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