ROBUST FEATURE LINE EXTRACTION
ON CAD TRIANGULAR MESHES
Vincent Vidal, Christian Wolf
Universit
´
e de Lyon, CNRS, INSA-Lyon, LIRIS, UMR5205, Villeurbanne F-69621, France
Florent Dupont
Universit
´
e de Lyon, CNRS, Universit
´
e Lyon 1, LIRIS, UMR5205, Villeurbanne F-69622, France
Keywords:
Triangle meshes, Feature lines, Crest lines, Feature extraction, Robust detection, Potts model, SVMs.
Abstract:
Feature lines are perceptually salient features on 3D meshes. They are of interest for 3D shape description,
analysis and recognition. Their detection is a necessary step in several feature sensitive mesh processing
applications such as mesh simplification, remeshing or non-photorealistic rendering.
In this paper, an estimator for the angle between tangent plane normals is introduced and a new automatic
method is proposed for robust detection of crest lines on 2-manifold triangular meshes, in particular Computer-
Aided Design models. The method integrates learning into a global minimization framework favoring geomet-
rically coherent solutions. We study our method in detail and compare it with other methods for the detection
of feature edges on 3D meshes. Our comparative results indicate that our method outperforms classical tech-
niques especially in the presence of noise.
1 INTRODUCTION
Feature line extraction consists in finding perceptu-
ally salient lines over 3D meshes that a human eye
will notice. Detection of feature lines on polygo-
nal surfaces is currently an area of intensive research
(Hildebrandt et al., 2005; Kim et al., 2009; Yoshizawa
et al., 2005; Zhihong et al., 2009). Feature line ex-
traction has several applications such as surface seg-
mentation (Stylianou and Farin, 2004), shape anal-
ysis, matching, and retrieval with geometric query
in 3D object databases. Feature line information is
also of interest in mesh simplification, remeshing (At-
tene et al., 2005) or surface smoothing. In non-
photorealistic rendering feature lines are highlighted
(DeCarlo et al., 2003).
Similar to edge detection in images, feature line
extraction on 3D surface meshes is an ill posed prob-
lem due to the lack of information on the sampling
process, the noise process and the signal geometry.
According to the situation at hand, the same geomet-
rical discontinuity might be due to a “true” feature in
the object, to insufficient sampling in a region of high
curvature, or to noise in vertex positions. Additional
difficulties in feature line extraction process are its
sensitivity to irregular mesh connectivity when ex-
tracted feature lines are composed of mesh edges, the
use of a parametric model (e.g snakes) when feature
line segments are independent of the mesh connectiv-
ity (Lee and Lee, 2002; Pauly et al., 2003), and the
selection of significant feature lines at the right scale.
Most of existing works tackle the problem of fea-
ture line extraction over dense triangulated meshes.
Some works deal with multi-scale feature extrac-
tion (Lee et al., 2005; Pauly et al., 2003) but re-
main oriented more towards the detection of fea-
ture vertices than feature lines. For smoothly vary-
ing natural meshes, the majority of approaches are
based on robust computation of principal curvature
extrema (Hildebrandt et al., 2005; Ohtake et al., 2004;
Yoshizawa et al., 2005; Zhihong et al., 2009) and
require the estimation of curvature derivatives. For
Computer-Aided Design (CAD) meshes, feature line
extraction often involves identifying all feature edges
using either the tensor voting theory (Kim et al.,
2009), dihedral angles or angles between best poly-
nomial fits (Hubeli et al., 2000).
In this paper, we propose a new robust method
for feature line extraction over 2-manifold triangular
CAD meshes. Our main contributions are the follow-
106
Vidal V., Wolf C. and Dupont F..
ROBUST FEATURE LINE EXTRACTION ON CAD TRIANGULAR MESHES.
