Michael M Bronstein

Iasonas Kokkinos

Scale-invariance in local heat kernel descriptors without scale selection and normalization

Keywords: shape retrieval, non-rigid shape similarity, intrinsic feature descriptor, heat diffusion, heat kernel signature, scale invariance

Today, only a small fraction of Internet repositories of geometric data is accessible through text search. Fast growth of these repositories makes content-based retrieval one of the next grand challenges in search and organization of such information. Particularly difficult is the problem of shape retrieval, as geometric shapes manifest a vast variability due to different scale, orientation, non-rigid deformations, missing data, and also appear in a variety of different formats and representations. One of the biggest challenges in non-rigid shape retrieval and comparison is the design of a shape descriptor that would maintain invariance under a wide class of transformations the shape can undergo. Recently, heat kernel signature was introduced as an intrinsic local shape descriptor based on diffusion scale-space analysis. In this paper, we develop a scale-invariant version of the heat kernel descriptor. Our construction is based on a logarithmically sampled scale-space in which shape scaling corresponds, up to a multiplicative constant, to a translation. This translation is undone using the magnitude of the Fourier transform. The proposed scale-invariant local descriptors can be used in the bag-of-features framework for shape retrieval in the presence of transformations such as isometric deformations, missing data, topological noise, and global and local scaling. We get significant performance improvement over state-of-the-art algorithms on recently established non-rigid shape retrieval benchmarks.

Invariance d'échelle dans les descripteur locaux du noyau de la chaleur sans sélection et normalisation d'échelle

Résumé : A ce jour, seulement une fraction des répertoires de données géométriques sur internet est accessible à la recherche contextuelle. Du fait de la rapide croissance de ces répertoires, les requêtes basées sur le contenu ont fait apparaître certains des grands défis à venir tels que la recherche et l'organisation de ces informations. Le problème de reconnaissance de forme est particulièrement difficile : les formes géométriques manifestent une vaste variabilité due à des différences d'échelle, à des différences d'orientation, à des déformations non-rigides ou à des données manquantes; elles apparaissent de plus dans une grande variété de formats et de représentations. L'un des plus grands défis dans la reconnaissance de formes non-rigides et leur comparaison, est la conception d'un descripteur de formes qui resterait invariant sous une large classe de transformations qu'une forme peut subir. Récemment, la signature du noyau de la chaleur a été introduite en tant que descripteur intrinsèque local de forme basé sur l'analyse de la diffusion dans l'espace d'échelle. Dans ce papier, nous développons une version du descripteur par noyau de la chaleur invariant par échelle. Notre construction est basée sur un espace d'échelle échantillonné logarithmiquement dans lequel les échelles de formes correspondent, à une constante multiplicative près, à une translation. Cette translation est annulée en utilisant le module de la transformée de Fourier. Les descripteurs invariants par échelle proposés peuvent être utilisés dans le cadre des "sacs de descripteurs" ("bags of features") pour la reconnaissance de formes dans la présence de transformations comme des déformations isométriques, de données manquantes, de bruit topologique, et de changement d'échelle global et local. Nous obtenons des améliorations de performance significatives, au dessus des algorithmes de l'état de l'art, sur des benchmarks récemment établis pour la reconnaissance de formes non-rigide.

Mots-clés : reconaissance de forme, similarité des formes déformables, descripteur intrinsèque, diffusion de la chaleur, signature du noyau de la chaleur, invariance par échelle

Introduction

Today, only a small fraction of Internet repositories of visual and geometric data is tagged and accessible through simple text search. Fast growth of these repositories makes content-based retrieval one of the next grand challenges in search and organization of such information. Particularly difficult is the problem of shape retrieval, as geometric shapes manifest a vast variability due to different scale, orientation, nonrigid deformations, missing data, and also appear in a variety of different formats and representations.

In principle, the common denominator of shape retrieval approaches is the creation of a shape descriptor or signature which captures the unique properties of the shape that distinguish it from shapes belonging to other classes on the one hand, and is invariant to a certain class of transformations a shape can undergo on the other [START_REF] Tangelder | A survey of content based 3D shape retrieval methods[END_REF][START_REF] Tangelder | A survey of content based 3D shape retrieval methods[END_REF]. In rigid shape analysis, different types of invariance were addressed. Rotation and translation invariance can be achieved using volume and area descriptors [START_REF] Zhang | Efficient feature extraction for 2D/3D objects in meshrepresentation[END_REF], spherical harmonics [START_REF] Kazhdan | Rotation invariant spherical harmonic representation of 3D shape descriptors[END_REF], geometric moments [START_REF] Tal | Content based retrieval of VRML objects -an iterative and interactive approach[END_REF], and distribution of pair-wise Euclidean distances [START_REF] Osada | Shape distributions[END_REF].

Dealing with non-rigid shapes requires compensating for the degrees of freedom resulting from deformations. Elad and Kimmel [START_REF] Elad | Bending invariant representations for surfaces[END_REF] and follow-up works [START_REF] Mémoli | A theoretical and computational framework for isometry invariant recognition of point cloud data[END_REF][START_REF] Bronstein | Generalized multidimensional scaling: a framework for isometry-invariant partial surface matching[END_REF] proposed modeling shapes as metric spaces with intrinsic (e.g. geodesic) distances, which are invariant to inelastic deformations. Ling and Jacobs [START_REF] Ling | Using the inner-distance for classification of articulated shapes[END_REF] and Bronstein et al. [START_REF] Bronstein | Analysis of two-dimensional non-rigid shapes[END_REF] used this framework with a metric defined by internal distances in 2D shapes. Reuter et al. [START_REF] Reuter | Laplace-spectra as fingerprints for shape matching[END_REF][START_REF] Reuter | Discrete Laplace-Beltrami operators for shape analysis and segmentation[END_REF] used the Laplacian spectra as intrinsic shape descriptors.

A particular type of intrinsic geometry is generated by heat diffusion processes on the shape. Coifman and Lafon [START_REF] Coifman | Diffusion maps[END_REF] popularized the notion of diffusion geometry, which is closely related to scale-space methods in image processing [START_REF] Sochen | Diffusions and confusions in signal and image processing[END_REF]. Rustamov [START_REF] Rustamov | Laplace-Beltrami eigenfunctions for deformation invariant shape representation[END_REF] was one of the first to use such distances in shape analysis, applying the method of Osada et al. [START_REF] Osada | Shape distributions[END_REF] to commute time distances (this method is similar to the recent work of Mahmoudi and Sapiro [START_REF] Mahmoudi | Three-dimensional point cloud recognition via distributions of geometric distances[END_REF] who used diffusion distances instead). In [START_REF] Bronstein | A Gromov-Hausdorff framework with diffusion geometry for topologically-robust non-rigid shape matching[END_REF], shapes were analyzed as metric spaces equipped with diffusion metrics.

