Shape Reconstruction Using Radial-Basis Functions and Partition of Unity
Ohtake Y.,
Belyaev A.,
Seidel H.-P.,
"3D scattered data approximation with adaptive
compactly supported radial basis functions",
Shape Modeling International 2004, Genova, Italy, IEEE Computer Society, 2004, pp. 31-39.
Electronic version: PDF (623K)
Abstract
In this paper, we develop an adaptive Radial-Basis Functions (RBF) fitting procedure
for a high quality approximation of a set of points
scattered over a piecewise smooth surface. We use compactly
supported RBFs whose centers are randomly chosen
from the points. The randomness is controlled by the
point density and surface geometry. For each RBF, its support
size is chosen adaptively according to surface geometry
at a vicinity of the RBF center. All these lead to a
noise-robust high quality approximation of the set. We also
adapt our basic technique for for shape reconstruction from
registered range scans by taking into account measurement
confidences. Finally, an interesting link between our RBF
fitting procedure and partition of unity approximations is
established and discussed.
Ohtake Y.,
Belyaev A.,
Alexa M.,
Turk G.,
Seidel H.-P.,
"Multi-level partition of unity implicits",
ACM Transactions on Graphics
(SIGGRAPH 03 Proceedings), vol. 22, No. 3, 2003, pp. 463-470.
Electronic version: PDF (627K)
Abstract
We present a shape representation, the multi-level partition of unity
implicit surface, that allows us to construct surface models from
very large sets of points. There are three key ingredients to our
approach: 1) piecewise quadratic functions that capture the local
shape of the surface, 2) weighting functions (the partitions of unity)
that blend together these local shape functions, and 3) an octree
subdivision method that adapts to variations in the complexity of
the local shape.
Our approach gives us considerable flexibility in the choice of
local shape functions, and in particular we can accurately represent
sharp features such as edges and corners by selecting appropriate
shape functions. An error-controlled subdivision leads to an adaptive
approximation whose time and memory consumption depends
on the required accuracy. Due to the separation of local approximation
and local blending, the representation is not global and can be
created and evaluated rapidly. Because our surfaces are described
using implicit functions, operations such as shape blending, offsets,
deformations and CSG are simple to perform.
More detalis can be found at the
Multi-level Partition of Unity Implicits page.
Ohtake Y.,
Belyaev A.,
Seidel H.-P.,
"A multi-scale approach to 3D scattered data interpolation
with compactly supported basis functions",
Shape Modeling International 2003, Seoul, Korea, IEEE Computer Society, 2003,
pp. 153-161.
Electronic version: PDF (3.55 Mb)
Abstract
In this paper, we propose a hierarchical approach to 3D
scattered data interpolation with compactly supported basis
functions. Our numerical experiments suggest that the
approach integrates the best aspects of scattered data fitting
with locally and globally supported basis functions.
Employing locally supported functions leads to an efficient
computational procedure, while a coarse-to-fine hierarchy
makes our method insensitive to the density of scattered
data and allows us to restore large parts of missed data.
Given a point cloud distributed along a surface, we first
use spatial down sampling to construct a coarse-to-fine hierarchy
of point sets. Then we interpolate the sets starting
from the coarsest level. We interpolate a point set of the hierarchy,
as an offsetting of the interpolating function computed
at the previous level. The figure above shows an original point
set (the leftmost image) and its coarse-to-fine hierarchy of
interpolated sets.
According to our numerical experiments, the method is
essentially faster than the state-of-art scattered data approximation
with globally supported RBFs and much
simpler to implement.
Other RBF reconstruction related pages:
Reconstruction from Surface Points and Contours
Haniwa: Reconstruction Case Study
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