voronoi-based extraction of a feature skeleton from noisy ... · pdf filevoronoi-based...

Voronoi-Based Extraction of a Feature Skeleton

from Noisy Triangulated Surfaces

Tilman Wekel and Olaf Hellwich

Computer Vision Group, TU-Berlin{t.wekel,olaf.hellwich}@tu-berlin.de

Abstract. Recent advances in 3D reconstruction allow to quickly ac-quire highly detailed and complex geometry. However, the outcome ofsuch systems is usually unstructured, noisy and redundant. In order toenable further processing such as CAD modeling, physical measurementor rendering, semantic information about shape and topology needs tobe derived from the data. In this paper, a robust approach to the extrac-tion of a feature skeleton is presented. The skeleton reflects the overallstructure of an object. It is given by a set of lines that run along ridgesor valleys and meet at umbilical points. The computed data is not justuseful for building semantic-driven CAD models in reverse engineeringdisciplines but also to identify geometrical features for tasks like objectrecognition, registration, rendering or re-meshing. Based on the meancurvature, a Markov random field is used to robustly classify each vertexeither belonging to convex, concave or flat regions. The boundaries of theregions are described by a set of points that are robustly estimated us-ing linear interpolation. A novel algorithm is used to extract the featureskeleton based on the Voronoi decomposition of the boundary points.The method has been successfully tested on real world examples and thepaper concludes with a detailed evaluation.

1 Introduction

A 3D model can be described in many different ways. Statistically speaking,better semantic knowledge leads to shorter descriptions [1]. Methods can be cat-egorized in two classes. Geometry-based approaches describe the object by acomposition of segments represented by basic geometric model such as spheres,cylinders or parametric surfaces. In feature based approaches, the general struc-ture of an object is captured by a set of lines running along the blending areasin between the segments. These feature lines are arranged in a network (featureskeleton). Locally, feature lines describe prominent characteristics such as val-leys, crests or ridges. The mathematical definition is given by local extrema ofthe principle curvatures. The theory of differential geometry cannot be appliedto triangulated surfaces in a straight forward manner due to their discrete nature[2]. Recently, there has been a lot of attention to this problem [3][4][5]. Manyapproaches are based on a surface patch that is locally fitted to a small environ-ment of a vertex. The patch is explicitly given by a function and the demanded

K.M. Lee et al. (Eds.): ACCV 2012, Part II, LNCS 7725, pp. 108–119, 2013.c© Springer-Verlag Berlin Heidelberg 2013

Extraction of a Feature Skeleton 109

differential properties can be calculated. One of the most significant problems ofthe estimation of feature lines arises from the fact that especially higher orderdifferential properties such as the principal curvature direction are very sensitiveto noise [6]. In this work, a new and robust approach to the extraction of a featureskeleton is presented. The algorithm only depends on the mean curvature andit is well suited for high noise levels. The paper is structured as follows. Closelyrelated approaches are briefly reviewed in the next section. It is shown in Section3, how a Markov random field (MRF) can be used to overcome the problem ofnoise and to estimate a stable classification for each vertex either belonging tosmooth areas or to critical regions. The computation of higher order differentialproperties is not required. In Section 4, a novel Voronoi-based method is intro-duced to build a consistent feature skeleton based on the vertex labeling. Evenif this is not the main focus of this paper, two possible methods to calculatethe Voronoi-diagram on a discrete surface are briefly reviewed in Section 5. Aqualitative and quantitative evaluation is given in Section 6. Finally, the paperconcludes with Section 7.