In image analysis, bottom-up approaches have become popular, notably due to the works of Zisserman et al. [START_REF] Sivic | Video google: A text retrieval approach to object matching in videos[END_REF][START_REF] Chum | Total recall: Automatic query expansion with a generative feature model for object retrieval[END_REF] and Schmid et al. [START_REF] Mikolajczyk | Scale & affine invariant interest point detectors[END_REF]. Using these approaches, an image is described as a collection of local features ("visual words") from a given vocabulary, resulting in a representation referred to as a bag of features. In shape analysis, such approaches have been introduced more recently by Ovsjanikov et al. [START_REF] Bronstein | Shape google: a computer vision approach to invariant shape retrieval[END_REF] and Toldo et al. [START_REF] Toldo | Visual vocabulary signature for 3D object retrieval and partial matching[END_REF] (see [START_REF] Mitra | Probabilistic fingerprints for shapes[END_REF][START_REF] Li | Shape topics: a compact representation and new algorithm for 3D partial shape retrieval[END_REF] for earlier similar ideas).

The bag of features paradigm relies heavily on the choice of the local feature descriptor that is used to create the visual words. In image analysis and 2D shape retrieval, typical features are blobs [START_REF] Matas | Robust wide-baseline stereo from maximally stable extremal regions[END_REF] and corners [START_REF] Mikolajczyk | Scale & affine invariant interest point detectors[END_REF], and the default choice for a local descriptor is the scale-invariant feature transform (SIFT) [START_REF] Lowe | Distinctive image features from scale-invariant keypoint[END_REF] or one of its varieties [START_REF] Bay | Surf: Speeded up robust features[END_REF][START_REF] Tola | A fast local descriptor for dense matching[END_REF]. Scale-invariant local descriptors can be constructed in two ways. First way is to use scale-space analysis of the image to locally estimate the scale [START_REF] Lindeberg | Feature Detection with Automatic Scale Selection[END_REF][START_REF] Lowe | Distinctive image features from scale-invariant keypoint[END_REF]. Descriptors are then extracted from appropriately scaled image patches. Second way is to use a combination of logarithmic sampling with Fourier analysis to compensate for the scaling effects [START_REF] Kokkinos | Scale invariance without scale selection[END_REF] (such an approach is also commonly used to compute a global image rotation and scaling in the context of registration [START_REF] Casasent | Position, rotation, and scale invariant optical correlation[END_REF][START_REF] Zokai | Image registration using log-polar mappings for recovery of large-scale similarity and projective transformations[END_REF]).

In shape analysis, on the other hand, there is no commonly agreed upon feature descriptor similar to SIFT. In non-rigid shape retrieval applications, an ideal feature descriptor should be first of all intrinsic and thus deformation-invariant. Second, it should cope with missing parts, and also be insensitive to topological noise and connectivity changes. Third, it should work across different shape representations and formats (e.g. point clouds and meshes) and be insensitive to sampling. Finally, the descriptor should be scale-invariant. The last two properties are especially important when dealing with shapes coming from Internet repositories such as Google 3D Warehouse, where shapes appear in a variety of representations and with arbitrary scales.

Different approaches such as contour and edge features [START_REF] Pauly | Multi-scale feature extraction on pointsampled surfaces[END_REF][START_REF] Kolomenkin | On edge detection on surfaces[END_REF], spin images [START_REF] Johnson | Using spin images for efficient object recognition in cluttered 3 d scenes[END_REF], local patches [START_REF] Mitra | Probabilistic fingerprints for shapes[END_REF][START_REF] Toldo | Visual vocabulary signature for 3D object retrieval and partial matching[END_REF], conformal factor [START_REF] Ben-Chen | Characterizing shape using conformal factors[END_REF], differential operators [START_REF] Zaharescu | Surface Feature Detection and Description with Applications to Mesh Matching[END_REF], and local volume properties [START_REF] Gelfand | Robust global registration[END_REF] were used as feature descriptors in shape retrieval literature. Unfortunately, none of them satisfy all of the above desired properties (for example, volumetric and patch-based methods are not intrinsic, and conformal factor is sensitive to topology).

Recently, a local feature descriptor based on multiscale heat kernels was proposed [START_REF] Sun | A concise and provably informative multi-scale signature based on heat diffusion[END_REF]. This descriptor satisfies all of the above properties except for scale invariance. Scale invariance poses an additional challenge, for a few reasons. Compared to images, shapes typically contain less features that would be roughly analogous to blobs or corners, and there is no clear generalization of such structures to 3D surfaces. Feature detection based on intrinsic scale-space analysis such as [START_REF] Sun | A concise and provably informative multi-scale signature based on heat diffusion[END_REF] would find a few reliable points (usually with high curvature), at which scale estimation can be done. In flat regions, no scale estimation is possible. For this reason, Ovsjanikov et al. [START_REF] Bronstein | Shape google: a computer vision approach to invariant shape retrieval[END_REF] avoided feature detection and used a dense feature descriptor computed at every point of the shape in combination with statistical weighting to reduce the influence of trivial points.

In this paper, we develop a scale-invariant version of the heat kernel signature by combining this descriptor with the recent approach of [START_REF] Kokkinos | Scale invariance without scale selection[END_REF] to scale invariance in images. Our construction is based on a logarithmically sampled scale-space in which shape scaling corresponds, up to a multiplicative constant, to a translation. This translation is then undone using the magnitude of the Fourier transform. Since our descriptor does not rely on local scale estimation, it is computable at every point including flat regions, and can be thus used in the shape retrieval framework of [START_REF] Bronstein | Shape google: a computer vision approach to invariant shape retrieval[END_REF], as well as for other applications such as dense correspondence between shapes.

Background

In the following discussion, we model shapes as Riemannian manifolds (possibly with boundary) and use the heat conduction properties as shape descriptors. Heat propagation on non-Euclidean domains is governed by the heat diffusion equation,

∆ X + ∂ ∂t u = 0, (1)
where, ∆ X denotes the positive semi-definite Laplace-Beltrami operator, a Riemannian equivalent of the the Laplacian. The solution u(x, t) of the heat equation with the initial conditions u(x, 0) = u 0 (x) (and respective boundary conditions if X has a boundary) describes the amount of heat on the surface at point x in time t. The solution of (1) with point heat distribution u 0 (x) = δ(x -z) as initial conditions is called the heat kernel and denoted by K X,t (x, z).

