Data-Dependent MLS for Faithful Surface Approximation Yaron Lipman Daniel Cohen-Or David Levin

Data-Dependent MLS for Faithful Surface Approximation Yaron Lipman Daniel Cohen-Or David Levin
Eurographics Symposium on Geometry Processing (2007)
Alexander Belyaev, Michael Garland (Editors)
Data-Dependent MLS for Faithful Surface Approximation
Yaron Lipman
Daniel Cohen-Or
David Levin
Tel-Aviv University
Abstract
In this paper we present a high-fidelity surface approximation technique that aims at a faithful reconstruction of
piecewise-smooth surfaces from a scattered point set. The presented method builds on the Moving Least-Squares
(MLS) projection methodology, but introduces a fundamental modification: While the classical MLS uses a fixed
approximation space, i.e., polynomials of a certain degree, the new method is data-dependent. For each projected
point, it finds a proper local approximation space of piecewise polynomials (splines). The locally constructed
spline encapsulates the local singularities which may exist in the data. The optional singularity for this local
approximation space is modeled via a Singularity Indicator Field (SIF) which is computed over the input data
points. We demonstrate the effectiveness of the method by reconstructing surfaces from real scanned 3D data,
while being faithful to their most delicate features.
1. Introduction
Approximating a surface from scattered data is a fundamental problem with various applications in computer graphics.
In particular, reconstructing or upsampling an unorganized
point set is an important problem in surface reconstruction
[HDD∗ 92, CBC∗ 01, OBA∗ 03]. The main challenge is to
faithfully reconstruct the unknown surface from a scattered
set of samples. A faithful reconstruction adheres to the data,
and aims to respect even its most delicate singularities or
sharp features. As such, the reconstruction should necessarily be piecewise-smooth.
Common techniques for surface reconstruction assume
that the unknown surface is smooth everywhere, and hence
make use of approximation spaces of a smooth functions
to reconstruct the surface. A powerful method to approximate an unknown surface from an unorganized noisy sample
set, is to fit local polynomials by the moving least-squares
(MLS) method [ABCO∗ 01]. The functional MLS approximation procedure [LS81, Lev98, Wen01] defines a smooth
approximant/interpolant for the unknown function, based
upon function samples at unstructured data sites. Combining
the MLS functional approximation scheme with a projection
operator [Lev03, AK04] then yields an efficient method for
smoothly approximating surfaces from 3D data sets. This
furnishes a highly generic and versatile tool for defining,
manipulating and reconstructing surfaces based on irreguc The Eurographics Association 2007.
lar point samples such as those obtained from 3D scanners
[PKKG03].
However, the MLS method is designed to reconstruct
smooth surfaces from data which is sampled from smooth
surfaces. This leads to an erroneous surface reconstruction at
regions where the smoothness assumption is invalid. One noticeable phenomenon of forcing smoothness is known as the
Gibbs phenomenon (see Figure 4 and 6(c)). In other cases,
the MLS might smooth out features. In both cases the reconstructed surface is clearly unfaithful to the original geometry
(see Figure 1). A main flaw of the MLS reconstruction stems
from the fact that its approximation space (i.e., polynomials)
does not respect singularities in the data.
In this paper, we present a data-dependent moving least
squares (DDMLS) method for surface reconstruction. It is
data-dependent in the sense that it is faithful to the data singularities. Hereafter, the terms sharp features or singularities
refer to discontinuities of arbitrary magnitudes in the function or its derivatives.
The technique we present here retains the simplicity of
the MLS machinery: For each point x, it fits a function
p ∈ F , from a class of functions F . Then, p(x) is the desired local approximation. However, here we use a varying, locally fitted, piecewise polynomial (splines) space as
F . The main challenge is to define the location (locus) of
the spline singularity, so that the spline approximation space
Lipman & Cohen-Or & Levin / Data-Dependent MLS
Figure 2: A closeup of projection of the black point. The
point cloud is colored from blue to red proportional to the
SIF at each data point. The green surface patch is the best
local approximant p from the data-dependent space F . The
yellow ball is the result of the projection. Note that although
the point cloud is sparse the local singularity is well estimated.
