Surface Mesh Qualities

Marco Attene∗

IMATI-GE / CNR, Italy

Keywords: Shape representation, triangle mesh, benchmark, data set

Abstract: 3D surface models are often stored as indexed face sets, whereas algorithms can typically treat only a subsetof the so-representable surfaces. Besides this gap between “potential” input and “allowed” input, more andmore algorithms specify their input requirements at a high level of abstraction (e.g. “natural” shapes), and thismakes it difficult to automatically assess the appropriateness of a specific model.In this article, surface meshes are analyzed in terms of indexed face sets and are characterized based on acollection of “qualities”. Qualities of a surface mesh represent the peculiarities of both the surface and itscombinatorial representation, can be combined to formally define input requirements of algorithms, can beexploited to effectively estimate high-level categorizations of 3D objects in a collection, and can be used toquickly produce benchmarks for geometry processing algorithms starting from unstructured digital libraries.

1 Introduction

While developing new algorithms to treat polygonmeshes, researchers make often several assumptionson the input. Unfortunately, checking the character-istics of a mesh may be complicated, it is often notthe focus of the main research and, since it wouldrequire a significant additional effort, it is typicallyleft unimplemented. Even worse, the application do-main of an algorithm might be defined at a very highlevel of abstraction (e.g. segmentation of man-madeobjects) for which even a preliminary formalizationbecomes a hard task. Furthermore, new algorithmsneed to be tested using relevent benchmarks, and evenif several repositories are available on the Internet,collecting a sufficient number of appropriate meshesmight become a time-consuming task. To the best ofour knowledge, no existing repository provides themeans to resolve queries such as “find meshes hav-ing a uniform distribution of vertices and which un-ambiguously enclose a polyhedron”. Thus, there isa need to characterize the input of geometric algo-rithms, which in turn requires to (1) define automaticprocedures to check the appropriateness of an inputmesh, and (2) enable the automatic creation of signif-icant benchmarks starting from uncharacterized meshcollections.

We provide a classification of a number of shapecharacteristics and show how to use them to createuseful benchmarks. We only deal with characteristicsthat can be formally defined in terms of elements of

∗e-mail: attene@ge.imati.cnr.it

a representative indexed face set, and term them “sur-face mesh qualities”. Even if accounting for all thepossibile qualities is clearly unfeasible, some charac-teristics are in widespread use, thus we identify a setof “atomic” qualities that can be combined to derivewhat the community mostly needs. The definition andclassification of these qualities is the main contribu-tion of the paper. Also, we have developed algorithmsto compute them and have shown how to use them toautomatically produce benchmarks (see fig. 1).

2 Background and Related Work

Quality criteria for meshes have been defined forrather specific contexts, such as finite element anal-ysis (Shewchuk, 2001) or surface remeshing (Alliezet al., 2008). Some quality values can be consideredto be actual “defects” (Attene et al., 2013) that preventthe mesh to be used in a specific application.

In AIM@SHAPE’s Digital Shape Workbench(AIM@SHAPE, 2004) some metadata can be used toquery the Shape Repository using a semantic searchengine. Though this is one of the most advancedshape query methods we are aware of, it still has sev-eral limitations: on the one hand, the set of metadataemployed is far from being exhaustive enough to en-able a systematic production of benchmarks; on theother hand, most of the available metadata must beentered manually while uploading a model, while justa few values can be computed automatically using theTriMeshInfo tool (Borgo et al., 2005).

Figure 1: Top row: the “10 most man-made objects” automatically extracted from the Princeton Shape Benchmark based ona combination of qualities forming a ranking function. Bottom row: the “10 most natural objects” extracted by inverting thesign of the same ranking function.

Definitions In the remainder terms such as abstractsimplicial complex, simplex and f ace of a simplexare freely used. For formal definitions we point thereader to (Attene, 2010). We denote a triangle meshas a pair M = (P,Σ), where P is a set of N vertex posi-tions pi = (xi,yi,zi) ∈ R3 with 1≤ i≤ N, and Σ is anabstract simplicial complex over these vertices. Evenwithout considering P at all, a number of topologicalcharacteristics of M can be defined based on Σ only.