On compact manifolds, the heat kernel can be presented as [START_REF] Jones | Manifold parametrizations by eigenfunctions of the Laplacian and heat kernels[END_REF] K

X,t (x, z) = ∞ i=0 e -λit φ i (x)φ i (z). (2)
INRIA where λ 0 , λ 1 , ... ≥ 0 are eigenvalues and φ 0 , φ 1 , ... are the corresponding eigenfunctions of the Laplace-Beltrami operator, satisfying ∆ X φ i = λ i φ i . Sun et al. [START_REF] Sun | A concise and provably informative multi-scale signature based on heat diffusion[END_REF] proposed using the heat kernel signature (HKS)

h(x, t) = K X,t (x, x) = ∞ i=0 e -λit φ 2 i (x) (3)
as local shape descriptors. The HKS is intrinsic and thus isometry-invariant (two isometric shapes have equal HKS), multi-scale and thus capture both local features and global shape structure, and also informative: under mild conditions, if two shapes have equal heat kernel signatures, they are isometric [START_REF] Sun | A concise and provably informative multi-scale signature based on heat diffusion[END_REF]. Ovsjanikov et al. [START_REF] Bronstein | Shape google: a computer vision approach to invariant shape retrieval[END_REF] used the HKS to construct global shape descriptors following the bag of features paradigm used in image retrieval applications [START_REF] Sivic | Video google: A text retrieval approach to object matching in videos[END_REF][START_REF] Chum | Total recall: Automatic query expansion with a generative feature model for object retrieval[END_REF]. First, the HKS descriptor is computed at every point of the shape (Figure 1, left). Next, using vector quantization, for each point on the shape, the HKS is replaced by the index of the most similar entry in a geometric vocabulary consisting of representative heat kernel signatures or "geometric words" (Figure 1, middle). The vocabulary is constructed offline by performing clustering in the HKS space. Finally, the distribution of geometric words on the shape is computed, resulting in a bag of features representation (Figure 1, right).

A notable disadvantage of the heat kernel signatures is their sensitivity to scale. Given a shape X and its scaled version X = αX, the new eigenvalues and eigenfunctions will satisfy λ = α 2 λ and φ = αφ. We therefore have the following equation:

h (x, t) = ∞ i=0 e -λiα 2 t φ 2 i (x)α 2 = α 2 h(x, α 2 t), (4)
relating the signature h at time t for X with the signature h at time α 2 t for X.

In some cases, the scaling effect can be undone using some global pre-normalization of the shape. Possible ways are normalizing the bounding box of the shape or its covariance (geometric moments), normalize the intrinsic diameter of the shape, i.e. the longest geodesic distance, or normalize the Laplace-Beltrami eigenvalues. The first approach will work only in rigid shapes, as non-rigid deformations change the bounding box. The second and the third approaches are insensitive to deformations, but would fail if the shape has missing parts. In the following, we describe an approach for local normalization of the heat kernel signature, which does not suffer from this problem.

Scale-invariant heat kernel signatures

In order to achieve scale invariance, we need to remove the dependence of h from α. This is possible through the following series of transformations applied to h. First, at each shape point x we sample the heat signature logarithmically in time and form the discrete function

h τ = h(x, α τ ). (5)
Based on Eq. 4, scaling the shape by α s/2 will result in a time-shift by s and amplitudescaling by α s (Figure 2, first row):

h τ = α s h τ +s . (6)
Second, we remove the multiplicative constant α s by taking the logarithm of h, and then the discrete derivative w.r.t. to τ (Figure 2, second row). The first step turns the multiplicative factor into an additive constant, s log α, which then vanishes in differentiation:

ḣ τ = ḣτ+s , (7)
(here, ḣτ = log h τ +1 -log h τ ). Finally, taking the discrete-time Fourier transform of ḣτ turns this shift in time into a complex phase;

H (ω) = H(ω)e 2πωs , (8)
where H and H denote the Fourier transform of ḣ and ḣ , respectively, and ω ∈ [0, 2π].

The phase is in turn eliminated by taking the Fourier transform modulus (FTM):

|H (ω)| = |H(ω)|. (9)
We thus have constructed the scale-invariant quantity |H(ω)| (denoted as SI-HKS and shown in Figure 2, third row) from the HKS at each point x, without performing scale selection. This allows us to compute descriptors at any point of our shape, where scale selection based on maxima detection could be impossible. Moreover, most of the signal information is contained in the low-frequency components of the FT, so we can build a compact descriptor by sampling |H(ω)| at a small number of low frequencies.

One caveat of our approach could be that scaling the shape and then resampling the function ḣτ makes the samples at the boundaries of the range of τ change. This can have dramatic effects if the signal information is concentrated at the boundaries of the scale-space. Fortunately, the HKS is typically smooth at low-and high-scales and therefore its derivative is equal to zero for a broad range of τ s at the beginning and end of ḣ.

INRIA

Numerical computation

Numerical computation of the HKS and the SI-HKS is done using formula [START_REF] Belkin | Constructing Laplace operator from point clouds in R d[END_REF], in which a finite number of terms is taken and the continuous eigenfunctions and eigenvalues of the Laplace-Beltrami operator are replaced by the discrete counterparts. The discretization of the Laplace-Beltrami operator depends on the representation on the shape. For shapes represented as point clouds, the Laplace-Beltrami operator can be approximated using [START_REF] Belkin | Constructing Laplace operator from point clouds in R d[END_REF]. For triangular meshes, one of the most common discretiza-tions is the cotangent weight scheme [START_REF] Pinkall | Computing discrete minimal surfaces and their conjugates[END_REF], defined for any function f on the mesh vertices as

(∆ X f ) i = 1 a i j w ij (f i -f j ), (10)
where w ij = cot α ij +cot β ij for j in the 1-ring neighborhood of vertex i and zero otherwise (α ij and β ij are the two angles opposite to the edge between vertices i and j in the two triangles sharing the edge), and a i are normalization coefficients proportional to the area of triangles sharing the vertex x i . This discretization preserves many important properties of the continuous Laplace-Beltrami operator, such as positive semidefiniteness, symmetry, and locality, and in addition it is numerically consistent [START_REF] Wardetzky | Discrete Laplace operators: no free lunch[END_REF].