Figure 1: Top row: original surface; Middle: MLS reconstruction; Bottom: DDMLS reconstruction. Note the faithful
reconstruction by the DDMLS at singularities as expressed
by the isophote lines (left column).
furnishes a local faithful approximant. Our approach to this
problem is based on the following rationale. The problem
of reconstructing a surface from discrete samples is an illposed problem since there is no unique surface that fits the
data. In particular, it is impossible to distinguish between
sharp and non-sharp data points (see Figure 3). Thus, for
each data point we define a continuous value that measures
its singularity potential. This defines a singularity indicator
field (SIF), which is used to construct the spline space F ,
from which the local approximation p is derived.
An important feature of the new method is that it automatically approximates the singularities in a threshold-free
manner. This enables the method to be faithful even to the
most delicate singularities, even in vicinity of abrupt singularities. This is demonstrated in Figure 4, where the height
function model (b) contains two types of circular features.
Across the outer circle there is an abrupt discontinuity, and
across the inner circle there is a delicate derivative discontinuity. Another example is shown in Figure 1.
We show that the DDMLS method allows the reconstruction of surfaces with high fidelity. All the examples that we
use exhibit some sharp features and fine details like the little scratch on the soap in Figure 8. We also show that this
technique is applicable to the enhancing of images.
1.1. Background and related works
In recent years many researchers have dealt with defining and manipulating point set surfaces following the MLS
methodology in [Lev03]. Alexa et al. [ABCO∗ 03] defined a
surface approximating a given 3D point set, and introduced
algorithms for representation and up-sampling. Amenta et al.
[AK04] have further deepened the understanding of the MLS
projection operator by expressing it as an extremal surface
and presenting an efficient algorithm to project a point on
the MLS surface and it variants. Pauly et al. [PKKG03] have
constructed a shape modeling framework based on point set
geometry.
Fleishman et al. [FCOS05] use the MLS methodology for
reconstructing surfaces with sharp features. They use a trialand-error process to define the neighborhoods of points in
the approximation. That is, for each point to be approximated, an initial group of nearby points is chosen by an exhaustive search. Then, other nearby points are added to the
initial group as long as a max-norm threshold is not violated.
This method, although robust to outliers, does not necessar-
Figure 3: Surface reconstruction from a discrete set of
points is an ill-posed problem. The figure demonstrates three
possible reconstruction procedures. One which smooths the
data, one interpolating the data, and one which allows sharp
features.
c The Eurographics Association 2007.
Lipman & Cohen-Or & Levin / Data-Dependent MLS
(a)
(b)
(d)
(f)
(h)
(c)
(e)
(g)
Figure 4: The height function in (b) contains two types of circular features. Across the outer circle there is a function discontinuity, and across the inner circle there is a delicate derivative discontinuity. The data set consists of the irregular point samples
(a). (d) and (f) demonstrate reconstruction by MLS and DDMLS, respectively. To better visualize the Gibbs phenomenon and
the over-smoothing, the corresponding isophote lines are presented at the bottom row. Note the oscillations near the boundaries
and the smoothing effect of the delicate inner circular feature caused by the MLS, compared with the faithfulness of the DDMLS.
To further illustrates the effect, 1D slices of the original, MLS, and DDMLS are displayed in (h) from top to bottom.
ily reconstruct a continuous surface near the sharp edges (see
Figure 13).
Reuter et al. [RJT∗ 05] have introduced an alternative projection operator to the MLS that respects sharp features.
However, in their work the user has to manually tag the sharp
features in the point cloud as an input to the algorithm.
Lipman et al. [LCOL06] looked for radial neighborhoods
which minimizes a bound on the error of the approximation.
In their work, the authors dealt with smooth surfaces only
and did not relate to the problem of surfaces with singularities and sharp features. However, in that paper the authors
provided some error estimates for the MLS which will be of
use in this paper.
The technique presented here is most related to the
work on essentially non-oscillatory (ENO) interpolation
[HEOC87]. ENO schemes originate and are in use in the
field of numerical methods for PDEs [SO89, JS96, JP00].
In rough terms, for each point x, the ENO reconstruction
scheme chooses a different stencil of samples for the reconstruction. In particular, ENO uses samples from the neighborhood of x which minimize the divided difference. This
yields an approximant which interpolates the data samples
and does not cross singularities, and hence does not suffer
from the Gibbs phenomenon and does not over smooth the
data.
