Folding meshes: Hierarchical mesh segmentation based on planar symmetry

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Folding meshes Hierarchical mesh segmentation
based on planar symmetry

• Patricio Simari, Evangelos Kalogerakis, Karan
  Singh

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Introduction and motivation

• Meshes may contain a high level of redundancy due
  to symmetry, either global or localized.
• We propose an algorithm for detecting approximate
  planar reflective symmetry globally and locally.
• Applications include
• Compression
• Segmentation
• Repair
• Skeleton Extraction
• Mesh processing acceleration

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Related work

• Perfect in polygons and polyhedra Atallah 85,
  Wolter et al. 85, Highnam 86, Jiang Bunke
  96.
• Approximate in point sets Alt et al. 88.
• 2D images/range images Marola 89, Gofman
  Kiryati 96, Shen et al. 99, Zabrodsky et al.
  95.
• Global 3D OMara Owens 96, Sun Sherrah 97,
  Sun Si 99, Martinet et al. 05.
• Global as shape desc. Kazhdan et al. 04.
• Local 3D Thrun Wegbreit 05, Podolak et al.
  06, Mitra et al. 06.

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Overview

• Property A symmetric surfaces planes of
  symmetry are orthogonal to the eigenvectors of
  its covariance matrix and contain its centre of
  mass.
• Leverage this fact iteratively re-weighted least
  squares (IRLS) approach with M-estimation to
  converge to a locally symmetric region.

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Solving for plane of symmetry

• Consider a candidate symmetry plane p and let di
  be the distance of vertex vi to the reflected
  mesh wrt p.
• Each vi is associated a weight wi according to

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Solving for plane of symmetry

• The plane of symmetry is estimated by the centre
  of mass m and the eigenvectors of the weighted
  covariance matrix C defined as

• These eigenvectors and centre of mass determine
  three planes.
• One with smallest sum cost is chosen.

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Support region motivation
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Controlling leverage
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Controlling leverage
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Controlling leverage
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Controlling leverage
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Controlling leverage
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Finding support region

• Given the current ? values we consider a face to
  be a support face if for all of its vertices di
  2s. Hampel et al. 86
• We find the largest connected region of support
  faces, and set weights for all vertices outside
  this region to 0.
• The plane finding and region finding steps are
  iterated until convergence.

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Initialization

• Initially, wi is defined to be the mesh area
  associated with vertex vi
• The initial support regions contains all faces.
• s 1.4826median(di) Forsyth and Ponce 02
  during initial iterations and then is fixed to
  2e.

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Convergence
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Convergence
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Convergence
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Convergence
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Convergence
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Convergence
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Convergence
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Finding other local symmetries

• Converge to symmetric region
• Segment out locally symmetric region
• Apply recursively to one half of the symmetric
  region (nested symmetries) and to each remaining
  connected component.

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Results Local symmetry detection
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Results Local symmetry detection
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Results Local symmetry detection
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Folding trees

• We introduce the folding tree data structure.
• Encodes the non redundant regions as well as the
  reflection planes.
• Created by recursive application of the detection
  method.
• Can then be unfolded to recover the original
  shape.

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Folding tree example
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Results Folding trees
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Results Folding trees
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Results Folding trees
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Results Folding trees
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Results Folding trees
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Conclusions

• We have presented a robust estimation approach to
  finding global as well as local planar
  symmetries.
• We have introduced a compact representation of
  meshes, called folding trees, and shown how they
  can be automatically constructed using the
  detection method.

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Future work

• Investigation alternate initialization schemes
• Extension to translational and rotational
  symmetries
• Exploration of other applications
• Repair
• Robust skeleton extraction
• Shape description/retrieval