In matrix notation, Eq. ( 10) can be written as

∆ X f = A -1 Lf, (11)
where A = diag(a i ) and L = diag l =i w il -(w ij ). The eigenvalues and eigenfunctions of the Laplace-Beltrami operator discretized according to 11 are computed by solving the generalized eigendecomposition problem [START_REF] Lévy | Laplace-Beltrami eigenfunctions towards an algorithm that "understands" geometry[END_REF] AΦ = ΛLΦ,

where Λ is the (k + 1) × (k + 1) diagonal matrix of the smallest eigenvalues λ 0 , ..., λ k , and Φ is an N × (k + 1) matrix of corresponding eigenfunctions φ 0 , ..., φ k such that φ il is the value of the lth eigenfunction at the point x i . Another way of approximating Laplace-Beltrami eigenfunctions on triangular meshes is using finite element methods (FEM) [START_REF] Reuter | Discrete Laplace-Beltrami operators for shape analysis and segmentation[END_REF].

The discrete heat kernel signature is approximated by

h(x l , τ ) ≈ k l=0 e -λ l α τ φ 2 il = Ψe -T Λ , (13)
where T = diag(α τ ) and Ψ = (φ 2 il ). Since the heat kernel depends only on the eigenfunctions and eigenvalues of the Laplace-Beltrami operator, at least in theory, one can compare shapes in different representations (e.g., point clouds to meshes). This property of heat kernel signatures is especially appealing in Internet shape retrieval applications, where the variety of shape representations and formats is enormous.

Results

We used the ShapeGoogle database [START_REF] Bronstein | Shape google: a computer vision approach to invariant shape retrieval[END_REF], consisting of 1061 shapes with simulated transformations. As of today, this is the largest non-rigid shape retrieval benchmark available. The database contained shapes from 469 different classes. For thirteen shape classes, the following transformations were simulated: 208 isometry, 208 global scale (varying approximately between 0.7 and 1.35), 128 local scale (local "swelling" of the shape), and 48 partiality+scale (missing parts in shapes with different global scaling). Examples of transformations are shown in Figure 4.

Heat kernel signatures (HKS) and the proposed scale-invariant heat kernel signatures (SI-HKS), respectively, were used as local shape descriptors. For the discrete computation of the heat kernels, we used the cotangent weight approximation of the Laplace-Beltrami operator and k = 200. For HKS, we used the parameters as in [START_REF] Bronstein | Shape google: a computer vision approach to invariant shape retrieval[END_REF] (six scales 1024, 1351, 1783, 2353, 3104 and 4096), which were experimentally found to give optimal performance on the ShapeGoogle database. In order to construct the SI-HKS, we used a logarithmic scale-space with base α = 2 and τ ranging from 1 to 25 with increments of 1/16. After applying the logarithm, derivative, and Fourier transform, the first 6 discrete lowest frequencies were used as the local descriptor.

Shape descriptors were constructed using bags of geometric words proposed in [START_REF] Bronstein | Shape google: a computer vision approach to invariant shape retrieval[END_REF]. For HKS and SI-HKS, a geometric vocabulary of size 48 was built using clustering in the signature space (six-dimensional in both cases). The HKS and SI-HKS at each point of the shape were replaced by the closest geometric word from the vocabulary using soft vector quantization. The distribution of geometric words (48-dimensional bag of features) was used as the shape descriptor. L 1 distance was used to compare the bags of features.

For comparison, we show the results of the ShapeDNA approach [START_REF] Reuter | Laplace-spectra as fingerprints for shape matching[END_REF], describing shapes by the vector of the first eigenvalues of the Laplace-Beltrami operator. We used first 15 eigenvalues to construct the ShapeDNA descriptors (this parameter was empirically selected to achieve optimal performance on the ShapeGoogle database). Eigenvalues were computed using the same cotangent weight discretization. L 2 distance was used to compare the ShapeDNA descriptors.

Shape retrieval performance was quantified using the precision-recall (PR) curve (Figure 5), plotting the tradeoff between precision (ratio of the number of relevant shapes retrieved and the total number of shapes retrieved) and recall (ratio of the number of relevant shapes retrieved and the total number of existing relevant shapes that could be ideally retrieved). We used the mean average precision (mAP) as a single number to quantify the retrieval quality (average precision is computed as the area below the precision-recall curve for each query, and the mAP is the average of AP over all queries).

Table 1 shows the performance of shape retrieval using bags of features built of HKS and SI-HKS. Our approach shows a dramatic improvement in the presence of varying scale (99.5% mAP compared to 61.32% with HKS) and also better performance for local scaling transformations (92.60% mAP compared to 85.83%). HKSbased bags of features produce negligibly (by 0.01%) worse results that SI-HKS on the class of isometric deformations. ShapeDNA shows similar nearly perfect performance on the class of isometries, but performs very poorly on scale and local scale transformations (36.72% and 72.17% mAP, respectively)

Figure 6 shows examples of first five matches retrieved using HKS and SI-HKS. With HKS, a scaled down centaur is confused with a dog (row a) and a scaled up horse is confused with an elephant (row c); while SI-HKS produces correct matches (rows b and d). In the presence of local scaling, because of the local nature of the descriptor, it remains unchanged far from the deformed parts. The SI-HKS shows better robustness to such scaling compared to HKS (e.g., in Figure 6 (g) local scaling transformations make HKS confuse between male and female shapes).

Conclusions

We presented an extension of the heat kernel signature allowing to deal with global and local scaling transformations. The use of Fourier transform magnitude to extract a scale-invariant quantity out of the heat kernel signature is advantageous over attempts to perform scale localization, which works only at prominent feature points. Our ap- proach allows to create a dense scale-invariant feature descriptor defined at every point of the shape. Besides invariance to global scaling, the scale-invariant HKS shows better resilience to local scaling transformations. Such transformations can arise, for example, due to locally-elastic deformations that stretch or shrink the shape surface. In future work, we intend to explore the proposed method in the context of part-based approaches in which the local descriptor is confined to a part of a shape and not computed across parts, and a separate descriptor is computed for each part. This way, inelastic deformations could be addressed.

INRIA

Figure 1 :

 1 Figure 1: Construction of a bag of features shape descriptor. Left: dense HKS local descriptor (shown three components as RGB colors); middle: local descriptor quantized in a geometric vocabulary of size 48 (each color represents a geometric word); right: bag of features counting the frequency of appearance of each geometric word.

