Augmented Reeb Graphs for Content-based Retrieval of 3D Mesh Models Tony Tung and Francis Schmitt Signal and Image Processing Department, CNRS UMR 5141 Ecole Nationale Sup´erieure des T´el´ecommunications 46, rue Barrault, 75634 Paris Cedex 13, France {tony.tung, francis.schmitt}@enst.fr Abstract This article presents an improved method of 3D mesh models indexing for content-based retrieval in database with shape similarity and appearance queries. The approach is based on the Multiresolutional Reeb Graph matching presented in [1]. The original method only takes into account topological information what is often not sufficient for effective matchings. Therefore we proposed to augment this graph with geometrical attributes. We also provide a new topological coherence condition to improve the graph matching. Moreover 2D appearance attributes and 3D features are extracted and merged to improve the estimation of the similarity between models. Besides, all these new attributes are user-dependent as they can be weighted by variable terms. We obtain a flexible multiresolutional and multicriteria representation called Augmented Reeb Graph (ARG). Good preliminary results have been obtained in a shape-based matching framework compared to existing methods based only on statistical measures. In addition, our study lead us to an innovative part matching scheme based on the same approach as our Augmented Reeb Graph matching.

1. Introduction The development of new 3D applications has become easier and tend to generalise due to the permanent progress of numerical technologies. Database of 3D objects has begun to proliferate for multimedia applications [23, 24, 25, 26] as well as for more industrial ones such as in CAD (e.g. motorcar companies). 3D objects indexing and content-based search appear as a natural answer for those who want to navigate in such huge database. As a consequence, queries have to be adapted to this new kind of data. Scientific literature only presents few methods dealing with 3D shape similarity queries. The techniques derived from the 2D, such as colour histogram or texture distribution taken from texture maps or from colour images of the ac-

quisition sequences of the object, are not well adapted to 3D queries. The approaches such as curvatures histograms [8] or EGI (Extended Gaussian Image) and its variants [9, 10] describe too locally the geometry of the objects. As a matter of fact they do not keep any topological information (mesh connectivity) and are highly sensitive to mesh modifications (noise on vertices position, decimation, etc...). In the other hand, global approaches such as cords histograms [5] or shape distributions [6] works quite well for the matching of primitive shapes but produce a too coarse description of the 3D objects shape to be a sharp retrieval tool when comparing similar complex shapes. Similar comments can be done for the Fourier based descriptors and moments calculations [3, 7]. To obtain an intuitive description of the shapes, the use of a skeletal or graph representation is very attractive. These approaches permit us to keep the topology of the objects which is a rich information for matching purpose. Unfortunately, difficulties lies in the construction of the skeleton which is either time consuming, too sensitive to noise on the shape surface, or need a source point which favour a direction. Besides, most of the time no matching scheme is provided [11, 12, 13], or no multiresolutional approach [14, 15, 16] whom the properties are useful for accurate similarity calculations and for various applications as presented in this article. The Multiresolutional Reeb Graph (MRG) [1] appears to be a very promising starting point of research, in particular because it is an efficient mean to exploit different techniques and merge the features as we will show in the next sections. The Reeb Graph of a 3D object is represented by a unidimensional skeleton. It is built using a function µ defined over the object surface considered as a compact manifold, and therefore relies on the mesh connectivity. The graph is robust in case of mesh modification thanks to the assumed integral function over the whole object [1]. The surface of the object is divided in regions according to value intervals of µ, and a node is associated to each connected component of the regions. The graph structure is then obtained by linking the nodes of the connected regions. Afterwards the MRG is constructed hierarchically from the finest level of resolution to the coarsest

level. And similarity calculation between MRG are done from the coarsest level to the finest level. Keeping the advantages of the multiresolutional representation offers by the MRG, we have provided topological, geometrical and visual (colour and texture) informations to the node graphs to obtain a hierarchical description of the models, which is both global and local and is called Augmented Reeb Graph (ARG). Therefore the 3D objects indexing in database will consist on coding in feature vectors the information associated to the nodes of the Reeb graphs. Tests of shape based retrieval were performed upon a database of high resolution 3D models [22, 23] and preliminary experiments were performed concerning the application of part matching. The MRG construction is presented in the next section. The graph matching procedure and the choice of the nodes attributes are shown in Section 3. Section 4 deals with the similarity calculation. The part matching scheme is presented in Section 5, and the results performed on our database are described in Section 6. We conclude in Section 7 with a discussion about the possible improvement of the methods.

