Plumber: a method for a multi-scale decomposition of 3D shapes into tubular primitives and bodies

Transcription

1 ACM Symposium on Solid Modeling and Appliations (2004) P. Brunet, N. Patrikalakis (Editors) Plumber: a method for a multi-sale deomposition of 3D shapes into tubular primitives and bodies M. Mortara G. Patanè M. Spagnuolo B. Falidieno J. Rossigna Abstrat Plumber is a speialized shape lassifiation method for deteting tubular features of 3D objets represented by a triangle mesh. The Plumber algorithm segments a surfae into onneted omponents that are either body parts or elongated features, that is, handle-like and protrusion-like features, together with their onave ounterparts, i.e. narrow tunnels and wells. The segmentation an be done at single or multi-sale, and produes a shape graph whih odes how the tubular omponents are attahed to the main body parts. Moreover, eah tubular feature is represented by its skeletal line and an average ross-setion radius. Categories and Subjet Desriptors (aording to ACM CCS): I.3.5 [Computer Graphis]: Curve, surfae, solid, and objet representations 1. Introdution Given a two-manifold losed surfae represented by a triangle mesh, Plumber automatially extrats the features that an be desribed as generalized ylinders or ones; we all these features, together with their onave ounterparts, i.e. narrow tunnels and wells, tubular features. The Plumber approah lassifies the verties of a given triangle mesh aording to their urvature and shape behaviour in neighbourhoods of inreasing size (see Figure 1, 2). Seed verties are loated on tubular features, and lustered to form andidate seed regions whih are then used to ompute the first reliable tube setion, alled the medial loop, whih is ensured to be around eah andidate tube and whih works as a generator of the feature. Then, the medial loop is moved in both diretions on the surfae, by using spheres plaed at the baryentres of the new medial loops, until the tube is ompletely swept. The size of the tube is related to the radius of the sphere, and the stop riterion is given by the abrupt variation of the medial loops lenght. The tube detetion is devised in order to work in a multi-sale fashion, where small tubes are deteted at first and larger ones at following steps. After the surfae segmentation, a geometri representation of eah tubular feature is onstruted by omputing its skeletal line. The onfiguration of eah feature, whose setion and length an arbitrarily vary, and its attahments to the body are hierarhially oded in a shape graph. Different appliation fields make the surfae segmentation an important task. For instane, while tubular strutures an be quite easily defined during design proesses their automati extration Istituto di Matematia Appliata e Tenologie Informatihe, Consiglio Nazionale delle Rierhe College of Computing, Georgia Institute of Tehnology, Atlanta (a) Figure 1: Tubular features reognized by Plumber on a omplex model: (a) tube axis and loops, (b) tubes olored with respet to their sale. from 3D meshes is not a trivial task. We believe that a variety of appliations, espeially shape reognition and analysis, will benefit if tubular features are identified and abstrated to a entreline and a set of setions. These abstrat models, may failitate the measurements of hanges over time in medial appliations (e.g. alifiation proess), or detet abnormalities suh as unnatural narrowing or ballooning. Finally, reliable ylindrial models are essential for proper design of prostheti tubular strutures. The basi idea of Plumber onsists of desribing a shape by using both loal point-wise, and global region-wise measures for shape deomposition and skeleton extration; in the following, we review the state of the art on those onepts used in the paper. (b)