These properties are most relevant to the reconstruction
of surfaces from irregular point samples. However, there
are two fundamental problems: First, ENO is an interpolac The Eurographics Association 2007.
tory scheme and thus cannot deal with noisy data. Second,
the ENO concept is defined for 1D data and does not seem
to have a straightforward generalization to irregular data in
higher dimensions. This is mainly due to the difficulty in recognizing well-posed sets for multivariate interpolation, especially in the case of irregular data points. In particular, there
is no natural generalization of divided difference to the multivariate case.
2. Overview
The core element of the present work is a new multivariate substitute for the divided difference check of the ENO
scheme. We compute a Singularity Indicator Field (SIF),
which intuitively, assigns to each point an estimate of its
proximity to a singularity. This serves as a basis for features detection and for the definition of the appropriate local approximating spline space. Singularities of surfaces are
typically 1-manifold, therefore, in order to reconstruct and
represent them, it is sensible to employ manifold approximation techniques. In this work we use the MLS machinery
again to approximate these 1-manifolds. The non-trivial part
is defining an indicator field over the data, on which the MLS
is applied. The benefit would be a consistent approximation
to the singularity manifold.
The SIF is illustrated in Figures 5 and 6(b) with pseudo
colors. In Section 4 we rigorously develop and define the
SIF and describe its construction. In Section 5 we describe
the local spline fitting, namely, using a piecewise polynomial
Lipman & Cohen-Or & Levin / Data-Dependent MLS
(a)
(b)
(a)
(b)
(c)
(d)
Figure 5: Visualization of the SIF of the surface from Figure
4 by a logarithmic scale where colors range from dark blue
to dark red. (b) shows a close-up view of the upper-right
corner. Note that the closer a point is to the singularity area,
the larger is its SIF relative to other points in its vicinity.
space that respects the (possible) local 1-manifold singularity. The surface is then reconstructed by projecting points
over these local (moving) splines, as illustrated in Figure 2.
Next we outline a pseudo-code of the method:
1. Preprocess. For each data point:
a. Find a local reference plane.
b. Calculate the SIF value for that point.
Figure 6: Surface reconstruction from a sparse point cloud
(a). Note for example the low density of points in the region
marked by a black circle. (b) presents the SIF values colored from blue to red. In (c) we exhibit that the MLS reconstruction smooths out the corners and suffers from the Gibbs
effect. In (d) we see that a DDMLS reconstruction is more
faithful to the sharp features of the input data.
2. Projecting a point s.
a. Find a local reference plane π. Denote by x the projection of s onto it.
b. Project the local data points on π to create a functional
settings.
c. Apply 2D univariate MLS to find the curve defined by
the SIF.
d. Construct the local spline space Sm such that its singularity line passes through and tangent to the univariate
MLS projection.
e. Find the best least-squares fit p ∈ Sm to the functional
data (Sm includes also the smooth polynomials).
f. p(x) is the desired DDMLS projection.
In the following section we review the basic MLS mechanism, which is employed here triply: first to define the SIF,
then to define the local 1-manifold singularity curve, and finally to reconstruct the 2-manifold in a data-dependent manner.
where F = Πm , the subspace of polynomials of total degree
m, φ(r) is a fast decreasing smooth function of finite support
size h, and h is a data parameter. In this paper we have used
2
the approximation φ(r) = e
mation at x is defined by
r
− (h/4)
2
. Then the MLS approxi-
M f (x) = p(x).
An equivalent definition of the same approximation is given
in terms of a polynomial reproduction property [Lev98]:
M f (x) = ∑ f (xi )LiX,m (x),
i∈I
where the “shape functions” {LiX,m }i∈I minimize the
quadratic form
Q = ∑ |LiX,m (x)|2 φ(kxi − xk)
(2)
i∈I
subject to the linear constraints of polynomial reproduction
3. Moving Least Squares
The key element in the MLS procedure for surfaces is the
MLS approximation of functions, which we describe below.
