Surface modelling - FAQ
Using Radial
Basis Functions to model 3D surface data
What do we mean by an RBF
model of an object?
We construct a
single function from surface data which can be considered as a
volume function. This function represents a signed `distance'
from the object's surface and is an explicit function of
position. Points inside the object have a negative `distance'
while points outside are positive. The object's surface is
defined implicitly as the zero set of this
function.
We can represent surface data with a
single 3D Radial Basis Function. This spatial function represents a signed `distance'
from the object's surface. Points inside the object have a negative `distance'
while points outside are positive. The object's surface is
defined implicitly as the zero set of this
function.
- A single
analytic function describes the signed-distance
function
- This function is
continuous and smooth (can be as smooth as one wishes by the
appropriate choice of the basic
function).
- We do not mean a
piecewise low-order algebraic surface, sometimes referred to as
an implicit patch or semi-algebraic set
- Unlike
constructive solid geometric (CSG) modelling, the object is not
decomposed into Boolean unions/intersections between primitives,
although RBF models could be used in this way.
 |
| An RBF implicit representation of a surface |
What advantages does an
analytic model offer?
- It can be
evaluated anywhere in 3D, on & off the surface, independent
of the locations of the original data points.
- Gradients and
higher derivatives are determined analytically. They are
continuous and smooth.
- The signed
distance function fitted to the surface data forms a solid model.
Iso-surfaces from the solid model are therefore guaranteed to be
manifold (i.e. manufacturable).
- Interpolation
and extrapolation are inherent in the functional representation.
Consequently, this approach can be applied to the problem of mesh
repair where incomplete meshes require hole filling and forming
closed, water-tight surfaces.
- An analytic
model can be used to produce a smooth filleted join between two
separate objects.
- An analytic
model offers new approaches to the problems of mesh
simplification, compression, morphing and reconstructing an
object from scattered point cloud data.
How is a function fitted to
incomplete surface data?
The
FastRBFTM tools construct additional
off-surface data consistent with a signed distance function.
This distribution is then approximated with a single analytic
function - the RBF interpolant.
How does data compression come
about?
Data compression can arise when fitting an RBF using
FastRBFTM's reduction option.
This option utilises a
greedy algorithm which
fits an RBF to the data using only a subset of the points
while still acheiving the desired accuracy at every data point.
- The RBF is
guaranteed to pass through ALL the data points to within the
user-specified precision.
- See the greedy surface fitting example in the main RBF FAQ.
How do I generate surface
meshes from an RBF model?
|
Our model of an
object is a volumetric function which explicitly describes
whether a point is inside or outside the object. The surface is
implicitly defined as the zero-valued iso-surface of this
function. An advantage of
this approach is that an iso-surface is guaranteed to be
manifold.
An iso-surfacing routine is used to generate an explicit mesh representations
from the implicit representation. In addition to its own implementation of the standard Marching Cubes iso-surfacing algorithm,
ARANZ has created a smart iso-surfacer which produces more optimal meshes and is particularly suited to efficient iso-surfacing of RBF signed distance fields. The new iso-surfacing algorithms also include a smoothing and anti-aliasing option.
|
An example of FastRBFTMmesh
optimisation |
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