• Nira Dyn
• Neil A. Dodgson
• IMA Preprint Series 1977
• (April 2004)

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Frame

• Introduction
• Related work
• Intrinsic distance mapping across point clouds
• Intrinsic meshless subdivision
• Intrinsic point cloud simplification
• Intrinsic proximity information
• An intrinsic subdivision operator for point
  clouds
• Geodesic centroid computation
• Experimental results

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Introduction

• Working with point cloud data directly
• Replace mesh connectivity by intrinsic proximity
  information
• Meshless subdivision rules using geodesic
  centroids
• Main contributions
• A first meshless geodesic subdivision operator
• A new method for the computation of geodesic
  weighted averages on manifold surfaces
• Benefits
• Not requiring any non-geometric preprocessing
  steps
• (costly surface reconstruction,simplificatio
  n, potential remeshing )
• Not vary with the particular type of data (mesh)
  representation used
• 
  
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4
Related work

• DYN N., LEVIN D. Subdivision schemes in
  geometric modelling . Acta Numerica 12 (2002),
  73144.
• LOOP C. Smooth surface subdivision based on
  triangles. In Masters th., Dep. of Math.,
  University of Utah, USA (1987).
• FLEISHMAN S., COHEN-OR D., ALEXA M., SILVA C. T.
  Progressive point set surfaces. ACM Trans. on
  Comp. Graph. 22, 4 (2003), 9971011.
• ALEXA M., BEHR J., COHEN-OR D., FLEISHMAN
  S.,LEVIN D., SILVA C. T. Computing and rendering
  point set surfaces. IEEE Trans. on Visual. and
  Comp. Graph.9, 1 (2003), 315.
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DL02

• Subdivision operator S recursively applied to
  control nets
• of
  arbitrary topology
• Refinement rule new control vertices are
  inserted and connected
• Geometric averaging ruleControl vertices are
  repositioned
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LOO87

• Mesh subdivision operator allows for such a
  simple distinction between a topological
  refinement and a geometric averaging rule
  applicable at all points.
• The refinement rule consists of the insertion of
  a new point at the midpoint of every edge
• The geometric rule, applicable at all points in
  the new mesh,smoothes the locations of the points
  by weighted averaging of their topological
  neighbours in the refined mesh
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FCOAS03

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Intrinsic distance mapping across point clouds

• r-offset
• then
• approximated by
• more details
  FastFPS

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FastFPS Fast Marching farthest point sampling

• Voronoi diagrams
• Farthest point sampling
• Fast Marching
• Farthest point sampling for point clouds
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Voronoi diagrams

• The perpendicular bisector
• The Voronoi cell
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Farthest point sampling

• The point farthest away from the current set
  of sample sites, S, is represented by the centre
  of the largest circle empty of any site si? S.
• The centre of such a circle is given by a
  vertex of the bounded Voronoi diagram of S,
  BVD(S).
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Fast Marching

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Farthest point sampling for point clouds

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Intrinsic point cloud simplification

• Simplify any highly dense input P to a base point
  set P0 still sufficiently dense to support
  meaningful geodesic centroid
• By letting the user control the radius of the
  largest empty circle in the domain of the
  simplified point set
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Intrinsic proximity information

• In the meshless case mesh connectivity
  information is replaced by local proximity
  information.
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An intrinsic subdivision operator for point clouds

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Geodesic centroid computation

• Minimize
• Back propagation with velocity field
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Experimental results
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Experimental results

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