We are given data, sampled from a function f , at some irregular set of data sites: (X, f (X)) ⊂ Ω × R, where Ω is a domain in Rd , and X = {xi : i ∈ I}, I = {1, 2, ..., N}. To define
the approximation at an arbitrary point x ∈ Ω the following
quadratic minimization problem is solved:
(
)
p := argmin
p∈F
∑ |p(xi ) − f (xi )|2 φ(kxi − xk)
i∈I
,
(1)
∑ LiX,m (x)b j (xi ) = b j (x),
j = 1, ..., K
(3)
i∈I
where b1 (x), ..., bK (x) is a basis for Πm . We shall make use
of both presentations of the MLS approximation in the development of the new tool presented here.
To apply the above functional MLS approximation to
surface-sampled data {pi }, a local reference plane is defined
for each projected point [Lev03]. In order to define a surface,
this plane should be chosen with care. In this work we define the reference plane using a local weighted PCA, where
c The Eurographics Association 2007.
Lipman & Cohen-Or & Levin / Data-Dependent MLS
X ∩ B(x, h)}, where B(x, h) is a ball of radius h centered at x,
and 0 ≤ ηi ≤ 1. From this equation we get
| f (x) − p(x)| ≤
max|ν|=m+1,y∈<Xh ∪x> |Dν f (y)| ∑i,ν
|xi −x|ν X,m
(x)|,
ν! |Li
(5)
where < Xh ∪ x > denotes the convex hull of the set of points
Xh ∪ x, where Xh = {xi }i∈Ih (x) .
After rearranging, we get
max
|ν|=m+1,y∈<Xh ∪x>
Figure 7: Scanned CAD model (top left, photograph). On the
top right is the DDMLS reconstruction. The two bottom rows
show enlarged parts, where the left most are photographs of
the scanned model (ground truth), the middle column is the
DDMLS reconstruction and the right-most column shows an
MLS reconstruction. Note the delicate feature marked by a
small arrow, which is smoothed in the MLS reconstruction.
4. Singularity indicator field
In this section we define the singularity indicator field (SIF),
and describe its construction. Singularities are related to unbounded high derivatives of the unknown function f , and
we would like to assign to each data point a value which
indicates its proximity to a singular point. Given a discrete
data set sampled from a function f , no upper bound to the
derivatives of f can be found, since, as mentioned above,
the reconstruction problem has no unique solution (see Figure 3). As we show below, it is possible to find a good local
lower bound to the absolute value of the derivatives of the
unknown function f . These lower bounds will be used to locate the points or curves of singularities of the surface.
To construct a lower bound of the derivative, we use the
error expression of the MLS approximation [LCOL06]. Denote by p the polynomial that is fitted to the data X, as defined in (1), then the error in the MLS approximation of x
is
f (x) − p(x) =
Dν f (ηi (xi −x)+x)
− ∑i,ν
(xi − x)ν LiX,m (x) ,
ν!
(4)
where ∑i,ν stands for ∑|ν|=m+1 ∑i∈Ih (x) , Ih (x) = {i | xi ∈
c The Eurographics Association 2007.
∑i,ν
|xi −x|ν X,m
(x)|
ν! |Li
. (6)
Taking x to be one of the data points, all the expressions
on the r.h.s. of the above innequality can be computed, and
we get a lower bound for the absolute values of the (m +
1)th order derivatives of the unknown function f near a data
point. Motivated by this inequality we define the singularity
indicator Λ j at point x j ∈ X by
Λj =
| f (x j ) − p(x j )|
∑i,ν
|xi −x j |ν X,m
|Li (x j )|
ν!
,
(7)
and we have an immediate property of Λ j , derived from (6):
Λj ≤
the weights are defined to be wi = φ(kpi − sk), where here
s is the projected point. Although this fails to define a projection operator in all cases [AK04], we found it practically
convenient and robust.
| f (x) − p(x)|
|Dν f (y)| ≥
max
|ν|=m+1,y∈<Xh ∪x>
|Dν f (y)|.
The values Λ j can be computed: The f (x j ) are known, and
the shape functions of the MLS |LiX,m (x)| are obtained via
the minimization of the quadratic form (2) subject to the linear constraints (3). The minimizer is computed by solving
a linear system derived by Lagrange multipliers [Lev98].
Note, that the cost of computing each Λ j is the same as a
single MLS fit.
