Farthest sampling segmentation of triangulated surfaces
Victoria Hernández-Mederos, Dimas Martínez, Jorge Estrada-Sarlabous, Valia Guerra-Ones
FFarthest sampling segmentation of triangulated surfaces
Victoria Hern´andez-Mederos a , Dimas Mart´ınez-Morera b ,Jorge Estrada-Sarlabous a , Valia Guerra-Ones c a Instituto de Cibern´etica, Matem´atica y F´ısica, ICIMAF, La Habana, Cuba b Departamento de Matem´atica, Universidade Federal do Amazonas, Manaus, Brazil c Technical University of Delft, TU Delft, Delft, The Netherlands
Abstract
In this paper we introduce
Farthest Sampling Segmentation (FSS) , a new method for segmentationof triangulated surfaces, which consists of two fundamental steps: the computation of a submatrix W k of the affinity matrix W and the application of the k-means clustering algorithm to the rowsof W k . The submatrix W k is obtained computing the affinity between all triangles and only a fewspecial triangles: those which are farthest in the defined metric. This is equivalent to select a sampleof columns of W without constructing it completely. The proposed method is computationallycheaper than other segmentation algorithms, since it only calculates few columns of W and it doesnot require the eigendecomposition of W or of any submatrix of W .We prove that the orthogonal projection of W on the space generated by the columns of W k coin-cides with the orthogonal projection of W on the space generated by the k eigenvectors computedby Nystr¨om’s method using the columns of W k as a sample of W . Further, it is shown that forincreasing size k , the proximity relationship among the rows of W k tends to faithfully reflect theproximity among the corresponding rows of W .The FSS method does not depend on parameters that must be tuned by hand and it is very flexible,since it can handle any metric to define the distance between triangles. Numerical experiments withseveral metrics and a large variety of 3D triangular meshes show that the segmentations obtainedcomputing less than the 10% of columns W are as good as those obtained from clustering the rowsof the full matrix W . Keywords: segmentation, 3D triangulations, farthest point, low rank approximation
1. Introduction
Mesh segmentation is an important ingredient of many geometric processing and computer graphicstasks, such as shape matching, parametrization, mesh editing and compression, texture mapping,morphing, multiresolution modeling, animation, etc. It explains why this subject has received alot of attention in recent years. In a review of mesh segmentation techniques, Shamir [Sha08]formulates the segmentation problem as an optimization problem and considers two qualitativelydifferent types of segmentation: the part-type, aiming to partition the surface into volumetric partsand the surface-type, attempting to segment the surface into patches. Segmentation techniquesare also classified in correspondence with general clustering algorithms, such as region growing,hierarchical clustering, iterative clustering, spectral analysis, etc.The most important tasks concerning shape segmentation is how to define a part of the surface.This is done by using various mesh properties or features such as area, size or length, curvature,
Preprint submitted to Elsevier December 2, 2020 a r X i v : . [ c s . G R ] D ec eodesic distances, normal directions, distance to the medial axis and shape diameter. In manysegmentation algorithms [KT03], [dGGV08], [LZ04], [LLSCS05], [PKA03], [ZL05], part analysis iscarried out by using surface based computations. For instance, geodesic and angular distances arecombined in [ZL05] to define a metric that encodes distances between mesh faces, while diffusiondistance is used in [dGGV08] to propose a hierarchical segmentation method for articulated bodies.The previous approaches are based on intrinsic metrics on the surfaces and do not capture explic-itly volumetric information. In [LZSC09] a volumetric part-aware metric is defined, combining thevolume enclosed by the surface with geodesic and angular distances. This metric is successfully ap-plied in various applications including mesh segmentation, shape registration, part-aware samplingand shape retrieval.According to the underlying technique, many segmentation algorithms [LZ04], [ZL05], [LJZ06],[LZ07], [dGGV08], [KLT05] belong to the class of the so called spectral methods. In these al-gorithms, an affinity or Laplacian matrix is constructed by using intrinsic metrics. The originalsurface is projected into low dimensional spaces, which are derived from the eigenvectors of theaffinity or Laplacian matrix. As a consequence of the Polarization Theorem, higher-quality cutboundaries may be obtained from these embeddings. For details about the spectral approachfor mesh processing and analysis, including mesh compression, correspondence, parameterization,segmentation, surface reconstruction, and remeshing, see the excellent survey [ZVKD10]. The main contribution of this paper is a new algorithm for segmentation of triangulated surfacesbased on the computation of few columns of the affinity matrix W . Given a distance betweenneighboring faces, the affinity among all faces and a sample of few distinguished faces is computed.The segmentation is performed applying a classical clustering algorithm to the rows of the ma-trix W k composed by the sample of the few k columns of W . The new method, called FarthestSampling Segmentation (FSS) does not require to compute the spectrum of W or of any of itssubmatrices. Hence, it is computationally cheaper than the segmentation algorithms based oneigendecompositions.From the theoretical point of view our first result is the proof that given a sample W k of thecolumns of W , the orthogonal projection of W in the space generated by the columns of W k is thesame as the orthogonal projection of W in the space generated by the approximated eigenvectorsof W , obtained using Nystr¨om’s method for the same sample. This theoretical result clarifies thepoint of contact between our method and the spectral approach and explains the success of thenew method FSS . Moreover, it is shown that if the columns of W k correspond to the k farthesttriangles in the selected metric, then for increasing size k , pairs of faces which are close in theselected metric project in pairs of rows of order n × k matrix W k that are close as points in R k andalso pairs of faces which are far-away in the selected metric project in pairs of rows of W k that arefar-away as points in R k .A wide experimentation that illustrates the quantitative and qualitative performance of the FSS method is also included. Through these experiments it becomes apparent the robustness of themethod for small samples of W . Furthermore, it is proposed an estimate of lower bound for thesize k of the sample that furnishes a good approximation to W . In section 2 we introduce the basic concepts that allow to define the similarity between any twotriangles of the mesh. Two low dimensional embeddings of the affinity matrix are described in2ection 3: the spectral and the statistical leverage. The main theoretical result of the paper isincluded in this section. The embedding based on the computation of the columns of the affinitymatrix that corresponds to the farthest triangles is introduced in section 4, where the coherenceand stability of
FSS method is shown and the associated algorithm is explained. Section 5 isdevoted to the numerical experiments, which show the quantitative and qualitative performance of
FSS . Moreover, this section also includes an experiment to prove that the immersion based on thefarthest triangles provides a good approximation of W . A comparison of FSS with the spectralmethod is finally included . The last section concludes the paper. We use capital letters to denotematrices and the same lower case letter to denote its elements. For example the element i, j of thematrix A is denoted by a ij . Moreover, A i, · and A · ,j represents the i -th row and the j -th columnof matrix A respectively, while A + denotes the Moore-Penrose inverse of matrix A . All meshsegmentations shown in this work are computed without including any procedure to improve thesmoothness of the boundaries of the segments or their concavity, such as proposed in [SSCO08],[WTKL14].