2 Related Work

The extraction of differential features has gained a lot of attention. Approachesbased on local feature extraction try to tackle the problem by estimating differ-ential quantities in order to identify points or edges on the mesh that are likely tobelong to a feature line. The potential candidates are then connected by a tracingalgorithm. Local approaches mainly differ in the computation of the differentialparameters and in the post-processing of the extracted feature lines. Hildebrandtet al. present a method to extract smooth feature lines [2]. The estimation ofthe differential properties is achieved by using a tensor that is based on locallyaveraged normal vectors. The largest eigenvectors correspond to the principlecurvatures. Laplacian smoothing is applied before tracing in order to improvethe quality of the feature lines. In a post-processing step, the detected featurelines are smoothed directly in 3D space independent of the surface. The resultsare qualitatively evaluated and compared to related approaches. The approachgiven by Ohtake et al. uses an implicit function to estimate curvature parameters[7]. The crossing of a feature line and an edge is accurately estimated by linearinterpolation. A quite similar approach is presented by Yoshizawa et al. A localneighborhood of a vertex is considered in order to improve the tracing and theconstruction of the feature lines [8]. Kim et al. apply tensor voting theory todetect features on triangular meshes [9]. A vote is a weighted covariance matrixwhich describes the correlation of different features in a local environment of avertex. A k-means-based algorithm identifies clusters that correspond to seman-tic classes such as edges or corners. The extracted feature lines correspond to thesegment borders. The work of Cazals et al. gives a theoretic analysis of ridges,umbilical points and their topology [10]. The derived definitions and hypothesesare evolved to perform feature extraction on triangulated surfaces. Some authorsuse Morse theory to analyze the topology. Morse theory studies smooth, real-valued functions over manifolds [11][6][12]. These approaches are very sensitive

110 T. Wekel and O. Hellwich

to noise and the result depends on the applied Morse function. Lavoue et al.present an MRF-based mesh segmentation algorithm driven by Gaussian curva-ture [13]. The MRF is initialized based on a k-means segmentation. Simulatedannealing is used to solve the resulting optimization problem. The result is givenby a labeling, geometric entities such as lines or points are not extracted. Themethod presented by Benhabiles et al. is quite similar. However, the authorsuse a conditional random field (CRF) to learn an edge-function according topre-labeled objects that belong to different semantic classes such as trees, carsor furniture [14]. Similar to the general idea of our approach, Rossl et al. extracta skeleton from a pre-labeled surface [15]. However, morphological operators areused to reduce the detected feature regions to a skeleton.

3 Classification

Consider a polyhedral surface S = {T,E, V }, consisting of triangles T , edgesE and vertices V . Before the feature skeleton is computed, the vertices areclassified either belonging to critical (lc) or flat (lf ) areas. A critical vertex caneither belong to a ridge (lr) or a valley (lv) . A vertex vi is associated witha normal vector ni and a hidden state xi, as it can be seen in the left plotof Figure 1. The labeling problem is stated by: f : x → l, where f describesa mapping, that assigns a label from l ∈ L,L = {lf , lr, lv} to each site x ∈X . The joint probability of a labeling X = X ′ depends on some observationV , which is given by the vertices and their geometric properties. The Markovproperty states, that given a local neighborhood N of a site x, the labeling of xis conditionally independent of all other sites P (x|X\x) = P (x|Ni). The labelingX∗ that maximizes P (X) is the solution to the overall inference problem X∗ =argmax

X∈X

P (X). The Gibbs distribution allows to define the overall probability

density function as: P (x) = Z−1 exp(− 1T U(x)). Z is a normalization constant.

The energy function U depends on the application and is composed of cliquepotentials which describe the probability of local configurations. Two differenttypes of clique potentials are considered:

U(x) =∑

{i}∈V

φ(xi, vi) +∑

{i,i′}∈E

ψ(xi, xi′ ) , (1)

where the first term represents the 1-vertex cliques and the second term the 2-vertex-cliques. The left sum corresponds to the local evidence and the right sumenforces regularization. The 2-vertex-clique potentials can be notated accordingto the Ising-model: ψ(x, x′) = Jδxx′ , where J is the regularization constant andδxx′ is defined as:

δxx′ =

{1, if x = x′

−1, otherwise. (2)

Equation (2) states, that adjacent vertices are likely to have the same class-labels.φ yields a-priori knowledge. The observation vi ∈ V is given by the polyhedral


−2 −1.5 −1 −0.5 0 0.5 1 1.5 2

0

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local field strength

p h

i

flatvalleyridge

Fig. 1. Topology of the MRF (left) and evidence function (right) with respect to sH

mesh. As it is shown in the evaluation, the curvature is a suitable evidence forthe given classification. The following evidence function is used to calculate φ:

φ(x, v) =

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

− tanh(Hs)2, if x = lv, H < 00, if x = lv, H ≥ 0

− tanh(Hs)2, if x = lr, H > 00, if x = lr, H ≤ 0

tanh(Hs)2 + 1, if x = lf .