Let us assume that the function f is piecewise Cm+1 , with
a jump-discontinuity in one of its (k + 1)th order derivatives,
k ≤ m + 1, near the point x j . Then, using a local Taylor
series approximation of total degree k, and using the polynomial reproduction property of the MLS, it can be shown
that Λ j = O(hk−m−1 ). For a non-degenerate distribution of
data points, we observe that Λ j = Θ(hk−m−1 ) for k = −1, 0
(the examples is the paper). This observation implies that
Λ j can be used as a local indicator to the presence of jumpdiscontinuities, or large magnitude of the derivatives of the
unknown function f at the vicinity of x j . Therefore, we define the Singularity Indicator Field (SIF) of the given data as
{xi , Λi }i∈I . Figure 5 shows an example of the SIF.
The above SIF should be computed for each projected
data point. However, to reduce computational cost, we approximate the SIF by precomputing the singularity indicator
of each data point once with respect to its own reference
plane. Since changing slightly the reference plane does not
change the relative magnitudes of the local derivatives this
approximation is good enough for our proposes. Figure 6(b)
shows the precomputed SIF for a specific example.
Lipman & Cohen-Or & Levin / Data-Dependent MLS
(a)
(b)
(c)
(d)
Figure 8: A scanned soap. (a) shows the ground truth (photograph). In (b) pct is taken as the 100th percentile, i.e., regular
MLS. In (c) and (d) pct is taken as the 90th and the 75th percentile, respectively. Note how the fine scratch on the soap (marked
by small arrow) is reconstructed by the DDMLS and smoothed out by the MLS. The small upper window at (a) shows a portion
of the scanned point cloud.
5. Local data-dependent spline space
frame v(x), n(x), where x p serves as the origin of axes:
Instead of using the polynomial space Πm in the approximation for x as is done in the standard MLS (1), we define the space from which the minimizer is sought in a datadependent manner. That is, we define the modified MLS operator, first, by solving the minimization problem (1), where
F is now taken as the space of piecewise polynomials fitted
to the point x. I.e., we let F = Sm , the space of splines of
total degree ≤ m, whose singularity locus is an approximation to the singularity in the data, near the point x, as derived
using the SIF. Thus we solve for
(
)
p := argmin
p∈Sm
∑ |p(xi ) − f (xi )|2 φ(kxi − xk)
,
(ti , yi ) = (hxi − x p , v(x)i, hxi − x p , n(x)i) .
Also we denote x in that coordinate system by (0, y). The
local approximation to the singularity is then defined by
(
)
q := argmin
q∈V
∑ |q(ti ) − yi |2 ψ(kxi − xk)Λi
,
(10)
i∈I
where V is a univariate polynomials space Πm (R), and ψ(r)
is a fast decreasing weight function. We now use the curve
(8)
i∈I
c f at x is defined by
Then, the modified MLS operator M
c f (x) = p(x).
M
(9)
For the definition of the local approximation spline space
F to be used at a given point x, we apply the MLS framework on the SIF data, and construct a local univariate (curve)
approximation to the singularities. I.e., we look for a curve
which is close, in a least-squares sense, to data points with
high SIF value. We explain this part of the procedure through
an example: In Figure 9 (a) the SIF at each data point is
drawn by a red circle whose radius is proportional to the SIF
value at that corresponding data point. The green point denotes the point of approximation x. We then use the MLS operator to find, now in a univariate setting, a univariate polynomial which fits high valued SIF points locally, e.g., the
green curve at Figure 9(a). More precisely, we first choose
a reference line `(x) (the thin black line in (a)). This can be
done in several ways, as mentioned before. Here, we have
used again a weighted PCA, where the weights are taken
as the singularity indicator values times the smooth radial
weights, i.e., φ(kxi − xk)Λi . Denote the projection of x on
`(x) by x p , and by v(x), n(x) the tangent unit vector and the
normal to `(x), respectively. Next, we project the data points
xi ∈ Xh on the reference line to represent the data in the local
(a)
(b)
Figure 9: Local data-dependent spline space for MLS the
projection. In (a) we demonstrate the construction by a discontinuous spline space, and in (b) by a continuous spline
space.
outlined by q as the locus of the local singularity of the space
Sm to be used in Eq. (8). We offer two possible different
constructions of the space Sm , one discontinuous and one
continuous:
Type-0 singularity. In this case the functions in Sm are
chosen to be discontinuous along the curve q(t) (illustrated
by the green line in Figure 9(a)). Thus, there are two independent polynomials each defined on each side of the curve
q. In this case the approximation defined by Eq. 9 is equivalent to solving Eq. (8) only for points xi which are on the
same side as x with respect to the approximated singularity
curve q. More formally, we use xi if and only if
sign(q(ti ) − yi ) = sign(q(0) − y).
c The Eurographics Association 2007.
