Adaptive Implicit Surface Polygonization using Marching Triangles

Transcription

1 Volume 20 (2001), Number 2 pp Adaptive Impliit Surfae Polygonization using Marhing Triangles Samir Akkouhe Eri Galin L.I.G.I.M L.I.G.I.M Eole Centrale de Lyon Université Claude Bernard Lyon 1 B.P. 163, Eully Cedex Villeurbanne Cedex samir@e-lyon.fr egalin@ligim.univ-lyon1.fr Abstrat This paper presents several improvements to the marhing triangles algorithm for general impliit surfaes. The original method generates equilateral triangles of onstant size almost everywhere on the surfae. We present several modifiations to adapt the size of the triangles to the urvature of the surfae. As raks may arise in the resulting polygonization, we propose a speifi rak-losing method invoked at the end of the mesh growing step. Eventually, we show that the marhing triangles an be used as an inremental meshing tehnique in an interative modeling environment. In ontrast to existing inremental tehniques based on spatial sudvision, no extra datastruture is needed to inrementally edit skeletal impliit surfaes, whih saves both memory and omputation time. Keywords: impliit surfaes, inremental meshing, marhing triangles, polygonization 1. Introdution Impliit surfaes have drawn a lot of attention for the past few years. They have been suessfully used in the design of smooth objets of arbitrary topology and geometry, as well as in desriptive and physially based animation systems. An impliit surfae is mathematially defined as a set of points in spae x that satisfy the equation f x 0. Thus, visualizing impliit surfaes typially onsists in finding the zero-set of f, whih may be performed either by polygonizing the surfae or by diret ray-traing. Despite several aelerated shemes , most impliit surfaes annot be ray-traed at interative rates, whereas polygonization algorithms provide a fast representation of the surfae that may be rendered with the now demoratized graphi hardware. As mentioned by some authors 20, polygonal meshes tend to beome the standard representation for surfae geometry in many omputer graphis appliations. Published by Blakwell Publishers, 108 Cowley Road, Oxford OX4 1JF, UK and 350 Main Street, Malden, MA 02148, USA. Ideally, a good polygonization of an impliit surfae should meet the following numerous riteria. The generated mesh should be onsistent with itself, i.e., without holes or disonneted verties, edges or faes. It should be topologially orret, whih implies that the polygonal approximation should be homeomorphi to the impliit surfae. The approximation should be aurate enough so that small features of the impliit surfae should not be missed. Triangles or polygons should not be distorted, i.e., very thin triangles should be avoided whenever possible. Eventually, the mesh should adapt to the urvature of the surfae, produing large polygons in flat regions and small ones in regions of high urvature. Spatial sampling tehniques are among the most popular methods. Those tehniques regularly or adaptively sample spae to find ells straddling the impliit surfae. Those ells are triangulated aording to their onfiguration type, whih is derived from the field funtion values at their verties. Although several attempts have been made to guarantee both the topologial onsisteny and the topologial orretness of the triangle mesh, those methods often generate numerous distorted triangles. Several triangle deimation tehniques as well as re-tiling algorithms 31 have been proposed to eliminate badly shaped triangles. Those methods need to be invoked as a post proessing step. In fat, very few methods diretly address the aspet of the generated triangles.

2 S. Akkouhe and E. Galin / Adaptive Impliit Surfae Polygonizationusing Marhing Triangles The reently presented marhing triangles algorithm 15 diretly generates impliit surfae meshes with almost equilateral triangles everywhere. The proposed method annot be diretly applied to losed general impliit surfaes however, as raks may appear in the polygonization. The very limitation of the original algorithm is the fixed step length used in the reation of triangles marhing on the surfae, whih may result in a loss of both topologial and geometrial features. In this paper, we propose a generalization of the algorithm by adapting the step length to the loal urvature of the impliit surfae. Craks that sometimes appear in the mesh growing step are losed by a speifi method. We show that the marhing triangles algorithm is an interesting alternative to the polygonization of impliit surfaes in the general ase. Our method an diretly reate a both topologially onsistent and topologially orret approximation of the surfae. Our experiments illustrate that this method performs as fast as spatial sampling tehniques, generates fewer and better shaped triangles. As shown in 12, a skeletal impliit surfae may be inrementally designed by adding or removing primitives, or by hanging some of their parameters. Most editing operations affet the field funtion loally. We show that the adaptive marhing triangles lends itself to the inremental polygonization of skeletal impliit surfaes. The triangles lying in regions of spae where hanges ourred are removed, and the algorithm may be invoked from the resulting open ontours to ompute the new mesh and adapt to the features of the modified surfae. The paper is organized as follows. Setion 2 presents an overview of existing polygonization methods, and ompares the quality of the generated mesh. Setion 3 deals with the adaptive marhing triangles algorithm. To make this paper self-ontained, we present the original method proposed in 15 and analyze the different steps that need to be modified or improved before presenting our own implementation. We ompare our method with some ellular surfae traking tehniques, both in terms of effiieny and mesh quality. Eventually, we present the inremental marhing triangles algorithm dediated to skeletal impliit surfaes in Setion 4. We show how to take advantage of shape oherene to adapt the algorithm to interative editing environment. 