2 M. Mortara, et al. / Plumber: a method for a multi-sale deomposition of 3D shapes into tubular primitives and bodies (a) (b) () (d) Figure 2: Plumber method: (a) identifiation of limb verties, (b) extration of their onneted omponents and medial loop, () iteration, (d) tube and a ap (blak) found at this sale. Deomposition methods based on the analysis of the shape boundary evaluate loal harateristis to identify pathes of the surfae that group verties with similar properties with respet to some measure. In most ases, surfae verties are lustered using the Gaussian urvature: for instane, the segmentation of a free form surfae into pathes of similar urvature is one of the key steps in reverse engineering [SJTH99, VMC97], and for the validation and verifiation of visualization produts to ontrol mesh quality [ZP01]. In [KT03], the segmentation method is defined as a fuzzy lustering of verties where the probability that a fae belongs to a path depends on its distane from the other faes of the path. The advantage of the method is the avoidane of over-segmentation, and that boundaries between adjaent regions are not jagged. The results show that the segmentation is meaningful, in the sense that the extrated omponents loate the main natural features of the objets. Skeletons suh as the Medial Axis Transformation (MAT) and the Reeb Graph assume that the surfae is the boundary of a volume, and analyse the shape aording to its interior, thus providing desriptions whih better highlight its global struture. The MAT is onstruted using the paradigm of the maximal enlosed spheres, whose entres define a lous of points whih desribes, together with the assoiated radius, the width variation of the shape. The MAT of a 3D surfae is generally a non-manifold omplex, omputationally heavy, and sensitive to noise beause tiny perturbations may produe a whole new ar. Furthermore, there is not a diret relation between tubular features and speifi omponents of the MAT, espeially when the tubes have an arbitrary shape and the ross setions do not exhibit any symmetry. More relevant for the identifiation of tubular features are methods for the extration of skeletons, whih provide an abstrat shape representation by a graph of lines that retain the onnetivity of the original shape. The Reeb graph [SKBT96, VL00, BMMP03] is a topologial struture whih odes a given surfae by storing the evolution of the level sets of a mapping funtion defined on its boundary. In [LTH01], tubular parts are identified using a sweeping tehniques along the ars of the skeleton whih is onstruted by joining the edges remaining after an edge ollapse proess on the whole mesh. These edges are linked in a tree struture, and it is used as a support for the sweeping proess where the mesh is interseted by a set of planes and tubes are identified by looking at the geometry of the ross-setions. The main differene between Plumber and segmentation methods previously disussed is that we extrat building primitives of the objet with a speifi struture, i.e. generalized ones and ylinders, and not only related to a urvature and onavity analysis [MPS 04, KT03]. Furthermore, while skeletal representations do not provide a sale-based deomposition of the shape and are usually unstable with respet to wripples or wrinkles, Plumber differentiates tubular features of different dimension. The reminder of the paper is organized as follows; in setion 2 the Plumber method is detailed, and the disussion on possible appliations onludes the paper. 2. The Plumber method Intuitively, ideal tubes are identified by parts of the shape whose intersetion with a sphere of appropriate radius produes two intersetion urves. The setion of the tube and its axis an be arbitrarily shaped, and the size of the tube is kept as a onstraint during the identifiation proess. Chosen a level of detail R, Plumber performs the following steps: 1. identify limb-regions assoiated with at least two loops on M (see Figure 2(a)); 2. shrink eah of the two seleted boundary omponents along the surfae to its medial-loop, whose points are nearly equidistant from the two border loops (see Figure 2(b)); 3. expand-bak the medial-loop by sweeping the extent of the shape in both diretions. More preisely, at eah iteration we plae a sphere of radius R in the baryentre of the new medial loops. If the intersetion between the sphere and the surfae generates two loops, mesh verties inside the sphere are marked as visited; 4. the proedure is iterated in both diretions until: no more loops are found, or more than one loop is found on not-visited regions; the new loop lies on triangles that are already part of another tube, or the length of the new loop exeeds a pre-defined threshold. 5. the tube skeleton is extrated by joining the loops baryentres. The previous steps are detailed in the following paragraphs. Vertex lassifiation Given an inreasing set of radii R i, i 1 n, Plumber haraterizes a 3D mesh M in a neighbourhood