Lipman & Cohen-Or & Levin / Data-Dependent MLS
This is illustrated in Figure 9(a), where only the orange
points participate in the approximation at x (the green point).
Since we only take an approximation to the singularity, in
order to better ensure we do not take points from the other
side of the singularity, we can take points which also satisfy
|q(ti ) − yi | > 14 |q(0) − y|.
Type-1 singularity. In case of surface reconstruction it is
usually required to force continuity. To that end, we would
use continuous spline functions as the approximation space.
Thus Sm is defined as the space of continuous piecewise
polynomials, of total degree ≤ m, with possible discontinuity in the derivatives across the line
(a)
(b)
(c)
(d)
γ(t) := (0, q(0)) + t(1, q0 (0)),t ∈ R.
γ is illustrated by the bold black line in Figure 9(b). For calculating the minimizer p ∈ F of Eq. (8) in this case, we augment the standard basis of Πm by the piecewise polynomials.
This can be done by first aligning the x-axis to γ and then
β
adding as basis functions xα (y)+ , where α ≥ 0, β ≥ 1 and
α + β ≤ m.
If x is far from a singularity the system might turn out to
be ill-conditioned. In such a case it is natural to turn back
to the old approximation space Πm : Practically, switching to
F = Πm for bad-conditioned system worked well for us.
Go recursively: singularities of singularities The same
argument as above can be applied in the univariate setting.
Instead of approximating the singularity using a smooth
polynomial, we can use a univariate spline with a point singularity. Now, the SIF is to be considered as a point cloud
with confidence values. The point of singularity can be deduced from a SIF of the SIF. In turn, the SIF of the singularity can be defined as described in section 4. See for example
Figure 12.
6. Noisy Data
In the presence of noise, one needs to compromise since in
the lack of any priors, one cannot distinguish noise from
features. To handle noise we can clamp the SIF based on
the following observation: The singularities in a surface (2manifold) are (usually) a lower dimension manifolds, that is,
curves (1-manifold) or points (0-manifolds). Hence, one can
assume that most of the data points are not near a singular
point. Define the modified SIF of the data Λ by:
Λnew
= H(Λ j − pct),
j
(11)
where pct is, for instance, 90th percentile of the Λ, and H is
the Heaviside function.
The practical effect of Eq. (11), is applying regular MLS
in areas with SIF values smaller than pct, which are generally the smooth areas of the surface. Areas with SIF values
larger than pct are approximated by a data-dependent MLS,
taking into account the (now non-zero) SIF. Hence, reconstructing features at areas with large SIF values, relative to
c The Eurographics Association 2007.
Figure 10: (a) A photograph of a Lilly flower. In (b) we see
the irregular sampled data ( 12% of the points), and in (c) a
Lucy-Richardson deblurring applied to the MLS reconstruction. (d) depicts the DDMLS reconstruction. Note the faithful reconstruction of the silhouette edges and delicate lines
in the background.
(a)
(b)
(c)
Figure 11: Noise direction analysis. (a) exhibits the data
points (10K points) colored by the SIF values, and (b) the
original data points rendered as a mesh to show the high
noise level. Note the delicacy of the sharp feature w.r.t. the
noise level. In (c) we see the reconstruction using DDMLS
with a directionality analysis to smooth noise and preserve
the features.
the rest of the surface. The effect of applying different pct
values is demonstrated in Figure 8.
Another way of dealing with noise is based on the observation that noise does not have a structured shape like, for
example, a direction, whereas features typically have. In the
simplest form, this can be integrated into the method during
weighted PCA construction of the local reference line. Let
us define the directionality score of the local SIF as the ratio
of the singular values in the weighted PCA: µ(x) := |s1 /s2 |
, i ∈ I, where |s1 | > |s2 | are the singular values in the PCA
analysis when considering point x. Then, fix a blending interval [µ1 , µ2 ], and a smooth blending function ψ(t) where
Lipman & Cohen-Or & Levin / Data-Dependent MLS
synthetic surfaces and images; for example Figures 1, 4, 6,
7, 8, 10. We have used mostly cubic piecewise polynomials
(m = 3).
