Constructing Hierarchies for Triangle Meshes

Transcription

1 IEEE TRANSACTIONS ON VISUALIZATION AND COMPUTER GRAPHICS VOL. 4 NO. APRIL-JUNE Construting Hierarhies for Triangle Meshes Tran S. Gieng Bernd Hamann Member IEEE Kenneth I. Joy Member IEEE Gregory L. Shussman Member IEEE and Issa J. Trotts Abstrat We present a method to produe a hierarhy of triangle meshes that an be used to blend different levels of detail in a smooth fashion. The algorithm produes a sequene of meshes 1... n where eah mesh i an be transformed to mesh i+1 through a set of triangle-ollapse operations. For eah triangle a funtion is generated that approximates the underlying surfae in the area of the triangle and this funtion serves as a basis for assigning a weight to the triangle in the ordering operation and for supplying the points to whih the triangles are ollapsed. The algorithm produes a limited number of intermediate meshes by seleting at eah step a number of triangles that an be ollapsed simultaneously. This tehnique allows us to view a triangulated surfae model at varying levels of detail while insuring that the simplified mesh approximates the original surfae well. Index Terms Mesh simplifiation triangle meshes level-of-detail representation shape approximation multiresolution. F 1 INTRODUCTION T HE rapid inrease in the power of omputer systems oupled with the inreasing sophistiation of geometri-modeling operations the inreasing preision of omputational simulations and the development of state-ofthe-art imaging systems an now produe models with inredible detail. The most ritial and fundamental researh problem enountered in the visualization of these omplex models is the development of methods for storing approximating and rendering very large data sets. The problem is to develop different representations of a data set eah of whih an be substituted for the omplete set depending on the requirements of the visualization tehnique. The data set may be represented by a few points or by several million points if neessary with eah of the data sets ontaining the essential features of the original data. A hierarhial or multiresolution representation [1] [] [3] [4] allows the study of large-sale features by onsidering the data set at a oarse resolution and the study of small-sale features by onsidering the data set at a fine resolution. Using a multiresolution data representation allows near real-time traveling through the data domain. The rendering algorithm an use a oarse resolution for image generation while the model moves and an add detail over time by adding finer levels when the model is stationary the progressive refinement approah to image generation. The rendering algorithm an also simultaneously display various levels of detail from the representation hanging among the levels ontinuously. This level-ofdetail problem is of great importane in visualization and omputer graphis. Multiresolution tehniques are useful for real-time data aess to support telepresene and ²²²²²²²²²²²²²²²² T.S. Gieng is with the Department of Computer Siene California Institute of Tehnology Pasadena CA gieng@s.alteh.edu. B. Hamann K.I. Joy G.L. Shussman and I.J. Trotts are with the Department of Computer Siene University of California Davis CA and with the Center for Image Proessing and Integrated Computing (CIPIC) University of California Davis. CA {hamann joy shussman trotts}@s.udavis.edu. For information on obtaining reprints of this artile please send to: tvg@omputer.org and referene IEEECS Log Number teleoperations and to enhane aessibility of the data sets in real-time deision proesses. We introdue a method to produe a hierarhial representation of large unstrutured triangle meshes. Given an initial mesh our algorithm redues the number of triangles through a series of triangle-ollapse operations. A triangle is seleted from the mesh and removed by ollapsing it to a point (see Fig. 1). A weight is assigned to eah triangle and is used as the riterion to selet triangles to be ollapsed. This weight is partially based on a urvature measure determined by the prinipal urvatures of a funtion that approximates the surfae in the area of eah triangle. To insure that the new mesh aurately approximates the underlying surfae we use the approximating surfae to supply the point to whih the triangle is ollapsed. This enables us to develop a simple algorithm based upon a triangle-ollapse strategy whih remains faithful to the underlying surfae. In a given mesh we an identify a number of triangles that an be ollapsed simultaneously and this allows our algorithm to output a sequene of meshes 1... n with the property that i an be smoothly ollapsed to i+1. By ollapsing a relatively large number of triangles in an intermediate triangulation simultaneously we ahieve signifiant memory