3 M. Mortara, et al. / Plumber: a method for a multi-sale deomposition of 3D shapes into tubular primitives and bodies of a vertex p at the sale R i by analysing the evolution of the onneted omponents of the urve γ i : M S p R i, where S p R i is the sphere of enter p and radius R i [MPS 04, MP02]. The following lassifiation is used: 1 boundary: the surfae around p is onsidered topologially equivalent to a dis (see Figure 3(a)). 2 boundary omponents: the surfae around p is tubular-shaped (see Figure 3(b)). Their lengths are used to distinguish between oni and ylindrial shapes, and p is lassified as a limb-vertex. n 3 boundary omponents: in a neighbourhood of p a branhing of the surfae ours (see Figure 3()). Therefore, the next step defines a riterion for judging if a limbregion is a good andidate for the tube identifiation; that is, the limb-region is around the tube. For instane, in the ase that the tube setion is ellipsoidal and its size is greater than the hosen sale, it may happen that the spheres used to lassify the verties produe only one intersetion urve on one side of the tube, and two on the other side (see Figure 5), thus giving rise to a limb-region not surrounding the tube. Therefore, the region is not tube-shaped at the given sale and it has to be disarded; it will be found at a larger sale. (a) (b) () Figure 3: Different ases of sphere to surfae intersetion. Interseting the mesh with a sphere with radius R i allows to identify limb-verties if they lay on a tube of diameter R i or smaller. At eah vertex p M, we onsider three spheres of radius R i ε, R i, and R i ε with ε given threshold proportional to the minimum edge in the triangulation. We onsider limb verties those ones whose urve γ i has two or more boundary omponents (see Figure 4). This lassifiation improves stability for identifying tubes of arbitrary ross setion where isolated limb-verties ould appear; a striter hoie onsists of onsidering as limb verties those ones lassified with the same label at all the three sales. Figure 5: Example of limb-regions (in yellow) whose verties on γ have one boundary omponent. Medial loop generation Seed tubular regions are used to onstrut a medial loop around eah andidate tube that will be used for the tube identifiation and its entreline onstrution. Beause we have already deteted all the andidate tubular regions, a seed point for eah tube is seleted; for instane, we ould hoose the entroid of eah region, i.e. the point with maximum distane from the region boundary, and then generate the loop with one of the methods proposed in [VF02, GW, LPVV01]. Instead, Plumber relies on the limb-region boundaries whih are loops surrounding the tube. The idea is to find the medial loop by moving the boundary loops in the middle of the limb-region; to this end, we perform a morphologial shrink by simultaneously invading the omponent from its two boundary omponents. (a) (b) () (d) Figure 4: In yellow limb-verties found at sale R ε (a), R (b), and R ε (). All the limb verties are depited in (d). The hoie of the set R i i 1 n is related to the sale of the features whih have to be extrated, and for performing a multi-sale analysis of the shape; small radii determine details, while bigger ones are used to analyse the global harateristis of the surfae. Further disussions are given in the paragraph Mutli-sale analysis. Identifiation of tube andidates from limb verties The seond step is the identifiation of the maximal edge-onneted omponents of limb-verties, using a depth-first searh. Note that while the analysis of the evolution of γ i produes a vertex-oriented lassifiation of M, regions omposed by limb-verties are not guaranteed to be tube-shaped as a whole (see Figure 2(a), on the handle). In partiular, limb regions may have not two boundary urves. (a) (b) () (d) (e) Figure 6: A tubular region affeted by small features, like the neel. The onfiguration of the sphere/mesh intersetion is depited, with spheres entred in verties of different feature types: (a) limb, (b) blend, (), (d) tip, (e) split. Firstly, the two boundary omponents of the limb-region (the two of greater length if the region has three or more border loops, as in Figure 6(e)) are omputed. Let R and L (for right and left respetively) be the two boundary omponents of the tube; at first, eah vertex p on L is assoiated with the ouple 0 that indiates that p has distane 0 (resp. ) from the boundary L (resp. R). The same initialization applies to R.