is easy to calculate and stable to small noise. In the other hand, this function provides a Reeb Graph which is not flexible enough for our study, as explained next. The chosen

2. Multiresolutional Reeb Graph overview

function µ in [1] has rotation invariance property and is defined as the sum of the geodesic distance g(v, p) from v to the others points {p|p ∈ S}: Z g(v, p)dS. µ(v) = p∈S

According to the Morse theory, a continuous function defined on a closed surface characterise the topology of the surface on its critical points [2]. Therefore, a Reeb graph can be obtained assuming a function µ calculated over the 3D object surface. The multiresolutional aspect results from the dichotomic discretization of the function values and from the hierarchical collection of Reeb graphs defined at each resolution.

2.1. The function µ Our study concerns objects which are 3D compact manifold meshes of triangles with vertices located in a Cartesian frame R(x, y, z). The authors of [15] present different continuous functions which can be used for the Reeb graph construction (cf. Figure 1). Assuming a point v(x, y, z) on the surface S of an object, the height function µ defined as µ(v(x, y, z)) = z is well adapted for elongated stand models as human representations. This function is easy to compute but by definition has a strong orientation dependence. Therefore it is not suited for objects for which the points dispersal is perpendicular to the axis z. The function µ defined as the geodesic distance from curvature extrema is based on growing regions associated to local Gaussian curvatures on seed vertices. The results highly depend on the position of the seeds and require the local curvature calculation which is not obvious on every objects, especially those coming from the Internet. The function defined as the distance of a surface point v to the centre of mass G of the object µ(v) = d(G, v), where d is the Euclidean distance,

Figure 1. Example of µ functions: height function (left), distance to centre of mass (middle), geodesic distance integral (right).

The authors [1] chose to normalise the function with maxp∈S µ(p) for the denominator. To exploit the full dynamic of µ(v) over [0, 1], we have normalised the function with its min and max values: µN (v) =

µ(v) − minp∈S µ(p) . maxp∈S µ(p) − minp∈S µ(p)

The Reeb graph is obtained by partitioning the object surface following regular intervals of µ values, and linking the connected regions to each other. In each interval, a node is associated to a set of connected triangles (cf. Figure 2). Based on geodesic distance calculation, the resulting Reeb graph of a human model with arms lifted up in the air and a human model with arms along the body will look similar as maximal values of µ stay at the extremity of the arms. On the other hand, the function µ defined as the distance to the centre of mass will lead to a different graph. As a matter of fact, it can be stated that the choice of the function µ depends on the user’s queries.

2.2. Multiresolutional aspect A graph of double or half resolution is respectively obtained by the division or the fusion of nodes [1]. A parent node generates n children nodes with n ≥ 1 at a higher level of resolution (cf. Figure 3).

• the parents of m and n have been matched at the previous level of resolution • m and n are in the same interval of the function µN • m and n have the same labels, meaning they both belong to branches which have been matched to each other. In deed, when two nodes are matched, a same label is propagated along the branches they belong to. Therefore neighbouring nodes belonging to a same branch have the same labels. In this way, we ensure that a branch is entirely matched to a branch [1]. Figure 2. Reeb graph: 3D model (left), function µN (middle), Reeb graph; edges in green; nodes in red (right) (model from [23]).