Figure 2 :

 2 Figure 2: Construction of the scale-invariant heat kernel signature. First row: heat kernel signatures h (red) and h (blue) computed at a corresponding point on a shape and its version scaled by the factor of 11, plotted on a logarithmic scale. h and h differ by scale and shift in τ . Second row: ḣτ and ḣ τ , where the multiplicative constant is undone and the change in scale corresponds to a shift in τ only. Third row: first 10 frequencies of |H(ω)| and |H(ω)| used as scale-invariant HKS; the two descriptors computed at the two different scales are virtually identical.

Figure 3 :

 3 Figure 3: Comparison of HKS (left) and the proposed scale-invariant HKS (right). First and third rows: three components of HKS and SI-HKS, represented as RGB color and shown for shapes differing by global (first row) and local (third row) transformations. Second and fourth rows: HKS and SI-HKS at three point on the head (blue), hand (green), and foot (red) of the human shape.

Figure 4 :

 4 Figure 4: Example of transformations used in our shape retrieval experiment (left to right): null, scale down, scale up, two examples of local scale, partiality+scale.

Figure 5 :

 5 Figure 5: Shape retrieval performance using HKS (first row), SI-HKS (second row) and Shape DNA (third row).

INRIA

Figure 6 :

 6 Figure 6: Shape retrieval results. Left: queries, right: first matches using HKS (a,c,e,g) and SI-HKS (b,d,f,h).

Table 1 :

 1 Shape retrieval performance (mAP in percents) using HKS and SI-HKS based bags of features and ShapeDNA[START_REF] Reuter | Laplace-spectra as fingerprints for shape matching[END_REF]. Best result is shown in bold.

	Transformation Queries	HKS	SI-HKS ShapeDNA
	Isometry	208	99.96% 99.97%	99.52%
	Scale	208	61.32% 99.95%	36.72%
	Local scale	128	85.83% 92.60%	72.17%
	Partiality+scale	48	54.67% 89.95%	27.42%
	All	1061	85.30% 97.25%	74.47%

RR n°7161


Ousama Malouf

On the classification of singular flat structures on surfaces

Keywords: Mathematics Subject Classifications (2010): 57M50, 32G15 Flat structure, Teichmüller space, conical singularity

We study in this work flat surfaces with conical singularities, that is, surfaces provided with a flat structure with conical singular points. Finding good parameters for these surfaces in the general case is an open question. We give an answer to this question in the case of flat structures on pairs of pants with one singular point. The question of decomposability of an arbitrary flat surface into flat pairs of pants is discussed.

Introduction

In this article, we consider flat surfaces with a finite number of conical singularities, that is, surfaces provided with a flat structure with conical singular points. A conical singular point has a total angle different from 2π.

Finding good parameters for these surfaces is still an open question (one should mention here that there are well-known good parameters for the subset of flat surfaces defined by quadratic differentials on Riemann surfaces). We treat here the question of classifying flat pairs of pants with one singularity. The decomposition of surfaces into pairs of pants is a common practice to study various structures on surfaces (see for instance [START_REF] Luo | Grothendieck's reconstruction principle and 2-dimensional topology and geometry[END_REF]). The idea, which is usually attributed to Grothendieck, and which is developed by Feng Luo in [START_REF] Luo | Grothendieck's reconstruction principle and 2-dimensional topology and geometry[END_REF], is to provide building blocks (generators) which permit to reconstruct any surface endowed with the studied structure following some specific rules (relations). It should be noted however that there exist flat surfaces which are not decomposable into flat pairs of pants by disjoint simple closed geodesics (for an example of flat surface of genus 3 with one singularity, see Example 5.1). But, it is possible to decompose such a surface into pairs of pants if we sacrifice some rules of decomposition, such as the simplicity of geodesics in the decomposition.

In Section 2, we give a definition of flat surface, we recall the formula of Gauss-Bonnet and we give some examples. In Section 3, we classify flat pairs of pants with one singularity and we study their space of parameters. In Section 4, we present the Teichmüller space of flat pairs of pants with one singularity and we study its topology. In Section 5, we discuss the decomposability of a surface into pairs of pants.

This work uses ideas from [START_REF] Charitos | Parameters for generalized Teichmüller spaces[END_REF][START_REF] Charitos | On the geometry of hyperbolic surfaces with a conical singularity[END_REF] in which the authors consider the similar question for hyperbolic surfaces with singular points.

Preliminaries

Definition 2.1. Let M g,k (denoted by M if there is no confusion) be a surface of genus g with k boundary components, provided with a flat metric d(., .) with finitely many conical singularities Σ = {s 1 , . . . , s n }. We assume all the boundary components of M are closed curves (homeomorphic to circles). We denote by ∂M the boundary of M and by Int(M ) = M \∂M the interior of M . More precisely:

• For each point x ∈ Int(M )\Σ there is a neighborhood U (x) isometric to a disc in the Euclidean plane.

• For each point s ∈ Σ ∩ Int(M ) there is a neighborhood isometric to a Euclidean cone of total angle 0 < θ < +∞, with θ = 2π.

• For each point x ∈ ∂M \Σ there is a neighborhood isometric to a Euclidean sector with angle measure π at the image of x.

• For each point s ∈ Σ ∩ ∂(M ) there is a neighborhood isometric to a Euclidean sector of angle 0 < θ < +∞, with θ = π.

The metric d(., .) will be called a flat structure on M , and M will be called a flat surface (with boundary if it exists).

Definition 2.2. The curvature κ at a conical singularity s of total angle θ is κ = 2πθ. The curvature at a singular point on the boundary of angle

θ is κ = π -θ.
This definition of curvature is motivated by the following formula: Proposition 2.1 (Gauss-Bonnet formula for closed surfaces [5, p. 113], [1, p. 190] or [7, p. 85-86] for a proof). Let M be a closed flat surface of genus g with n conical singularities. Let θ i , i = 1, . . . , n be the total angles at the singularities. The formula of Gauss-Bonnet which connects the number of singularities with their total angles and the genus is:

n i=1 (2π -θ i ) = (4 -4g)π (2.1)
The Euler characteristic of this surface is given by χ(M ) = 2 -2g, and the formula of Gauss-Bonnet in terms of the Euler characteristic is:

n i=1 (2π -θ i ) = 2πχ(M ) .
Corollary 1 (Gauss-Bonnet formula for a disc with b holes). Let M be a flat surface of genus 0 with boundary, with n singularities in the interior and m singularities on the boundary, and let b be the number of boundary components. Let θ i , i = 1, . . . , n be the total angles of conical singularities in the interior, and let τ j , j = 1, . . . , m be the total angles of singularities on the boundary. Then,

n i=1 (2π -θ i ) + m j=1 (π -τ j ) = (4 -2b)π .
(2.2)

Proof. By taking the double of M (in this operation, each singularity on the boundary is glued to its copy) we obtain a closed flat surface of genus g = b -1 with 2n + m singularities. By the formula of Gauss-Bonnet (2.1) we get the result.