DOI: 10.5220/0003361701060112
In Proceedings of the International Conference on Computer Graphics Theory and Applications (GRAPP-2011), pages 106-112
ISBN: 978-989-8425-45-4
Copyright
c
 2011 SCITEPRESS (Science and Technology Publications, Lda.)

ing:
• introduction of a robust estimator for the angle be-
tween tangent plane normals (in section 3),
• design of relevant geometric features for feature
edge learning (in section 4),
• globally consistent detection of feature edges over
triangular surface meshes (in section 5).
The remainder of this paper is organized as follows.
In section 2 relevant definitions are presented. Section
3 describes the computation of our estimator for the
angle between tangent plane normals. Section 4 tack-
les feature edge learning and gives a feature vector
characterizing feature edges. In section 5 our method
for globally consistent feature line extraction is de-
tailed. In section 6 experimental results are analyzed.
Finally, section 7 concludes the paper and presents
some future work.
2 DEFINITIONS
The following definitions hold for 2-manifold trian-
gular meshes with borders.
Dihedral angles are angles between the two nor-
mals of two adjacent triangles to an edge. θ
e
denotes
the dihedral angle at edge e. The angle between tan-
gent plane normals is defined as the angle between
the two normals of two estimated tangent planes to an
edge. The estimation of tangent planes depends on
a scale parameter.
ˆ
θ
r
e
stands for an estimation of the
angle between tangent plane normals at edge e and at
scale r. Details on its computation are given in section
3. A mesh edge e is a feature edge if e is a boundary
edge or if there is a high discontinuity of the normal
direction along e. A boundary edge is shared by one
triangle and its dihedral angle and estimator for the
angle between tangent plane normals are equal to π.
A mesh edge e is a normal edge if e is not a feature
edge.
3 ESTIMATING THE ANGLE
BETWEEN TANGENT PLANE
NORMALS
This section introduces the computation of the angle
between the two normals of two estimated tangent
planes to an edge. A local tangent plane fitting (first
order approximation) is proposed and the angle be-
tween the two unit normals of tangent planes is com-
puted afterwards.
3.1 Robust Estimation of Tangent
Planes
The estimation of the local tangent planes is obtained
through region growing. However, it is not a classical
region growing in sense that a new triangle can be
added to one region even through it is not adjacent
to any region triangles. Moreover, once a triangle is
added to a local region, it remains in that region till
the region growing ends.
Robustly estimating the two neighboring tangent
planes involves:
• taking into account a scale parameter resulting in
a bounding sphere (centered at the edge midpoint)
in which a plane is estimated,
• the detection of outlier triangles due to noise,
• the identification of additional discontinuities
in the bounding sphere due to multiple region
changes (e.g. at additional feature lines close to
the edge or at areas of high curvature).
As in classical fitting problems, the plane parameters
(such as the normals) depend on the classification of
the triangles into inliers, outliers and out of region tri-
angles (see figure 1), and the latter classification de-
pends on the plane parameters.
3.2 Our Approach
This section describes our approach for the computa-
tion of the two unit normal estimates of tangent planes
to an edge. The final tangent planes are forced to go
through the edge midpoint.
Algorithm Description. At the beginning of the al-
gorithm, the two triangles adjacent to the edge are
added to their corresponding regions. Thus, first es-
timates of tangent plane normals are set to corre-
sponding first region triangle normals. Then, a greedy
traversal of triangles within the bounding sphere is
done: triangles closer to the edge separating plane
(see figure 1) are tested first for addition (using a pri-
ority queue with euclidean point-to-edge separating
plane distance as priority). That permits to get an es-
timation of tangent plane normal which depends on
the close-to-far away triangle order of traversal. The
left and right regions, and estimated normals of tan-
gent planes are updated as follows:
• triangles, whose signed distance from its barycen-
ter to the separating plane is positive (resp. strictly
negative), are candidates for right (resp. left) re-
gion;
• if a candidate triangle has a normal close to the
currently estimated plane normal (the angle in be-
ROBUST FEATURE LINE EXTRACTION ON CAD TRIANGULAR MESHES
107

Figure 1: Estimator for the angle between tangent planes:
dotted lines represent the mesh surface (with inliers, outliers
and out of region triangles). The estimation of left and right
tangent planes depends on a bounding sphere centered at
the edge midpoint.
tween normals must be less than a threshold), it is
added to the region;
• after each addition of triangle in a region, the esti-
mated normal of the tangent plane associated with
this region is updated as the area weighted averag-
ing of normals over the associated triangles;
• when the region growing ends, all previously re-
jected triangles are tested again for acceptance in
the same region: the plane normal has changed
during the region growing and thus some previ-
ously rejected triangles may be accepted this time.