It is also assumed that a node can be matched to a group of topologically consistent nodes [1]. However the simple statement of these criteria is not sufficient to ensure that two branches will still be matched together when the resolution gets higher (cf. Figure 4). As a consequence, we add a new topological consistency criterion: if the parents of the neighbours of m and n have been matched, then they must have been matched together.

Figure 3. MRG construction by merging µN intervals. Nodes from a level of resolution r merge in parent nodes at a higher level of resolution (from left to right: r = 2, 1 and 0).

3. Augmented Reeb Graph matching The matching strategy of two MRGs is based on the detection of nodes with similar topological configurations in term of connections to their neighbours nodes and parents/children relations. The aim is to maximise the number of corresponding nodes. For this purpose a set of topological consistency criteria were stated. The pairs of nodes which verify the criteria are candidates for the matching. And a loss function determines the best matching nodes among the candidates. To improve the matching of the original MRG method, we have introduced an additive topological consistency criteria and added geometrical attributes to the nodes. Finally a function estimates the similarity between the two Augmented Reeb Graphs.

3.1. Addititional topological consistency We consider the graph comparison of two objects M and N . Topological consistency criteria permit to find the pairs of nodes (m, n), with m ∈ MRG(M ) and n ∈ MRG(N ), which could possibly be matched afterwards in the procedure. The nodes m and n are topologically consistent if:

Figure 4. Matching nodes (left). At a higher level of resolution, branches mixing can happened. To avoid this, the matching of the parents of the neighbours have to be tested (right).

3.2. Geometrical attributes From a topological point of view, the matching of the graphs could be acceptable. But in our framework these matching are not well adapted, because topologically speaking an arm worth a leg (they are limbs linked to the body), whereas we would like to get more accuracy in the matching to efficiently compare complex 3D models. The topological consistency criteria are such defined that if two nodes are not matched together at the coarsest level of resolution, whereas they should be, then there is no chance that their children nodes will be matched together at a finer resolution. Therefore it is crucial that nodes are matched the best they can from the beginning of the consistency tests. For this reason, we propose to provide localisation information to the nodes. A node m is associated to a set of connected triangles, so it can be located at their barycentric position (cf. Figures 2 and 3). We choose a spherical coordinate representation (r, θ, ϕ):

• r(m) ≥ 0 is the distance to the node at the centre of the frame,
• θ(m) = arctan xy ∈ [− π2 , π2 ],
• ϕ(m) = arccos zr ∈ [0, π].

Using the example of a human model (cf. Figure 5) whom the axis x would get from the bottom to the top of the body, axis y oriented from left to right and axis z from behind to front, we can observe that θ allows to identify the left from the right, ϕ distinguish arms from legs and r could be decisive in a ambiguous configuration (an arm up in the air could be taken as an head). It becomes necessary that ob-

Figure 5. Nodes located in spherical coordinates (model from [23]).

jects are oriented towards the principal directions (x, y, z). For this purpose, we decide either to use the natural axis of the object centred to the centre of mass with a coordinates normalisation, or to apply a classical spatial alignment technique such as the Principal Component Analysis to turn the object towards the principal axis [3].

3.3. The loss function The loss function loss(m, n) quantifies the difference between two nodes m and n. It is applied on every groups of topologically consistent nodes and returns those which will be effectively matched by minimising the function: loss(m, n) = αl · |a(m) − a(n)| + βl · |l(m) − l(n)|+ γl · |r(m) − r(n)| + δl · |θ(m) − θ(n)| + l · |ϕ(m) − ϕ(n)|. a is the relative area of the set of triangles associated to a node. l stands for the complexity of a node and is linked to the number of its children as described in [1]. αl , βl , γl , δl and l are weighting terms which influence the attributes according to the type of desired queries. This aspect gives lots of freedom to the matching of nodes. Associated to the chosen function µ defined as the integral of geodesic distance, the flexibility of the method can be illustrated by the

possibility to choose either to match models having exactly the same shapes, or to let more freedom by allowing the matching between approximative similar shapes (typically two human models in different positions). And this can be done by simply adapting the weighting terms to the desired queries. Besides, it is also possible to replace the absolute values used to calculate the differences by quadratic distances, or any other dedicated functions to emphasise the differences or similarities between two nodes. Figure 6 shows the gain obtained in the matching of two graphs by introducing the localisation information in the nodes.