2. Distance and affinity matrices
Denoting by T the triangulation composed by a set F of faces f i , i = 1 , ..., n , the segmentationproblem consists in defining a partition of F . In most of the segmentation algorithms, an importantstep to group the elements of F consists in introducing pairwise face distances and constructing anaffinity matrix by using them. In the literature, several face distances have been considered, see[KT03, GSCO07, SSCO08, LZSC09]. For instance, in [KT03] the distance between two adjacentfaces is defined as a convex combination of their geodesic and angular distances. Other metrics havebeen specially designed to capture parts of the volume enclosed by the surface, such as the part-aware distance [LZSC09] and the shape-diameter function ( SDF ) [SSCO08]. The part-aware metricin [LZSC09] happens to be expensive, since its computation requires to perform two samplings ofthe triangular mesh by using ray-shooting. On the other hand, as remarked in [LZSC09], the
SDF function does not capture well the volumetric context.
Denote by f i and f j two triangles of T sharing an edge. Assume that we have already defineda distance d ij between faces f i and f j . For instance d ij could be the angular distance defined as η (1 − (cid:104) n i , n j (cid:105) ), where (cid:104) n i , n j (cid:105) denotes the scalar product between the normalized normal vectors n i , n j to the triangles f i and f j respectively and η is a weight introduced to reinforce the concavityof the angles. Another distance very common in the literature is the geodesic distance , that in thecase of neighboring triangles is defined as the length of the shortest path between their barycenters b i and b j , see the details on these two distances in [KT03]. Our third test distance is introducedin [LZSC09] and is based on a scalar function defined on the triangulation, the so called SDF -function (see [SSCO08]). The sdf distance between any two adjacent faces f i and f j of T is definedas | SDF ( b i ) − SDF ( b j ) | . Each of these test distances captures different features of the triangulation.The distance d ij between any pair of faces f i and f j is computed using the weighted dual graph G d defined from the selected distance. The i -th knot of the graph G d represents the triangle f i in T for i = 1 , ..., n , and there is an edge between the i -th and the j -th nodes of the graph G d if thefaces f i and f j share an edge on the triangulation T . The weight of the edge joining the i -th andthe j -th nodes in G d is d ij if d ij > ε , with 0 < ε very small. Thedistance d ij between faces f i and f j , which are not necessarily neighboring faces, is defined as the ength of the shortest path between the i -th and the j -th knots in the graph G d . This length maybe computed using Dijkstra algorithm. Observe that the distance d ij satisfies the axioms of metric.We denote by D = ( d ij ) , i, j = 1 , ..., n the matrix of the distances between each pair of faces of thetriangulation. Given a suitable metric that allows to compute the pair-wise distance between faces of the trian-gulation, the affinity matrix W encodes the probability of each pair of faces of being part of thesame cluster and can be considered as the adjacency matrix of the weighted graph G d previouslyintroduced.Assume that the distance d ij between any pair of faces f i and f j of the triangulated surface hasbeen already computed. Then the affinity w ij between faces f i and f j which are closer should belarge. In the literature it is customary to use a Gaussian kernel to define w ij , i, j = 1 , ..., n as w ij = e − d ij / (2 σ ) (1)where σ = n (cid:80) i (cid:80) j d ij . Observe that 0 < w ij ≤ w ii = 1 for all i = 1 , ..., n . Moreover, W = ( w ij ) , i, j = 1 , ..., n is a symmetric matrix. Denote by M the diagonal matrix M = diag ( m ii ),where m ii = (cid:80) nj =1 w ij .Many papers in the literature deal with normalized versions of affinity matrix, which are calledLaplacians in the more general context of clustering of data for exploratory analysis, see [vL06].For instance, in [SM97] the (nonsymmetric) affinity matrix M − W is used in spectral image seg-mentation, while the symmetric normalized affinity matrix Q = M − / W M − / is used in [NgW02]for data clustering and in [LZ04] to segment triangular meshes. In applications, the affinity matrix W of order n is huge, therefore segmentation methods requiring the computation of all matrixentries are very expensive. To overcome this problem, the segmentation algorithm proposed in thispaper computes only few columns of matrix W .
3. Low dimensional embeddings for clustering
In geometric processing community, low dimensional embeddings are frequently used to transformthe input data from its original domain to another domain. The main purpose of these embeddingsis to reduce the dimensionality of the problem, preserving the information of the original data insuch a way that the solution of the new problem is cheaper and easier. The segmentation problemcan also be considered as a clustering problem, which is frequently solved applying k-means method,[Lloyd82]. In this context, dimensionality reduction for k-means is strongly connected with low rankapproximation of the matrix containing the data to be clustered [BZMD15]. In our problem, thematrix containing the information about “data points” is the affinity matrix W . Each row of W represents the affinity between a triangular face and the rest of faces. Hence, a valid strategy tosolve the segmentation problem consists in computing a low rank approximation of the affinitymatrix W and clustering its rows. The most popular low dimensional embeddings in the literature are the spectral ones, which areconstructed from a set of eigenvectors of a properly defined linear operator. They have beensuccessfully applied in mesh segmentation [LZ04],[LJZ06], [LZ07], [dGGV08] and also in other4eometric processing applications, such as shape correspondence [JZK07] and retrieval [EK03] andmesh parametrization [Got03], [MTAD08].From the theoretical point of view, spectral embeddings are supported by a classical linear algebraresult, the Eckart-Young theorem [EY36]. It establishes that the best rank k approximation in theFrobenius norm of a real, symmetric and positive semi-definite matrix W of dimension n is thematrix E k = (cid:101) U k ( (cid:101) U k ) t (2)where (cid:101) U k is the matrix with columns √ λ u , √ λ u , ..., √ λ k u k and u , u , ..., u k are the eigenvectorsof W corresponding to its largest eigenvalues λ ≥ λ ≥ ... ≥ λ k . It means that the followingequality holds for the Frobenius norm of the error W − E k (cid:107) W − E k (cid:107) F = min X ∈ R n × k , rank ( X ) ≤ k (cid:107) W − XX t (cid:107) F (3)If we denote by U k the matrix with columns u , u , ..., u k , then (cid:101) U k = (Λ k ) U k , where Λ k is theorder k diagonal matrix with diagonal elements λ , λ , ..., λ k . Moreover, it not difficult to provethat ( U k ) + = ( U k ) t . Hence, the orthogonal projection U k ( U k ) + W of W on the space generatedby columns of U k satisfies U k ( U k ) + W = U k ( U k ) t W = U k (Λ k )( U k ) t = ( (cid:101) U k (Λ k ) − )(Λ k )( (cid:101) U k (Λ k ) − ) t = (cid:101) U k ( (cid:101) U k ) t = E k The equality E k = U k ( U k ) + W means that the best rank k approximation of W is the projectionof