4 LEARNING FEATURE EDGES
To make the edge classification process more robust,
it is based on several measures and on complex inter-
actions. Learning a statistically representative model
of feature edges from ground-truth data permits to
do that. In this paper, Support Vector Machines
(SVMs), which is a supervised learning method, are
used. More details on SVMs can be found in (Vapnik,
1998). In the following, the feature vector associated
with an edge for learning is presented.
Feature Vector Construction. Let F
e
be a feature
vector associated with an edge e, i.e. it characterizes
e in terms of geometrical and contextual measures. A
small number of features for F
e
is selected, e.g. less
than 50 features, according to their F-score defined in
(Chen and Lin, 2006). The larger the F-score is, the
more likely this feature is discriminative.
The four first features in feature vectors F
e
are the
following measures:
• cosine of the dihedral angle cos(θ
e
),
• cosine of the angle between the unit normals of
the vertices opposite to the edge (-1 is taken for
border edges),
• mean of the two edge vertices’ minimum of
cosines of their adjacent edges’ dihedral angles,
• mean of the two edge vertices’ normal variation
(Lee and Lee, 2002).
Then, three scale-dependent features are added into F
e
(depends on a radius r):
• cosine of the angle between tangent plane normals
cos(
ˆ
θ
r
e
) (
ˆ
θ
r
e
is defined in section 2),
• five curvature measures such as the mean of edge
vertices curvature (Kmin and Kmax principal cur-
vatures (Cohen-Steiner and Morvan, 2003), Gaus-
sian curvature: Kmin ∗ Kmax, mean curvature:
0.5(Kmin + Kmax), Kmax − Kmin),
• mean of the two edge vertices’ similarity measure
(Lee et al., 2005).
r is set as a percentage of the mesh bounding box
minimal dimension (min BB). The following scales
are worth identifying feature edges in the presence of
noise and are thus included in F
e
:
• for the estimator for the angle between tangent
plane normals: r = 4, 7, 10, 13, 16, 19, 22, 25,
28 % of min BB,
• for principal curvatures and similarity measure: r
= 1, 5, 9, 13, 17 % of min BB.
F
e
is composed of 43 features. Each feature in F
e
is
normalized to [−1, 1].
5 GLOBALLY CONSISTENT
FEATURE EDGE DETECTION
In the previous section, feature edge detection has
been improved by selecting more measures for char-
acterizing them and by learning a prediction model
capable of taking into account all informative features
in the training set. However, all final decisions are lo-
cal for each edge. Local decisions suffer from not
taking into account the label distribution of neighbor-
ing edges. Conversely, global decisions over all mesh
edges may improve the detection result quality by fa-
voring consistent labelling.
Post-processing can introduce a slight dependence
on the neighboring edges’ results. However, this is far
from optimal, as the true circular dependence is not
taken into account: the result for each edge depends
on the results of neighboring edges and vice versa.
