Approximating Bounded, Non-Orientable Surfaces from Points.

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Approximating Bounded, Non-orientable Surfaces from Points
Anders AdamsonMarc Alexa
Department of Computer Science, Darmstadt University of Technology
Fraunhoferstr. 5, 64283 Darmstadt
aadamson@gris.informatik.tu-darmstadt.de
Abstract
We present an approach to surface approximation from
points that allows reconstructing surfaces with boundaries,
including globally non-orientable surfaces. The surface is
defined implicitly using directions of weighted co-variances
and weighted averages of the points. Specifically, a point
belongs to the surface, if its direction to the weighted av-
erage has no component into the direction of smallest co-
variance. For bounded surfaces, we require in addition that
any point on the surface is close to the weighted average of
the input points. We compare this definition to alternatives
and discuss the details and parameter choices. Points on
thesurfacecanbedeterminedbyintersectioncomputations.
We show that the computation is local and, therefore, no
globally consistent orientation of normals is needed. Conti-
nuity of the surfaces is not affected by the particular choice
of local orientation. We demonstrate our approach by ren-
deringseveralbounded(andnon-orientable)surfacesusing
ray casting.
Keywords: point-sampled geometry, ray-surface inter-
section, manifold, boundary, non-orientable
1Introduction
Point sets without additional topological information are
a popular representation of surfaces (see [3, 16, 32]). Real
world shapes are sampled using scanners resulting in an
initial surface representation consisting of points [21, 27].
Points are also a suitable primitive for the display of sur-
faces, especially as the complexity of the shape results in
triangles being rendered into less than a pixel for the final
display [26, 28, 18, 11, 10, 33].
For many modeling tasks, however, a (local) approxima-
tion of the surface based on the points is required. A variety
of methods for the reconstruction of surfaces from points
exist. Some of these are approaches are global in nature,
others allow locally computing surface parts but are based
on locally approximating a global implicit function.
All approaches seem to assume that the represented sur-
face is a solid and, thus, unbounded and globally orientable.
This might stem from the fact that point clouds are mostly
acquired by scanning real-world (solid) objects, so that ar-
eas lacking points are considered inadequately sampled and
to be fixed. However, with the rising interest in unstruc-
tured point sets as a general surface representation, we feel
it is important to consider surfaces with boundary explic-
itly. Furthermore, enhancing reconstruction to faithfully
deal with bounded, possibly non-orientable surfaces, in-
creases the robustness against sampling errors and makes
the algorithm more versatile.
We extend our framework for approximating surfaces
from scattered points [1] to handle smooth boundaries and
show that the surface could be (globally) non-orientable.
Specifically, we make the following contributions:
Boundary definition: The surface is defined implicitly us-
ing compactly supported functions. We show how to
compute and define a smooth and natural boundary us-
ing only the quantities we compute during surface ap-
proximation.
Boundary behavior and parameters: We prove several
properties about the behavior of the computations in-
volved in the estimation of the boundary. We use these
approximations to solve the problem of parameter es-
timation and provide the user with an interval of rea-
sonable parameters to choose from.
Orientation: The computation procedure for points re-
quires only the intersection of a curve/ray with a plane.
As the orientation of the plane’s normal is irrelevant, a
global orientation of normals is not required. We show
that the continuity of the surface is not affected.
Despite these features, the point set could still define a
solid and our implicit definition of the surface would allow
defining inside and outside. Note that we cannot handle
non-manifold surfaces, e.g., surfaces with self intersections.
Inthefollowing, webrieflyreviewsomeoftherelatedwork,
define the surface and explain how to compute it efficiently,

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before we explain in detail the definition of the boundary
and how to choose the necessary parameters.
2Related Work
A variety of works tackle the (ill-posed) problem of re-
constructing a surface from an unstructured set of points.
Hoppe et al. [17] define a signed distance field from the
points. For each point a normal direction is estimated and
the normal is oriented. The signed distance to the surface
is defined as the normal component of the distance to the
closest point.
In a sense the surface definition of Hoppe et al. is related
to Voronoi-based reconstruction techniques [8, 13, 4, 15, 6,
5] as the surface is defined by a closest point relationship.
These techniques define the surface as a subset of the De-
launay triangulation of the point set. As such, they are all
global methods and generate a C0surface.
Surfaces with higher order continuity from points are
typically implicit. For the specific purpose of consolidat-
ing registered range images Curless et al. [12] used a dis-
crete representation (see also [23] for a comparable ap-
proach). Radial basis functions with global [29, 9, 31] or
local [22, 19, 25] support are a more general technique for
approximating functions from scattered constraints. Note
that all of these techniques are inherently global in nature
– some of them explicitly consider the case of incomplete
point sampling and explain how, despite the missing points,
a globally unbounded and oriented surface would be recon-
structed.