W on the space generated by the eigenvectors of W corresponding to its largest eigenvalues.Spectral clustering algorithms also rely on the polarization theorem [BH03] which suggests that asthe dimensionality of the spectral embeddings decreases the clusters in the data are better defined.In practical applications, it is necessary to choose a value of k representing a good compromisebetween these two apparently conflicting results. This value should be small enough to obtain agood polarization of the embedding data, but at the same time large enough to reduce the distortionof the relationships among the data due to the embedding.Spectral methods of segmentation are in general expensive, since they require the computationof eigenvalues and eigenvectors of the so called Laplacian matrix. In some cases [LZ04],[LJZ06],the Laplacian matrix is obtained introducing a normalization of the affinity matrix. In other cases[RBGPS09], it arises from a discretization of the Laplace-Beltrami operator. In geometry processingcontext the Laplacian matrix used in segmentation is a dense and usually very large matrix. Toface this problem, [LJZ06] uses Nystr¨om’s method, since it only requires a small number of sampledrows of the affinity matrix and the solution of a small scale eigenvalue problem. More precisely, theset of faces of F is partitioned in two: a set X of the faces of the “sample” of size k << n and itscomplement Y of size n − k . Let be p X the set of k indices of the faces contained in the sample X , p Y the set of n − k indices of the faces contained in Y and p = ( p X , p Y ) the vector representationof the permutation matrix P , then the permuted affinity matrix W P := P W P t has the followingstructure W P := P W P t = (cid:20) A BB t C (cid:21) (4)where A is the order k affinity matrix of the elements in X and B is the order k × n matrix of thecross-affinities between elements in X and Y . The eigenvectors of W corresponding to the k largest5igenvalues, i.e. the columns of U k , may be approximated [FBCM04], [LJZ06] by the columns ofthe n × k matrix P t N k , with N k = (cid:20) U A B t U A Λ − A (cid:21) (5)where A = U A Λ A U tA is the spectral decomposition of A . The orthogonal projection F k of W onthe space generated by the columns of matrix P t N k provided by Nystr¨om’s method is given by F k = ( P t N k ) ( P t N k ) + W (6)The accuracy of the eigenvectors computed by using the Nystr¨om’s method strongly depends onthe selection of the k columns of W corresponding to sample X . Therefore, different schemes havebeen considered in the literature, for instance random sampling, uniform sampling, max-min far-thest point sampling and greedy sampling [FBCM04], [KMA12], [Mah11],[ST04],[LJZ06].Nystr¨om’s method has associated an approximation to W P given by (cid:102) W P := N k Λ A ( N k ) t (7)It is straightforward [FBCM04] to check that it holds (cid:102) W P = (cid:20) A BB t B t A − B (cid:21) . (8)Since (cid:102) W P ≈ W P , after 4, 6 and 7 we have that F k = P t N k ( N k ) + P W ( P t P ) = P t N k ( N k ) + W P P ≈ P t N k ( N k ) + (cid:102) W P P = P t N k ( N k ) + ( N k Λ A ( N k ) t ) P = P t N k Λ A ( N k ) t P = P t (cid:102) W P P ≈ P t W P P = P t ( P W P t ) P = W Hence, F k may be considered as an approximation to the affinity matrix W with similar quality asthe Nystr¨om approximation P t (cid:102) W P P to W . The accuracy of the approximation F k also dependson the selection of the k columns of W corresponding to sample X . Proposition 1.
Let W be a symmetric order n matrix and p = ( p X , p Y ) a permutation vector ofindices 1,2,...,n, where p X has size k . Let be P the order n permutation matrix represented byvector p and W k the n × k matrix whose columns are the columns of W with indices in p X . Then,it holds, ( W k ) ( W k ) + = ( P t N k ) ( P t N k ) + (9) where N k is given by (5). Proof .From A = U A Λ A U tA it follows U A = AU A Λ − A . Hence, N k = (cid:20) AU A Λ − A B t U A Λ − A (cid:21) = (cid:20) AB t (cid:21) U A Λ − A (10)6ut (cid:20) AB t (cid:21) = P W k , thus it holds N k = P W k U A Λ − A and we get( P t N k )( P t N k ) + = ( W k U A Λ − A )( W k U A Λ − A ) + = ( W k )( W k ) + . (cid:4) Remarks
From the previous result it holds that: • The orthogonal projection H k of W on the space generated by the columns of W k given by H k = W k ( W k ) + W (11)coincides with the orthogonal projection F k of W on the space generated by the columns ofmatrix P t N k , which is provided by Nystr¨om’s method. • Since H k = F k , the accuracy of the approximation H k to W is similar to Nystr¨om approxi-mation P t (cid:102) W P P to W . It also depends on the selection of the k columns of W correspondingto sample X . • If we associate the i th-face of T with the i th-row of W , then clustering the rows of W maybe replaced by clustering either the rows of W k or the rows of P t N k . Mahoney proposes in [Mah11] a method to select a “sample” of the columns of a matrix W ofdimension n in such a way that the space generated by the selected columns provides a goodapproximation of W . Given k , with k << n , the method assigns to the j -th column of W a“leverage” or “importance score” π j that measures the influence of that column in the best rank k approximation of W .The use of the leverage scores for column subset selection dates back to 1972, [Jol72]. However,the introduction of the randomized approach has given an essential theoretical support, [HIW15],to the leverage scores in the role of revealing the important information hidden in the underlyingmatrix structure.More precisely, if v , v , ..., v k are the right singular vectors of W corresponding to the largestsingular values, the leverage π j is defined as π j = 1 k k (cid:88) i =1 ( v ji ) (12)where v ji denotes the j -th component of v i . The normalization factor k is introduced in (12)to consider π j as a probability associated to the j -th column of W . By using that score as animportance sampling probability distribution, the algorithm in [Mah11] constructs a n × m matrix C , composed by m ≥ k columns of W . With high probability, the selected columns are those thatexert a large influence on the best rank k approximation of W . In our experiments in section 5, weuse a slight modification of Mahoney’s algorithm to obtain a matrix C , that we denote by C k , thatis composed exactly by k columns. More precisely, if we arrange in decreasing order the leverages π j ≥ π j ≥ ... ≥ π j n then, the i − th column of matrix C k is the column j i of W , for i = 1 , ..., k .The orthogonal projection of W on the space generated by the columns of C k is given by, G k = C k ( C k ) + W (13)7 . Farthest sampling mesh segmentation method ( FSS ) Computing distances from every node to a subset of nodes of a graph (landmarks or referenceobjects) is a well known method to efficiently provide estimates of the actual distance. In thiscontext, this distance information is also referred to as an embedding. Landmarks have been usedfor graph measurements in many applications, such as roundtrip propagation, transmission delayor social search in networks, but the optimal selection of the location of landmarks has not beenexhaustively studied [PBCG09]. Kleinberg et al. [KSW04] address the problem of approximatingnetwork distances in real networks via embeddings using a small set of landmarks