(a)
(b)
(c)
(d)
Figure 12: Singularities of singularity curves. (a) shows the
SIF on the jittered grid. In (b) the original function is drawn
as an image, where the singularity curve has a point singularity. (c) exhibits reconstruction using a polynomial approximation to the singularity curve, and (d) reconstruction
using a piecewise-polynomial to approximate the singularity
curve.
Our method has the important property of being threshold free, which is important for faithful surface reconstruction as well as for processing and representing piecewise
smooth geometry. The local singularity structure is realized through a spline approximation space used in the MLS
method, where the piecewise polynomial singularity locations are defined in a relative, local, threshold free manner.
Note however, that as we described in Section 6, there is a
threshold parameter that is used in cases of noisy data, in
order to distinguish features from noise.
In Figure 13 we compare our method to the method of
Fleishman et al. [2005]. We applied the two methods to a surface with a sharp feature of a magnitude decreasing to zero.
The detection of such a sharp feature using a threshold based
method will necessarily fail. The surface is sparsely sampled with a jittered grid of 2.5K samples. The faithfulness
of the reconstruction is demonstrated by isophotes curves.
Moreover, note that the iterative construction of Fleishman
et al. [FCOS05] does not consider the singularity as manifold. Hence, it cannot guarantee a consistent approximation,
nor a continuous reconstruction across sharp edges.
Figure 13: Comparison with Fleishman et al. [2005]. The
original surface (left) has a sharp feature with a vanishing
magnitude. The detection of a sharp feature using a threshold will necessarily fail. The surface is sparsely sampled (top
left) with a jittered grid of 2.5K samples. The reconstruction
with Fleishman et al. [2005] is shown in the middle, and
with the DDMLS on the right. The faithfulness of the reconstruction is demonstrated with the isophotes (bottom row).
ψ(t) = 0 for t < µ1 , and, ψ(t) = 1 for t > µ2 . Now define the
operator as
c f (x).
MBf (x) = ψ(µ(x))M f (x) + (1 − ψ(µ(x)))M
Figure 11 demonstrates this technique for resampling a relatively sparse noisy data with a delicate sharp feature. In this
context, we note that using an enhanced SIF field can be useful, for example we have used (Λi )4 .
7. Discussion
The main contribution of this work is the use of an adaptive
spline space in the MLS approximation via a novel feature
detection method in a scattered data setting. An interesting
observation is that we use the inability of the MLS to reconstruct discontinuities in order to detect them, as expressed by
the SIF. We have applied DDMLS to real scanned models,
Another advantage of the new method is the ability to
reconstruct sharp features using relatively sparse point set.
For example, the models in Figures 4, 6, 13 consist of only
5K,12K,2.5K input data points, respectively. Generally, we
retain the computational complexity of the standard MLS
since we perform two MLS projections for each point. In
our MATLAB implementation the method projects around
200 points per second, and the preprocess calculates 500 Λ
values per second. This is two orders of magnitude faster
than the method of Fleishman et al. [FCOS05]. For example, in Figure 13, Fleishman’s method is 70 times slower
than DDMLS.
Currently, the limitation of the technique is that it requires
careful tuning in case of noisy data. Otherwise, it may amplify the noise or suppress delicate features. Another limitation is that it does not deal well with non-manifold singularity structure, like the corner of a cube. An interesting
research direction is to extend the method to construct such
non-manifold singularities. Note that the presented method
can be applied to data of arbitrary dimension. In our setting,
it is applied to a 2D surface while respecting 1D singularity.
However, it can also be applied to volumetric data with discontinuities. Another interesting direction, which is demonstrated by a simple example in Figure 12, is the detection
of breakpoints in curves of sharp features and using this to
further adapt the local spline space.
8. Acknowledgements
This work was supported by the Israel Science Foundation.
c The Eurographics Association 2007.
Lipman & Cohen-Or & Levin / Data-Dependent MLS
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