Transcript Folding meshes: Hierarchical mesh segmentation based on planar symmetry

Folding meshes:
Hierarchical mesh segmentation
based on planar symmetry
Patricio Simari, Evangelos Kalogerakis, Karan Singh
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:
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Compression
Segmentation
Repair
Skeleton Extraction
Mesh processing acceleration
P. Simari, E. Kalogerakis, K. Singh – University of Toronto
Folding Meshes: Hierarchical mesh segmentation based on planar symmetry
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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: O’Mara & 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.
P. Simari, E. Kalogerakis, K. Singh – University of Toronto
Folding Meshes: Hierarchical mesh segmentation based on planar symmetry
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Overview
• Property: A symmetric surface’s
planes of symmetry are orthogonal to
the eigenvectors of its covariance
matrix and contain its centre of mass.
• Leverage this fact: iteratively reweighted least squares (IRLS)
approach with M-estimation to
converge to a locally symmetric
region.
P. Simari, E. Kalogerakis, K. Singh – University of Toronto
Folding Meshes: Hierarchical mesh segmentation based on planar symmetry
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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:
P. Simari, E. Kalogerakis, K. Singh – University of Toronto
Folding Meshes: Hierarchical mesh segmentation based on planar symmetry
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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.
P. Simari, E. Kalogerakis, K. Singh – University of Toronto
Folding Meshes: Hierarchical mesh segmentation based on planar symmetry
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Support region: motivation
P. Simari, E. Kalogerakis, K. Singh – University of Toronto
Folding Meshes: Hierarchical mesh segmentation based on planar symmetry
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Controlling leverage
P. Simari, E. Kalogerakis, K. Singh – University of Toronto
Folding Meshes: Hierarchical mesh segmentation based on planar symmetry
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Controlling leverage
P. Simari, E. Kalogerakis, K. Singh – University of Toronto
Folding Meshes: Hierarchical mesh segmentation based on planar symmetry
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Controlling leverage
P. Simari, E. Kalogerakis, K. Singh – University of Toronto
Folding Meshes: Hierarchical mesh segmentation based on planar symmetry
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Controlling leverage
P. Simari, E. Kalogerakis, K. Singh – University of Toronto
Folding Meshes: Hierarchical mesh segmentation based on planar symmetry
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Controlling leverage
P. Simari, E. Kalogerakis, K. Singh – University of Toronto
Folding Meshes: Hierarchical mesh segmentation based on planar symmetry
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Finding support region
• Given the current ρ values we consider
a face to be a support face if for all of
its vertices di ≤ 2σ. [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.
P. Simari, E. Kalogerakis, K. Singh – University of Toronto
Folding Meshes: Hierarchical mesh segmentation based on planar symmetry
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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.
• σ = 1.4826*median(di) [Forsyth and
Ponce ‘02] during initial iterations and
then is fixed to 2ε.
P. Simari, E. Kalogerakis, K. Singh – University of Toronto
Folding Meshes: Hierarchical mesh segmentation based on planar symmetry
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Convergence
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Folding Meshes: Hierarchical mesh segmentation based on planar symmetry
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Convergence
P. Simari, E. Kalogerakis, K. Singh – University of Toronto
Folding Meshes: Hierarchical mesh segmentation based on planar symmetry
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Convergence
P. Simari, E. Kalogerakis, K. Singh – University of Toronto
Folding Meshes: Hierarchical mesh segmentation based on planar symmetry
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Convergence
P. Simari, E. Kalogerakis, K. Singh – University of Toronto
Folding Meshes: Hierarchical mesh segmentation based on planar symmetry
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Convergence
P. Simari, E. Kalogerakis, K. Singh – University of Toronto
Folding Meshes: Hierarchical mesh segmentation based on planar symmetry
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Convergence
P. Simari, E. Kalogerakis, K. Singh – University of Toronto
Folding Meshes: Hierarchical mesh segmentation based on planar symmetry
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Convergence
P. Simari, E. Kalogerakis, K. Singh – University of Toronto
Folding Meshes: Hierarchical mesh segmentation based on planar symmetry
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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.
P. Simari, E. Kalogerakis, K. Singh – University of Toronto
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Results: Local symmetry detection
P. Simari, E. Kalogerakis, K. Singh – University of Toronto
Folding Meshes: Hierarchical mesh segmentation based on planar symmetry
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Results: Local symmetry detection
P. Simari, E. Kalogerakis, K. Singh – University of Toronto
Folding Meshes: Hierarchical mesh segmentation based on planar symmetry
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Results: Local symmetry detection
P. Simari, E. Kalogerakis, K. Singh – University of Toronto
Folding Meshes: Hierarchical mesh segmentation based on planar symmetry
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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.
P. Simari, E. Kalogerakis, K. Singh – University of Toronto
Folding Meshes: Hierarchical mesh segmentation based on planar symmetry
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Folding tree example
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Results: Folding trees
P. Simari, E. Kalogerakis, K. Singh – University of Toronto
Folding Meshes: Hierarchical mesh segmentation based on planar symmetry
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Results: Folding trees
P. Simari, E. Kalogerakis, K. Singh – University of Toronto
Folding Meshes: Hierarchical mesh segmentation based on planar symmetry
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Results: Folding trees
P. Simari, E. Kalogerakis, K. Singh – University of Toronto
Folding Meshes: Hierarchical mesh segmentation based on planar symmetry
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Results: Folding trees
P. Simari, E. Kalogerakis, K. Singh – University of Toronto
Folding Meshes: Hierarchical mesh segmentation based on planar symmetry
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Results: Folding trees
P. Simari, E. Kalogerakis, K. Singh – University of Toronto
Folding Meshes: Hierarchical mesh segmentation based on planar symmetry
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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.
P. Simari, E. Kalogerakis, K. Singh – University of Toronto
Folding Meshes: Hierarchical mesh segmentation based on planar symmetry
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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
P. Simari, E. Kalogerakis, K. Singh – University of Toronto
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