and obtain the-oretical bounds for choosing landmarks randomly. More recently, in [PBCG09] is shown that inpractice, simple intuitive strategies outperform the random strategy.The previous ideas and the result and discussion at the end of section 3.1 inspired us to propose amesh segmentation method based on the computation of a small sample of columns of the affinitymatrix W . It is well known that the quality of the approximation P t (cid:102) W P P to W given by Nystr¨om’smethod (see (8)), strongly depends on the selection of the sample. Consequently, our segmentationmethod FSS consists in two steps: first we use the farthest point heuristic to select which columnsof W are computed and then, the clustering method is applied to the rows of this submatrix of W .The rationality behind these steps is the following. A representative subset of the columns of W must have maximal rank. Since the entries of the j -th column of W are the affinity values betweenall faces of T and the j -th face, the sample should not include columns corresponding to faceswhich are very close between them. In this sense, the farthest point heuristic is a good strategyto avoid redundancy in the sample. On the other hand, if two faces f i and f j of T are close inthe selected metric d , i.e., d i,j is small, then the corresponding row vectors D i, · and D j, · of thewhole distance matrix D are approximately equal, since according to the triangular inequality thedifference between the r -th components of D i, · and D j, · , is bounded by d i,j for r = 1 , ..., n . Hence,due the continuity of the Gaussian kernel, the row vectors W i, · and W j, · of the affinity matrix W are also approximately equal with a difference greater or equal than the difference between the i -thand j -th rows of W k . In other words, the proximity among rows of W and consequently betweenthe faces of T , is well reflected by the proximity of the same rows of W k . Our algorithm
FSS is deterministic and greedy in the sense that at each iterative step, it makesa decision about which column to add according to a rule that depends on the already selectedcolumns. As we already mentioned, the sample of W is derived from a sample of columns of thedistance matrix D . To obtain a good approximation of D it is enough to select a set X ⊂ F ofdistinguished faces that can be considered as landmarks, in the sense that the distance betweenany pair of faces f i and f j can be approximated in terms of the distance of f i (respectively f j ) tothe landmark faces.The method iteratively computes the columns of a matrix X , which contains a sample of k columnsof the distance matrix D . More precisely, in the first step we choose randomly a value j with1 ≤ j ≤ n and define the first column of matrix X as the vector built up with the distances of allfaces to the j -th face, X i, = d i,j , i = 1 , ..., n . Then, we search the index j of the face farthestto the j -th face and assign to the second column of X the vector of the distances of all faces tothe face j . In general, in the step l, k ≥ l ≥
2, we have the l − j , j , ..., j l − previously8elected and a matrix X of order n × ( l −
1) with columns D · ,j , D · ,j , ..., D · ,j l − , which contains thedistances of all faces f i , i = 1 , ..., n to the faces f j , f j , ..., f j l − . In this step we look for the index j l of the face that maximizes the minimal distance to the faces f j , f j , ..., f j l − , j l = arg { max ≤ i ≤ n { min ≤ r ≤ l − x i,r }} (14)where x i,r = d i,j r , i = 1 , ..., n, r = 1 , ..., l − X in the position ( i, r ), i.e, thedistance between faces f i and f j r . Once we have j l , we compute the l -th column of X as the vectorof distances of all faces to the j l -th face.The i -th row of X are the coordinates of a point in R k , which could be considered as a k dimensionalembedding of the point in R n given by the i -th row of D (which represents the distances of all facesto the face f i ). Given k , by using the matrix X we compute a matrix W k = ( w kij ) of order n × k ,which is composed by a sample of columns of the affinity matrix W , w kij = e − x ij / (2 σ k ) , i = 1 , ..., n, j = 1 , ..., k (15)where σ k = 1 nk n (cid:88) i =1 k (cid:88) j =1 x i,j (16)Finally, it remains to explain how we compute the size k of the sample. In this sense, several optionsare possible. The simplest one is to define a priori the value of k , for instance as the integer part ofprescribed percent of the total n of faces. In this case, the sampling algorithm may be summarizedas follows. Procedure Sampling input : triangulation T , number of clusters n c , size of sample k . Choose randomly an index j with 1 ≤ j ≤ n . for l = 1 to k do Compute the distance of all n faces to the face j l . Assign the distances to the column l ofmatrix X . Compute j l +1 by using (14). end for output : sampling distance matrix X .Another option for computing the size k of the sample is the following. A value β l ≥ , l ≥ j l in (14) is introduced defining, β l := max ≤ i ≤ n { min ≤ r ≤ l x i,r } (17)Recall that for all l ≥ β l ≥
0. Furthermore, from the definition (17) it is clear that the sequence { β l , ≤ l ≤ n } is monotonic decreasing with β n = 0. In fact, while new faces are included into thesample, the distance of the new face that is simultaneously farthest away to all actual members ofthe sample decreases. If the sample contains all faces, i.e. if l = n , then matrix X is a permutationof columns of the distance matrix D and for all i = 1 , ..., n it holds min ≤ r ≤ n x i,r = 0, thus we get β n = 0. Hence, the value of β k may be also interpreted as a measure of the error introduced when9he “original” data in a n -dimensional space are substituted by their k -dimensional embedding.Given an upper bound (cid:15) >
0, the size k of the sample may be computed as k = min ≤ l ≤ n { l such that β l β < (cid:15) } (18)For 1 > (cid:15) >
0, the value of k computed by using (18), depends on (cid:15) and it is usually much smallerthan n . A pseudocode of the previous Sampling algorithm is included below. Procedure Sampling input : triangulation T , number of clusters n c , (cid:15) > Choose randomly an index j with 1 ≤ j ≤ n , assign l ← δ ← (cid:15) . while δ > (cid:15) and l < n do Compute the distance of all n faces to the face j l . Assign the distances to the column l ofmatrix X . Compute β l and j l +1 by using (17) and (14) respectively. Assign δ ← β l β Assign l ← l + 1. end while output : sampling distance matrix X . Now we are ready to explain why
FSS is coherent in the sense that clustering the rows of W k happens to be consistent with clustering the rows of the full matrix W and finally segmenting thetriangulated surface. That is very important, since it is well known that in general points far-awaymay project in very close points. This is not the case in the FSS embbeding and consequently, noartifacts appear in the segmentation process.From now on we call farthest point (FP) sample of size k to the set of indices X F k := { j , j , ..., j k } such that j is randomly chosen and for 2 ≤ l ≤ k , j l is the index of the face maximizing the distanceto faces j , j , ..., j l − . The FP sample X F k is associated to the n × k matrix X , submatrix of D ,and the n × k matrix W k , whose columns are (approximately) the columns j , j , ..., j k of the fullmatrix W . Lemma 1.
