folding meshes: hierarchical mesh segmentation based on planar symmetry patricio simari, evangelos...

Folding meshes: Hierarchical mesh segmentation based on planar symmetry

Patricio Simari, Evangelos Kalogerakis, Karan Singh

P. Simari, E. Kalogerakis, K. Singh – University of TorontoFolding Meshes: Hierarchical mesh segmentation based on planar symmetry

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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: 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.


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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 re-weighted least squares (IRLS) approach with M-estimation to converge to a locally symmetric region.


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• 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:

Solving for plane of 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.


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Support region: motivation


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Controlling leverage


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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.


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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.


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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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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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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

folding meshes: hierarchical mesh segmentation based on planar symmetry patricio simari, evangelos...

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