4 M. Mortara, et al. / Plumber: a method for a multi-sale deomposition of 3D shapes into tubular primitives and bodies The distane of a vertex p from one boundary is omputed as the shortest edge path onneting p with a boundary vertex. Then, the distane values of all points are updated, propagating from L and R towards the interior of the region. The distane propagation from L will update the first value of the distane vetor, while that from L will affet the seond field; at the end of the proess, verties are lassified as nearer to L or R (see Figure 7). Edges onneting verties of different type are ut by the medial loop we are looking for; that is, we join the mid-points of those medial edges to produe a medial loop. This onstrution ahieves two good effets with respet to other methods [Kar99, VF02]: the medial loop is guaranteed to be non-trivial and inside the region. The non-minimality of its length does not affet the growth of the tube, and the onstrution of the skeleton. In the ase of three or more boundary omponents, the hoie of starting from the two loops of greater length is to guarantee a stronger reliability to the tube extration with respet to smaller intersetion urves whih may be due to loal undulations of the shape. setion and its baryentre ontributes to the skeleton as a new node. Otherwise (see Figure 8(b), in the oval), the growth stops. the intersetion ounts two, or more loops; that is, a bifuration ours (see Figure 8(b)). The growing of the tube in this diretion stops, and the last visited triangles are unmarked. Finally, the baryentres of the medial loops are joined to define the tube skeleton. (a) Figure 8: (a) No new loop is found on the snake tail (in the box), and a loop disarded after the length hek on the head (in the oval). (b) A branhing ours on the dolphin tail. (b) (a) (b) () Figure 7: (a) Limb verties, (b) onneted omponent of the limb verties with two boundary omponents, and medial loop (marked urve), () medial sphere entred in the baryentre of the medial loop, and tube growing. Loop expansion and skeleton onstrution The loop expansion is ontrolled by a verifiation proedure whih, at eah step, extends the enter-line and at the same time ensures that the surfae is tubular around it. A first medial sphere is drawn, whose entre p is the baryentre of the medial loop, and whose radius is R. If M S p R has not two boundary omponents, the growing stops and the andidate tube is disarded. Otherwise, a new sphere with the same radius is entred in the baryentre of the two loops; the proess is then split into two parts, trying to grow the tube in both diretions. Now we fous on the sphere moving in one of the two diretions, sine the other ase is symmetri. At eah iteration, the sphere rolls to the baryentre of the next loop, and the triangles laying ompletely or partially inside the sphere are marked as belonging to that tube. Then, the intersetion between the sphere in the new position and the mesh is again omputed, taking into aount only the intersetion urves through non visited triangles (all the spheres exept the medial one have always a bakward loop, passing on the already marked triangles). During the loop expansion, the following ases may arise: no intersetion urves are found. This is the ase of a tubular protrusion terminating in a tip; visited triangles loate a ap (see Figure 8(a), in the square); the intersetion urve onsists of one loop (see Figure 8(a)). If its length is less than a pre-defined threshold, the size of the tube setion is not varying too muh; the loop beomes a new ross Multi-sale analysis The extration of tubes at sales R 1 R n adopts a fine-to oarse strategy, marking triangles as visited while the tube grows and whih are not taken into aount during the following steps (see Figure 9). Analogously, the medial loop omputation simply does not take into aount smaller tube verties, propagating distane values only on not-visited verties (Figure 9(d)). Deisions are taken when the loop passes partially on not-visited and tube triangles. For example, in the ase depited in Figure 9(e), the two smaller loops fall on tube triangles, and are not ounted; therefore, this is the ase of two intersetion loops, and not that of a branhing. The tube is grown, and the result of the two iteration steps is shown in Figure 9(f). This set of radii is seleted by the user, or assigned by uniformly sampling the interval from the minimum edge lenght in M to that of the diagonal of its bounding box. At the end of the whole proess, tubes are labelled with respet to the sale at whih they were found. The onneted omponents of the shape parts whih are not lassified as tubes or aps define body parts of the objet, and the resulting deomposition is oded in a tube-body onnetivity graph whih represents the spatial arrangement of the tubular features onto bodies. A smooth transition of radii ensures a meaningful growth of the tube at a sale R i, while disarding smaller features and analysed at the previous levels of detail R j j 1 i 1. Strit/ non strit mode Together with the size of the sphere rolling over the entreline, the other parameter to be fixed in the tube growing step is the threshold in the loop length hek. To this end, we stop the growing when the tube beomes too large, i.e. the length of the intersetion loops varies too muh. If we onsider a natural objet, we probably do not want to deompose natural limbs into piees; on the other hand, in the ase of a manufatured model, we may want to be preise with respet to the tube size, eventually splitting a tube into omponents of