Given a triangulation T and a selected metric d , for fixed initial face index j , let X F k be the FP sample of size k . It exists k ∗ such that, if two faces of T are far-away (very close,respectively) in the metric d , then for all k > k ∗ , the corresponding rows of matrix W k are alsofar-away (very close, respectively) as points in R k . Proof .First we prove that for any sample X , it holds that pairs of faces which are close in the selectedmetric d project in pairs of rows of order n × k matrix W k , associated to X , that are close as pointsin R k . Indeed, if two faces f i and f j of T are close in the selected metric d , i.e., if d i,j is small, thenthe corresponding row vectors D i,. and D j,. of the full distance matrix D are approximately equal,since according to the triangular inequality the difference between the r -th components of D i,. and D j,. , is bounded above by d i,j , i.e., | d i,r − d j,r | ≤ d i,j for r = 1 , ..., n. X ⊂ { , ..., n } holds the same upper bound for the corresponding rowvectors X i,. and X j,. of the order n × k matrix X . Hence, pairs of faces which are close in theselected metric d project in pairs of rows of order n × k matrix W k that are close as points in R k ,independently of the choice of sample X .On the other hand, assume that faces f i and f j of T are far-away in the selected metric d , i.e. d i,j is large. Then for any ε with d i,j (cid:29) ε >
0, there exits k ∗ such that β k < ε for all k > k ∗ . Moreover,it is clear from the definition of β k that given face f i , we can find an index i ∗ ∈ X F k such that d i,i ∗ ≤ d j,i ∗ ≤ β k for 1 ≤ j ≤ n . Hence, d j,i ∗ + β k ≥ d j,i ∗ + d i,i ∗ ≥ d j,i From the previous inequalities we obtain that for r ∈ X F k , the maximum difference between the r -th components of D i,. and D j,. , is bounded below bymax r ∈X F k { | d j,r − d i,r | } ≥ d j,i ∗ − d i,i ∗ ≥ d j,i − β k − d i,i ∗ ≥ d j,i − β k > d j,i − ε Thus, for k > k ∗ , rows i and j of matrix W k are far-away, i.e., pairs of faces which are far-away inthe selected metric d project in pairs of rows of matrix W k associated to FP sample X F k , that arefar-away as points in R k . (cid:4) In the step 3 of
Sampling Algorithm the initial face j is chosen at random. Now we show thatchoosing at random different indices of the initial triangle, the differences between the i -th and l -th rows of the corresponding affinity matrices associated to the FP samples tend to be equal forincreasing size k . Thus, the segmentation result is stable with respect to the selection of the initialface j . Lemma 2.
Given a triangulation T and a selected metric d , let be X F k and X k F two FP samples ofsize k , with different initial faces j and j and associated affinity matrices W k and W k , respectively.Then for increasing size k it holds (cid:107) W kl,. − W ki,. (cid:107) − (cid:107) W kl,. − W ki,. (cid:107) → for any pair of indices l, i ∈ { , , ..., n } . Proof .Denote by β k and β k the β values (17) corresponding to X F k = { j , ..., j k } and X k F = { j , ..., j k } ,respectively. Set (cid:101) β k = max { β k , β k } . Given ε >
0, if k is sufficiently large, we may assume that (cid:101) β k < ε . Then, for 1 ≤ r ≤ k it holds d i,j r ≤ (cid:101) β k and d l,j r ≤ (cid:101) β k with 1 ≤ l, i ≤ n . Hence, | d i,j r − d i,j r | ≤ (cid:101) β k and | d l,j r − d l,j r | ≤ (cid:101) β k Thus, from the previous inequalities it follows for 1 ≤ l, i ≤ n and 1 ≤ r ≤ k , | | d i,j r − d l,j r | − | d i,j r − d l,j r | | ≤ | d i,j r − d i,j r − d l,j r + d l,j r | ≤ | d i,j r − d i,j r | + | d l,j r − d l,j r | ≤ (cid:101) β k < ε Hence, even choosing at random different indices of the initial triangle, for increasing size k of thecorresponding FP samples, the difference between the l -th and i -th rows of affinity matrix W k andthe differences between the l -th and i -th rows of affinity matrix W k tend to be equal. (cid:4)
11n section 5.4 we show that choosing fixed initial triangle j , most of the segmentation resultsobtained with FP samples of relatively small size k are very close to the segmentation obtainedfrom the full affinity matrix. As previously pointed out, the
FSS method firstly computes a small sample W k of columns of W .Based on the identification of the rows of W with the faces of T , the segmentation of the mesh isobtained clustering the rows of W k . Below we include a pseudocode of the algorithm FSS , whichreceives as input the triangulation T and the number n c of desired clusters. The number k ofcolumns of the affinity matrix to be computed depends on the sampling procedure. In any case,it is assumed that k << n . The algorithm FSS calls to the procedure
Sampling , which returnsthe matrix X . The l -th column of X is the vector of distances of all faces to the j l -th face in theselected sample. Procedure FSS input : triangulation T ,number of clusters n c Call to the procedure
Sampling to compute the sampling distance matrix X . Assign k as the number of columns of X . Compute j l +1 by using (14). Compute the normalization factor σ k given by (16). for i = 1 to n do for j = 1 to k do w kij = e − x ij / (2 σ k ) end for Compute normW i := (cid:107) W ki, · (cid:107) , where W k = ( w kij ) , i = 1 , ..., n, j = 1 , ..., k for j = 1 to k do w kij = w kij /normW i end for end for Apply k-means to the n points given by rows of W k to obtain n c clusters. Construct the segmentation vector s = ( s , ..., s n ), where s i with 1 ≤ s i ≤ n c , is the index ofthe cluster assigned to i -th face of T , which is defined as the cluster assigned by k-means to i -th row of W k . output : Segmentation vector s = ( s , ..., s n ).Observe that algorithm FSS normalizes the rows of W k (as suggested in [LZ04]), hence theserows may be interpreted as points on the ( k − S k − . The segmentation iscarried out applying the classic k-means clustering algorithm to these points. Compared with otheralgorithms reported in the literature, the new algorithm has several advantages. First, like [LJZ06]only a submatrix W k of W has to be computed. On the other hand, unlike as in [LZ04] and[LJZ06], algorithm FSS does not require to compute the spectrum of W or the spectrum of anysubmatrix of W .The construction of matrix W k requires the computation of( n −
1) + ( n −
2) + ... + ( n − k ) = 1 / k (2 n − k −
1) = O ( kn )12istances between faces. In comparison, obtaining the whole affinity matrix W is much moreexpensive and would require the computation of O ( n ) distances between faces.The total computational cost C t of algorithm FSS is the sum of the cost C k of computing theapproximation W k of the affinity matrix W plus the cost C s of clustering the rows of W k in n c clusters. If m is the cost of computing the distance d between two faces of a given triangulationwith n faces (this cost strongly depends of the underlying metric, for instance if the metric is thegeodesic or the angular metric, then m = O ( n log( n )) ), then C k = O ( knm ). Since k , n c and thenumber n i of Lloyd [Lloyd82] iterations are bounded, it holds that C s = O ( n i n c kn ). Thus, if only k << n columns of the affinity matrix W are computed, then the total cost C t is dominated by C k ,i.e., C t = O ( nm ). In the numerical experiments of the next section we show that the immersionbased on the farthest triangle provides a good approximation of W and therefore it is useful toperform the segmentation.
5. Numerical experiments
To prove the performance of the mesh segmentation algorithm proposed in this paper we wrotetwo main codes. The first code is the basis for the experiment developed in section 5.1, whichillustrates the advantages and limitations of the low dimensional embeddings previously considered.The second code is an implementation of the algorithm
FSS whose results are reported in sections5.2 and 5.3.We recall that the mesh segmentations shown in this section are computed without including anyprocedure to improve the quality (smoothness of the boundaries of the segments or their concavity),such as proposed in [SSCO08], [WTKL14].