M is homogeneous (or pure) if all the edges andvertices in Σ are faces of at least one triangle in Σ. Asimplex belongs to Star(σ) if it is incident to σ. Asimplex belongs to Link(σ) if it is not incident to σ

but it is a face of a simplex in Star(σ). M is combi-natorially manifold if Σ is a combinatorial manifold,which means that the Link of each of its vertices ismade of a single ring of edges (Glaser, 1970). If theLink of all vertices is a single chain of edges (eitheropen or closed to form a ring), then Σ is a manifoldwith boundary, and we say that M is combinatori-ally manifold with boundary. A combinatorially man-ifold mesh M is orientable if all its triangles can beoriented consistently (Attene, 2010). Based on thenotion of genus of a graph (Goodman and Joseph,2004), we say that the combinatorial genus of an ori-entable mesh M = (P,Σ) is the minimal integer n suchthat Σ can be drawn without crossing itself on a spherewith n handles (i.e. Σ can be PL-embedded onto anoriented manifold of genus n).

The geometric realization |M| of M (Attene, 2010)is a subset of R3 for which a Euclidean topology ex-ists, and we say that M is geometrically manifold iffthe neighborhood of each point in |M| is homeomor-phic to a disk in this topology. A triangle mesh may bemanifold in the combinatorial sense and not in the Eu-clidean one, e.g. when the mesh self-intersects. Also,a geometrically manifold mesh may be not combina-torially manifold, e.g. when a topologically singularedge has three incident facets, but two of them coin-cide geometrically.

3 Surface mesh qualities

Qualities must be exactly defined and measureablewithout ambiguities. Thus, we must be able to saythat in a given mesh M a quality Q has a unique valueV . In other words, each quality is a function over thespace of all the possible meshes. To express that, wewill make use of predicates of the form Q(M) = V .The set of values that a quality Q can assume iscalled the range or codomain of Q. For example, forQ = number o f f acets the range is the set of natu-ral numbers N, for Q = minimum triangle angle therange is the real interval [0,π], for Q = is orientablethe range is the set B = {true, f alse}.

We deal with three classes of characteristics thatwe call mesh qualities, surface qualities, and shapecategory indicators. Mesh qualities quantify featuresof the mesh such as, for example, the number of ver-tices or the average length of edges. Surface qual-ities characterize the surface, and thus are indepen-dent of the specific represention used: the topologicalgenus or the total area are examples of surface qual-ities. If the shape model is represented by a mesh,surface qualities may be derived from (a combina-tion of) mesh qualities. Shape category indicatorsare also related to the shape, but they are indepen-dent of the model used and characterize objects interms of high-level classifications such as, for exam-ple, models which are articulated or architectural.Normally, shape category indicators cannot be com-puted by simply combining mesh qualities or surfacequalities. Nonetheless, in some cases it is possible tocompute useful estimates (see section 3.3).

3.1 Mesh qualities

Most graphic formats encode surface meshes throughindexed face sets, where a first block specifies the ver-tex positions and a second block represents polygonsas sequences of indices. Clearly, files of this type

are not guaranteed to represent a well-defined poly-hedron, while they may easily encode non-manifoldand/or non-orientable sets of polygons. Thus, here-with we compute mesh qualities to provide usefulinformation about the connectivity and geometry ofthese indexed mesh representations. Computing thesequalities involves counting elements, measuring theirattributes and performing statistical analysis.

Connectivity Besides counting the basic elementsconstituting the mesh (i.e. vertices, edges and facets),some applications need to ensure that their input isan actual simplicial complex, possibly after triangu-lation of non-simplicial facets. Thus, it is importantto verify that all the indexed facets and edges areactually valid (i.e. facet indices must be different andstay within the limited range of vertices), and that nosimplex is defined more than once. Also, it mightbe important to know if these defects are sparse orsystematic within the file (e.g. to decide if it is worthto attempt a repairing). For these reasons, our listof mesh qualities includes the number of duplicatededges and facets and the number of invalid edgesand facets. Note that a vertex should be considereda duplication only if its has the same coordinatesof another vertex, thus it implies an analysis of theembedding. Conversely, for indexed edges and facetsthere are purely combinatorial possibilities whichinclude: several instances of the same set of indexes;different orderings of the same set of indexes. Inorder to account for applications that can only dealwith homogeneous (or pure) simplicial complexes,we also count isolated vertices and naked edgeshaving no incident facets. Finally, we also take intoaccount the large number of algorithms that assumethat the input is a two-manifold. Here it is importantto distinguish between two classes of algorithms:those that require the mesh to be combinatoriallymanifold (e.g. because their internal data structurecan only treat these objects) and those that requiregeometrical manifoldness (e.g. because they need tounambiguously separate the embedding space intointerior and exterior). To account for combinatorialmanifoldness, our list of mesh qualities includesthe number of singular vertices and the number ofsingular edges. We remind that a vertex is singularif its Link is not a single chain of edges, while anedge is singular if it has more than two indicentfacets. Being a characteristic of the representedsurface, geometrical manifoldness falls in the domainof surface qualities (see section 3.2). The followinglist summarizes mesh qualities characterizing theconnectivity; the range is the set of natural numbersN in all the cases, and the symbol “#” means “Num-

ber Of”: # duplicated edges, # duplicated facets,# invalid edges, # naked edges, # invalid facets,# isolated vertices, # singular vertices,# singular edges, # vertices, # edges, # facets.