, (3)

H = (κ1 + κ2) is the mean curvature at vertex v and s is a scaling parameter.The local field strength can be seen in the right plot of Figure 1. The plottedfunction maps the mean curvature to φ(x = lf , v) (green), φ(x = lv, v) (blue)and φ(x = lr, v) (red). There are various algorithms for discrete optimizationproblems, a state-of-the-art loopy belief propagation method turns out to bemost suitable for the given application [16]. The algorithm computes a nor-malized marginal probability vector p(vi) for each vertex. It is composed as:p(vi) = (p(xi = lf), p(xi = lv), p(xi = lr))

T . The result of the classificationstep for the Fandisk model is shown in Figure 2. The coloring is according tothe marginal probability vectors. The images in the lower row demonstrate thenoise sensitivity of the curvature estimation and how the spatial regularizationhelps to overcome this problem. Each vertex is assigned with a vector containingthe marginal probability values of all three classes. The propability vector p(vi)of a vertex vi is assumed to be a sample from a smooth, continuous vector fieldp(v). The boundary curve b(m,n) between the two classes lm,ln is then implicitly

given by the set of all boundary points: {∀v ∈ b(m,n) : p(v)(m)

= p(v)(n)}. Let

e(i,j) be an edge that connects the vertices {vi, vj} of two different class labels.e(i,j) is then referred as border edge. The corresponding boundary curve mustcross that edge and the intersection point qi,j is found by linear interpolation.The curve is approximated by a set of edges that connects all intersection pointsin a correct order. The situation is visualized in (a) of Figure 3.


Fig. 2. Result of the MRF classification with different scaling and regularization pa-rameters. The images in the first row show how the scaling parameter s affects theclassification. The scaling depends on the model as well as on the application. The cur-vature estimation is very sensitive to noise as it can be seen in the lower right image,the classification is based on local inference only (J = 0.0). The regularization of theMRF allows to overcome the problem, as it is shown in the lower row. J ≈ 1.0 seemsto be suitable for all considered models. If the value for J is to high, important detailsof the structure will not be reflected properly, as it can be seen in the lower left image.

4 Construction of the Feature Skeleton

Vertices which belong to ridges and valleys are properly identified by the MRF-based classification. The corresponding feature lines can be seen as the under-lying skeleton, plotted in (b) of Figure 3. In the following, it is shown how thedemanded feature lines can be defined by the Voronoi diagram of the boundarypoints. S is assumed to be 2D manifold and the inner geometry of S is consideredin the following.

4.1 Voronoi-Based Extraction of a Topological Skeleton

The feature skeleton F is defined as a subset of the Voronoi diagram {F ⊂ D}, asit can be seen in (b) of Figure 3. Given a 2D manifold surface S, all intersectionpoints qi,j ∈ S yield the sites of a Voronoi diagram D. The Voronoi diagram

consists of vertices, edges, sites and cells: D = {V , E, S, C}. All elements of theVoronoi diagram are indicated by a tilde. Let su and sv be two sites, then allpoints which are closer to su than to sv can be defined as:

cuv = {p ∈ R2 : δ(p, su) < δ(p, sv)} , (4)

where δ(., .) is the distance function. The corresponding Voronoi cell cu is thenthe set of all points which are closer to cu than to any other cell: cu =

⋂u�=v

cuv .


(a) Boundary curves (b) Voronoi decomposition

Fig. 3. Result of the MRF optimization and Voronoi decomposition. The left plotshows the underlying surface (grey, dashed lines), where faces and vertices are eitherclassified as belonging to class lr (red) or lf (green). The boundary points (bicolored)lie on the class boundaries and represent the sites of the corresponding Voronoi diagram(b). The right plot illustrates the local topology of a Voronoi edge eo and its relationto the underlying surface.

Consider a Voronoi edge eo which is a bisector of two adjacent sites, given by theintersection points su = qi,j and sv = qk,l. A Voronoi edge eo can be associatedwith its corresponding source and target vertex {vs(o), vt(o)} and its right andleft site {su(o), sv(o)}. eo is classified as belonging to the feature skeleton eo ∈ F ,if the following conditions are fulfilled:

1. The Voronoi edge must have a source and a target vertex. Note, that theVoronoi diagram might contain edges with one vertex only. A feature edgemust have a finite length.

2. The source and the target vertex must lie on faces that belong to the sameclass {lz, z �= f}, which is straight forward, since the feature skeleton repre-sents the centric line of ridge or valley regions.