More generally, the Euler characteristic of a surface M of genus g with b boundary components is given by χ(M ) = 2 -2gb, and the formula of Gauss-Bonnet in terms of the Euler characteristic is:

n i=1 (2π -θ i ) + m j=1 (π -τ j ) = 2πχ(M ) .
Example 2.1. Let M be a pair of pants (a surface homeomorphic to a disc with two holes), equipped with a flat structure with one conical singularity s in its interior and such that each component of the boundary

∂M = c 1 ∪ c 2 ∪ c 3 is geodesic without singularities.
From s we take three geodesic segments d i , i = 1, 2, 3 which realize the distances between s and c i respectively, Figure 3. We denote by l i , r i the lengths of c i , d i respectively. We cut and open M along the geodesic segments d i . Then we obtain a connected surface P which can be decomposed into three rectangles R i , i = 1, 2, 3 whose lengths of sides are l i , r i respectively, and one triangle whose sides have lengths l i , as shown in Figure 4.

Following the inverse procedure, we consider a triangle T of side lengths l i , and three rectangles R i each of which shares a side with T and the other side being of length r i as shown in Figure 4. Now, by identifying the sides of equal length r i , we obtain a flat pair of pants with one conical singularity. Note that the total angle of the singularity is θ = 4π. This example will be studied later in Section 3.

Example 2.2. We cannot build a flat pair of pants with no singularities at all. Indeed, if we assume that such a pair of pants exists, by the formula (2.2) we find 0 = -2π which is impossible.

Finally, by gluing along their boundaries 2g -2 flat pair of pants, each one with a single singularity, we obtain a closed flat surface of genus g with 2g -2 conical singularities.

The geometry of a flat pair of pants with a conical singularity

In this section we assume that M is a flat (singular) surface with boundary homeomorphic to a disc with two holes. We assume further that M has only one conical singularity s. We call such a surface a flat pair of pants with one singularity (or just flat pair of pants). An example of such structure is obtained by taking the metric associated to two transverse measured foliations with one singularity on M (see Figure 1). It is clear from this picture that the parameter space for the transverse measures of these two transverse foliations has dimension four, and it produces a space of flat structures of dimension four on the pair of pants, with one singular point. It will follow from the discussion below The first natural guess is that the lengths of boundary geodesics might be good parameters, in analogy with the case of hyperbolic pairs of pants (described for instance in [START_REF] Fathi | Travaux de Thurston sur les surfaces[END_REF]). This is easily seen to be false, as one can glue Euclidean cylinders to boundaries of flat pairs of pants, changing the isometry type, without changing the boundary component length (see Figure 2). Thus, the parameter space for flat structures on pair of pants is more complicated than the parameter space for hyperbolic structure on such surfaces.

Let ∂M = c 1 ∪ c 2 ∪ c 3 denote the boundary and denote by d(., .) the distance function on M . Let d i be a geodesic segment which realizes the distance d(s, c i ) between the singularity s and the boundary component c i . This segment intersects the boundary component with a right angle. We denote by l i the length of c i and by r i the length of d i . Obviously, the parameters r i , i = 1, 2, 3 depend on the position of s. We have the following proposition. larity is on a boundary component, since by the formula of Gauss-Bonnet, the angle at s will be 3π. By our assumption on the uniqueness of the singularity, d i does not meet another singularity on the component c i of the boundary, and so, since c i is geodesic, d i is orthogonal to c i . For the same reasons, d i does not share any of its interior points with boundary components.

First case: We assume that s ∈ Int(M ). This implies that the lengths of d i are different from zero, r i = 0, i = 1, 2, 3. We cut M along d 1 ∪ d 2 ∪ d 3 , see Figure 3, to obtain a polygonal flat surface P homeomorphic to a disc without singularities in the interior, see Figure 4.

None of the parameters l i , i = 1, 2, 3 can be zero, because if one of them is zero, the surface M looses one of its boundary components. This fact implies that we always have three copies of s after the cut. Let us denote by s i , i = 1, 2, 3 the copy which lies between d (i+1) mod 3 and d (i+2) mod 3 , Figure 4. We can draw in P the geodesics between these copies. These geodesic segments bound a triangle T = (s 1 s 2 s 3 ) , Figure 5, which can be degenerate, with none of its angles greater than π. For instance, in Figure 6 the angle ∡(s 2 s 1 s 3 ) = π. This angle cannot be greater than π, because this implies that d 1 , along which the cut was made, was not a geodesic segment which represents the distance between s and c 1 . This contradicts the assumption.

We recognize easily, Figure 4, in addition to T , three rectangles R i , i = 1, 2, 3. A rectangle R i , i ∈ {1, 2, 3} is bounded by a side of T , c i and the two copies of d i . P is composed of T and R i , i = 1, 2, 3.

A rectangle is uniquely determined up to isometry by its length and height, and a triangle is uniquely determined up to isometry by its lengths of sides. Then, given the parameters l i , r i , i = 1, 2, 3, where the l i , i = 1, 2, 3 satisfy the triangle inequalities, we can construct a unique pair of pants with one singularity by gluing together the three rectangles R i and the triangle T given by the parameters l i , r i , i = 1, 2, 3. The triangle could be degenerate, and this case lies on the boundary of the space of triangles. This happens only when

l i = l (i+1) mod 3 + l (i+2) mod 3 (3.1)
for some i ∈ {1, 2, 3}. Figure 6.

Second case: We assume that s ∈ ∂M . Then at least one of r i , i = 1, 2, 3 equals zero. More precisely, s belongs to only one c i , i ∈ {1, 2, 3}, and so only one r i equals zero, because if s belongs to more than one boundary component, it does not have a neighborhood homeomorphic to a disc. In such a case we call this a degenerate pair of pants. We will outline these cases later. The geodesic segments between the copies of s bound a triangle T = (s 1 s 2 s 3 ) which can be degenerate, with none of its angles greater than π (for the same reason as in the first case). It is easy to recognize, in addition to T , two rectangles R i , i = 2, 3. Figure 8.

