Extraction%20and%20remeshing%20of%20ellipsoidal%20representations%20from%20mesh%20data

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Extraction and remeshing of ellipsoidal
representations from mesh data

• Patricio Simari
• Karan Singh

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Overview

• Input surface data in mesh form.
• Output ellipsoidal representation approximating
  input
• Ellipsoidal representation surface defined
  piecewise by a set of ellipsoidal surfaces
• Ellipsoidal surface ellipsoid plus boundaries
• Used as is or remeshed if desired.

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Motivation

• Efficient rendering and geometric querying
• Compact representation of large curved areas
• Can also be used to represent volumes
• Direct parameterization of each surface
• Objects perceptually segmented along concavities

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Related work

• Bischoff et al., Ellipsoid decomposition of
  3D-models.
• Hoppe et al., Mesh optimization.
• Cohen-Steiner et al., Variational shape
  approximation.
• Katz et al., Hierarchical mesh decomposition
  using fuzzy clustering and cuts.

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Approximation error

• Total approximation error
• Mesh region (connected set of faces)
• Mesh face

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Error metrics defined on vertices
Radial Euclidean distance
vi
?P(vi)
P
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Error metrics defined on vertices
Angular distance
ni
nP(vi)
P
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Error metrics defined on vertices
Curvature distance
Hi
HP(vi)
P
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Combining error metrics

• Combined vertex error
• Weights serve dual purpose
• linearly scale metrics to comparable ranges
• Allow user to adjust for relative preference of
  one metric over another

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Negative ellipsoids

• Ellipsoids have positive curvature so they would
  not capture surface concavities
• Negative ellipsoids remedy this

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Ellipsoid segmentation algorithm

• Extension of Lloyds algorithm (k-means)
• Fitting step compute Pi that minimizes E(Ri,Pi)
• Classification step assign each face fj to a
  region Ri that minimizes E(fj,Pi)
• Added constraint regions must remain connected.
• Use flooding scheme (implies losing convergence
  guaranty.)
• Also include teleportation to avoid local
  minima.

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Remeshing ellipsoidal representations

• Parametric tessellation of surfaces
• unit sphere is sampled, cropped and tessellated
• Iterative vertex addition
• Boundary points are tessellated
• Faces are split at centre with highest error
• Edges are flipped

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Error metric for ellipsoid volume

• Ellipsoids, being closed surfaces, can also be
  used to represent volume.
• Same algorithm can be used by adapting error
  metric
• Regions are approximated by an ellipsoid of
  similar volume.

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Future work

• Segmentation boundaries reduction or do away
  with explicit representation
• Initialization scheme that decides number of
  ellipsoids and gives a good initial placement

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Using ellipsoidal boundaries

• Each primitive is a polygon which lies on an
  ellipsoidal surface
• Determine if a point is on the polygon
• Reduce to planar polygon using stereographic
  projection.

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Smoothing segmentation boundaries
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Impact of different metrics
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Volume vs. surface fitting