Geometry Bad embeddings may easily lead tomeshes which are not appropriate for several applica-tions. For this reason, our mesh qualities include thenumber of duplicated vertices, but also the number ofduplications in the realized edges and facets whichare induced by duplications of their vertices. Dupli-cated vertices may easily lead to degenerate facetswith zero area, which are clearly an issue for severalapplications that need, for example, to computesurface normals. Degenerate facets, however, are notonly due to coincident vertices, while they can beproduced when the embedding aligns the vertices. Inany case, it is important to count degenerate facetsand edges. It is also important to count possible inter-secting facets. Here it is worth to distinguish betweenthe mesh quality number of intersecting factes andthe surface quality is self-intersecting, even thoughthe former implies the latter: when a surface model isself-intersecting, indeed, the number of intersectingfacets may vary depending on the specific discretiza-tion used to represent the model. So, the numberof intersecting facets is a mesh quality, whereas thefact that the surface is self-intersecting is a sur f acequality. Knowing the number of such intersectingelements may help in deciding if it is worth toattempt a repairing to make the mesh exploitable.Besides counting things, depending on the specificembedding other interesting qualities assume varyingvalues within continuous domains. To start with,we define a number of qualities that characterize thevertex sampling, which include uniformity, isotropy,and gradation. Indeed, several algorithms discussedin the literature are claimed to be robust againstnon-uniform sampling densities. In this regard,we observe that there is no standard definition forconcepts such as sampling uniformity when appliedto meshes. To avoid forcing the adoption of anyspecific definition, we tackle the problem indirectlyby measuring things that can be exactly defined, butthat can be also easily combined to represent the var-ious possible definitions. In this case, for triangularmeshes we consider the lengths of all the edges andmeasure their standard deviation and their averagevalue. These two qualities can be later combinedto represent the sampling non-uniformity as, forexample, the ratio between the edge length standarddeviation and the average edge length, which will be0 for perfectly uniform samplings. If there are nontriangular facets, they are first triangulated through a

scheme that minimizes the total edge length (Atteneand Falcidieno, 2006). In some contexts (e.g. someremeshing algorithms), meshes are said to be uniformin relation to the uniformity of triangle areas, thus weproceed as in the case of edge lengths: we computethe standard deviation of these areas and encode it asa quality of the mesh. Note that the average trianglearea does not need to be encoded explicitly becauseit can be derived as the ratio between the totalsurface area (see section 3.2) and the total numberof facets, which are both encoded qualities. Amongthe algorithms that work well on non-uniformlysampled meshes, some might have problems in caseof extreme sampling anisotropy (e.g. because theydiscard the available connectivity and use k-nearestneighbors to reproduce a local topology). Thus, wecompute the sampling anisotropy around each vertexand encode the average, minimum and maximumvalues as mesh qualities. The sampling anisotropyaround a vertex v is computed as the ratio between the2nd and 3rd eigenvalues of the 3×3 coveriance matrixC(v) = ∑i(vi− v)(vi− v)T , where the vis are all thevertices in Link(v). To account for all the applicationsthat expect mesh triangles to be well-shaped (e.g.FEM), we measure all the triangle angles and encodethe minimum and maximum as mesh qualities.Similarly, we also compute all the dihedral anglesformed by pairs of adjacent triangles and encode theirminimum and maximum values. To make it possibleto assess the presence of other extreme configurationssuch as e.g. spikes, for each vertex v we calculatethe excess angle as the sum Kv = 2π−∑i αi, wherethe αis are the incident triangle angles at v, and weencode both the maximum and minimum valuesas mesh qualities. The following list summarizesmesh qualities characterizing the geometry; therange is indicated in parentheses, and SDev means“Standard deviation”: # duplicated vertices(N), # multiply realized edges (N),# multiply realized facets (N), # degenerate edges(N), # degenerate facets (N), # intersecting facets(N), average edge length (R), edge length SDev (R),triangle area SDev (R), min sampling anisotropy(R), averaage sampling anisotropy (R),max sampling anisotropy (R), min triangle angle(R), max triangle angle (R), min diherdal angle(R), max dihedral angle (R), min excess angle (R),max excess angle (R).