5 M. Mortara, et al. / Plumber: a method for a multi-sale deomposition of 3D shapes into tubular primitives and bodies Table 1: Plumber timings (m:s:ms) performed on an Athlon 1000 MHZ. Model n V Vert. lassif. Medial loop Tube grow (a) (b) () Cylinder se Pot (1 iter.) se Pot (2 iter.) se Shale se (d) (e) (f) Figure 9: Iteration of Plumber at inreasing sales. (even slightly) different setions. For this reason, two alternatives are available (see Figure 10): a strit mode, useful in CAD and medial appliations, a non-strit mode, for other appliations where a ontinuous variation of the tube size does not require to split the tube. In the strit mode, eah time a new loop is generated its length is ompared with that of the intersetion loops assoiated to the medial sphere at the beginning of the proess, and not with the length of the medial loop tube setion whih an be non minimal and misleading. Other hoies were also taken into aount, suh as average, minimum, and maximum tube length; the one adopted is a ompromise between the required stritness and robustness. In the non strit mode, a loop is aepted if its length is less or equal to twie the length of the previous loop. In both ases, the user an selet values on the base of a-priori information or speifi needs. (a) (b) () (d) (e) (f) (g) (h) (i) (l) Figure 10: (a) Initial level of detail, (b) limb-region, () tube growing in non-strit mode, and (d) tube extration. In (e) tube growing from the same limb-verties in strit mode, (f) the extrated tube, (g), (h) (i) next iterations, (l) ahieved segmentation at the hosen sale. Shape graph Throughout the previous paragraphs we have detailed a method for identifying and lassifying tubes of different size and bodies ahieving a segmentation of the input objet. We enrih this geometri lassifiation with an expliit representation of the struture of the model whih odes the relations between primitives in a hierarhy of tubes and bodies. This strutured representation is a shape-graph whose nodes are the extrated primitive shapes, while the ars ode the adjaeny relation among the previous ones, i.e. their relative position and orientation. Eah node is a tube, whose labels are the medium radius and the axis length, a body, whose labels are the number of holes and the approximate volume, or a ap, whose labels are the basis setion, the axis length and the urvature extrema. Eah ar between two adjaent nodes falls into one of these ases (see Figure 11): tube-body, tubetube, ap-tube. The tube-body or tube-tube adjaeny is alled H- juntion (i.e. handle-juntion) if both boundaries of the tube lay on the same body or tube respetively; in this ase, the ar is a loop and the tube loates a handle on the input objet. In the ase that only one boundary of the tube belongs to the tube-body the adjaeny is alled a T-juntion. Computational omplexity The predominant ost of the method is represented by the initial surfae haraterization [MPS 04] to detet limb verties, whih is O n 2 V with n V number of mesh verties. The following tube extration proedure is muh faster. The lustering of limb verties into regions is treated triangle-wise; starting from a first seed triangle having three limb verties, the region is onstruted adding neighbouring limb triangles through a breadth-first searh. The boundary omputation of a region is linear in the number of verties of the region: all the verties are visited, and when a seed boundary vertex is found, the boundary loop whih it belongs to is omputed moving by adjaeny. The medial loop omputation is in the worst ase very expensive: the problem of omputing the minimal distane between two verties an be solved by the Dijkstra s algorithm in O nlog n, where n is the region ardinality, but in our ase the minimum distane from all the boundary points takes O n 2 log n. In pratie boundary verties are muh less than n, about n 1 2, thus reduing time omplexity in the average ase. The tube growing proedure onsists at eah step in a triangle visit, starting from those laying on the previous medial loop, until a triangle interseted by the sphere is found. Eah triangle inside the sphere is visited one, and the omputation of the intersetion urve itself is linear in the number of interseted triangles, determined by adjaneny. So eah tube is grown in linear time with respet to the number of triangles it inludes. Timings are reported in Table 1.