3. If the adjacent sites {su(o), sv(o)} belong to the same boundary curve bc,

the arc length of the curve segment bc(u,v) defined by su(o) and sv(o) on bc

must be sufficiently large, which is stated by: ‖bc(u,v)‖ > dmin.

dmin can be seen as a scaling parameter. The problem boils down to the questionof when does a small bump starts to be a branch, as it can be seen in Figure4. Consider the Voronoi edge (a) and its corresponding boundary points (c).The thick dashed lines indicate the segment of the respective boundary curve(b). dmin defines the maximum length of that segment. The third condition

is fulfilled if the arc length ‖bc(u,v)‖ is above the given threshold dmin (Figure4(a)). If the arc length of the segment (b) is too small, the corresponding Voronoiedge is not included in the feature skeleton, shown in Figure 4(b). The resultingparameter indirectly controls the minimum size of a branch.


(a)

(c) (b)

(c)

(a) Branch

(a) (c)

(b) (c)

(b) Bump

Fig. 4. Identification of a branch. Two types of branches are displayed. Consider aVoronoi edge (a) in both plots. The adjacent boundary points (c) belong to the sameboundary and define a curve segment (b). In the left plot, (b) is sufficiently long andthe considered edge of the Voronoi diagram (a) is classified as belonging to the featureskeleton F . If the curve segment is too short, the edge is not included in F , as it canbe seen in the right plot.

5 Voronoi Diagrams on Surfaces

Assuming a 2D manifold S embedded in R3, the computation of Voronoi dia-

grams is conceptually similar to the 2D version but requires a different measureof distance. In the following, two possible approaches are briefly discussed. Liuet al. construct a Voronoi diagram using the geodesic distance. The approach isbased on the MMP-algorithm which has been developed by Mitchell et al. [17].It computes a function D(p) : S → R that returns the minimum distance fromp to a fixed source point. The resulting path is called a geodesic. It is stated,that a geodesic must be a straight line inside every crossed triangle. The pathcan now be established by unfolding sequences of adjacent faces to a planarstructure. Surazhsky et al. show that a ray of adjacent geodesics sharing thesame face can be efficiently represented by a single tuple of eight parameters[18]. The demanded function is computed by propagating these rays along thetriangulated surface. The Voronoi diagram can now be constructed based onthe geodesic distance functions of all sites. In order to avoid the explicit com-putation of the geodesic distance, an alternative approach is proposed in thecontext of this work. Here, the Voronoi diagram is computed in the parameterspace of the surface. Consider that there exists a (piecewise) linear embeddingof m : S → S, where S ∈ R

2 . m should preserve the topology of the mesh aswell as the shape and angles of the triangles. There are various parametrizationmethods but most of them require a fixed border parametrization, which mapsthe boundary to a convex shape in the parameter space [19]. Even if a solution isthen always guaranteed, this method causes high angular and metric distortion.The conformal mapping presented by Levy et al., however, allows a free form bor-der [20]. Unlike other approaches, it avoids nonlinear-optimization. A conformalmapping-function locally preserves shapes and angles. Areas and distances are


scaled only. More details about this method can be found in the correspondingpaper of B. Levy et. al. The Voronoi diagram is then calculated on S and repro-jected to the original Surface S. If S is not topologically equivalent to a disk,S is decomposed into compact patches S = {S0, ..., Sk}. A standard segmen-tation algorithm based on an Euclidean cost function is used to hierarchicallydecompose the surface until all resulting patches are topologically equivalent toa disk. The construction of the feature lines is done for each patch Sa in R

2. Themethod has been implemented based on the parametrization and 2D Voronoipackages of the CGAL-library [21][22].

6 Evaluation

In the following, the presented method is tested and compared on real worldexamples at different levels of noise. Second, a quantitative evaluation is doneby investigating two complex CAD-models, where ground-truth is available. Ournovel method (V) is compared to two competitive and quite popular algorithmsfor the extraction of crest lines. The first algorithm (C) is part of the CGAL-library, the implementation is based on the work of F. Cazals et al. [10][23].Yoshizawa et al. provide an implementation (Y) that comes with the correspond-ing paper [8]. All algorithms have some parameters which need to be adjustedmanually. Both algorithms, (C) and (Y) allow to neglect crest lines of insufficientlength and sharpness. A smoothing parameter controls the size of the local envi-ronment which is taken into account for the curvature estimation. Our algorithmis mainly parametrized by the scaling and the regularization factor of the MRF(s, J). They are chosen such that the results are as equal as possible in termsof scaling and smoothness. Three real world models are taken from the AIMshape repository: Fandisk (a,b), Oilpump (c,d) and Rockerarm (e,f) [24]. Themodels are acquired by scanning devices and have a simple shape. Edge regionscan be clearly identified. Each model is evaluated at two different noise levels toinvestigate the robustness. The results can be seen in Figure 7. Uniform noisewith zero-mean is applied and the ratio r is given by: r = 10dmaxw