savings. Thus the transition from mesh i to i+1 is haraterized by ollapsing many triangles in parallel instead of ollapsing just one. The sequene of meshes along with the triangle-ollapse operations an be used to reate a smooth visual transition between levels in the hierarhy. In Setion we disuss related work. We examine the topology of the triangle-ollapse operation in Setion 3. We define what it means for a triangle to be ollapsible and exhibit the effet of the triangle-ollapse operation on the mesh. In Setion 4 we onstrut a funtion that approximates the underlying surfae in the area of a triangle. This approximating surfae is used in two ways: 1) to define the point to whih a triangle will ollapse and ) to assign weights to the triangles /98/$ IEEE

2 146 IEEE TRANSACTIONS ON VISUALIZATION AND COMPUTER GRAPHICS VOL. 4 NO. APRIL-JUNE 1998 (a) (b) () Fig. 1. Collapsing a triangle: The shaded triangle is seleted from the mesh in (a) ollapsed toward the entroid of the triangle in (b) reating a new mesh in () whih has four fewer triangles than the original mesh. The algorithm assigns a weight to eah triangle whih determines its priority in the mesh redution proess. In Setion 5 we give a omplete desription of the algorithm whih generates a sequene of triangle-ollapse operations and a sequene of meshes. The alulation of the fators that make up the weights is disussed in Setion 6. Results of the algorithm s use are given in Setion 7. RELATED WORK Algorithms that simplify meshes are an important omponent of researh in visualization and omputer graphis. State-of-the-art graphis workstations whih now allow the display of many thousands of shaded or textured polygons at interative frame rates are not suffiient to render the truly massive data sets ontaining millions of triangles. Rendering systems today require real-time level-of-detail representations to adaptively display the models at onstant frame rates or to ontinuously vary the detail on the model depending on the amera position. Three lasses of algorithms exist that diretly pertain to our work and that deal with the triangle meshes diretly: algorithms that simplify the mesh by removing verties; algorithms that simplify the mesh by removing edges; and algorithms that simplify the mesh by removing faes. Shroeder et al. [5] have developed an algorithm that simplifies the mesh by removing verties. Verties are identified through a distane-to-plane riterion where an average plane is formed through a vertex and its adjaent verties. If the vertex is within a speified distane of the average plane it an be deleted; otherwise it is retained. Removing a vertex from the mesh reates a hole that must be retriangulated and several strategies may be used. Shroeder et al. use a reursive loop splitting proedure to generate a triangulation of the hole. Renze and Oliver [6] have published a similar algorithm whih is extended to meshes ontaining n-dimensional simplies. Rather than a loop splitting proedure they fill the hole by using an unonstrained Delaunay triangulation algorithm. Hoppe [7] [8] and Popovi and Hoppe [9] desribe a progressive-mesh representation of a triangle mesh. This is a ontinuous-resolution representation based upon an edge-ollapse operation whih selets and ollapses individual edges of the mesh. Hoppe et al. [1] formulate the data redution problem in terms of a mesh optimization problem ordering the edges aording to an energy minimization funtion. Eah edge is plaed in a priority queue by the expeted energy ost of the edge ollapse. As edges are ollapsed the priorities of the edges in the neighborhood of the transformation are reomputed and reinserted into the queue. The result is an initial oarse representation of the mesh and a linear list of edge-ollapse operations eah of whih an be regenerated to generate finer representations of the mesh. The geometrially ontinuous edge-ollapse operation allows the development of a smooth visual transition between various levels of the representation. Other edge-ollapse algorithms have been desribed by Zia and Varshney [11] who use the onstruted hierarhy for view-dependent simplifiation of models and Garland and Hekbert [1] who utilize quadrati error metris for fast alulation of the hierarhy. Hamann [13] has developed an algorithm that attempts to simplify the mesh by removing triangles. His algorithm removes triangles from a mesh by first ordering the triangles aording to the urvature at their verties (see [14]). The urvature values are preomputed based on the original triangulated mesh. Triangles are then inserted into a priority queue and removed iteratively. Modified triangles reeive new urvatures at their verties and are inserted bak into the priority queue. The user an speify a perentage of triangles to be removed or an error tolerane. Eah of