2. Related work Existing tehniques may be lassified in three ategories. Spatial sampling tehniques regularly or adaptively sample spae to find ells straddling the impliit surfae, and tesselate ells to reate the overall polygonization. Surfae traking approahes iteratively reate a triangulation by marhing along the surfae. Eventually, surfae fitting tehniques progressively adapt and deform an initial mesh to onverge to the impliit surfae. The quality of a mesh may be addressed both in terms of geometrial auray and topologial onsisteny and orretness. Topologial onsisteny requires that the polygonal approximation should be onsistent with itself, for instane without holes or disonneted verties, edges or faes, whereas topologial orretness implies that the polygonal approximation should be homeomorphi to the impliit surfae. The geometrial quality of a triangulated mesh not only involves the detetion and the aurate polygonization of small features of the impliit surfae, but also the avoidane of distorted triangles. Spatial sampling tehniques Those methods subdivide spae into ells and searh for only those ells that interset the impliit surfae. In general, ells are either ubes or tetrahedra. The sign of the potential field at the verties of the ells generates a onfiguration type 3 that haraterizes the polygonization approximating the impliit surfae. Tetrahedra generate topologially onsistent meshes 1, yet with numerous, often over distorted, triangles. The omputation of the mesh may be performed unambiguously for tetrahedra, whih results in a topologially onsistent triangulation 1. Triangle deimation tehniques 27 18, vertex reloation 1 or generi re-tiling 31 may be invoked as a postproessing step to eliminate badly shaped triangles, whih slows down the overall meshing algorithm. Ambiguous onfigurations represent the main drawbak of ubi ell based polygonization shemes. Ambiguities arise with ubi ells as several meshes may be proposed for the same onfiguration type. They are in general taken as unaeptable for mesh topologial onsisteny reasons. Several disambiguation strategies have been addressed 10. They inlude, and are not restrited to, simplex deomposition 3, modified look-up table disambiguation 23, gradient onsisteny heuristis and quadrati fit 10, tri-linear interpolation tehniques 25 24, and reursive subdivision 4. Exhaustive enumeration methods use a voxel deomposition of spae to generate ubi ells This approah is omputationally expensive and does not adapt to the loal geometry of the impliit surfae, possibly missing parts or generating a poor approximation. Adaptive subdivision tehniques onverge to the surfae by reursively subdividing spae into smaller sub-ells 3. Some ellular polygonization shemes an guarantee that the surfae is ontained within the union of a set of small ells. Both the Lipshitz ondition 17 and interval analysis 28 provide riteria that disard some ells that do not straddle the impliit surfae. The remaining ells where the riteria fail are reursively subdivided. As demonstrated in 3, a onventional otree deomposition of spae may introdue artifats in the polygonization

3 S. Akkouhe and E. Galin / Adaptive Impliit Surfae Polygonizationusing Marhing Triangles suh as raks that are reated between the triangles of adjaent ells of different size. Several authors have proposed tehniques to solve the rak problem, a good overview may be found in 32. Surfae fitting tehniques In ontrast to spatial sampling tehniques, surfae fitting methods do not rely on a lassifiation of spae into ells. They reate a seed mesh that roughly approximates the impliit surfae, and progressively adapt and deform it to the impliit surfae. Element driven tehniques 9 8 are dediated to impliit surfaes reated from skeletal elements. In this approah, eah element is haraterized by a set of seed points loated on its bounding volume. Seeds migrate towards the surfae along predefined diretion vetors to generate a mesh approximating the impliit surfae. The main strength of this strategy is its effiient primitivedriven approah that operates on elements instead of meshing the impliit surfae as a whole. It offers interative updates when adding or removing primitives, whatever the topology and the omplexity of the impliit surfae. The main diffiulty onsists in attahing the triangle pathes together. In pratie, the algorithm sometimes fails at tesselating regions where numerous primitives blend, whih results in holes or even dangling faes in the polygonization. In ontrast, the shrink-wrap method 19 6 is a global approah. Starting from an embedding mesh, it progressively onverges towards the surfae. Therefore, it fails at finding finding holes hidden within the impliit shape. As shown in 29, traking the ritial points provides suffiient bakground for deteting topologial features and reating a polygonization whose topology is homeomorphi to the impliit surfae. Instead of shrinking the surfae, an inflation proess orretly detets and handles holes in the objet. Although this method performs well with a few simple primitives with an gaussian basis, it beomes omputationally ineffiient for more omplex models as the number of ritial points grows exponentially with the number of primitives. A major improvement has been proposed 14 by using a polynomial approximation of the kernel that loalizes the support of the primitives ausing the number of ritial points to grow linearly with the number of primitives. Still, omplex skeletal elements, suh as lines segments, polygons or even polyhedra annot be handled as they generate regions in spae where ritial points annot be traked. Surfae traking tehniques Those methods (also known as ontinuation methods) start from a seed element on the surfae and rely on a mesh growing sheme to iteratively reate a polygonal approximation of the surfae. Starting from a straddling ell, ellular surfae traking tehniques 34 1 iteratively find straddling ells among its neighbors until all neighbors have been heked. They suffer from the same limitations as general spatial sampling methods. They tend to generate numerous triangles that are often over distorted. Sine ells are of onstant size, the algorithm may miss small features. Moreover, surfae traking tehniques require a seed ell for eah surfae omponent whih may be diffiult to reate in the general ase. An alternative tehnique onsists in onstraining a partile system to the impliit surfae A Delaunay triangulation has to be invoked as a post-proessing step to reate