Topology Other interesting mesh qualities charac-terize the overall topology of the simplicial complex.Counting the number C of connected components isuseful while analyzing meshes for applications thatexpect a connected input, but also to challenge tools

that are claimed to work with multiple components.Furthermore, such a count helps in distinguishingbetween an actual polygon soup and a mesh which isintentionally made of several pieces. We also countthe number B of connected components of boundaryedges, which we shortly call number of boundarycomponents. A mesh is combinatorially manifoldif it is homogeneous (i.e. there are neither isolatedvertices nor naked edges) and there are no singularelements; in this case each boundary componentis actually a boundary loop, and their number Bcan be used to derive the combinatorial genus g ofthe mesh through the generalized Euler-Poincareequality V − E + F = 2(C − g) − B, where V , Eand F are the numbers of vertices, edges and facetsrespecitively. Orientation and orientability are alsoimportant qualities as they may help in derivingother characteristics (see section 3.2). The follow-ing list summarizes mesh qualities characterizingthe overall topology: # connected components(N), # boundary components (N), combinato-rial genus (N), is orientable (B), is oriented (B),is combinatorial manifold (B).

3.2 Surface qualities

Some surface qualities can be derived based on meshqualities and existing theorems, e.g. a mesh which iscombinatorially manifold and orientable is also geo-metrically manifold if there are no self-intersections.Also, if such a mesh has no boundary then it enclosesa polyhedron. This is often required by advancedapplications that, e.g., need to tetrahedrize the innershape. On the contrary, a non-orientable closed meshcannot be realized without self-intersections even ifit is combinatorially manifold (e.g. a Klein bottle)and, therefore, can be declared to be not geometri-cally manifold even without performing a direct anal-ysis of the geometry. Similar strategies can be em-ployed for other topological characteristics of the sur-face, e.g. to count the number of connected compo-nents of the realized mesh we start from the number ofcombinatorial components and look for intersectionsrepresenting connections that have no counterpart inthe abstract complex.

In several shape matching and similarity applica-tions, algorithms use some forms of normalizationto perform scale-invariant comparisons. For poly-hedra, the total volume is mostly often used, but inother cases the total surface area or the radius ofthe smallest bounding sphere can be employed effec-tively. Thus, it is worth encoding these three qualities.

The analysis of the Gauss map provides interest-ing information about the represented surface. For a

polygon mesh M we can model a “discrete” Gaussmap as a finite set of points on the unit sphere S2 rep-resenting the facet normals. We loosely call this set ofpoints the Gauss map of M, and indicate it with N(M).To start with our analysis, we calculate the standarddeviation of the sampling density of N(M). To do this,we cannot rely on a useful connectivity within N(M),thus we uniformly subdivide the sphere S2 in k regions(k = 128 in our prototype), and just count how manypoints of N(M) belong to each region. The standarddeviation of the resulting histogram is an indicatorof how the sampling density of N(M) varies aroundthe sphere, and can be exploited to derive higher-levelcharacteristics of the shape (see section 3.3). To makethis indicator less sensitive to the mesh sampling, in-stead of simply counting the number of points in eachregion, we use the areas of the facets that originatedthe points as weights in the sum.

Also, it is interesting to know if all the surfacenormals point towards a unique direction d, such asin the case of Digital Terrain Models or of digitizedrange images. Unfortunately such a direction d mayexist while being unknown, thus we must employ ageneralized approach that does not rely on such infor-mation. However, we observe that if such a d actu-ally exists, then the Gauss map of the surface is lim-ited within a hemisphere. Thus, as a solution, we justcheck whether the convex hull of N(M) contains theorigin: it it does, then M has at least two facets whosenormal vectors point towards opposite directions (i.e.their dot-product is negative).

Finally, we compute a quality which measureshow normal vectors change after having moved allthe vertices to the center-of-mass of their neighbors.For each triangle t we calculate its normal no, then wemove its three vertices to the centers-of-mass of theirneighbors, and calculate the so-modified triangle’snormal nm. The average over all the triangles ofthe quantity 1− < no,nm > is what we call averagenormal instability, and can be exploited to estimatethe amount of noise in a digitized mesh. The follow-ing list summarizes the discussed surface qualities:total surface area (R), enclosed volume (R), bound-ing ball radius (R), is geometric manifold (B),is self intersecting (B), # geometric components(N), Gauss map density SDev (R),is origin in Gauss map CH (B), aver-age normal instability (R).