Figure 6. Left: without geometrical information, legs can be matched with arms as there are topologically equivalent (blue links denote matching nodes). Right: by adding geometrical information, arms and legs are well matched (models from [23]).

4. Similarity calculation Assuming known the list of matching nodes, a function is required to measure the similarity between two 3D models. First, a function calculates the similarity between each matched nodes. The original function introduced by [1] has been enriched with various features extracted from the mesh models. Using the same approach as in Section 3.2 for the geometrical attributes, these 2D/3D extracted features are weighted to allow different types of queries and to significantly refine the similarity calculus between two objects. This results in the Augmented Reeb Graph representation. The next sections present the proposed nodes similarity function, the introduced 2D/3D object features (relative volume, cords lengths, curvatures and colour) and the resulting graphs similarity function.

4.1. Similarity of nodes The calculation of the similarity function between two nodes m and n is defined as in [1] :

sim(m, n) = αs ·min(a(m), a(n))+βs ·min(l(m), l(n)). To improve the retrieval of similar objects with more flexible queries and a larger set of global and local characteristics we introduce the following geometric and visual features extracted from the section of the object associated to each node m and at each level of resolution: • v(m), the relative volume associated to m, • cords(m), a statistic measure of the extent of the surface region associated to m, • curvm (m) and curvg (m), a statistic of the mean and Gaussian curvatures estimated on the triangulated surface associated to the node m, • colorR,G,B (m), a statistic of the texture/colour associated to the node. Features are extracted at the finest level of resolution and associated to the corresponding nodes. The multiresolutional aspect of the measures is ensured by the hierarchical relation between parents and children nodes. Our idea is to provide to the nodes all of the relevant characterisations that could be calculated on a set of triangles. Our choices are not exhaustive but reflect the current frequently used techniques. Moreover, it is possible to favour one or several measures by adaptating the weighting terms. This again, let a lot of freedom to the users for their queries.

4.2. Volume calculation Let’s consider the interval [µ1 , µ2 ] of the function µ associated to the node m. The relative volume v(m) is defined as: 1 vol(m) · , v(m) = R vol(S) where vol(S) represents the total volume of the model, and vol(m) stands for the volume of the section of m. vol(S) is the sum of the tetrahedra formed by the oriented triangles of the mesh and the centre of mass of the object [4]. vol(m) is obtained by summing 1) the signed volume of the tetrahedra formed by the triangles associated to m and the barycenter of these triangles, 2) the signed volume of the tetrahedra formed by this barycenter, an edge of the mesh corresponding to the surface edge µ = µ1 associated to m, and the barycenter of the points associated to these edges, and 3) the symmetric volume for the surface edge µ = µ2 (cf. FigR level of resoluure 7). The factor R1 corresponds to the P tion of ARG(M ), leading to the relation m∈M v(m) = 1 as used for the relative area calculation a in [1]. Obviously, this volume calculation method is valid only if triangles can be oriented. Besides, an additive feature of the object sec3 tion associated to the node m could be the measure a(m) v(m)2 which has no dimension.

Figure 7. Volume associated to a node obtained by summing the volume of the three types of tetrahedra included in the section.

