Extraction and remeshing of ellipsoidal representations from mesh data Patricio Simari Karan Singh.
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Transcript of Extraction and remeshing of ellipsoidal representations from mesh data Patricio Simari Karan Singh.
![Page 1: Extraction and remeshing of ellipsoidal representations from mesh data Patricio Simari Karan Singh.](https://reader036.fdocuments.in/reader036/viewer/2022070411/56649f385503460f94c551a2/html5/thumbnails/1.jpg)
Extraction and remeshing of ellipsoidal representations from mesh data
Patricio SimariKaran Singh
![Page 2: Extraction and remeshing of ellipsoidal representations from mesh data Patricio Simari Karan Singh.](https://reader036.fdocuments.in/reader036/viewer/2022070411/56649f385503460f94c551a2/html5/thumbnails/2.jpg)
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.
![Page 3: Extraction and remeshing of ellipsoidal representations from mesh data Patricio Simari Karan Singh.](https://reader036.fdocuments.in/reader036/viewer/2022070411/56649f385503460f94c551a2/html5/thumbnails/3.jpg)
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
![Page 4: Extraction and remeshing of ellipsoidal representations from mesh data Patricio Simari Karan Singh.](https://reader036.fdocuments.in/reader036/viewer/2022070411/56649f385503460f94c551a2/html5/thumbnails/4.jpg)
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.”
![Page 5: Extraction and remeshing of ellipsoidal representations from mesh data Patricio Simari Karan Singh.](https://reader036.fdocuments.in/reader036/viewer/2022070411/56649f385503460f94c551a2/html5/thumbnails/5.jpg)
Approximation error
• Total approximation error
• Mesh region (connected set of faces)
• Mesh face
![Page 6: Extraction and remeshing of ellipsoidal representations from mesh data Patricio Simari Karan Singh.](https://reader036.fdocuments.in/reader036/viewer/2022070411/56649f385503460f94c551a2/html5/thumbnails/6.jpg)
Error metrics defined on vertices
Radial Euclidean distance
P
vi
∏P(vi)
![Page 7: Extraction and remeshing of ellipsoidal representations from mesh data Patricio Simari Karan Singh.](https://reader036.fdocuments.in/reader036/viewer/2022070411/56649f385503460f94c551a2/html5/thumbnails/7.jpg)
Error metrics defined on vertices
Angular distance
P
nP(vi)ni
![Page 8: Extraction and remeshing of ellipsoidal representations from mesh data Patricio Simari Karan Singh.](https://reader036.fdocuments.in/reader036/viewer/2022070411/56649f385503460f94c551a2/html5/thumbnails/8.jpg)
Error metrics defined on vertices
Curvature distance
P
HP(vi) Hi
![Page 9: Extraction and remeshing of ellipsoidal representations from mesh data Patricio Simari Karan Singh.](https://reader036.fdocuments.in/reader036/viewer/2022070411/56649f385503460f94c551a2/html5/thumbnails/9.jpg)
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
![Page 10: Extraction and remeshing of ellipsoidal representations from mesh data Patricio Simari Karan Singh.](https://reader036.fdocuments.in/reader036/viewer/2022070411/56649f385503460f94c551a2/html5/thumbnails/10.jpg)
Negative ellipsoids
• Ellipsoids have positive curvature so they would not capture surface concavities
• Negative ellipsoids remedy this
![Page 11: Extraction and remeshing of ellipsoidal representations from mesh data Patricio Simari Karan Singh.](https://reader036.fdocuments.in/reader036/viewer/2022070411/56649f385503460f94c551a2/html5/thumbnails/11.jpg)
Ellipsoid segmentation algorithm
• Extension of Lloyd’s 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.
![Page 12: Extraction and remeshing of ellipsoidal representations from mesh data Patricio Simari Karan Singh.](https://reader036.fdocuments.in/reader036/viewer/2022070411/56649f385503460f94c551a2/html5/thumbnails/12.jpg)
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
![Page 13: Extraction and remeshing of ellipsoidal representations from mesh data Patricio Simari Karan Singh.](https://reader036.fdocuments.in/reader036/viewer/2022070411/56649f385503460f94c551a2/html5/thumbnails/13.jpg)
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.
![Page 14: Extraction and remeshing of ellipsoidal representations from mesh data Patricio Simari Karan Singh.](https://reader036.fdocuments.in/reader036/viewer/2022070411/56649f385503460f94c551a2/html5/thumbnails/14.jpg)
Future work
• Segmentation boundaries: reduction or do away with explicit representation
• Initialization scheme that decides number of ellipsoids and gives a good initial placement
![Page 15: Extraction and remeshing of ellipsoidal representations from mesh data Patricio Simari Karan Singh.](https://reader036.fdocuments.in/reader036/viewer/2022070411/56649f385503460f94c551a2/html5/thumbnails/15.jpg)
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.
![Page 16: Extraction and remeshing of ellipsoidal representations from mesh data Patricio Simari Karan Singh.](https://reader036.fdocuments.in/reader036/viewer/2022070411/56649f385503460f94c551a2/html5/thumbnails/16.jpg)
Smoothing segmentation boundaries
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Impact of different metrics
![Page 18: Extraction and remeshing of ellipsoidal representations from mesh data Patricio Simari Karan Singh.](https://reader036.fdocuments.in/reader036/viewer/2022070411/56649f385503460f94c551a2/html5/thumbnails/18.jpg)
Volume vs. surface fitting