these algorithms fouses on an individual vertex edge or triangle in the mesh. Our algorithm has a similar fous but reates a hierarhy of meshes not just a list of elements. The hierarhy of meshes an be used to reate a ontinuous-redution algorithm that enables us to smoothly vary the detail over the set of meshes. Others have used different approahes to address these problems. Turk [15] uses a retiling strategy to plae new points on the original mesh. These points are then retriangulated resulting in a new oarser mesh that represents the original data set. Cohen et al. [16] use a generalization of offset surfaes to produe a hierarhy of inner and outer meshes that surround the original mesh. Lindstrom et al. [17] address the ase of triangulations in the form of uniformly gridded height fields. With this speialized struture it is possible to segment the grid into a quad-tree

3 GIENG ET AL.: CONSTRUCTING HIERARCHIES FOR TRIANGLE MESHES 147 (a) (b) () (d) Fig.. Stenils of triangles. (a) The shaded triangle has a onneted ayli stenil. (b) The shaded triangle has a disonneted ayli stenil. () The shaded triangle has a onneted yli stenil. (d) The shaded triangle has a omplete stenil. In the last ase the stenil boundary polygon is outlined in bold. We note that the boundary polygon of the triangle in (a) ontains a vertex of the triangle and therefore the stenil is not omplete. representation and then provide triangle redution within the quad-tree nodes. Our algorithm is based on a triangle-removal strategy that reates a hierarhy of meshes not just a hierarhy of triangles. These meshes an be used to perform a ontinuous level-of-detail rendering whih enables us to smoothly blend the various levels. This paper is an expansion of our work in [18]. We now provide full details of the topology onsiderations in the triangle-ollapse operation present an algorithm to handle meshes with boundaries and desribe additional fators used to order the triangles for ollapse. 3 TRIANGLE-COLLAPSE OPERATIONS In the rendering ontext we define a surfae to be a pieewise linear surfae defined by a mesh of triangles. We require that the triangle mesh be onneted and that eah edge in the mesh be shared by at most two triangles. Meshes should not be self-interseting that is no triangle of the mesh should have an intersetion with the interior of another triangle. Our objetive is to ollapse triangles in the mesh. If we examine Fig. 1 we see that ollapsing a triangle affets other triangles in the area. These affeted triangles defining the stenil are modified in the triangle-ollapse operation. (a) (b) Fig. 3. Edge swapping to remove yles in the stenil. (a) The seleted triangle ontains a yle in the stenil. (b) The yle is removed by swapping the ommon edge between the triangle and the neighboring triangle that belongs to the yle. To determine if a triangle is ollapsible we must examine the stenil and its effet under the ollapse operation. 3.1 Stenils of Triangles If we are to ollapse a given triangle T it is the triangles surrounding T that influene the resulting mesh after the ollapse. This set of triangles the stenil 6 T of T is the set of triangles T i where T i T and T i shares a vertex with T (see Fig. 1: The stenil of the ollapsing triangle is outlined in bold). The three triangles of the stenil that share an edge with T the neighbors of T are eliminated in the ollapse operation and the remaining triangles of the stenil are modified strething them to a ommon vertex. The stenil is alled onneted if for eah pair of triangles T i1 and T ik in the stenil a sequene of triangles Ti Ti K T 3 i exist in k 1 the stenil suh that T ij and T ij+1 are neighbors 1 for j = 1... k 1. Fig. shows examples of onneted and disonneted stenils. Three triangles T 1 T and T 3 form a yle in the mesh if they are pairwise neighbors (see Fig. ). Eah triangle of a yle must have a vertex of valene three. A stenil 6 is alled yli if it ontains a yle otherwise it is alled ayli. To be in the stenil a yle must ontain a neighbor triangle of T. Cyles an be eliminated in the original mesh by edge swapping (see Fig. 3). Repeated swapping may be neessary to eliminate the three yles that ould possibly our in the stenil. If a triangle T has a onneted ayli stenil 6 T we an order the triangles of the stenil and obtain a polygon the stenil boundary polygon that desribes the outer boundary of the stenil. If this polygon ontains no verties of the original triangle T the stenil is alled omplete. Examples of various triangles and their stenils are shown in Fig.. We an ensure that omplete stenils are defined on the boundary by adding a single vertex a point-at-infinity to the mesh. This point p is then onneted to eah vertex on the boundary of the mesh (see Fig. 4). 1. A neighboring triangle of T shares an edge with T.