Hierarchical methods based on blending individual local
implicitapproximationssuchas[24]aswellasmovingleast
squares type surface approximation [20, 3, 14] are local rep-
resentations. While the MLS-based approaches explicitly
require the surface to be sampled sufficiently dense from a
closed manifold, hierarchical methods are rather concerned
with the problem of adequately filling un-sampled regions.
Schaufler and Jensen [30] define the intersection of a ray
and the point set as follows: In each point a disk is con-
structed using the point normal. A cylinder around the ray
is intersected with the disks. The intersection is computed
as a weighted average of disks whose centers are inside the
cylinder. Note that the reconstructed surface could have
boundary, because not every cylinder contains points. On
the other hand, the boundary of the surface would depend
on the particular rays chosen.
Our approach is also based on intersecting rays with the
surface [1], however, it could be understood as a robust
computation of a particular implicit surface (which is iden-
tical to the MLS definition [20]). Implicit surfaces typi-
cally represent solids. According to Bloomenthal et al. [7],
boundaries are generally represented by defining additional
functions on the space and requiring that these functions
are, e.g., positive. Our approach could, indeed, be inter-
preted in this way.
3Surface definition
We first give a brief definition of the surface as defined
in [1] and establish several notations for later use.
Given points P = {pi ∈ R3,i ∈ {1,...,N}} on or
close to a surface S. The point set surface SPis defined as
the zero-set of the implicit function
f(x) = n(x) · (x − a(x)),
(1)
which describes the distance of x to a plane with normal
n(x) through the weighted average of points a(x). For the
surface definition to be reasonable (i.e. the surface is con-
tinuously differentiable), we require the mappings defin-
ing normal direction n : R3→ S2and weighted average
a : R3→ R3to be smooth in their input at least in a suf-
ficient neighborhood Ω around P. We will give particu-
lar choices for the mappings and the neighborhood below.
These choices are useful for efficient and local computation
of the surface, and they lead to natural definition of bound-
aries.
Let θ : R → R be a monotone decreasing function with
localsupportrθ, whichisstrictlypositiveinsidethesupport,
i.e. θ(χ) = 0 ⇔ χ > rθ. We will use θ as a weight function
determiningtheinfluenceofapointpionapointinspacex.
The neighborhood Ω is consequently defined as the region
in space that is affected by the points:
?
i
Alternatively, we can define the distance of a point x to the
point set as the minimum distance of x to any point in P:
dP(x) = min
i
Using this definition, Ω could be defined as the set of points
with distance dP less than rθ, which also describes the
union of rθ-balls around the points in P. Note that all these
definitions of Ω are equivalent because we assume θ to be
strictly positive in its support.
The weighted average of points a is defined (for points
in Ω only) as
?
Normal directions are defined as directions of smallest
weighted co-variance, using the same weights as before.
This could be understand as a least squares fit of a plane
with unit normal n through x, i.e. the minimizer of
?
Ω =
x|
?
θ(||pi− x||) > 0
?
(2)
||x − pi||.
(3)
a(x) =
ipiθ(||pi− x||)
?
iθ(||pi− x||)
,x ∈ Ω.
(4)
min
||n||=1
i?n,pi− x?2θ(||pi− x||)
?
iθ(||pi− x||)
.
(5)
2

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It is well known that this constrained minimization prob-
lem can be solved using the eigenvectors of the matrix
W(x) = {wjk} of weighted co-variances in the directions
of an orthonormal basis of R3. Let ei,i ∈ {0,1,2} be such
a basis, then the coefficients of the matrix are given as
?
Let {vi} be the eigenvectors of W(x) corresponding to the
eigenvalues λ0≤ λ1≤ λ2, we set n = v0for any x ∈ Ω.
Using this representation it can be deduced that n(x) is a
smooth function on Ω if λ0 < λ1for all x ∈ Ω. We call
point sets for which this is true well sampled (see [1] for
more information).
Note that the continuity arguments are independent of
the orientation of the normal: The computation is local and
inside a ball with sufficiently small radius we could orient
the normals consistently so that n(x) is indeed a smooth
vector-valued function of x inside the ball.
mals are now inverted inside this ball, the surface defined
as n(x) · (x − a(x)) = 0 is unaffected. Since we made no
specialassumptionsonthelocationoftheballthisargument
extends to the whole surface.
For θ we typically assume a Gaussian-shaped function
wjk=
i
?ej,pi− x??ek,pi− x?θ(||pi− x||).
(6)
If the nor-
θ(d) = e−d2/h2,
(7)
where h is the anticipated feature size, which roughly cor-
responds with spacing among points in P. In practice, we
compute h as the average Euclidean k-nearest neighbor dis-
tance, where we use k = 6. To get a locally supported
weight function, we approximate the Gaussian by a spline
with appropriate local support and continuity. We will, nev-
ertheless, sometimes use the feature size h in our argu-
ments.
