Shape Approximation using Spherical Quadric Error Metrics

Shape approximation algorithms aim at computing simple geometric
descriptions of dense surface meshes. Many such algorithms are
based on mesh decimation techniques, generating coarse triangulations
while optimizing for a particular metric which models the distance
to the original shape.
This approximation scheme is very efficient when enough polygons are
allowed for the simplified model. However, as coarser approximations
are reached, the
intrinsic piecewise linear point interpolation which defines the
decimated geometry fails at capturing even simple structures. We
claim that when reaching such extreme simplification levels,
highly instrumental in shape analysis, the
approximating representation should explicitly and progressively model
the volumetric extent of the original shape.
In this paper, we propose Sphere-Meshes, a new shape representation
designed for extreme approximations and substituting a
sphere interpolation for the
classic point interpolation of surface meshes. From a technical
point-of-view, we propose a new shape approximation algorithm, generating
a sphere-mesh at a prescribed level of detail from a classical polygon mesh. We also
introduce a new metric to
guide this approximation, the *Spherical Quadric Error Metric* in
R^4, whose minimizer finds the sphere that best approximates
a set of tangent planes in the input and which is sensitive to surface orientation,
thus distinguishing naturally between the inside and the
*outside* of an object. We evaluate the performance of our algorithm on
a collection of models covering a wide range of topological and
geometric structures and compare it against alternate methods. Lastly,
we propose an application to deformation control where a
sphere-mesh hierarchy is used as a convenient rig for altering the input
shape interactively.

@article{TGB:2013:SM, author = {Jean-Marc Thiery and Emilie Guy and Tamy Boubekeur}, title = {Sphere-Meshes: Shape Approximation using Spherical Quadric Error Metrics}, journal ={ACM Transaction on Graphics (Proc. SIGGRAPH Asia 2013)}, year = {2013}, volume = {32}, number = {6}, pages = {Art. No. 178}, }