4 148 IEEE TRANSACTIONS ON VISUALIZATION AND COMPUTER GRAPHICS VOL. 4 NO. APRIL-JUNE 1998 Fig. 4. A single point-at-infinity is added to the mesh and onneted to eah vertex on the boundary. This enables us to assoiate omplete stenils to triangles on the boundary. 3. Collapsing a Triangle Collapsing a triangle and removing it from the mesh affets both the triangle itself and the triangles of its stenil. As an be seen in Fig. 5 when a triangle is ollapsed the triangle and its neighbors are eliminated from the mesh. The triangle is replaed by a single point whih is onneted to all points of the stenil boundary polygon reating a new triangulation of the region. Geometrially this transition is smooth; topologially it is disontinuous when the three verties eventually beome one. To make this speifi onsider a triangle T and let its neighboring triangles be denoted by TN T 1 N and T N3. For eah i let p Ni be the vertex of T Ni that is not a vertex of T (see Fig. 6). If p 1 p and p 3 are the verties of the triangle and if we denote val(p) to be the valene of the point p in the triangulation then after the triangle T is ollapsed the following holds: We have removed four triangles from the mesh T T T and T N3. N1 N From the n triangles in the stenil the area is now retriangulated with n 4 triangles. These triangles all share a new vertex whih has valene equal to val(p 1 ) + val(p ) + val(p 3 ) 9. Eah point p Ni is a vertex of one fewer triangle: Its valene is redued by one. The Euler-Poinaré harateristi [19] is preserved. Consider a triangle T whose stenil ontains v verties e edges and t triangles. After the triangle is ollapsed the resulting set of triangles will have v verties e 6 edges and t 4 triangles. The Euler-Poinaré harateristi of both sets of triangles is the same:. The number of edges that emanate from a vertex define its valene. Fig. 5. A triangle and its stenil undergoing the ollapse operation. As the triangle is ollapsed a new mesh is reated ontaining four fewer triangles. Geometrially this transition is smooth; topologially it is disontinuous. Fig. 6. Topology of the triangulation before the ollapse (a); and after the ollapse (b). In the proess the triangles TT T and T N1 N N are eliminated from the triangulation. The new triangulation has a vertex at and the valene of eah of the verties p p and p is redued by one. N 3 3 N N 1

5 GIENG ET AL.: CONSTRUCTING HIERARCHIES FOR TRIANGLE MESHES 149 (a) (b) Fig. 7. Introduing yles into the triangulation. (a) The vertex p N has valene four and when triangle T is ollapsed in (b) a yle is introdued in the stenils of the triangles T 1 and T. (a) (b) () Fig. 8. Folds an our in the modified triangles of the stenil if a star-shaped property does not hold. In this planar example the triangle in (a) has a non-star-shaped stenil. As we ollapse in (b) the potential onflit an be seen and in () the fold an be seen. (t 4) (e 6) + (v ) = t e + v. The ollapsing proess an introdue triangles with yli stenils. If any of the points pn p 1 N or p N3 has valene four ollapsing the triangle T redues the valene of this vertex by one thus reating a vertex of valene three and a yle. This yle will ause three triangles of the mesh to have ayli stenils if the yle is in the interior of the mesh (see Fig. 7). We define a ollapsible triangle as one that does not introdue additional yles as a result of the ollapse operation. Oddly shaped stenils also pose a potential problem. If the stenil is oddly shaped the ollapse proess an potentially reate folds in the resulting triangles (see Fig. 8). To avoid this problem we require the stenil boundary polygon when projeted to the plane of the triangle to be starshaped with respet to the entroid of the triangle. This implies that for any point q on the stenil boundary polygon the line q must be ontained in the interior of the polygon [19]. With these observations we define a triangle T to be ollapsible if 1) it has a omplete stenil 6 T ; ) the valene of the vertex of eah neighboring triangle that is not a vertex of the triangle T is not equal to four; and 3) its projeted boundary polygon is star-shaped with respet to the entroid of the triangle. When ollapsing triangles of a mesh verties hange valenes and triangles disappear. Thus a triangle may not be ollapsible in one level of the hierarhy but may be ollapsible in other levels. To ollapse a triangle we only need to identify the stenil boundary polygon and the point to whih the triangle is ollapsed. 