the orresponding polygonization. This step involves extra omputations that slow down the overall proess. The polygonization an be inrementally updated by adding or removing elements 26. However, disjoint piees of the impliit surfae as well as holes may be missed if the initial set of partiles was wrongly sattered. Reently, Hilton 15 has proposed a mesh growing tehnique that diretly reates surfae triangles. Starting from a seed triangle on the impliit surfae, the marhing triangles algorithm iteratively reates new triangles from the boundary edges so that the mesh should eventually over the entire impliit surfae. In ontrast to the spatial sampling and surfae fitting tehniques, only the partile system approah and the marhing triangles method diretly generate a set of homogeneously distributed verties on the impliit surfae, and avoid distorted triangles. The algorithm relies on a relaxed Delaunay onstraint to loate new verties on the surfae that will be linked to the boundary edges of the existing mesh. The proposed method annot be diretly applied to losed general impliit surfaes however, as raks may appear in the polygonization. The algorithm is limited by the fixed step length used in the reation of triangles marhing on the surfae. This results in failure of the mesh growing sheme to guarantee the orret topology and the geometrial auray if the surfae urvature is greater than 1 r where r denotes the step length. Those drawbaks will further detailed in the next setion. 3. The marhing triangles algorithm In this setion, we present the adaptive marhing triangles algorithm for general impliit surfaes. Let us reall that this ontinuation method iteratively reates new triangles from a list of boundary edges, initialized with the three edges of a seed triangle, so that the growing mesh should eventually over the entire impliit surfae. The marhing triangles method proposed by Hilton 15 relies on a predition orretion step to ompute new mesh verties on the impliit surfae. As mentioned in 4, although predition orretion methods work well for ontours in 2, diffiulties arise when applied to surfaes in 3. In partiular, predition orretion methods may fail to detet global

4 S. Akkouhe and E. Galin / Adaptive Impliit Surfae Polygonizationusing Marhing Triangles surfae overlap, and therefore annot generate losed surfaes. This setion is organized as follows. In the first part, we reall some theoretial bakground and present the original marhing triangles algorithm as desribed in 15. Speifially, we explain why this algorithm may fail at generating losed meshes for general impliit surfaes. In the seond part, we show that the reation of new surfae verties is a key step in the mesh growing algorithm. Therefore, we present some implementation details, and propose a tehnique for eliminating the raks that may appear in the polygonization Bakground In this setion, we present the marhing triangles algorithm desribed by Hilton 15. Let X x 0 x n denote a set of points in 3. The Delaunay triangulation of X is omposed of tetrahedral volumes V suh that there exists a sphere passing through the verties of V that does not ontain any other point of the set X. Boissonnat 5 has demonstrated the following important property : Property Whenever the set of points X lie on a manifold surfae, the Delaunay triangulation is haraterized by the ondition that it is omposed of triangles T suh that there exists a sphere passing through the verties of T that does not ontain any other point of the set X. 2 x i T Figure 1: Delaunay onstraint for the reation of a new triangle T x p Hilton 15 has proposed a surfae generation riterion derived from this property. Given a partial triangulated approximation of the surfae, the proposed tehnique relies on the loal Delaunay property to reate new triangles and generate the triangulation of the surfae. Starting from a seed triangle on the impliit surfae, the marhing triangles algorithm iteratively reates new triangles on the surfae from the boundary edges of the triangulated surfae. Notations Let us define some notations that will be used throughout this setion. A triangle T x y z will denote a triangle with verties x, y and z. The orientation of the triangle will be defined by its normal n T. Two triangles T and T will said to have the same orientation if the dot produt of their x p normals n T n T is positive. Let T denote the irumenter of the triangle T. The following Delaunay surfae onstraint guarantees that eah new triangle T uniquely defines the loal surfae. Triangle reation onstraint A triangle T x p an be added to the mesh boundary at edge e if no part of the surfae of the existing mesh, i.e., no existing triangle, intersets the sphere entered at T irumsribing the triangle T x p with the same orientation (see Figure 1). The orientation riterion of the triangle generation onstraint allows the polygonization of impliit surfaes that exhibit thin folding setions (see Figure 2). n T n T x i 2 S x p n T Figure 2: Creation of a new triangle T : the empty sphere riterion does not apply as the sphere S interset another part of the mesh (at vertex x i ) whose surfae normals n T and n T exhibits a different orientation than n T Starting from a seed triangle on the impliit surfae, the marhing triangles algorithm iteratively reates new triangles from the boundary edges so that the growing mesh should eventually over the entire impliit surfae. The new edges generated by adding new triangles to the polygonization are appended to the end of a list of boundary edges, referred to as Le. The general algorithm detailed in 16 addresses both losed and open surfaes. When dealing with open surfaes, the algorithm requires an expliit representation of the boundary of the surfae provided by a boundary funtion denoted as b x, whih is defined as false if the point x is internal to the surfae and true if x lies on the boundary. In the following paragraphs, we do not take this boundary funtion into aount as we address the triangulation of general impliit surfae manifolds. The algorithm proeeds as follows, iteratively analyzing eah edge e in the list : 1. Create a new vertex x in the plane of the triangle T x i that ontains the edge e. This point will be used as a first guess in the omputation of the surfae vertex x p.