A more powerful solution is to integrate the full
set of dependencies directly into the decision pro-
cess and to combine it with the local geometrical
measures. This can be achieved by minimizing a
global objective function (energy) over all edge la-
bels, which gives lower scalar values for better solu-
tions (labelling), both in terms of geometry and con-
sistency:
˜
ω = argmin
ω
E(ω, Θ)
GRAPP 2011 - International Conference on Computer Graphics Theory and Applications
108

E(ω, Θ) =
∑
e
E
d
(ω
e
)
| {z }
unary term
−µ
∑
{e,e
0
}∈N
E
h
(ω
e
, ω
e
0
, Θ)
| {z }
pairwise term
(1)
ω (resp.Θ) is the set of all edge label variables (resp.
edge geometrical measures). N is the set of ordered
pairs of adjacent edges. The unary term E
d
(ω
e
) (cf.
equation 2) is a data term which evaluates the lo-
cal performance of assigning the binary label ω
e
to
the edge e. Binary labels 0 and 1 correspond to
normal edge and feature edge. The pairwise term
E
h
(ω
e
, ω
e
0
, Θ) (cf. equation 3) is devoted to region
homogeneity and is based on an improved Ising/Potts
model (Geman and Geman, 1984) which locally en-
courages consistent region labellings. The positive
weight µ sets the relative strength of the regularizing
pairwise term compared to the unary term.
The data term is given as follows:
E
d
(ω
e
) = exp((−1)
ω
e
d(e)) (2)
E
d
(ω
e
) directly incorporates classification and confi-
dence of the SVM — which depends on the obtained
prediction model — through the signed distance d(e)
of the mapped feature vector associated with edge e
to the separating hyperplane (in feature space). It is
positive for predicted feature edges and negative oth-
erwise.
The regularizing pairwise term is given as:
E
h
(ω
e
, ω
e
0
, θ
e
, θ
e
0
, T
ee
0
) =



0 if w
e
6= w
0
e
β if w
e
= w
0
e
= 0
α
ee
0
(θ
e
, θ
e
0
, T
ee
0
) else
(3)
β is a positive constant which controls the homogene-
ity of normal edges. α
ee
0
(cf. equation 4) is a pos-
itive functional which favors consistent labelling of
aligned (within a tolerance) neighboring edges if they
have similar dihedral angles.
α
ee
0
(θ
e
, θ
e
0
, T
ee
0
) =
exp









−λ





|cos(θ
e
)−cos(θ
0
e
)|
σ
| {z }
similarity
+1−cos(T
ee
0
)
| {z }
alignment














(4)
In equation 4, λ is a positive constant which moni-
tors the neighboring edge alignment tolerance. The
greater λ is, the more two neighboring edges must
be aligned. σ handles the tolerated variance of
λ|cos(θ
e
)−cos(θ
0
e
)| along feature lines. T
ee
0
is the
turning angle (tangential angle) between e and e
0
.
The similarity term avoids detecting adjacent feature
edges with a very different dihedral angle. The align-
ment term predominates over similarity term, for fa-
voring non-jagged feature line extraction. Note that
the 00 and 11 labelling cases are favored compared to
01 and 10 cases: therefore homogeneous labellings
are preferred in region composed of a majority of
normal edges and along feature lines to fill in some
gaps. Mathematically speaking, this intuitively ex-
plained notion is called sub-modularity. It allows to
calculate the exact global minimum of (1) using graph
cuts technique (Kolmogorov and Zabih, 2004).
6 EXPERIMENTATION AND
DISCUSSION
In this section, the mesh data set is presented as well
as the chosen method for comparing edge classifica-
tion approaches. Then, experiment parameters are de-
fined. Finally, experimental results are analyzed.
6.1 CAD Mesh Data Set
Our data set is made of 181380 edges grouped into 18
CAD mesh models (9 normalized meshes with their
9 noisy versions obtained by adding 0.5% Gaussian
noise intensity to mesh vertex positions). The nor-
malization process consists in a mesh scaling to make
the maximal dimension of a mesh bounding box equal
to one. 0.5% refers to the bounding box maximal di-
mension.
Among 181380 edges there are 4468 ground-truth
feature edges (cf. table 2). All meshes have fea-
ture lines composed of mesh edges. Ground-truth for
the 9 non-noisy meshes has been manually set, and
noisy meshes ground-truth has been deduced from it.