4Computing points on the surface
A ray-surface (or, more generally, curve-surface) inter-
section approach is used to efficiently compute points of the
surface (see [1] for more details). The method is iterative:
For a point xion the ray obtained in the i-th iteration, a
normal direction n(xi) and a local constant surface approx-
imation through a(xi) are computed. The local surface ap-
proximation is then intersected with the ray yielding xi+1.
In practice, this iteration quickly converges, if we start with
points in the proximity of SP.
To determine a good starting point for the iterations effi-
ciently, a set of enclosing balls {Bi} is constructed by plac-
ing a ball Biof radius rBaround each input point pi. The
union of the balls
?
B = {x|dP(x) < rB} =
i
Bi,Bi= {x : ||x−pi|| < rB}
(8)
x
x
1
2
Figure 1. The rationale of defining bounded
surface:While all points are in the same
plane, we expect x2not to part of the recon-
structed surface.
is expected to contain the surface, i.e. SP ⊂ B, which is
achievedbychoosingtheglobalradiusrB≈ 1.5h(notethat
this value has been determined experimentally). All poten-
tial ray-surface intersections can be found by first intersect-
ing B (a spatial data structure should be used to accelerate
the computation necessary to find the right balls). To evalu-
ate whether a ball Bicontains a ray-surface intersection, the
following three steps are repeatedly applied, starting with
the intersection x0of the ray and Bi:
1. Support plane: The matrix of weighted co-variance
w.r.t. xiis constructed and the eigenvectors are used
as a local frame in xi; the eigenvector corresponding to
the smallest eigenvalue is the normal direction n(xi).
2. Evaluation: The weighted average a(xi) is computed
and f(xi) is evaluated. If |f(xi)| is sufficiently small,
an intersection is found.
3. Approximation and intersection: If |f(xi)| is large,
a planar approximation of the surface is constructed
by a plane with normal n(xi) and an offset f(xi) rel-
ative to xiin normal direction. The next point on the
ray xi+1is computed by intersecting the ray with that
plane. If xi+1is outside the ball Bithe iterations are
stopped and the next intersected ball is examined.
In a typical setting, 3 to 5 iterations are sufficient to deter-
mine a point on the surface. Note that due to our choice of
weight function there can only be one intersection inside a
ball Bi[2].
While we have used the term normal in the description
of the algorithm, the orientation of this normal is irrelevant.
The local constant surface approximation is independent of
3

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Figure 2. A set of points representing a planar
disk with a hole.
the normal orientation. Because the ray-surface intersec-
tion is computed inside a ball Biwith small radius rB, the
computation is local as required in Section 3.
5Defining the boundary
For defining a boundary, we illustrate our reasoning us-
ing Figure 1: The input points are distributed equally over a
half-plane. The two points of evaluation x1and x2as well
as all points in P are assumed to be co-planar. The direction
of smallest weighted co-variance of x1and x2is, thus, the
constant normal to the plane. Thus, x1is part of the surface,
and x2is part of the surface if x2∈ Ω.
Remember that the surface is defined only inside Ω,
which is bounded because weight functions are locally sup-
ported. Using ∂Ω for the definition of the boundary is not
a good idea in practice: the computations involved in eval-
uating Eq. 1 break down near the boundary of Ω, because
the sum of weights in the denominators tends to zero (cf.
Eqs. 4 and 5). For illustration purposes, we will use an ir-
regular point set representing a planar disk with a hole (see
Figure 2). The numerical breakdown of our current imple-
mentation close to ∂Ω can be clearly observed in Fig. 3,
showing a rendering of the surface generated by ray cast-
ing.
The numerical problem could be avoided by requir-
ing the sum of weights to exceed a certain threshold, i.e.
?
that this is a bad idea: The boundary would depend on the
sampling density. To illustrate this, imagine any point set
iθ(||pi− x||) > ?. This might lead to a definition of
boundaries that are robustly computed, however, we feel
Figure 3. A rendering resulting from ray cast-
ing the points shown in Fig. 2. The support
radius of the weight functions rθ is chosen
relatively small; the breakdown of the eval-
uation of points close to ∂Ω can be clearly
observed.
representing a surface with holes. A given threshold on
the sum of weights defines the boundary. Note that this
boundary could be arbitrarily shifted towards ∂Ω by sim-
ply adding points in the locations of the existing points.
The use of an enclosing ball structure B for the compu-
tation with radius rB < rθhas been motivated mainly by
efficiency arguments in Sec. 4. The numerical breakdown
close to ∂Ω is another reason. In practice, all points in the
set of enclosing spheres could be robustly evaluated. One
might use ∂B to define the boundary of SP, however, this
results in piecewise circular boundary that is only C0where
circular arcs meet.