Knowing the parameters l i , r i , i = 1, 2, 3 we can determine if the singularity is on the boundary and on which component. The rectangles R i and the triangle T are uniquely determined by the parameters and so is the flat structure.

If Relation (3.1) is satisfied, the triangle is degenerate. In this case we notice that only one r i , i ∈ {1, 2, 3} that correspond to the boundaries on the right hand side of this identity can take the value zero. The other cases are degenerate.

We proved that to each flat structure on M with one singular point we may associate unique set of values l i , r i , i = 1, 2, 3. Conversely, to each choice of values, we may associate a unique flat structure on M .

We deduce that each flat pair of pants M is uniquely determined, up to isometry, by the parameters l i end r i . And vice versa.

Let us give the limits of parameters in view of the last proof. The length parameters can take any values satisfying these conditions:

• 0 < l i < ∞, i = 1, 2, 3, • l i ≤ l (i+1) mod 3 + l (i+2) mod 3 , i = 1, 2, 3, • 0 ≤ r i < ∞, i = 1, 2, 3, • 0 < r i + r (i+1) mod 3 , i = 1, 2, 3 • If l i = l (i+1) mod 3 + l (i+2) mod 3 , for some i ∈ {1, 2, 3} then 0 < r i < ∞
Remark. Let M be a flat surface with one singularity s, which is homeomorphic to a disc with n holes, n ≥ 3. We denote by c i the boundary components of M and by d i the geodesic segments from s to c i . If we set l i the lengths of c i and r i the lengths of d i , then the parameters l i , r i do not characterize M in the sense of Proposition 3.1. This follows because if we cut M along d 1 ∪ d 2 ∪ • • • ∪ d n then instead of a triangle (s 1 s 2 s 3 ) we get a (n + 1)-gon which, of course, is not uniquely determined by the lengths of its edges. We return to the case where M is a pair of pants and ∂M = c 1 ∪ c 2 ∪ c 3 . By the previous discussion, where we saw how a flat pair of pants with one singularity can be cut into pieces, three rectangles and one triangle, we find it convenient to introduce these terms: We say that the triangle is degenerate if the condition:

l i = l (i+1) mod 3 + l (i+2) mod 3 , for some i ∈ {1, 2, 3}
is satisfied. If not, we say that the triangle is non-degenerate. We say that the rectangle R i , i ∈ {1, 2, 3} is degenerate if r i = 0 for some i ∈ {1, 2, 3}. If not, we say that the rectangle R i is non-degenerate.

Remark (Degenerate cases). A degenerate pair of pants appears when the singularity does not have a neighborhood homeomorphic to a disc in the Euclidean plane. From the previous proof, we can conclude the following degenerate cases:

• In the case when the triangle is non-degenerate, if two or three rectangles R i , i ∈ {1, 2, 3} are degenerate we have a degenerate pair of pants. A neighborhood of the singularity is similar to one of the forms in Figure 9.

• In the case when the triangle is degenerate, that is, for some i ∈ {1, 2, 3} identity (3.1) is satisfied, if both rectangles R (i+1) mod 3 , R (i+2) mod 3 are degenerate we have a degenerate pair of pants. A neighborhood of the singularity is similar to Figure 10.

We should finally pay attention that in this case when R i is degenerate, the resulting surface is not a pair of pants since it has four holes. Figure 11.

After the previous discussion, we will introduce new parameters for flat pairs of pants. These parameters seem to be more convenient. For this, let now k i be a geodesic segment which realizes the distance between c (i+1) mod 3 and c (i+2) mod 3 , i = 1, 2, 3, see Figure 12. Denote by l i the length of c i and by a i the length of k i . Definition 3.1. The six non-negative parameters l i , a i , i = 1, 2, 3 will be called the distance parameters of M . We have the following theorem.

Proposition 3.2. The distance parameters l i , a i , i = 1, 2, 3 determine a unique flat pair of pants M with one singularity.

Proof. By the proof of Proposition 3.1 we see that flat pairs of pants with one singularity are uniquely determined by the parameters l i , r i , i = 1, 2, 3, and correspond to one of the following cases:

1. If the triangle is non-degenerate, we distinguish two cases.

(a) If all the rectangles R i are non-degenerate, it is easy, after the cut, to see that geodesic segments between boundaries pass all by the singularity. Figure 13. So we have: These relations prove that knowing l i , a i , i = 1, 2, 3 we can determine r i , i = 1, 2, 3, and so, all the parameters needed to determine the flat pair of pants in this case, by Proposition 3.1.

r i + r (i+1) mod 3 = a (i+2) mod 3 , i = 1, 2, 3 . (3.2)
(b) If only one rectangle is degenerate then the pair of pants can be cut into two rectangles and one non-degenerate triangle as in Figure 8. By the same relations (3.2) we can determine the flat pair of pants. The singularity, in this case, is on the boundary component which corresponds to the degenerate rectangle.

2. If the triangle is degenerate, we also distinguish two cases.

(a) If all the rectangles R i , i = 1, 2, 3 are non-degenerate, we see that the relations (3.2) also hold here and so the flat pair of pants is well determined, in this case, by the distance parameters. Figure 6.

(b) If one of R (i+1) mod 3 , R (i+2) mod 3 is degenerate then the pair of pants can be cut into two rectangles. Again, it is determined by relations (3.2). Figure 14.

Here we give the limits of the new parameters in view of the last proof. The distance parameters can take any values in the limit of these conditions:

1) 0 < l i < ∞, i = 1, 2, 3, 2) l i ≤ l (i+1) mod 3 + l (i+2) mod 3 , i = 1, 2, 3, 3) 0 < a i < ∞, i = 1, 2, 3, 4) a i ≤ a (i+1) mod 3 + a (i+2) mod 3 , i = 1, 2, 3, (Using Relations (3.2)), 5) If l i = l (i+1) mod 3 + l (i+2) mod 3 , for some i ∈ {1, 2, 3}, then a i < a (i+1) mod 3 + a (i+2) mod 3 . (Using Relations (3.2)).
It is easy to see that these conditions are equivalent to preceding ones. In addition, it is interesting to see that these conditions are equivalent to the fact that we have two Euclidean triangles (first four conditions) for which a degenerate case of one triangle prevents a degenerate case of the other (fifth condition).