−1, wheredmax is the maximum random displacement and w is the length of the diagonalof the bounding box. The Algorithm (Y) and (C) deliver satisfying results atlow noise levels. Most prominent features are detected and the tracing computessmooth and properly connected lines (a,c,e). However, even if extremely simplemodels are considered, some outliers which are spread randomly on smooth sec-tions seem to be unavoidable (e). Both algorithms produce a set of isolated lineswhich are not connected. This does not allow to construct intersections of morethan two line-segments and produces gaps a priori. At higher noise levels, theestimated feature lines become scattered and the number of outliers increasessignificantly (b,d,f). As expected, (C) is most sensitive to noise since it incorpo-rates higher order differential quantities. The feature skeleton extracted by thepresented algorithm clearly represents the most significant ridges and valleys,as it is shown in (b). The MRF-based classification minimizes the number ofoutliers. The general structure of the skeleton is very invariant to noise although


Fig. 5. Two CAD-models are used for quantitative evaluation (Pump Carter (pc) andStub Axle (sa)). The colored feature lines represent the ground-truth. Only featurelines of sufficient quality are included to simplify the parameter configuration of thetested algorithms.

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Fig. 6. ROC-curves for the synthetic models at different noise levels. The result ofour algorithm (V) is compared to the work of Cazals et al. (C) and Yoshizawa et al.(Y). The more the curve hugs the upper left corner, the better the detection rate.The quality of all three algorithms drops with increasing noise. The second model ismore challenging since it has a higher geometric detail which leads to feature lines withdifferent sharpness levels.

the smoothness of the lines is slightly affected. The interesting aspect of thisapproach is, that the topology as well as the connectivity of the line segments is im-plied by the Voronoi decomposition. There is no need for tracing or an explicit con-struction of branches or intersections. A possible drawback of this method is that


Fig. 7. Evaluation on real world examples at different noise levels


adjacent ridge or valley regions might get merged due to the spatial regulariza-tion of the MRF. Even if a consistent Voronoi-based construction is not affected,it might lead to inaccurate results, as it can be seen in the lower left area of theOilpump model (d). The second experiment quantifies the robustness to noise ofeachmethod. Two CAD-models, taken from the AIM shape repository are consid-ered: Pump Carter (pc) and Stub Axle (sa), shown in Figure 5. The colored edgesrepresent the ground-truth data. The algorithms are not directly applied to theCAD-model. Instead, a surface reconstruction algorithm is used to compute a newand dense polyhedral mesh. This ensures, that the triangulation as well as the res-olution is completely decoupled from the original CAD-models. The comparisonis now interpreted as a vertex classification problem which allows to quantify theperformance as receiver-operator-curves (ROC). Each vertex of the reconstructedmesh is either classified as belonging or not belonging to a feature line. All linesare represented as dense point-clouds and the labeling is realized by an Euclideanpoint-to-point distance. The score of each vertex is given by the negativeminimumdistance to the feature skeleton. Our algorithm outperforms the others at all threenoise levels, as it is shown in Figure 6. It can clearly be seen, that the drop in per-formance at higher noise levels is relatively low.

7 Conclusion

A robust approach to extract the feature skeleton from noisy meshes is pre-sented. The work holds several contributions. The MRF-based classification is apowerful way to overcome the noisy and error prone characteristics of discretecurvature estimation. Second, curve tracing is explicitly avoided since it is ex-tremely sensitive to noise and tends to produce scattered feature lines. Instead,the novel Voronoi-based construction of the feature skeleton implicitly deals withtopological problems such as branches or intersections. Performance as well asreliability will be investigated in future work. The computation of the MRF iscomputationally expensive and the loopy belief propagation might be replacedby more advanced methods. Concerning the reliability, optimal values for thescaling as well as for the regularization parameter should be inferred from thedata.

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