4 APPROXIMATING THE UNDERLYING SURFACE It is our goal is to faithfully approximate the underlying surfae of the triangulation that is stay as lose as possible to the original mesh. For geometri models it is the urvature of the model that has the primary effet on the mesh redution proess. In general we would expet triangles to fit the underlying surfae well in the areas where the urvature is low. Therefore we should be most willing to redue the number of triangles in the areas where the urvature of the underlying surfae is low and be least willing to do it when the urvature is high. In this setion we define a bivariate funtion that approximates the surfae in the area of a triangle T. We use this surfae in many ways: We approximate the prinipal urvatures of the surfae by the prinipal urvatures of the graph of this funtion; we ol-

6 15 IEEE TRANSACTIONS ON VISUALIZATION AND COMPUTER GRAPHICS VOL. 4 NO. APRIL-JUNE 1998 lapse our triangles to points on this surfae; and we find the error between our triangles and their projetions on this surfae. Let p 1 p... p n be the verties of the triangles that make up the stenil of T and let be the entroid of T. We establish a loal oordinate system in the plane of T whose origin is. This oordinate system uses any two orthonormal vetors u r and v r and uses the unit normal vetor n r to the plane of the triangle T (see Fig. 9). For eah vertex p j of a stenil triangle we onvert the oordinates of p j to oordinates in the loal system defined by P. These oordinates an be alulated by first projeting p j onto the plane P. This new point p P j is obtained by where P r p = p d n d j j j j 4 j 9 r = n p is the distane of the point p j from the plane P. Projeting p j P onto the onstruted axes defines its loal oordinates u j and v j : j P 4 j 9 P 4 j 9 r uj = u p and r v = v p (see Fig. 1). If eah p j is onverted into (u j v j d j ) in the new oordinate system we an use the points (u 1 v 1 d 1 ) (u v d )... (u n v n d n ) to onstrut a least-squares degree-two polynomial f T (u v) = u + 11 uv + v + 1 u + 1 v + (1) whih we use to approximate the original surfae in the area of the triangle. We an substitute the oordinates of the points (u j v j d j ) into (1) defining the linear system u u v v u v d u u v v u v d 1 U M M M M M M u u v v u v d n n n n n n 1 = 1 1 = The resulting normal equations are T U U = U T d1 d M d n 1 n. () and provided the determinant of U T U does not vanish 3 this system an be solved and the oeffiients of the funtion f T (u v) be determined. 3. If the determinant does vanish we onsider additional points outside the stenil. (3) Fig. 9. Establishing a oordinate system in the plane P. Use the entroid of the triangle T as the origin and use any two orthonormal vetors u r and v r in the plane P as basis vetors. The vetors u r v r and the unit normal n r form an orthonormal oordinate system. Fig. 1. To establish oordinates of the stenil points eah point is projeted onto P. The oordinates u j and v j along with the distane d j from the plane define the oordinates of the point in the loal oordinate system. 4.1 Curvature Estimates The two prinipal urvatures of the graph of f T (u v) are and κ 1 = H + H K (4) κ = H H K (5) where K is the Gaussian urvature of the surfae at (u v) and H is the mean urvature at (u v) (see []). The Gaussian urvature is defined by K = and the mean urvature by fuufvv fuv 41 + fu + f v9 (6)

7 GIENG ET AL.: CONSTRUCTING HIERARCHIES FOR TRIANGLE MESHES 151 H = fu + fv9 u vv u v uv v uu 1+ f f f f f + 1+ f f. (7) In our ase f T (u v) is a bivariate polynomial and its partial derivatives are fu = u + 11v + 1 fv = 11u + v + 1 fuu = fvv = and f = uv 11 defining the oeffiients that we an substitute diretly into (6) and (7) to obtain the Gaussian and mean urvatures of f T at (u v). These an then be substituted into (4) and (5) to obtain the two prinipal urvatures. 4. Calulating the Image of an Edge on the Approximating Surfae Given an edge of a triangle T defined by the loal oordinates (u v ) and (u 1 v 1 ) using the equation of the approximating surfae we an express the equation of the triangle edge as ut 5 u + t u u 1 u ta = vt 5 v + tv v7 = + v + tb 1 where a = u 1 u and b = v 1 v. Inserting this into (1) we obtain f u v = f u t v t T T = u t + v t + u t v t + u t + v t = u + at + v+ bt + 11 u + at v + bt + u + a t + v + b t > 11 C > C + > u + v + 11uv + 1v + 1v + C = a + b + a b t + u a + v b + u + v a + a + b t = At + Bt + C (8) whih is learly a quadrati in t. Thus for eah edge of a triangle we an alulate an assoiated parabola on the approximating surfae. 