4.3. Cords measurement The cords measure introduced in [5] is represented by a normalised histogram of the distance from the triangles centres to the centre of mass of the object. Applied to a node n at the finest level of resolution, the histogram cord(n) is the histogram of the distances between the triangle centres associated to the node n, and the position of their barycenter. Each contribution to the histogram is weighted by the triangle area associated to the cord. Therefore, the histogram is 1 where area(S) is the normalised by the factor R1 · area(S) total surface area of the object. The histogram cord(m) of a parent node m at coarser resolution is thus the histogram of the cords to the barycenter of the node m obtained using the barycentres of the children of m. At the coarsest level of resolution r = 0, the histogram cord(m) is equivalent to the approach in [5] applied on the entire mesh. This statistical measure gives the information on the extent of a set of triangles. Hence it is a signature of the mesh geometry associated to a node. The use of statistical methods based on histograms to describe an object, as the cords [5] or the shape distribution [6], is well adapted for simple shapes. In our context and from a certain level of resolution, nodes are mainly locally associated to shapes such as cylinders or paraboloids which justify the relevant use of these geometrical measures.

4.4. Curvatures measurement Local geometric properties of the surface are represented by mean and Gaussian curvatures. We can see on Figure 8 that objects contains meaningful local informations to characterise the geometrical variations. The mean curvature is calculated on every points P of the mesh, according to the usual formula, by summing the angles (θN )i of the normals of the dihedral formed by the adjacent faces to P (i = 1, ..., Ne where Ne is the number of edges adjacent to P). Angles are weighted with the third of the mean area (Sm )i of the two faces (barycentric area):   1 X (θN )i , curvm (P) = 1 Ne i 3 (Sm )i

Figure 9. Colour histograms of the set of triangles associated to an ARG node.

Figure 8. Mean curvature calculated on each vertex of the mesh (model from [25]).

cally for each node of the ARG at the coarsest level of resolution, and then hierarchically and globally for each level of resolution: X color∗ (m) = color∗ (n), n∈{children of m}

The Gaussian curvature at a point P is calculated using the angular defect formula: P 2π − j θj , curvg (P) = 1 P j Sj 3

Sj being the area of the j th triangle having P as one of its vertices, and θj is the angle at vertex P in this triangle. The two histograms of these curvatures are calculated for each node at the finest resolution and afterwards propagated to the parents: X curv∗ (m) = curv∗ (n), n∈{children of m}

where ∗ represents m or g, and curv∗ (n) is the histogram of the curvatures of the triangle surface associated to n and 1 . As for the cords hisnormalised by the factor R1 · area(S) togram, each contribution to curv∗ is weighted by the associated triangle area. It is also possible to use only one histogram to represent the local curvature by combining the two curvatures (cf. Koenderink shape index [19, 8]) or to use any other characterisation of local curvatures [20, 21].

4.5. Texture extraction When browsing in a database of 3D textured objects, it can be also useful to make queries upon the texture or the colour of the models. The joint use of shape and texture is a highly discriminative criterion, but is still not much exploited in 3D. For this purpose, we use the information included in the texture maps associated to the 3D objects. These 2D features will naturally be merged to the 3D features previously presented. Our colour quantification is done using three normalised histograms on (R, G, B) (cf. Figure 9). The quantification of the texture is obtained lo-

where ∗ represents R, G, or B and color∗ (n) is the colour histogram of the node n normalised as in Sections 4.3 and 7.4.

4.6. Objects similarity Assuming two objects M and N , the function used to calculate their similarity is obtained by summing the similarity function of the matching pairs of nodes {m, n}: X sim(m, n), SIM (M, N ) = 1 − {m,n}

where SIM is minimal for similar objects. The 2D/3D features presented in the previous sub-sections have been introduced in the function sim(m, n), with weight parameters (λ1 , λ2 , λ3 , λ4 ) in accordance with the desired queries: • λ1 · min(v(m), v(n)), • λ2 · min(cord(m), cord(n)), • λ3 · min(curv∗ (m), curv∗ (n)), • λ4 · min(color∗ (m), color∗ (n)). We define the min calculation between two histograms hist(m) and hist(n) as: min(hist(m), hist(n)) =

N bin X

min(histi (m), histi (n)),

i=1

where Nbin is the size of the histogram and histi is the ith value. For all objects M , N and O, the function SIM has the following properties: 1. Self-identity: SIM (M, M ) = 0,