5 S. Akkouhe and E. Galin / Adaptive Impliit Surfae Polygonizationusing Marhing Triangles Figure 3: Craks may appear during the mesh growing step if no speifi step is performed to lose the mesh 2. Create a new surfae vertex x p by projeting x onto the impliit surfae following the gradient of the field funtion f. 3. Apply the Delaunay surfae onstraint to the new triangle T x p and proeed as follows : 3.1. If T x p passes the onstraint, then add the triangle to the mesh and stak the edges e x p and e x p to the list of edges Le that need to be proessed If T x p does not pass the onstraint, hek if one of the triangles T and T 2 satisfy the Delaunay surfae onstraint, and modify the mesh aordingly if needed Otherwise, step over the edge e to the next andidate edge. The method is implemented as a single pass through the edge list Le. Whenever the mesh growing sheme fails, the edges are left in the edge list. At the end of the algorithm, Le forms an open ontour in the polygonization. Therefore, although the marhing triangle tehnique performs well in the ase of open impliit surfaes, it may generate raks for losed impliit surfaes (see Figure 3). The first two steps of the algorithm aim at evaluating a new vertex loation x p on the impliit surfae to reate a andidate triangle T x p. The third step heks if a new triangle may be reated from the boundary edge, preferably using the new vertex x p, and using only existing boundary verties otherwise. Projetion step Let us reall the projetion algorithm desribed in 15. The reation of the new surfae vertex x p is a key step of the algorithm. Let x denote the projetion perpendiular to the mid-point of the boundary edge in the plane of the triangle T by a onstant distane denoted as d. The new vertex x p is defined as the nearest point on the impliit surfae to x. This method exhibits one major drawbak. In pratie, the point x that lies in the plane of the triangle T x i is always reated at a onstant distane from the edge e. Therefore, the algorithm does not adapt to the urvature of the surfae. The point x may be reated too far away from the existing mesh, espeially in regions of spae where the impliit surfae exhibits high urvature, whih slows down the omputation of the surfae vertex x p. Triangle reation step In the general ase, when the growing mesh starts folding, the triangle reation method keeps failing at generating new triangles that satisfy the Delaunay onstraint. The algorithm 15 an no longer reate a valid andidate triangle from the boundary edge list Le. This leads to unwanted raks in the polygonization as shown in Figure 3. As pointed out in 16, the Delaunay onstraint needs to be relaxed so as to lose the mesh on the fly during the mesh growing sheme, whih requires an extra step in the original algorithm. Unfortunately, unwanted raks may still appear in the polygonization. In pratie, the raks propagate around the surfae in apparently random diretions. We may explain the phenomenon by realling that the propagation of the triangulated surfae is not ontrolled. The boundary edges that need to be further proessed are simply staked in a list, and new triangles are heked and iteratively added in a random way. The following two major reasons explain why the triangulation algorithm annot always terminate. The triangle riterion onstraint only ensures that the sphere S of the newly reated triangle T x p does not interset the existing mesh. Thus, newly reated triangles only satisfy a somewhat relaxed Delaunay onstraint. Although a projeted vertex x p might appear valid (its orresponding sphere S does not interset previously reated triangles), it should be rejeted as it lies inside another sphere S of a previously reated triangle T (see Figure 4). Moreover, the sphere S used in the triangle reation onstraint is always omputed in the same way, although the De-

6 S. Akkouhe and E. Galin / Adaptive Impliit Surfae Polygonizationusing Marhing Triangles S x p T 2 Figure 4: The new vertex x p seems valid aording to the triangle reation onstraint applied to T, although it does not satisfy the Delaunay riterion for the bounding sphere S of the triangle T launay property states that suh a sphere should exist. Therefore, situations our when andidate triangles are rejeted, although they ould have been preserved by arefully seleting another sphere Implementation In this setion, we present the different steps of our implementation of the marhing triangles algorithm in the ase of general impliit surfaes. We assume that the impliit surfae is simply onnex out of larity. Let us reall that in our interative editor, the impliit surfae is reated from elements whih enables us to find a seed triangle for eah disonneted omponent of the surfae. The algorithm keeps trak of a list of boundary edges Le, whih is initialized with the edges of the seed triangle. At eah step, new triangles are tentatively reated from andidate edges of the list Le. The overall algorithm may be written as follows : 1. Starting from the list Le generated by the seed triangle, iteratively reate new triangles from the boundary edge list Le until no more triangles may be omputed this way or until the list is empty. 2. Close the raks that may appear in the polygonization. Those steps are further detailed in the next paragraphs Triangle reation step We aim at reating a new point x p on the impliit surfae to haraterize a andidate triangle T x p that will be heked against the Delaunay surfae onstraint. The new triangle T should be lose to equilateral. In 15 the point x p is omputed by projeting a point x on the surfae, where x is reated at a onstant distane d from the edge e in the plane of the triangle T. A better approah onsists in adapting the parameter d to the loal urvature of the surfae. Antiipating the loal urvature of the field funtion T e S proves to be diffiult. A straight forward, however expensive tehnique, might onsist in evaluating the urvature at the verties of the boundary edge e, using the Hessian of the field funtion to adapt the distane d. We propose another approah that avoids the omputation of the urvature, and that seems to be a fair effiieny-auray tradeoff. Our algorithm proeeds in three steps. Geometry orretion step Let x m denote the mid