We denote by C the set of all flat structures defined by the parameters l i , a i , i = 1, 2, 3 under the previous conditions, and by B the set of all 6-tuples convex in R 3 . The same can be said about the set of parameters a i , i = 1, 2, 3. Thus, B is convex in R 6 .

The boundary ∂B has six components which are non bounded convex subsets of R 5 . They correspond to cases of equality in the triangle inequality. Let us denote by ∂ i B l , i ∈ {1, 2, 3} the boundary component defined by

l i = l (i+1) mod 3 + l (i+2) mod 3 .
The sets ∂ i B l , i = 1, 2, 3 are pairwise disjoint subsets of ∂B. Similarly, we denote by ∂ j B a , j ∈ {1, 2, 3} the boundary component defined by

a j = a (j+1) mod 3 + a (j+2) mod 3 .
The sets ∂ j B a , j = 1, 2, 3 are also pairwise disjoint subsets of ∂B.

The intersection Then, by the homeomorphism Φ, we have the following description of the Teichmüller space: Corollary 2. T(M ) is homeomorphic to a non-compact submanifold of R 6 of dimension 6, with a natural cell-structure, having six cells of codimension one on its boundary.

∂ i,j (∂B) = ∂ i B l ∩ ∂ j B a , is homeomorphic to R 4 if i = j,
We know that any convex subset of Euclidean n-space E n is contractible. This gives us the following result: Theorem 4.2. T(M ) is a contractible space.

In the next section we start a discussion of the decomposition into pairs of pants of closed surfaces with one singularity. Details will be given in subsequent work.

Closed flat surfaces with one singularity

In general, a flat surface with one singularity is not decomposable by disjoint simple closed geodesics into pairs of pants. For this, we have the following example:

Example 5.1. Take a closed flat surface M 3 of genus 3 with one singularity s. Suppose that we are able to decompose it by disjoint simple closed geodesics into pairs of pants. By the fact that there is no flat pair of pants without any singularity, we conclude that every pair of pants resulting from the decomposition has the singularity s on its boundary. We know that a decomposition of M 3 by disjoint simple closed geodesics gives rise to four pairs of pants. This means that four boundary components should share the singularity. This is impossible under the present rules of identification (one boundary component is identified to one boundary component). Then, the decomposition is impossible.

The decomposition becomes possible if we change the rules of composition. For example, admitting geodesics to be non-simple or non-disjoint. That is, admitting the identification of parts of boundary components rather than boundary components entirely. I will not discuss here these rules, their study can be dealt with a separate work later.

Figure 1 :Figure 2 :

 12 Figure 1: The two transverse measured foliations on the pair of pants induce a flat structure with one singular point

Proposition 3 . 1 .Figure 3 :Figure 4 :

 3134 Figure 3: Flat pair of pants with one singularity in the interior l 1

Figure 5 :Figure 6 :

 56 Figure 5: Here we see the triangle on the pair of pants l 1

Figure 7 :Figure 8 :

 78 Figure 7: Singularity on the boundary

Figure 9 :Figure 10 :

 910 Figure 9: Neighborhood of singularity

sFigureFigure 12 :

 12 Figure 11: Impossible case

3 Figure 13 :Figure 14 :

 31314 Figure 13: After cut, l i , a i are the lengths of c i , k i respectively

  and empty if i = j. This means that each boundary component of B has two convex connected boundary components homeomorphic to R 4 .

Acknowledgements

I would like to thank Athanase Papadopoulos for all the corrections and discussions during the preparation of this article. I also thank Daniele Alesandrini for his time.

(l 1 , l 2 , l 3 , a 1 , a 2 , a 3 ) ∈ R 6 which satisfy these conditions. We can define a oneto-one mapping Φ : C → B such that to a flat structure f ∈ C corresponds the unique 6-tuple (l 1 , l 2 , l 3 , a 1 , a 2 , a 3 ) of B which determines f. Obviously we have Φ(C) = B.

The Teichmüller space of flat pairs of pants with one singularity

Let us denote by F(M ) the space of all flat structures with one singularity on a pair of pants M . We fix an orientation on M and let Homeo + (M, ∂) be the set of homeomorphisms of M which preserve the orientation and each boundary component of M (setwise). It is well known that each element of Homeo + (M, ∂) is isotopic to the identity (see Exposé 2 in [START_REF] Fathi | Travaux de Thurston sur les surfaces[END_REF]). The space Homeo + (M, ∂) acts on F(M ) as follows:

Definition 4.1. We define the Teichmüller space T(M ) of M as the quotient

Obviously, T(M ) consists of all flat structures which belong to C, the set of all flat structures defined by the parameters l i , a i , i = 1, 2, 3. To each flat structure with one singularity on M we may associate a unique configuration, which consists of a triangle T and three rectangles R i , i = 1, 2, 3, glued as in Figure 4. Conversely, to each configuration we may associate a unique flat structure with one singularity on M . This defines a mapping Φ : T(M ) → B which is one-to-one, and Φ(T(M )) = B.

Remark (Due to Athanase Papadopoulos). Let T M (M ) be the Teichmüller space of flat structures with one singularity on a pair of pants defined by a pair of transverse measured foliations. This space is of dimension 4. Since T(M ), the Teichmüller space of all flat structures with one singularity on a pair of pants, is of dimension 6, as we showed here, then T M (M ) T(M ). Thus, the space of flat structures we counter here is larger than the space of flat structures induced by quadratic differentials with one zero. Proof. As explained in Proposition 3.2, Φ is a bijection. If we consider the Euclidean distance between tuples of parameters (l 1 , l 2 , l 3 , a 1 , a 2 , a 3 ) and the following distance between two flat structures f 1 , f 2 ∈ T(M ):

we can see easily that both Φ and Φ -1 are continuous.

Obviously, B is a non bounded convex subset of R 6 . In fact, the set of parameters l i , i = 1, 2, 3 can be seen as the space of all triangles which could be degenerate, but without those which have a side of length zero. This is, a non bounded pyramid of three faces with its sides and apex deleted. This space is The result in Example 5.1 can be generalized as follows: Let M be a closed flat surface of genus g ≥ 3 with a single conical singularity s. Then M cannot be decomposed into pairs of pants by disjoint simple closed geodesics.

Author's address: Ousama Malouf Institut de Recherche Mathématique Avancée 7 rue René Descartes F-67084 Strasbourg Cedex France malouf@math.u-strasbg.fr