2. Positivity: SIM (M, N ) ≥ 0, 3. Symmetry: SIM (M, N ) = SIM (N, M ), As defined in [16], the function SIM is a semi-metric. The triangle inequality is not verified as the function SIM takes into account the pairs of matching nodes between two objects, and they can be different when comparing two different objects to a same one. A new function SIM satisfying the triangle inequality would probably perform better classification and searching [17] but remains to be found. Finally, we have obtained a flexible multiresolutional 3D indexing scheme, both global and local, and using various criteria illustrated by the different features extracted (topological, geometrical, and colorimetric).

5. Application to part matching Our shape matching method has both the properties of a high-level descriptor thanks to the global representation with the multiresolutional Reeb graph, and a low-level descriptor with the local 2D and 3D features extracted at each level of resolution and for each node, especially at the finest level. This multiresolutional aspect can be exploited for many different purposes. Our current research work lead us to a promising application which consist on using the enriched nodes for part matching as illustrated in Figure 10. As a matter of fact, the information included in the graph nodes are discriminant enough to be retrieved in a database. Moreover, the different representations of the nodes at each level of resolution, which include their children nodes, can ensure the precision of the matching. An approach of part

could match anywhere). We then propose a part matching scheme which relies on the ideas presented in this article and which exploits the multiresolutional aspect and the topological consistency criteria of the nodes for matching shape parts, and for extracting 2D/3D features to retrieve the most similar node. Part matching scheme Initialisation: 1. Choose an object M . 2. Choose a level of resolution. 3. Choose the node m at this level of resolution which corresponds to the part to be matched with other parts in the database. Matching: 4. Compute the similarity function between m and every nodes n at the same level of resolution (or eventually at other close levels) among all the objects N in the database. 5. Test topological consistency on nodes similar to m with respect to a predefined threshold for the similarity function. 6. We consider now the subgraph of m and of all its children as the ARG of the part of interest, that is the ARG of the truncated object region associated to the node m. We consider similarly the subgraphs of all the consistent nodes n as the ARGs of the parts corresponding to nodes n. Then we apply the general matching algorithm described in section 3 to these subgraphs in order to determine the most similar parts to the one chosen in object M as query.

6. Results

Figure 10. Nodes taken from Reeb graphs for part matching (models from [23]).

matching was presented in [14], but the skeletal representation of the objects was not well adapted for this purpose. The lack of a multiresolutional strategy is obvious as a part of a skeleton could be a too coarse description (a cylinder

For each object, the ARG structure with the node attributes are coded in a binary file. Its size is proportional to the number of nodes and increases with the complexity of the object. ARG files are computed once and require ∼10s for an object with 10000 triangles, the most timeconsuming being the computation of the function µ over the object surface. For that we use the Dijkstra algorithm with a complexity in O(N log(N )), N being the number of vertices of the mesh. The ARG files stand for the feature vectors in the indexing framework, and the comparison of two objects takes between 1 to 2 millisecond with a PC P4-2GHz. The 3D model database was created by reconstructing museum objects [18, 22], and downloading objects from the Internet [23, 24, 26]. Most of the models have a high resolution (10000 to 50000 triangles) and closed surface with oriented triangles. To show the advantages of the Augmented Reeb Graph, we have made our similarity tests

between eight different classes of objects (cf. Figure 11). Several of these classes could be easily confused by com-

other objects. For our classification test (cf. Figure 12), we used the values αl = 0.3, βl = 0.15, γl = 0.15, δl = 0.2 and l = 0.2 with L1 norms for the loss function, and αs = 0.65 and βs = 0.35 for the sim function. A test using the original approach of [1] was performed to illustrate the effectiveness of our approach (cf. Figure 13). We

Figure 12. Results of the similarity function on 50 models using the ARG matching method: similar models are retrieved and classes are well separated.