4.3 The Error Between an Approximating Surfae and a Triangle Edge Given an edge of a triangle T defined by the loal oordinates (u v ) and (u 1 v 1 ) we an alulate the maximum error between the loal surfae approximation and the edge. The parabola f T (t) = At + Bt + C B from (8) has a loal extremum for t = If t 1 then A we define the maximum error e as otherwise % &K 'K ( )K *K B e = max f T 5 ft f A T5 1 Fig. 11. On the approximating surfae the image of a triangle edge is a quadrati polynomial passing through p and p 1. Fig. 1. The Maximum error e between an edge of a triangle and the approximating surfae an our at either at a loal maximum between and 1 or at the endpoints. e = max{ f T () f T (1) }. 4.4 Calulating the Collapse Point We use the approximating funtion f T to determine the point to whih a triangle T is to be ollapsed. For triangles on the interior of the mesh the funtion f T is evaluated at (u v) = ( ) and the ollapse point is defined to be 7n r (9) + f T where n r is the unit normal vetor to the plane of the triangle. This is the point where the approximating surfae intersets a line through the entroid in the diretion given by n r (see Fig. 13). To preserve the boundary of the mesh in the various hierarhies the triangles on the boundary must be ollapsed arefully. Enumerating the types of triangles on the boundary (see Fig. 14) we obtain 1)A type-1 triangle has one edge on the boundary. To ollapse this triangle we find the approximating biquadrati surfae develop the quadrati polynomial from this edge and use a point on the approximating polynomial that orresponds to the midpoint of the edge (see Fig. 15). )A type- triangle has a single vertex on the boundary. In this ase the triangle is ollapsed to a point on the approximating surfae that orresponds to the boundary vertex (see Fig. 16).

8 15 IEEE TRANSACTIONS ON VISUALIZATION AND COMPUTER GRAPHICS VOL. 4 NO. APRIL-JUNE 1998 Fig. 13. To establish the point to whih a triangle will ollapse we find the intersetion of the line through the origin of the loal oordinate system in the diretion of the normal vetor to the triangle and the approximating surfae. Fig. 15. Type-1 triangles are ollapsed to a point that represents the edge of the mesh. This point an be found by finding the line that passes through the midpoint of the edge in the diretion of the normal and interseting this line with the approximating surfae. Fig. 14. Four types of boundary triangles an be identified in the mesh: 1) Triangles that have an edge on the boundary; ) triangles that have a single vertex on the boundary; 3) triangles that have two edges on the boundary (representing orners); and 4) triangles that have two verties on the boundary. Eah triangle has stenil triangles that ontain the point-at-infinity p. 3)A type-3 triangle has two edges on the boundary. This implies a orner vertex and the triangle is ollapsed to the vertex shared by the two boundary edges. 4)A type-4 triangle has two verties on the boundary but no edges. We find these triangles in two situations: They are neighbors of a orner triangle (a type-3 triangle). The triangles have a yle in their stenil the orner triangle and two triangles ontaining the point-at-infinity and so they are not ollapsible. The two verties touh two boundary urves of the mesh. If we ollapse these triangles the boundary of the mesh will be ompromised. In general triangles with more than one vertex and no edge on the boundary are not ollapsible. If suh a triangle were ollapsed the boundary ould not be preserved. Our algorithm may not ollapse these Fig. 16. Type- triangles are ollapsed to the point on the approximating surfae that orresponds to the triangle boundary vertex. type-4 triangles diretly but allows them to be neighbors of ollapsing triangles. 5 THE MESH REDUCTION ALGORITHM With the triangle-ollapse operation the mesh redution algorithm is straightforward to desribe. Given an initial triangle mesh we alulate a weight for eah triangle T of the mesh and plae the triangle on a priority queue ordered by inreasing weight. The weight assigned to T represents the effet that the ollapse of T will have on the mesh. We then repeatedly perform the following steps: 1) A triangle T is removed from the front of the queue ollapsed and a new mesh is generated. ) For eah modified triangle of the stenil of T realulate the weight of the triangle and reposition it in the queue. This proess is repeated until a oarse mesh is generated with a speified number of triangles.

