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Extraction and remeshing of ellipsoidal representations from mesh data
Patricio SimariKaran Singh
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.
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
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.”
Approximation error
• Total approximation error
• Mesh region (connected set of faces)
• Mesh face
Error metrics defined on vertices
Radial Euclidean distance
P
vi
∏P(vi)
Error metrics defined on vertices
Angular distance
P
nP(vi)ni
Error metrics defined on vertices
Curvature distance
P
HP(vi) Hi
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
Negative ellipsoids
• Ellipsoids have positive curvature so they would not capture surface concavities
• Negative ellipsoids remedy this
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.
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
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.
Future work
• Segmentation boundaries: reduction or do away with explicit representation
• Initialization scheme that decides number of ellipsoids and gives a good initial placement
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.
Smoothing segmentation boundaries
Impact of different metrics
Volume vs. surface fitting