Folding meshes: Hierarchical mesh segmentation based on planar symmetry Patricio Simari, Evangelos...
-
Upload
beryl-waters -
Category
Documents
-
view
216 -
download
0
description
Transcript of 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
2
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
P. Simari, E. Kalogerakis, K. Singh – University of TorontoFolding Meshes: Hierarchical mesh segmentation based on planar symmetry
3
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 TorontoFolding Meshes: Hierarchical mesh segmentation based on planar symmetry
4
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.
P. Simari, E. Kalogerakis, K. Singh – University of TorontoFolding Meshes: Hierarchical mesh segmentation based on planar symmetry
5
• 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
P. Simari, E. Kalogerakis, K. Singh – University of TorontoFolding Meshes: Hierarchical mesh segmentation based on planar symmetry
6
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 TorontoFolding Meshes: Hierarchical mesh segmentation based on planar symmetry
7
Support region: motivation
P. Simari, E. Kalogerakis, K. Singh – University of TorontoFolding Meshes: Hierarchical mesh segmentation based on planar symmetry
8
Controlling leverage
P. Simari, E. Kalogerakis, K. Singh – University of TorontoFolding Meshes: Hierarchical mesh segmentation based on planar symmetry
9
Controlling leverage
P. Simari, E. Kalogerakis, K. Singh – University of TorontoFolding Meshes: Hierarchical mesh segmentation based on planar symmetry
10
Controlling leverage
P. Simari, E. Kalogerakis, K. Singh – University of TorontoFolding Meshes: Hierarchical mesh segmentation based on planar symmetry
11
Controlling leverage
P. Simari, E. Kalogerakis, K. Singh – University of TorontoFolding Meshes: Hierarchical mesh segmentation based on planar symmetry
12
Controlling leverage
P. Simari, E. Kalogerakis, K. Singh – University of TorontoFolding Meshes: Hierarchical mesh segmentation based on planar symmetry
13
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 TorontoFolding Meshes: Hierarchical mesh segmentation based on planar symmetry
14
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 TorontoFolding Meshes: Hierarchical mesh segmentation based on planar symmetry
15
Convergence
P. Simari, E. Kalogerakis, K. Singh – University of TorontoFolding Meshes: Hierarchical mesh segmentation based on planar symmetry
16
Convergence
P. Simari, E. Kalogerakis, K. Singh – University of TorontoFolding Meshes: Hierarchical mesh segmentation based on planar symmetry
17
Convergence
P. Simari, E. Kalogerakis, K. Singh – University of TorontoFolding Meshes: Hierarchical mesh segmentation based on planar symmetry
18
Convergence
P. Simari, E. Kalogerakis, K. Singh – University of TorontoFolding Meshes: Hierarchical mesh segmentation based on planar symmetry
19
Convergence
P. Simari, E. Kalogerakis, K. Singh – University of TorontoFolding Meshes: Hierarchical mesh segmentation based on planar symmetry
20
Convergence
P. Simari, E. Kalogerakis, K. Singh – University of TorontoFolding Meshes: Hierarchical mesh segmentation based on planar symmetry
21
Convergence
P. Simari, E. Kalogerakis, K. Singh – University of TorontoFolding Meshes: Hierarchical mesh segmentation based on planar symmetry
22
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 TorontoFolding Meshes: Hierarchical mesh segmentation based on planar symmetry
23
Results: Local symmetry detection
P. Simari, E. Kalogerakis, K. Singh – University of TorontoFolding Meshes: Hierarchical mesh segmentation based on planar symmetry
24
Results: Local symmetry detection
P. Simari, E. Kalogerakis, K. Singh – University of TorontoFolding Meshes: Hierarchical mesh segmentation based on planar symmetry
25
Results: Local symmetry detection
P. Simari, E. Kalogerakis, K. Singh – University of TorontoFolding Meshes: Hierarchical mesh segmentation based on planar symmetry
26
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 TorontoFolding Meshes: Hierarchical mesh segmentation based on planar symmetry
27
Folding tree example
P. Simari, E. Kalogerakis, K. Singh – University of TorontoFolding Meshes: Hierarchical mesh segmentation based on planar symmetry
28
Results: Folding trees
P. Simari, E. Kalogerakis, K. Singh – University of TorontoFolding Meshes: Hierarchical mesh segmentation based on planar symmetry
29
Results: Folding trees
P. Simari, E. Kalogerakis, K. Singh – University of TorontoFolding Meshes: Hierarchical mesh segmentation based on planar symmetry
30
Results: Folding trees
P. Simari, E. Kalogerakis, K. Singh – University of TorontoFolding Meshes: Hierarchical mesh segmentation based on planar symmetry
31
Results: Folding trees
P. Simari, E. Kalogerakis, K. Singh – University of TorontoFolding Meshes: Hierarchical mesh segmentation based on planar symmetry
32
Results: Folding trees
P. Simari, E. Kalogerakis, K. Singh – University of TorontoFolding Meshes: Hierarchical mesh segmentation based on planar symmetry
33
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 TorontoFolding Meshes: Hierarchical mesh segmentation based on planar symmetry
34
Future work• Investigation alternate initialization
schemes• Extension to translational and
rotational symmetries• Exploration of other applications
• Repair• Robust skeleton extraction• Shape description/retrieval