Moving Least Squares Multiresolution Surface Approximation

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Moving Least Squares Multiresolution Surface
Approximation

• Boris Mederos
• Luiz Velho
• Luiz Henrique De Figueirdo

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Overview

• About Authors
• Introduction
• Related Works
• This work
• Results
• Conclusion

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About Authors

• Boris Mederos
• PhD of Instituto Nacional de Matematica Pura
  e Aplicada,
• Rio de Janeiro, Brasil, Advisor Luiz Velho
  and Luiz Henrique de Figueiredo research
  interests lie in surface reconstruction and CG

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Luiz Velho

• Full Researcher at IMPA and Leading Scientist of
  the VISGRAF Laboratory.
• various aspects of computer graphics and related
  areas. the central focus of his work is
• the investigation of multiscale models
• and hierarchical computational
• methods associated with them

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Luiz Henrique De Figueirdo

• Associate Researcher at IMPA and a member of its
  Visgraf laboratory.
• Research interests include computational
  geometry, geometric modeling, and interval
  methods in computer graphics, specially
  applications of affine arithmetic.

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Introduction
The problem of surface reconstruction and
refinement from scatted points without normals
has received a growing amount of attention in
computer graphics. and there are several
algorithms known for this problem
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Related Works

• 3D Delaunay triangulation
• a new Voronoi based surface algorithm
  .SIGGRAPH98
• Greedy approach
• The ball-pivoting algorithm for surface
  reconstruction 99
• Incremental algorithms
• computes a set of representative points and
  triangulates these representative points.
• Curve and surface reconstruction from
  regular and non regular point sets 101-126 2001

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This Work

• Clustering
• Reduction
• Triangulation
• Refinement

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Clustering
This step is to partition the original set of
points Q into a finite set of clusters. This
method is based on a BSP tree,Each node contains
a sub set First define subdivision criteria
for BSP tree
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Clustering

• Since C is a symmetric positive semi-definite
• 33 matrix,its three eigenvalues are real and
  order them as

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Clustering

• Hence the ratio
• can be used for the curvature of S around p
• Now the subdivision criteria is
• 1 the ratio is larger than a tolerance
• 2 the number n is larger than

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Clusters
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Reduction

• This algorithm uses a new method based on
  moving least squares theory (MLS) to find a
  representative point for each cluster.

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Reduction

• Assuming the centroid of the set of points in the
  cluster is c,now
• we abtain the point
• where the weighted covariance matrix M is a 33
  matrix whose
• entries are

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Reduction

• To compute the vector nc, we can start with t
  0,and the minimization problem can be rewritten
  in the form

While B is the matrix of weight covariance
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Reduction

• Using the direction as above ,we can compute

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Trianulation

• The algorithm computes a sequence of triangulated
  surface with border. At each step, it choose a
  border edge, finds a new triangle associated to
  this edge, and updates the current surface.
• it maintains a half-edge data structure H and a
  list L of half-edge.

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Pseudo-code
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Pseudo-code
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Triangulation
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Refinement

• Refine the initial coarse triangulation ,first
  refine an edge uv

For each edge uv,its mid-point is m,
And its normal
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Refinement

• Minimize the following functional with respect to
  t

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Refinement
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Results
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Coclusion

• The new method computes representative points
• Triangulation algorithm does not need to compute
  3D Delaunay triangulation
• Refinement method is fast and gives a fine
  triangular mesh.

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The end
THANK YOU!
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MLS

• The introduction

First find a local reference domain (plane) for
r, then use the domain to compute a local
bivariate polynomial approximation to the surface
in a neighborhood of r
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Reference domain

• The local plane

is computed so as to minimize a local weighted
sum of squared distances of the points pi to the
plane
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Local map

• To compute local bivariate polynomial
  approximation first Let