Ground-truth is needed for learning feature edges and
permits to quantitatively compare all methods pre-
sented in this paper.
6.2 Comparison by ROC Curves
Comparison between two feature-normal edge clas-
sification methods is done using Receiver Operating
Characteristic (ROC) curves. A ROC graph depicts
relative trade-offs between benefits (true positive on
the Y axis) and costs (false positives on the X axis)
(Fawcett, 2006). To determine the best method over
an interval, the Area Under Curve (AUC) of the ROC
curve in that interval is computed.
ROBUST FEATURE LINE EXTRACTION ON CAD TRIANGULAR MESHES
109

Table 1: Minimum, maximum, mean and standard devi-
ation of average of f-scores (Chen and Lin, 2006) com-
puted as follows: Firstly, features’ f-scores are computed
for each mesh model (except sphere and torus). Then, the
mean of these features’ f-scores are computed for non-noisy
and noisy mesh models. Finally, some statistics (min, max,
mean and standard deviation) are computed on similar sub-
sets of features (i.e. estimator for the angle between tangent
planes, curvatures, and first four features in F
e
).
Features
in F
e
Models Min Max Mean sdev
ˆ
θ
r
e
non-noisy 9.196 1249.950 277.934 523.248
ˆ
θ
r
e
noisy 2.618 3.202 2.910 0.208
curv. non-noisy 0.143 0.699 0.360 0.152
curv. noisy 0.056 0.456 0.238 0.127
4 first non-noisy 0.533 1428.810 357.705 714.068
4 first noisy 0.405 2.569 0.969 1.068
6.3 Experimental Settings
Tangent Plane Estimation. A triangle is accepted
into a region during the region growing process if the
angle between its normal and the currently estimated
normal of the tangent plane is less than 23 degrees.
Feature Vector. The estimator for the angle be-
tween tangent planes
ˆ
θ
r
e
has a quite high informative
power as can be seen in table 1. It is much more
significant than curvature measures according to f-
scores (Chen and Lin, 2006) and than the four first
features in the feature vector for noisy meshes. How-
ever, for non-noisy mesh models, the four first fea-
tures are slightly more informative. By increasing the
size of the support, the angle between tangent plane
normals has become more robust to the presence of
noise, at the cost of being less sensitive to small fea-
tures in non-noisy data.
Learning Feature Edge. For learning with SVMs,
the LIBSVM library (Chang and Lin, 2001) has
been used with RBF kernel. The best model hyper-
parameters have been selected using grid search with
cross-validation and maximization of the AUC which
is able to cope with unbalanced training datasets.
All duplicated entries in the training set have been
removed and a subsampling taking at maximum 5000
training samples per mesh model has been done. The
subsampling process keeps the class distributions un-
changed. To compensate for unbalanced training data,
the error weighting factor associated with the feature
edge class is set 9 times greater than the one used for
normal edge class.
Globally Consistent Feature Edge Detection. To
evaluate the benefits of the global minimization of
equation 1 (Potts model) and those of the learning
Table 2: Statistics for mesh models used for edges classi-
fication (a: non-noisy models ; b: models with Gaussian
noise GN): nb. of edges, nb. of feature edges, Area Under
Curve (%) for 4 methods: thresholding, hysteresis thresh-
olding, globally consistent edge detection with data term
based on dihedral angle, and with data term based on SVM.
Mesh #t.e. #f.e. thres. Hys. dih+G SVM+G
1232 joint 9024 660 100.0 100.0 100.0 100.0
cone 14850 50 100.0 100.0 99.8 100.0
cup 17010 381 99.7 99.7 99.7 98.2
cut cone 864 144 100.0 100.0 100.0 100.0
cylinder 3540 40 100.0 100.0 100.0 100.0
fandisk 19479 743 99.9 99.9 100.0 97.3
screw 3723 216 100.0 100.0 100.0 100.0
sphere 14700 0 - - - -
torus 7500 0 - - - -
total 90690 2234 99.9 99.9 99.9 99.4
a) Non-noisy mesh models.