point of the boundary edge e. We first projet x m onto the impliit surfae in the diretion of the gradient f x m so as to fit to the loal geometry of the impliit surfae (see Figure 5). This step reates a new point on the surfae denoted as x s. As x m is fairly lose to the surfae, only a few iterations are needed at this point of the algorithm. x m f x m x s t x f x Figure 5: Charaterization of the surfae point x s and the projeted point x p Computation of the starting point We rely on a diretion vetor denoted as t to define the point x (see Figure 5). We define t as the unit tangent vetor to the surfae at the surfae vertex position x s. Let denote the ross produt of two vetors : t e f x s e f x s The point x may be written as x x s dt where d is a variable distane parameter. Sine we wish to set T x x 1 x p almost equilateral, the stepping length d should be set to the height of an equilateral triangle, therefore : d 3 2 e In some ases, adjaent boundary edges may have ompletely different lengths. For instane step 3 2 of the marhing triangles algorithm skips the Delaunay onstraint and sometimes reates thin triangles to lose the mesh. Using the average length of the neighboring edges enables us to avoid the reation of new triangles of small size where this might not be neessary. Let e k 1 e and e k 1 e 2 the two edges neighboring e k e. x p

7 S. Akkouhe and E. Galin / Adaptive Impliit Surfae Polygonizationusing Marhing Triangles The stepping length derived from the average length, denoted as ē, may be written as follows : 3 d 2 ē ē e k 1 e k e k 1 3 This tehnique present a severe drawbak however. As the size of the edges of the triangles dereases in regions of high surfae urvature, triangles gets smaller and smaller. They do not grow however when the marhing triangles algorithm explores regions of low or moderate urvature. Thus, flat regions ould be meshed with numerous small triangles after traversing a region of high urvature. 2 e x p 2 e Figure 6: Adapting the distane to the size of the neighboring edges We address this problem by onstraining the stepping distane d to a minimum value denoted as d min. The distane d min refers to the minimum size of the side of a generated triangle. Whenever d gets too small, we adjust it aording to the bound d min. Our experiments show that simply lamping d to d min may produe artifats in the polygonization suh as triangles of small aspet ratio. We prefer to use a weighted average between d and the bound d min. We proeed as follows : if d d min then set dnew 3 4d 1 4d min. Whenever the stepping length drops to 0, the adjusted length onverges to the minimum value 1 4d min. Computation of the new surfae vertex Eventually, the algorithm omputes the projetion x p of x by marhing along the gradient of the field funtion. This proess may be split into two steps. The first step loates a pair of points inside and outside the surfae. The seond step omputes x p by performing an iterative bisetion of the previous line segment. Let x and y denote the two points that onverge to the surfae by following the gradient of the field funtion. The algorithm may be written as follows : 1. Initialize y with the starting point x. 2. While both points are on the same side of the impliit surfae, i.e., f x and f y are of the same sign, perform the following sub-steps Evaluate an approximation of the distane to the surfae as : f x d f x x p 2.2. Compute the new loation for point y, marhing from x along the diretion of the gradient vetor. Let u the unit diretion vetor : u f x f x The stepping length is derived from this equation by saling the distane d by a salar fator α for some reasons that will be detailed in the next paragraph. The iterative sheme presents as follows : y x α f x f x f x If f x and f y are of opposite signs, then exit loop, otherwise store y in x and restart loop at step When the algorithm reahes this step, x and y are on opposite sides of the surfae, so perform bisetion over the line segment x y. The fration f x f x provides us with an estimation of the distane between the point x and the impliit surfae. As f x drops to zero when x gets loser to the surfae, the onvergene may beome very slow. Therefore, we sale the stepping length by a salar fator α to speed up the overall proess and avoid onverging to the surfae without rossing it. In general, α 1 5 works well. At the end of this iterative sheme, we find two points, one on eah side of the impliit surfae, and invoke a bisetion step to find x p Tesselating the raks At the end of the mesh growing proess, the triangulation of the impliit surfae is not losed. The remaining boundary edges of the list Le form raks in the polygonization. Although the geometry of those raks is often omplex (the hole is geometrially distorted all over the surfae), the Delaunay ondition implies that the width of the rak on the surfae never exeeds the size of the boundary triangles. At this point of the algorithm, we need to bridge the gap between boundary triangles to lose the polygonization. We invoke a rak fixing algorithm that reates new triangles from the remaining boundary edge list Le. This proess only onnets verties of the existing mesh and does not reate new verties (see Figure 7). Contour splitting Our algorithm relies on a divide and onquer approah. Starting from the boundary edge list Le, we reursively subdivide the omplex ontour into two simpler and smaller ontours denoted as Le and L e. We perform a reursive splitting until a minimum size is reahed, for instane when the urrent edge list Le has less than six elements. Larger ontours are split by reating new triangles in the mesh whose verties are taken among the boundary verties of Le (see Figure 8). The reursive splitting algorithm may be outlined as follows :