We propose a different approach to obtain boundaries,
that are smooth, easy to compute, and almost insensitive to
variations in the sampling density. Refer again to Figure 1:
To distinguish the location of x1and x2we inspect the rel-
ative location of the weighted average a in the points. Note
that the weighted average of points is necessarily contained
in their convex hull. Thus, for points far away from the
point set the distance ||x − a(x)|| increases, while we ex-
pect this distance to be rather small for locations close to (or
“inside”) the points. For further use we will call
c(x) = ||x − a(x)||
(9)
the off-center value of x.
The main idea for defining a boundary is to require the
off-center value to be less than a user-specified threshold.
More precisely, the surface SPis defined as
SP= {x ∈ Ω|f(x) = 0 ∧ c(x) < ?c}
(10)
4

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x - a(x)
x
Figure 4. A plot of c(x) = ||a(x) − x|| for a
slightly noisy point set with a gap.
In the following section we will discuss reasonable choices
for ?c.
6 Choosing the off-center limit
For the following discussion we use a series of plots of
c(x) in one dimension and corresponding renderings pro-
duced with ray casting and color coding c(x) of the disk
point set shown in Figure 2. The input points in one dimen-
sion could be interpreted as a one-dimensional slice through
the disk point set: The points result from regularly sampling
the line, adding Gaussian noise to the locations, and remov-
ing some points to generate a gap. We assume the space
inside the gap/hole is in Ω, i.e. rθis sufficiently large. The
graph of c(x) in one dimension is shown in Figure 4.
As expected, the off-center value c(x) is small in-
between the points (in fact, for equidistant input points far
away from the boundary of Ω it is c(x) = 0) and it increases
almost linearly far away from the points. This is also re-
flected in the color coded renderings of the disk (see Fig-
ures 3, 5, and 6), where blue indicates small values and red
indicates large values of c(x). Note that the off-center value
is particularly small in the barycenter of points – including
the barycenter of holes. Figure 6 shows what this behavior
entails if the radius of the sphere structure rBis relatively
large: Inside the hole a small island appears. Other potential
problems are
• holes in areas of the surface that seem to be adequately
sampled. This results from a value for ?cthat is too
small (see Fig. 5a),
• piecewise circular boundaries resulting from a too
large value for ?c(see Fig. 5b).
It is clear that we have to define ?crelative to rB. Before we
can do so we need to establish a connection between c(x)
and the distance of x to the point set dP(x). We can show
(see Appendix A) that
1. c(x) >
from the points and
?
d2
P(x) − h2(2 −√3) as x moves away
2. limdP(x)→∞dP(x) = c(x) (assuming that θ has very
large radius of support).
In other words, c(x) behaves roughly like the distance to
the point set, however, is larger when x moves away from
the surface (a notable exception is around the barycenter of
holes, where dPis maximal, while c(x) is small).
For an exact characterization of ?cwe consider two dif-
ferent case for the radius rBof the enclosing ball structure.
First, consider rBto be chosen in a natural way, i.e. so that
a portion of the hole/gap is not covered by B (Fig. 5. In
order to avoid unwanted holes in the surface ?cneeds to be
chosen large enough, however, ∂B defines an upper bound
on ?c: Because c(x) >
d2
?
P(x) − h2(2 −√3) and rBre-
?
flects a bound on dPwe require ?c<
To bound ?cfrom below, we need to find a reasonable
maximum of c(x) in cases where we would consider the
surface closed. In our setting, the only reasonable defini-
tion of a closed surface is to assume it is closed where it
is contained in B. Thus, the case where the enclosing ball
structure just covers a hole is the relevant limiting case (see
Fig. 6 for an illustration). In order to avoid the island (which
could as well be seen as an unwanted hole in the surface),
?chas to be larger than the maximum of c(x) in this region.
Note that because c(x) is larger than dP(x) as we move
away from the boundary of a hole, it is not necessary that
an ?c< rBthat is larger than c(x) in small gaps (i.e. those
that are covered by B) exists. To show that such an ?cex-
ists in practice, we need to consider the practical choice of
rBbased on h, and the way we determine h from the dis-
tribution of the points. In particular, by assuming Gaussian
weights of the form θ(d) = e−d2/h2we estimate c(x) to be
bounded by h (see Appendix B). Given our practical choice
of rBthis leads to the following bounds for ?c:
1 + 4√3
9
r2
B− h2(2 −√3).
rB> ?c>2
3rB
(11)
Figures 5c and 6b show reasonable values for ?cfor both
setting of radii rB.
7Results
We have tested our approach by ray casting several sur-
faces represented by unstructured point sets using the ray
intersection method described in Section 4. The point sets
all represent surfaces with boundaries, and some are glob-
ally non-orientable.
In all our examples we have estimated the feature size
h by finding the 6 nearest neighbors for all points and then
5