Mesh+GN #t.e. #f.e. thres. Hys. dih+G SVM+G
1232 joint 9024 660 93.1 92.5 90.3 96.9
cone 14850 50 93.0 92.8 94.1 93.9
cup 17010 381 98.2 98.8 97.3 95.0
cut cone 864 144 100.0 100.0 99.3 100.0
cylinder 3540 40 99.8 99.8 99.8 100.0
fandisk 19479 743 99.4 99.6 99.1 98.3
screw 3723 216 97.8 97.9 100.0 98.6
sphere 14700 0 - - - -
torus 7500 0 - - - -
total 90690 2234 97.3 97.3 97.1 97.5
b) Noisy mesh models.
term separately, a data term depending on the edge
dihedral angle alone is proposed:
E
d
(w
e
, θ
e
) =

(2θ
true
− |θ
e
|)
2
if w
e
= 1
θ
2
e
otherwise
(5)
E
d
(w
e
, θ
e
) is an even positive smooth function over
θ
e
. Note that E
d
(0, θ
e
) = E
d
(1, θ
e
) when |θ
e
| = θ
true
(θ
true
∈ [0, π[). The chosen parameters for the pairwise
term have been set as a good trade-off between all
models parameters using grid search. For the SVM
based data term (cf. equation 2), we experimentally
set µ to 0.1 (cf. equation 1), β to 0 (cf. equation 3), λ
to 15 and σ to 10 (cf. equation 4). For dihedral angle
based data term (cf. equation 5), we experimentally
set µ to 2 (cf. equation 1), β to 10
−3
(cf. equation 3),
λ to 15 and σ to 10 (cf. equation 4). To generate the
ROC curves of equation 1 with SVM based data term,
the bias/constant term of the prediction model vary
in [−2, 1] (200 samples at all). To generate the ROC
curves of equation 1 with dihedral angle based data
term (cf. equation 5), θ
true
vary in [0, π] (200 samples
at all).
GRAPP 2011 - International Conference on Computer Graphics Theory and Applications
110

Ground-truth. Our method.
Simple thresholding. Hysteresis thresholding.
Figure 2: Noisy fandisk model: feature line extraction ob-
tained for a false positive rate of 10
−4
: as can be seen our
method is capable of extracting most of feature lines with
almost no false positives.
6.4 Experimental Results and
Discussion
Results are presented in table 2 and in figure 2.
Thresholding. As can be observed, thresholding
techniques based on dihedral angle work quite well
for non-noisy classical CAD models. However, they
are very sensitive to the presence of noise as can be
seen for noisy cone and noisy 1232 joint models in
table 2. Indeed, due to noise the false positive rate
rises quickly and the AUC is decreased.
Globally Consistent Feature Edge Detection. Re-
sults for the global model (cf. equation 1) with data
term set from equation 5 are similar to those obtained
for thresholding techniques (cf. table 2). Results with
data term set from learning (cf. equation 2) are better
than all other methods for noisy mesh models. How-
ever, they are slightly worse than other techniques for
non-noisy mesh models. The robustness against noise
results in less sensitivity too small geometric features.
7 CONCLUSIONS AND FUTURE
WORK
We have introduced the estimator for the angle be-
tween tangent planes which has a high informa-
tive power, in particular for noisy mesh models.
Moreover, we described a globally consistent feature
line extraction technique over 2-manifold triangular
meshes: it globally finds the best solution taking into
account all edge dependencies with the edge local fea-
tures (either based on dihedral angle or learnt). This
globally homogeneous decision makes the classifica-
tion results less sensitive to the presence of noise. As
future work, we will investigate automatic parameter
learning for parameters in equation 1.
ACKNOWLEDGEMENTS
This work has been supported by the French National
Research Agency (ANR) through MDCO program
(project MADRAS No. ANR-07-MDCO-015).
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