As we previously mention a valid strategy to solve the segmentation problem is based on computinga low rank approximation of the affinity matrix W . In the following experiment we compare theapproximation power of the low dimensional embeddings described in sections 3 and 4. Givena mesh with n triangles we compute the complete affinity matrix W of order n given by (1).Furthermore, the projection A k of W on different spaces for k = 1 , ..., n is also computed. Moreprecisely, given a value of k , four projections are computed, each one obtained when A k is thematrix E k given by (2), the matrix F k given by (6), the matrix G k given by (13) and the matrix H k given by (11).For each approximation A k , the absolute error in Frobenius norm, error abs = (cid:107) W − A k (cid:107) F (19)is computed and compared to the error (3) of the best approximation E k .Figures 1 and 2 show the results obtained for two 3D triangulation representing an octopus and ahand ( models 125.off and 200.off of the Princeton Segmentation Benchmark [CGF09]). In theseexamples, the distance matrix D was computed using the geodesic metric to measure the distancebetween triangles. In Figure 1 we plot the absolute error (19) as function of k . The red curvecorresponds to the error (19) obtained when A k is the matrix E k of the best rank k approximationof W . Similarly, magenta and blue curves are computed with A k = G k , H k respectively. Since H k = F k (see Proposition 1) the curve corresponding to Nystr¨om projection agrees with the bluecurve corresponding to the embedding proposed in this paper. Observe that this curve is the closest13 igure 1: Absolute errors (19) computed for increasing values of k using different approximations A k of the affinitymatrix W : best (red), leverage (magenta), ours and Nystr¨om (blue). From left to right: errors curves for the octopusmodel with 2682 faces, zoom of a section of the first image, error curves for the hand model with 3000 faces, zoomof a section of the third image. to the curve obtained for the embbeding corresponding to the best approximation of the affinitymatrix.Figure 2 shows curves log(1 + γ k ) and log(1 + β k ), where γ k is the k -th singular value of W (indescending order) and β k is given by (17) (with l = k ). These curves correspond to the octopusand the hand models and all decrease very fast. Figure 2: The curves log(1 + γ k ) and log(1 + β k ), both plotted as functions of k . From left to right: the curvesfor the octopus model with 2682 faces and k = 1 , ...,
500 and the curves for the hand model with 3000 faces and k = 1 , ..., In general, in all our experiments with several triangulations we observe that:1. The absolute error curves show that the matrices H k and F k (recall that H k = F k ) providesthe approximation to W closest to the optimal E k for k ≤ n/
2. In practice, we are interested ina good rank k approximation of W with k << n . Hence, the embedding corresponding to thefarthest point sampling provides the better approximation with the lower computational cost.It explains experimentally why the mesh segmentation algorithm FSS compares favorably toanother methods reported in the literature, which are based on the spectrum of W and happento be more expensive.2. The method proposed in this paper cannot be considered as a random method, since exceptthe first column of the sample, the rest of the columns are selected deterministically. However,one may think of 1 − β l /β , l > conditional probability of selecting the j l -th columnof W given that the columns j , ..., j l − have been previously selected. In this context, ourfarthest point sampling scheme may be considered as an algorithm to select the k columnswith highest conditional probability (see more details in section 5.4). In this section we show the performance of our mesh segmentation algorithm
FSS with several 3Dtriangulation models. In the experiments reported in this section and in the next one, the kmeans + [AV07] algorithm is applied to the normalized rows of the rectangular matrix W k , composed bythe k selected columns of W . These rows are considered as points in S k − . To measure the distancebetween two vectors we use the cosine distance , which means that the distance is defined as oneminus the cosine of the included angle between them. Several replicates of kmeans ++ algorithmare applied and for each replicate the seeds of the n c clusters are selected randomly. In general, thealgorithm FSS works very fast since in all segmentations k is at most 10% of the total number offaces. Our goal here is just to evaluate visually the quality of the segmentations produced by thealgorithm.The first example that we have considered is the model of a cube defined by a triangulation with10 800 faces. This model is ideal to check how the algorithm works when the distance betweentriangles is measured in terms of the angular distance . To segment the model we computed only1% of the columns of the affinity matrix W . Figure 3 left shows that the results are excellent,since the faces of the cube correspond exactly with the 6 clusters produced by the automaticsegmentation. In the second example, we use the geodesic distance between triangles to define theaffinity matrix W of the eight model. In Figure 3 center we show the segmentation in 2 clustersof the model, obtained computing only 2% of the columns of W . Observe that each cluster agreesapproximately with one handle of the eight. In our third example, we use the sdf distance tosegment the pliers model in 5 clusters. In Figure 3 right we show the results obtained computing5% of the columns of W . As in the previous examples, the clusters are natural partitions of themodel. Figure 3: Left: segmentation with the angular distance of the cube model. The clusters are obtained computing 1%of the 10800 columns of W . Center: segmentation of the eight model with the geodesic distance obtained computing2% of the 1536 columns of W . Right: pliers model segmentation with the sdf distance obtained computing 5% of the8970 columns of W . In our next examples we use a product metric to compute the distance between neighboring tri-angles. More precisely, if the faces f i and f j share an edge of the triangulation, then the productdistance d ij between them is defined as d ij := d gij d aij , where d gij , d aij are the geodesic and angulardistances between f i and f j respectively. As usual, the distance between no adjacent faces is de-fined as the length of the shortest path in the dual graph. Figure 4 left shows the segmentation in8 clusters of the bunny model. This result was obtained computing the 10% of the total numberof columns of the affinity matrix W . Observe that the segmentation produced by the productdistance distinguishes well not only the big ears but also the small tail. The hand model with 3000faces is more challenging. As we observe in Figure 4 right, there is some leakage in the clusterscorresponding to the fingers, even when this leakage is substantially smaller than the one shownin the hand model in [LZ04]. Moreover, the palm and the back of the hand belong to differentclusters since the combined metric is not enough to capture all the volumetric information. Theselimitations could be overcome if we include in the definition of the combined metric a part-awaredistance, such as the part-aware distance in [LZSC09].15 igure 4: Two segmentations obtained with the metric defined as the product of geodesic and angular distances.Top: two views of the segmentation of the bunny model. The segmentation was computed using 10% of the 3860columns of the affinity matrix W . Bottom: two view of the segmentation of the hand model, segmentation obtainedcomputing 1% of the 3000 columns of W . In this section we study the behavior of our mesh segmentation algorithm
FSS through severalexamples of the Princeton Segmentation Benchmark [CGF09] for evaluation of 3D mesh segmen-tation algorithms. This benchmark comprises a data set with 380 surface meshes of 19 differentobject categories. It also provides a ground-truth corpus of 4300 human segmentation.In the next examples we observe that the algorithm proposed in this paper, based on the compu-tation of few columns of the affinity matrix, produces segmentations which compare to the resultsobtained using the full affinity matrix. Recall that it doesn’t mean that the segmentation obtainedwith few columns of the affinity matrix W is always good, but that it is as good as the one obtainedcomputing all columns of W . In other words, if we select carefully which columns of W are com-puted, then the quality of the results essentially depends on how good the selected metric reflectsthe features of the triangulation. In our experiments we compute the distance between trianglesusing several metrics: the geodesic distance, the angular distance, the product of them and the sdf distance [SSCO08].Usually, the quantitative evaluation of a segmentation algorithm is done by comparing the au-tomatic segmentation with one or more reference segmentations of the ground-truth corpus. Inthe literature one can find several metrics to evaluate quantitatively the similarity between twosegmentations of a triangulated surface [BH03], [CGF09]. In this section we employ two differentnon-parametric measures: the Jaccard index J I [FM83] and the Rand index RI [Rand71]. Thesegmentation of a mesh with n triangles may be described by a vector s = ( s , ..., s n ), where s j is the index of the cluster to which the j -th triangle belongs. Given two segmentations s a and s b of the same mesh, we denote by J I ( s a , s b ) and RI ( s a , s b ) the similarity between them accordingto the Jaccard and Rand indexes respectively. For both indexes it holds: 0 ≤ J I ( s a , s b ) ≤ ≤ RI ( s a , s b ) ≤
1, where the value 1 corresponds to the maximal similarity, i.e.