6 M. Mortara, et al. / Plumber: a method for a multi-sale deomposition of 3D shapes into tubular primitives and bodies (a) (b) () (d) Figure 11: (a) Centrelines on a tea-pot with respet to two levels of detail, (b), () Segmentation of the tea pot into ap, body, tubes and adjaeny relations, (d) shape graph. 3. Appliations and Conlusions The Plumber algorithm provides a multi-sale method to deompose a omplex shape into its tubular features and bodies. The segmentation onsiders as bodies those regions that are not tubular shaped; therefore, a sub-lassifiation of these primitives is neessary. Main diffiult tasks are their identifiation, general onfiguration and the identifiation of a basi shape for the abstration. The interpretation and ategorization of tubular features has the drawbak of introduing heuristi thresholds to make deisions on the tube size, or for distinguishing branhing parts from ompliated onfigurations of tubes as in Figure 1, and 9. The redution of the influene of these parameters and the abstration of tubular features with generalized ylinder and ones for ollision detetion appliations are the further improvements of Plumber. 4. Aknowledgements This work has been supported by the Researh Agreement Surfae Analysis between GVU/Gateh and IMATI-GE/CNR. Thanks are given to the Shape Modelling Group at IMATI-GE/CNR. Referenes [BMMP03] BIASOTTI S., MARINI S., MORTARA M., PATANÈ G.: An overview on properties and effiay of topologial graphs in shape modelling. In Shape Modeling International (2003), pp [GW] [Kar99] GUSKOV I., WOOD Z.: Topologial noise removal. In Graphi Interfae, pp KARTASHEVA E.: The algorithm for automati utting of three dimensional polyhedrons of h-genus. In Shape Modeling International (1999), pp [KT03] KATZ S., TAL A.: Hierarhial mesh deomposition using fuzzy lustering and uts. Transations on Graphis 22, 3 (2003), [LPVV01] LAZARUS F., POCCHIOLA M., VEGTER G., VER- ROUST A.: Computing a anonial polygonal shema of an orientale triangulated surfae. In Symposium on Computational Geometry (2001), pp [LTH01] [MP02] [MPS 04] [SJTH99] LI X., TOON T. W., HUANG Z.: Deomposing polygon meshes for interative appliations. In Symposium on Interative 3D graphis (2001), pp MORTARA M., PATANÈ G.: Shape overing for skeleton extration. International Journal of Shape Modeling 8, 2 (2002), MORTARA M., PATANÈ G., SPAGNUOLO M., FAL- CIDIENO B., ROSSIGNAC J.: Blowing bubbles for the multi-sale analysis and deomposition of triangle meshes. Algorithmia, Speial Issues on Shape Algorithms 38, 2 (2004), , 3, 5 SACCHI R., J.F. P., THOMAS P., HAFELE K.: Curvature estimation for segmentation of triangulated surfaes. In 3-D Digital Imaging and Modelling (1999), pp [SKBT96] SHINAGAWA Y., KUNII T., BELAYEV A., TSUKIOKA T.: Shape modeling and shape analysis based on singularities. International Journal of Shape Modeling 2, 1 (1996), [VF02] VERROUST A., FINIASZ M.: A ontrol of smooth deformations with topologial hange on a polyhedral mesh based on urves and loops. In Shape Modeling International (2002), pp , 4 [VL00] VERROUST A., LAZARUS F.: Extrating skeletal urves from 3d sattered data. The Visual Computer 16, 1 (2000), [VMC97] VÁRADY T., MARTIN R. R., COX J.: Reverse engineering of geometri models - an introdution. Computer-Aided Design 29, 4 (1997), [ZP01] ZHOU L., PANG A.: Metris and visualization tools for surfae mesh omparison. In Symposium on Eletroning Imaging: Siene and Tehmology (2001). 2