Figure 11. Eight classes of objects were used for our tests. There are different numbers of objects per class for a total of 50 objects (models from [22, 23, 24, 26]).

mon approaches: statues, human models, elongated vases, etc. Some statues are representations of human beings but don’t look like the models of the human class, some others are very unique in the database and should not match with

Figure 13. Results of the similarity function using the original approach in [1]: more unwanted similarity are found between objects of different classes.

use the weight αl = αs = 0.5 and βl = βs = 0.5. As said previously, the lack of geometrical information compared to our extension leads to less nodes matching and then sim-

ilar objects could be missed when the topology matching is wrong. These preliminary results reveal that the ARG method is more efficient to retrieve object classes. The six other tested methods make a strong confusion between statues and other objects (cf. Figure 14). Those with the best results (combined histograms of cord lengths and cord angles with the 1st principal axis [5] and 3D Hough Descriptor [8]) are more noisy than the ARG and still present some class mixing and wrong matching. Moreover, the results concerning the classification of the vases show that they are not efficient enough to distinguish the difference between vases with one or two handles, whereas the ARG is. The D2 shape distribution [6], the complex EGI [10], the 3D shape spectrum descriptor [8] and the surface/volume ratio [4] were not efficient on our database. The Figure 15 illustrates the Reeb graph for objects of different classes and obviously its hability to catch the genus of the object and then its specific power for graph matching.

Figure 14. Results of the six other tested methods: there are lots of noise and classes are mixed.

Figure 15. Reeb graphs of different objects at level of resolution 4: the graphs keep the genus (models from [22, 23]).

7. Conclusion Keeping in sight our problematic of textured 3D object indexing and content-based retrieval, our study leads us to enhance the approach presented in [1] by introducing the Augmented Reeb Graph. Geometrical and colorimetric attributes are added to the topological consistency criteria which are also extended. The loss function calculation is consequently modified, and the similarity function is improved by merging the different 2D/3D features (geometrical and visual). Then, we obtain a multiresolutional and multicriteria shape descriptor. Our results demonstrate the ability of the ARG to retrieve similar objects classes in a database, and particularly objects with long appendices, or with the same genus. However this method seems not so well adapted for compact objects as busts or faces. And actually the function µ works with meshes that have only one connected component, thus the objects should have been correctly modelized. Further works could be done to improve the characterisation of the texture. For example by unfolding the texture associated to each node (patch), and reparametrizing it to extract spatial or frequency informations. Our method aims the interactivity and the flexibility, but requires lots of parameters (weight terms of the different features). It could be then interesting to apply optimisation techniques on parameters to define default values in case of specific applications for which learning sets could be provided. Perspectives of research also concern the choice of the function µ and the construction of the Reeb graph. It seems that associating methods using a robust graph (which allows topological perturbation) such as the Extended Reeb Graph [15] with our approach of Augmented Reeb Graph (with its multiresolutional aspect), could be very efficient. The similarity function can be a useful tool to compare objects with similar shapes, especially when associated to 2D and 3D features. For example, they could establish measurements between two objects depending on their vol-

ume, curvatures or visual differences. Moreover, it is also possible to use these informations to make classifications with respect to any of the criteria offer by the extracted features (colour,size,...). In fact, as the technology of the computer keeps on growing (CPU, memory and HD size), Augmented Reeb Graph will allow us to integrate and merge all of the future most relevant shape descriptors. Also, other applications such as part matching seems accessible with our approach but has not been developed yet and remains to be fully tested. Finally, we would like to point out again the flexible and robust aspects of our approach, its effectiveness and accuracy relying on the fusion of different techniques.

Acknowledgements This work has been partially supported by the SCULPTEUR European project IST-2001-35372 [22]. The authors would like to thank Pierre-Alexandre Pont for having coded the algorithm of [1] and for the additional topological consistency criterion, and Bianca Falcidieno and Marco Attene for their precious advice concerning the function µ.

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