J I ( s a , s b ) = 1or RI ( s a , s b ) = 1 means that segmentations s a and s b are identical. In the experiments we com-pute Jaccard and Rand distances between s a and s b given by, d J ( s a , s b ) := 1 − J I ( s a , s b ) and d R ( s a , s b ) := 1 − RI ( s a , s b ).In Figure 5 we show the results obtained for three models of the Princeton Segmentation Bench-mark: the sunglasses (model 42), the octopus (model 121) and the bird ( model 243). For allmodels, Rand and Jaccard distances are computed comparing the automatic segmentation (leftcolumn) with and the ground truth segmentation (right column). Table 1 below shows the valuesof Rand and Jaccard distances as well as the percent of columns of the affinity matrix used toobtain the automatic segmentation and the metric employed for computing the distance betweentriangles. In these examples a low percent of columns provides good segmentations.Example distance % d R d J glasses geodesic 2% 0 .
062 0 . .
052 0 . .
048 0 . Table 1: Rand and Jaccard distances ( d R and d J , respectively) between the automatic segmentation and the groundtruth segmentation.Figure 5: Left column: segmentations obtained computing few columns of the matrix W , right column: groundtruth segmentations. First row: segmentation of the sunglasses based on geodesic distance and obtained computing2% of the 8324 columns of W . Second row: segmentation of the octopus based on the angular distance and obtainedcomputing 1% of the 11888 columns of W . Third row: segmentation of the bird based on the sdf distance andobtained computing 1% of the 6312 columns of W .
17n Figure 6 we illustrate that the quality of the automatic segmentation with few columns of W strongly depends on the capability of the selected metric to capture the features of the mesh. Inthis example we show that even if we compute the whole affinity matrix W , the resulting automaticsegmentation is far a way from the ground truth segmentation, as it is shown by the values of d R and d J between these segmentations ( d R = 0 . d J = 0 . .
5% and 100% of the columns of W are very similar, since thevalues of d R and d J between them are small ( d R = 0 .
010 and d J = 0 . k-means algorithm. Figure 6: Three segmentations of a fish (model 225 of the Princeton Segmentation Benchmark). Left and center:segmentations based on the geodesic distance obtained using the 0 .
5% and the 100% of the 12148 faces. Rand andJaccard distances between them are d R = 0 .
010 and d J = 0 .
064 respectively. Right: ground truth segmentation.Rand and Jaccard distances between the segmentation obtained computing all the columns of W and the groundtruth segmentation are d R = 0 . d J = 0 . In the experiments of this section we use a different approach to measure the quality of the seg-mentation. Instead of comparing the automatic segmentation, obtained computing k columns ofthe affinity matrix, with the ground-truth of the corpus, we compare it with the segmentationobtained using all columns of the affinity matrix. In our opinion, this comparison is fairer, sincethe simple metrics ( geodesic , angular and sdf distances) that we have used to compute the distanceand affinity matrices are not always enough to produce good segmentations. Hence, the comparisonof the automatic segmentations with ground-truth corpus segmentations does not help in the senseof proving that the results with few well selected columns of the affinity matrix are of quality quitesimilar to the results obtained computing all affinity matrix.Applying our segmentation method, 18 meshes of the Princeton Segmentation Benchmark [CGF09]are segmented using k columns of the affinity matrix W , where k = 0 . , , , , %10 and%25 of the total number n of faces. The Rand distance d R between the segmentations obtainedfor k columns and the segmentation obtained with the same distance for the whole matrix W iscomputed. Three different metrics are considered, the geodesic , the angular and the sdf metrics.For each of these metrics, in Figure 7 it is shown the histogram of relative frequency of the Randdistance values between the segmentations obtained for k columns and the segmentation obtainedwith the same metric for the whole matrix W . This experiment shows that with high probability,the segmentations with small number of columns k are very close to the corresponding segmenta-tion obtained for the whole matrix W . 18 igure 7: Relative frequency histograms of the Rand distance values, d R , between the segmentations obtainedwith the whole matrix W and the segmentation obtained with the same metric for k columns of W , with k =0 . , , , , , geodesic , angular and sdf metric. β curve The graph of β k (17) as function of k has a “ L ”-shape, similar to the singular values curve, whichdecreases very fast for small values of k , see Figure 2. It suggests that β curve could be usedto propose an lower bound for the size k of the sample that furnishes a projection H k providinga good approximation to W . In the first row of Figure 8 it is shown a section of the β curve( k, β k β ) , ≤ k ≤ n for three models of the Princeton Segmentation Benchmark (hand, bearing,octopus), considering different distances ( geodesic, angular, sdf , respectively). Observe that thesecurves decrease very fast for small values of k and tend slowly to 0 when k goes to n .For several values of k , Table 2 shows the Rand and Jaccard distances between the automaticsegmentation s k and the ground truth segmentation of the Princeton Segmentation Benchmark.The smallest number of columns for which the slope of the β curve may be considered as very smallare marked with ∗ . Observe that for any fixed model and all considered values of k , the valuesof the Rand and Jaccard distances between the automatic segmentation and the ground truthsegmentation are very similar. In the second row of Figure 8 the segmentations corresponding to k ∗ are shown. Figure 8: First row: normalized curve β k /β curve. Second row: segmentation of the models with k ∗ columns of W . k d R d J hand geodesic ∗ .
124 0 . .
124 0 . .
123 0 . angular ∗ .
033 0 . .
033 0 . .
028 0 . sdf ∗ .
043 0 . .
040 0 . .