8 S. Akkouhe and E. Galin / Adaptive Impliit Surfae Polygonizationusing Marhing Triangles Figure 7: Closed meshes obtained after tesselating the raks produed by the marhing triangles algorithm 1. Starting from an edge e of Le, find a splitting vertex x s in Le suh that the triangle T x s splits the ontour into two smaller ontours Le and L e. 2. If x s exists, reate the splitting triangle and reursively apply the algorithm on the sub-lists Le and L e. 3. Otherwise, step to the next edge of the ontour. If all the edges of the ontour have been heked, then leave the ontour unhanged. The haraterization of the splitting vertex x s involved in the deomposition of the argument list of boundary edges Le is performed as follows. For eah edge e of Le, we try to find another vertex x s in the boundary edge list Le so that the triangle T x s should split the ontour and have a orret orientation. Thus, we searh the nearest point of Le to the edge e. This point should be faing the open edge e, and needs to be different than the two verties and 2 next to the edge e so as to avoid a non splitting onfiguration. Contour meshing The seond step of the algorithm aims at meshing all the small ontours. Those ontours fall into two different ategories. Some ontours form simple holes on the surfae, whereas some are faing eah other. Faing ontours need to be paired as they ut the surfae into two parts and may prove to be end ontours of a tubular setion in the mesh. Suh ases may our when polygonizing shapes haraterized by thin tubular setions, suh as the horns of the elk shape (see Figure 7). The lassifiation of ontours proves to be a diffiult hallenge. We need to find whih ontours are to be paired, and deide whether paired ontours should be meshed separately or merged by a tubular polygonization. This may be ahieved by haraterizing the topology of the impliit surfae. As shown in 29, traking the ritial points of the field funtion f provides suffiient knowledge for reating a polygonization whose topology is homeomorphi to the impliit surfae. This involves omputationally expensive algorithms and data-struture however. L e S 2 e T Le x s Figure 8: Divide and onquer approah in the generation of triangles bridging the gap between the remaining boundary edges Le In our implementation, we rely on a simple algorithm to generate a topologially onsistent mesh diretly, whih might not be topologially orret. This is a typial tradeoff between topologial orretness and speed. Contours that have six or less boundary edges are treated as simple holes in the surfae, and are meshed by onneting the verties of the ontours. In pratie, we have never found ases where suh ontours should have been paired and merged by a tubular setion. When all small holes in the surfae have been fixed, we proeed as follows. If only one ontour exists, then we lose it as a simple hole ontour. Otherwise, pairs of ontours Le L e are merged by a fixing triangle into a single new ontour Le Comparison with existing tehniques In this setion, we present a omparison between the marhing ubes method and the marhing triangles method. Let us reall the sope of this analysis. Different riteria may be used to evaluate the relative performane of those methods. Meshing tehniques an be ompared in terms of speed or against the auray of the resulting mesh. In general,

9 S. Akkouhe and E. Galin / Adaptive Impliit Surfae Polygonizationusing Marhing Triangles Figure 9: Impliit bird model polygonized with marhing ubes (left, enter) and marhing triangles (right) Model Marhing ubes (left) Marhing ubes (enter) Marhing triangles (right) Time Triangles Aspet Time Triangles Aspet Time Triangles Aspet Bear Bird Elk Cuboid Table 1: Timings (in seonds) for tesselating several impliit surfae models. The marhing ubes approah invokes either the tri-linear interpolation disambiguation method (left) or a tetrahedral deomposition of ells (enter) to produe a topologially onsistent mesh the overall quality of a polygonized impliit surfae an be measured in terms of geometrial auray, topologial onsisteny, topologial orretness and adaptivity to the loal urvature of the surfae. The number of generated triangles, their aspet ratio and onnetivity are also very important parameters. As the marhing triangles algorithm generates a topologially onsistent triangulation of the surfae, we have invoked ell disambiguation algorithms to avoid raks in the surfaes generated by the marhing ubes approah, so that both tehniques should generate topologially onsistent meshes. We have implemented both the tetrahedral deomposition of ambiguous straddling ells 3 4 and the tri-linear interpolation sheme proposed by Natarajan 24. Figure 9 shows the same impliit surfae model polygonized with different methods. The first two meshes on the left have been reated by the marhing ubes algorithm with a tetrahedral deomposition and a tri-linear interpolation sheme respetively. The rightmost mesh was generated by the marhing triangles tehnique. Our experiments demonstrate that it is diffiult to ompare the marhing triangles and the marhing ubes algorithms as the marhing triangles adapt to the urvature of the surfae, whereas the marhing ubes relies on fixed size ells to polygonize an impliit surfae. Table 1 reports several statistis for meshing the bird model and other shapes displayed in Figure 7. The marhing ubes rely on a voxel deomposition of spae, and the size of the seed ube was defined as 1 50 th of the size of the bounding box of the impliit model. This small ell size is needed to produe a topologially orret and geometrially aurate mesh. In ontrast, the marhing triangles was invoked with a seed triangle whose orresponding bounding box featured a 1 20 th relative size. That size proved to be small enough to apture the fine details of the surfae as the algorithm adapts to the urvature and reates smaller triangles wherever needed. Timings show that the marhing triangles and the marhing ubes polygonize an impliit surfae in omparable times. The fastest tehnique is the marhing ubes ombined with a tri-linear disambiguation tehnique, whih generates fewer triangles than the tetrahedral deomposition. The other reported statistis favor the marhing triangles however. For all impliit models, the marhing triangles gen-