3.3 Shape category indicators

All the qualities described above are at a rather lowlevel of abstraction, and indeed have an exact mean-ing and can be computed automatically. The defini-

tion of higher level characteristics (e.g. we want totest an algorithm on ”man-made” objects) is muchharder to define and compute exactly, but since we cancompute a number of low-level qualities, we still havetwo possibilities to tackle the problem: (1) Declara-tive approach: the low-level qualities are combinedto form predicates that, based on the experience ofan expert, have high probability of representing thehigh-level characteristic. E.g. a high-resolution meshwith a high Gauss map density SDev is likely to bea man-made object. (2) Machine-learning approach:a user appropriately tags a sufficiently large trainingset of meshes (e.g. as being man-made or not) and,for each of them, the values of a number of qualitiesare stacked to form a feature vector. These vectors areused to train a learning system (e.g. a Support VectorMachine) which will be eventually able to provide anestimate of the high-level classification for the queryobjects. Clearly, the accuracy of such an estimate de-pends on a number of factors, including the qualityof the training set (number of meshes and discrimi-natory power) and the actual correlation between thequalities in the feature vector and the high-level char-acteristic.

4 Benchmark creation

When extracting a specific dataset (e.g. a bench-mark) from a large repository we can collectmeshes responding to combinations of query termsof two types: Requirements Q1 = V 1 (or Q1 >V 1, or Q1 < V 1). All the meshes in thedataset must respond to these query terms. Forexample, # singular vertices = 0 must be truewhen creating a set of manifold meshes. Pref-erences Q1 As Close As Possible To V 1 (or Q1As High As Possible). If R is the set of all the meshesthat satisfy the requirements, these query terms areused to actually rank elements in R. If the objec-tive is creating a benchmark made of 100 manifoldmeshes preferably with a high-resolution vertex sam-pling, a ranking based on the # o f vertices qualitywould make it possible to simply cut the sorted list ofretrieved models to have the most appropriate bench-mark.

5 Results and Discussion

We have implemented a tool that calculates all thepossible mesh and surface qualities described, andhave run experiments on three public repositories: aSHREC dataset (Biasotti and Attene, 2008), the Mesh

Segmentation Benchmark (Chen et al., 2009), and thePrinceton Shape Benchmark (Shilane et al., 2004).The considered SHREC dataset was designed to chal-lenge the robustness of shape matching algorithmsagainst various forms of “defects”; it is made of 1500meshes with variations in shape, smoothness, sam-pling density/uniformity, topology, etc. After havingcomputed all the quality files (in about 30 minutes),we could successfully extract both the “10 most noisymeshes” and the “10 least noisy meshes” through thepreference query term average normal instability =As High As Possible and its inverse respectively.Similarly, we could extract the 10 “less uni-formly sampled” meshes through the queryterm edge length SDev/average edge length =As High As Possible and all the meshes that“enclose a polyhedron and that do not havespikes” (i.e. is geometric mani f old AND# boundary components = 0, −max excess angle =As High As Possible).

The Mesh Segmentation Benchmark (MSB) col-lects meshes from other repositories, and its ob-jective is to evaluate segmentation algorithms. Allthe 380 meshes in the MSB enclose valid polyhe-dra, and computing the quality files took 12 min-utes. Here we could extract the “10 models havingthe highest genus”. Note that, since we can assumethat meshes enclose polyhedra, the usual notion ofgenus and that of combinatorial genus are equiva-lent, so we could simply use combinatorial genus =As High As Possible. Meshes in the PrincetonShape Benchmark (PSB) are collected from the Web,thus are rather heterogeneous from several pointsof view. Here we computed the quality files inabout 16 minutes (note that meshes in the PSBare less complex than those in SHREC). We couldsuccessfully extract up to “10 range maps” us-ing the requirement is origin in Gauss map CH =f alse, the “10 most man-made objects” and the“10 least man-made objects” (figure 1)throughthe preference term Gauss map density SDev =As High As Possible and its inverse respectively.

See (Attene, 2012) for additional figures and de-tails about the aforementioned experiments.

6 Conclusions

Surface mesh qualities make it possible to automat-ically analyze the input of an algorithm to assessits suitability and/or appropriateness. Atomic qual-ities can be combined to express derived properties,to estimate high-level classifications of a shape andto automatically produce benchmarks out of unclas-

sified mesh collections. The importance of shapebenchmarks is demostrated by the continuous birthof application-specific examples (e.g. (Chen et al.,2009) for segmentation, (Biasotti and Attene, 2008;Shilane et al., 2004) for matching and retrieval, (Wanget al., 2010) for watermarking).

Acknowledgements

This work is partly supported by the EU FP7 ProjectN. 26204 (VISIONAIR) and by the MIUR-PRINProject N. 2009B3SAFK 002. Thanks are due to theSMG members at IMATI for helpful discussions.

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