040 0 . Table 2: Rand and Jaccard distances ( d R and d J respectively) between the proposed automatic segmentation andthe ground truth segmentation. The automatic segmentation was obtained using the metric indicated in the secondcolumn and computing the number of columns of the affinity matrix indicated in the third column. Since the method
FSS proposed in this paper has some points of contact with the one introducedin [LZ04] and improved in [LJZ06], in this section we compare the segmentations obtained withboth approaches. The algorithm introduced in [LJZ06] applies Nystr¨om’s method to approximatethe spectral embeddings of faces of the triangulation. To avoid the expensive computation of thenormalized matrix Q = M − / W M − / and its largest eigenvectors, Nystr¨om’s method computesapproximately the largest eigenvectors of Q , from a small sample of its rows (or columns) and thesolution of a small scale eigenvalue problem (see section 2.2). The final step consists in applying k-means to the rows of Q . The selection of the sample has a strong influence on the accuracy ofthe approximated eigenvectors.No comments on the recommended relationship among the size of the sample and the number ofeigenvectors are included in [LJZ06]. In the numerical experiments reported here, the same sampleof max-min farthest faces is used to select the columns of W to be computed by our method FSS and also for the Nystr¨om approximation of the largest eigenvectors of W . Further, in the resultsobtained with the spectral method, we set the number of eigenvectors equal to the number ofclusters. Moreover, as suggested in [FBCM04], the Nystr¨om approximated eigenvectors of Q areorthogonalized before applying k-means clustering.Figure 9 shows the segmentation based on the sdf distance of several models, using the spectralmethod with Nystr¨om approximation and our method FSS . Both methods compute the samepercent of columns of W corresponding to the farthest triangles. Table 3 shows the values of Randand Jaccard distances between the corresponding segmentations. In general we observe that evenwhen the segmentation are different, the quality of them is similar. The Rand distance betweensegmentations are very small in all cases, but Jaccard distances are larger reflecting better thevisual differences.In our last example we show an unexpected artefact that we have observed. Sometimes the spectralsegmentation obtained using Nystr¨om approximation produces clusters which are non connected .This undesirable effect is eliminated when we increase the size of the sample of columns of theaffinity matrix W . In contrast, our segmentation approach using the same sample of columns of W always produces connected clusters (recall lemma 1). In Figure 10 left we show the spectralsegmentation of the woman model obtained with the normalized matrix Q . Center and right images20 igure 9: First row: Spectral segmentations with Nystr¨om approximation, the number of eigenvectors is equal to thenumber of clusters. Second row: segmentations obtained with our method FSS . The sample of columns of the affinitymatrix is the same in both approaches: 0 .
5% for the hand, 25% for dog and 5% for fawn.
Example % d R d J hand 0 .
5% 0 .
054 0 . .
082 0 . .
084 0 . Table 3: Rand and Jaccard distances between our segmentation and the segmentation produced by the spectralapproach using Nystr¨om approximation. The sample of columns of W is the same for both methods. The secondcolumn of the table indicates the size of the sample which is the indicated % of the total number of triangles. of this figure show the spectral segmentation with Nystr¨om approximation and with the methodproposed in this paper, both using 5% of columns of W corresponding to the farthest triangles inthe sdf distance. The described artefact becomes evident in the spectral segmentation with Nys-trom approximation. Figure 10: Left: spectral segmentation computing all columns of the affinity matrix, center: spectral segmentationwith Nystr¨om approximation computing 5% of columns of the affinity matrix W , right: our segmentation based inthe same sample of columns of W . . Conclusions and future work We have proposed a segmentation method for triangulated surfaces that only depends on a metricto quantify the distance between triangles and on the selection of a sample of few triangles. Theproposed method computes the weighted dual graph of the triangulation with weights equal tothe distance between neighboring triangles. The k farthest triangles in the chosen metric are usedto compute a rectangular affinity matrix W k of order n × k , where the number k of columns ismuch smaller than the total number of triangles n . Rows of W k encode the similarity betweenall triangles and the k triangles of the sample. Thus, clustering the rows of W k happens to beconsistent with the results of clustering the rows of the whole matrix W and no artefact appears.Hence, a valid strategy to solve the segmentation problem consists in clustering the rows of W k byusing for instance the k-means algorithm.From the theoretical point of view the problem of reducing the dimensionality for clustering isstrongly connected with the low rank approximation of the matrix containing the data to beclustered, which in our context is the affinity matrix W . In this sense, we have proved that forany sample X of k indexes, the rank k approximation of W obtained projecting it on the spacegenerated by the columns of W with indexes in X , coincides with the rank k approximation obtainedprojecting W on the space generated by its approximated eigenvectors, computed by Nystr¨om’smethod with the same sample of columns of W . Moreover, it is shown that if the columns of W k correspond to the k farthest triangles in the selected metric, then the proximity relationship amongthe rows of W k tends to faithfully reflect the proximity among the corresponding rows of W .In practice, our experiments have confirmed that this occurs even for relatively small k , resultingin a low computational cost method. Multiple experiments with a large variety of 3D triangularmeshes were performed and they have shown that the segmentations obtained computing less thanthe 10% of columns of the affinity matrix W are as good as those obtained from clustering the rowsof the full matrix W . We have also observed that the quality of the results, objectively measured interms of Rand and Jaccard distances between the automatic and the ground truth segmentations,depends strongly on the capability of the selected metric of capturing the geometrical features ofthe mesh. Our experiments with geodesic , angular and sdf distances show that none is enough toproduce good segmentations in all cases. A combination of two o more metrics usually leads tobetter results.Compared to other segmentation methods considered in the literature, the segmentation methodproposed in this paper has several advantages. First, it does not depend on parameters that mustbe tuned by hand. Second, it very is flexible since it can handle any metric to define the distancebetween triangles. Finally, it is very cheap, with a computational cost of O ( nm ), where m is thecost of computing the distance between two faces of the triangulation. In this sense, the proposedmethod is cheaper than spectral segmentation methods, which in the best case ( when Nystr¨omapproximation is used ) compute additionally the eigenvectors of an order k matrix, with an extracost of O (( n − k ) k ) + O ( k ) operations.In the present work, we intentionally focus on simple single segmentation fields on 3 D meshes andthe clustering is obtained applying a well know clustering algorithm, k-means++ , in order to achievestraightforwardly a fair comparison of our segmentation method with the spectral method. Never-theless, our farthest point based segmentation FSS method may be extended to more complicatedscenarios, where several attributes are combined in a single segmentation field, such as in [LZSC09],[WTKL14] or where several multi-view clustering methods have been proposed to integrate withoutsupervision multiple information from the data, [HCC12], [CNH13]. Therefore, in the future, we22lan to investigate the advantages of replacing Nystr¨om’s method with ours, for instance, comparewith [YLMJZ12], [EFKK14], in order to make the proposed algorithms more efficient in terms ofcomputational and memory complexity as well as to obtain segmentations without artifacts, suchas nonconnected clusters. Finally, it is worth mentioning that the basic ideas of our segmentationmethod - to compute only few columns of the affinity matrix and to apply a clustering algorithmto the rows of this submatrix - could be straightforwardly used in other segmentation or clusteringproblems, for instance in segmentation of digital images [MiCh16].