10 S. Akkouhe and E. Galin / Adaptive Impliit Surfae Polygonizationusing Marhing Triangles erates fewer triangles that have a better aspet ratio (Table 1). Moreover the overall quality and visual aspet of the resulting mesh is definitely better as the size of the triangles adapts to the urvature of the impliit surfae. 4. Inremental tehnique In this setion, we show that the marhing triangles may be easily integrated as an effiient polygonization tool in an interative impliit surfae editing environment. Skeletal impliit surfaes benefit from speifi mathematial properties that make them suitable both for interative modeling, animation and fast polygonization. Let us reall some bakground out of larity. Blobs 34 form a restrited lass of skeletal impliit surfaes. Blobs are haraterized by a salar field f x y z generated by summing the influenes of n salar field elements f i x y z. The surfae is defined as points in spae whose potential field equals a threshold value denoted as T. The field ontributions f i may be written as dereasing funtions of the distane to a skeleton f i g i d i where g i : "! is the potential funtion, and d i : 3! refers to distane to the skeleton. The skeleton and the distane funtion haraterize the shape of the element, whereas the potential funtion g i, also referred to as the blending funtion, defines the way elements blend together. The region of influene of an element, denoted as Ω i, is defined as the ompat support of the funtion f i. Whenever blending funtions have a finite support, the regions of influene of the elements are bounded in spae. Field funtions f i have no influene over f outside Ω i. Therefore, a modifiation of the parameters haraterizing an element i will hange the potential field f i and the hanges will be propagated to f but limited to Ω i Previous work In an editing environment, the skeletal impliit surfae is inrementally designed by adding or removing elements, or hanging some of their parameters. Editing operations fall into two ategories. In general, adding, removing or modifying elements only hange the field funtion f in a restrited region of spae, and therefore only imply loal hanges over the resulting impliit surfae. In ontrast to those editing operations, the modifiation of the threshold parameter has a global influene over the whole shape. A fast inremental polygonization tehnique has been presented in 12. The proposed algorithm fouses on regions where hanges in the potential field f ourred so as to interatively update the mesh only wherever needed. The algorithm relies on an otree deomposition of spae ombined with Lipshitz onditions 17 to onverge to the surfae and find terminal ells straddling the impliit surfae. The otree struture automatially adapts itself to the size of shape by an inflating-deflating sheme with a view to handling shape growth and shrink during interative editing. Terminal straddling ells are polygonized with a tri-linear interpolant approximation of the field funtion so as to handle ambiguous ases and generate a onsistent polygonization 24. This tehnique exhibits several drawbaks. Firstly, numerous triangles are generated, often over distorted, even in relatively smooth parts of the surfae. The very reason for this is that surfae straddling ells need to be of equal size so as to avoid raks. Therefore, the otree generates as many terminal straddling ells as a voxel deomposition of spae. The seond drawbak deals with memory. Let us reall that the otree struture stores the potential field value at the verties of its ells and the Lipshitz onstants so as to keep up as muh information as possible and speed-up the updating proess when adding or removing elements. Terminal straddling ells also store referenes to mesh triangles. Suh information is needed to ahieve interative rates in the inremental updating of the model. Therefore, maintaining the otree data-struture proves to be memory demanding when applied to omplex models that require deep otrees. Eventually, editing operations need to be oded separately, invoking the inflating and deflating shemes as neessary Inremental algorithm In ontrast to the otree based inremental algorithm, the inremental marhing triangles method is simpler. Adding, modifying or removing elements may be handled by a single simple algorithm. We proeed as follows. We eliminate triangles in the region of spae where hanges in the field funtion ourred. This step generates a set of boundary edge lists, denoted as Le. The new mesh may be diretly reated by invoking the marhing triangles method from the ontours of Le. Overview of the algorithm At eah editing step, the mesh is haraterized by a list of triangles denoted as L T. Triangles are onneted by an adjaeny graph, whih is implemented as a winged edge data-struture. Let Ω 0 denote the region of spae where the potential field f is modified, the algorithm hek whether the triangles of the existing mesh interset Ω 0, and reate a list of boundary edge lists Le. 1. Proess eah triangle T of the existing mesh as follows : 1.1. If T does not interset Ω 0, then keep T and step over to the next triangle Otherwise, starting from T, iteratively destroy the triangles that lie inside Ω 0 and generate a new boundary edge list Le that will be added to the set of edge lists Le. 2. Apply the marhing triangles algorithm starting from the set of edge lists Le.

11 S. Akkouhe and E. Galin / Adaptive Impliit Surfae Polygonizationusing Marhing Triangles Figure 10: Inremental modeling of the horns of an impliit elk model. Step 1 2 of the algorithm uses T as a seed triangle for eliminating a path of triangles that lie inside Ω 0. This may be performed effiiently by traversing the adjaeny graph data-struture of the existing mesh. This step may be taken as the inverse proess of the marhing triangles algorithm. Starting from a seed triangle in an existing mesh, we iteratively destroy triangles and keep trak of growing boundary edge lists Le. In pratie, the deimation of the mesh an be performed at interative rates thanks to the onnetivity information provided by the adjaeny graph Results The marhing triangles tehnique lends itself for the inremental polygonization of skeletal impliit surfaes in an interative editing environment. Timings ompare favorably to the results obtained by the inremental otree method presented in 12. The main reason for this is that the inremental marhing triangles method is simpler. We need not update an otree data-struture, whih proves to be memory exhausting when applied to omplex models that require deep otrees. Triangles are effiiently removed in the region of spae where hanges in the field funtion ourred, whereas the new mesh is diretly reated by invoking the marhing triangles method from the resulting open ontours in the mesh. To ompare the different methods, we have inrementally built different impliit models and ompared the elapsed time for both the inremental update and a whole polygonization. Figure 10 illustrates an interative modeling step in the reation of an impliit elk like model. The skeletal elements are of any dimension, for instane the head involves one triangle skeleton whereas the legs and the horns are built from segment skeletons. During the editing step, the horns of the elk have been modified to ahieve the desired final shape. Several piees of the surfae were re-polygonized beause of the propagation of the influene of the modified elements. As expeted, the aeleration provided by the inremen- Step Inremental Global Triangles Time Triangles Time Body Head Left ear Right ear Nose Left arm Right arm Left leg Right leg Table 2: Timings (in seonds) for global and inremental polygonization of the bear, the number of generated triangles is also reported tal algorithm are similar to those obtained with the otree sheme presented in 12. Table 2 reports timings for the inremental polygonization of the bear. Unsurprisingly, aelerations are the greater as the objet beomes more and more omplex and involves more and more primitives. Experiments show that the inremental polygonization reates slightly more triangles than a global polygonization. Table 2 reports that this overhead may range from 10% to 20%. This larger number of generated triangles omes from the suessive alls to the rak losing algorithm that generates extra small triangles at eah modeling step. Moreover, our method adapts to the urvature of the surfae by dereasing or inreasing the size of the triangles in

12 S. Akkouhe and E. Galin / Adaptive Impliit Surfae Polygonizationusing Marhing Triangles regions of high and low urvature respetively. The triangle growing sheme is more diffiult to handle than the triangle shrinking sheme however. Therefore, triangles tend to get smaller faster than they an enlarge. 5. Conlusion In this paper, we have presented a generalization of the marhing triangles method for losed impliit surfae manifolds. By adapting the step length to the loal urvature of the impliit surfae, we an guarantee a both topologially orret and geometrially aurate triangle mesh. The marhing triangles proves to be an interesting approah to address the polygonization of simply onnex impliit surfaes. We have ompared this tehnique with different implementations of the marhing ubes method. Experiments arried out on several impliit models demonstrate that topologially onsistent meshes an be generated in omparable time. For all impliit models, the marhing triangles generates fewer and better shaped triangles than the marhing ubes. The overall quality and visual aspet of the resulting mesh is definitely better as the size of the triangles adapts to the urvature of the impliit surfae. Only the triangles generated during the rak fixing step may be sometimes distorted. One drawbak, whih is inherent to all ontinuation methods, is the determination of seed triangles on the onnex parts of the impliit surfae. This problem disappears when using skeletal impliit surfaes, suh as blobs 34. In the sope of interative modeling, we have shown that an inremental implementation of the marhing triangles outperforms other inremental methods, suh as the inremental otree sheme desribed in 12. We are urrently trying to adapt this method to the Blob- Tree model 36 that ombines impliit primitives in a hierarhial tree struture whose nodes may be blending, warping and boolean operators. Beause of the boolean operators, this model may produe sharp surfae features. One hallenge would onsist in meshing the BlobTree by deteting gradient disontinuities on the fly during the mesh growing step. Referenes 1. E. L. Allgower and S. Gnutzmann. Simpliial Pivoting for Mesh Generation of Impliitly Defined Surfaes. Computer Aided Geometri Design, 8 : , C. Blan and C. Shlik. Extended field funtions for soft objets. Proeedings of Impliit Surfaes 95, 1 : 21-32, April J. Bloomenthal. Polygonization of impliit surfaes. Computer Aided Geometri Design, 5(4) : , J. Bloomenthal, C. Bajaj, J. Blinn, M. P Cani-Gasuel, A. Rokwood, B. Wyvill and G. Wyvill. Introdution to Impliit Surfaes, Morgan Kaufmann J. D. Boissonat. Geometri strutures for threedimensional shape representation. ACM Transations on Graphis, 3(4) : , A. Bottino, W. Nuij and K. VanOverveld. How to Shrinkwrap through a Critial Point : an algorithm for the adapative triangulation of iso-surfaes with arbitrary topology. Proeedings of Impliit Surfaes 96, 2 : 53-72, Otober C. C. Lin and Y. T. Ching. A note on omputing the saddle values in isosurfae polygonization. The Visual Computer, 13(7) : , B. Crespin, P. Guitton and C. Shlik. Effiient and aurate tessellation of impliit sweeps. Proeedings of Construtive Solid Geometry 98, April M. Desbrun, N. Tsingos and M. P. Gasuel. Adaptive Sampling of Impliit Surfaes for Interative Modeling and Animation. Computer Graphis Forum, 15(5) : , Deember A. VanGelder and J. Wilhelms. Topologial onsiderations in Isosurfae Generation. ACM Transations on Graphis, 13(4) : , Otober E. Galin. Métamorphose et Visualisation de Blobs à Squelettes. Ph.D. Thesis, Université Claude Bernard Lyon 1, July E. Galin and S. Akkouhe. Inremental Polygonization of Impliit Surfaes. Graphi Models and Image Proessing, 62 : 19-39, J. D. Gasuel. Impliit Pathes : An Optimized and Powerfull Ray Intersetion Algorithm for Impliit Surfaes. Proeedings of Impliit Surfaes 95, 1 : , April J. C. Hart, A. Durr and D. Harsh. Critial Points of Polynomial Metaballs. Proeedings of Impliit Surfaes 98, 3 : 69-76, June A. Hilton, A. Stoddart, J. Illingworth and T. Windeatt. Marhing triangles : range image fusion for omplex objet modelling. International Conferene on Image Proessing, A. Hilton and J. Illingworth. T. CVSSP Tehnial Report, D. Kalra and A. H. Barr. Guaranteed Ray Intersetions with Impliit Surfaes. Computer Graphis, 23(3) : , July A. Kalvin and R. Taylor. Superfaes : Polygonal Mesh Simplifiation with Bounded Error. IEEE Computer Graphis and Appliations, 16(3) : 64-77, 1996.