9 GIENG ET AL.: CONSTRUCTING HIERARCHIES FOR TRIANGLE MESHES 153 Fig. 17. Several triangles of the mesh an be ollapsed simultaneously. The triangle stenils of T 1 and T must not interset. The algorithm generates a series of triangle-ollapse operations & & 1 &... & m and a sequene of meshes 1... m eah of whih differs from the previous mesh by one triangle ollapse. Sine eah of the triangle-ollapse operations is reversible we an store the oarse mesh m and the sequene & m & m 1... & 1 (in similar way to [1]) and reate desired meshes of various resolutions by reversing the triangle-ollapse operations expanding the verties to triangles in sequene. We an make a straightforward modifiation to this algorithm. Instead of ollapsing just a single triangle in a mesh the algorithm identifies a ertain perentage of triangles that an be ollapsed in parallel reognizing that two triangles an be ollapsed simultaneously if their stenils do not overlap (see Fig. 17). In this ase we remove a triangle T from the queue if the following two onditions hold: T has a weight less than a speified value. If T T 1 T... T k represents the triangles already removed form the mesh then the stenil of T annot interset the stenil of T i for i =... k. One the sequene of triangles T T 1 T... T k has been seleted the triangles are ollapsed and a new mesh 1 is generated. The weights of eah of the triangles in the stenils of T i i =... k are realulated and the triangles are repositioned in the queue. A new sequene of triangles is then seleted from the queue and a mesh is reated. This proess ontinues until a oarse mesh is obtained. The result is a sequene of triangle ollapse operations & & 1... & ij... & mn and meshes 1... with the property that any mesh j an be transformed to mesh j+1 by simultaneously performing the triangle ollapses & j. Sine eah of these operations is reversible we an store the oarse mesh n and the reversed set of triangle-ollapse operations & n & n 1 K & & m n K & i j K & with markers indiating whih ollapses an be done simultaneously. This approah leads to a signifiantly smaller number of triangulation levels in the final hierarhy. The sequene of meshes allows a smooth blending algorithm to be implemented by defining a partial triangleollapse operation between onseutive meshes. If we define a parameter t t 1 we an define a triangle mesh i (t) with the property that i () = i and i (1) = i+1. i (t) is onstruted by partially ollapsing all seleted triangles of i : If T is a seleted triangle of i with p 1 p and p 3 as its verties and is the point to whih the triangle is ollapsing then the mesh i (t) is the result of linearly interpolating p i and that is p15 t = p1 + t p17 p t = p + t p and p t = p + t p The meshes i (t) provide a geometrially ontinuous method to vary smoothly between different levels in the hierarhy despite the fat that two levels are topologially different. Fig. 18 illustrates the smooth transitions possible between the meshes. Fig. 18a shows a portion of the mesh on a torus data set. The triangles that have been seleted for ollapse are shown in yellow and the stenils of these triangles are shown in blue. Fig. 18b shows the mesh (.75) where the triangles have been partially ollapsed. 6 TRIANGLE WEIGHTS The algorithm assigns a weight to eah triangle of the original mesh. The weights are used to determine the order in whih triangles of the mesh will be ollapsed. The triangles with the smallest weight should be removed first. The weight of a triangle T depends on four riteria: 1) the absolute urvature 4 κ(t) of the approximating funtion in the area about T; ) a shape measure α(t) whih assigns a higher weight to triangles that are near-equilateral; 3) a topologial measure V(T) whih penalizes triangles that will produe high-valene verties when ollapsed; 4) an error measure E(T) that indiates the error between the ollapsed stenil of a triangle and its approximating surfae; and 5) the triangle area A(T). Weights are defined by W(T) = A(T) (w κ κ(t) + w α α(t) + w v V(T) + w e E(T)) where w κ w α w v and w e are user-speified weights with w κ w α w v w e 1. Triangles with small weight will have the least impat on the mesh when ollapsed and are ollapsed first. The urvature weight κ(t) is the absolute urvature of the graph of the approximating funtion f T (u v) of T at u = v = normalized by the maximum absolute urvature observed in the data set. When we multiply this weight by the area of the triangle large triangles in areas of high urvature have large weights and small triangles in flat areas have small weights. The angle weight α(t) is given by 4. We define the absolute urvature κ(t) as the sum of the absolute values of the two prinipal urvatures i.e. κ(t) = κ 1 + κ.

10 154 IEEE TRANSACTIONS ON VISUALIZATION AND COMPUTER GRAPHICS VOL. 4 NO. APRIL-JUNE 1998 (a) (b) Fig. 18. Smooth transition between meshes. (a) (b) () Fig. 19. Produing verties of high valene. If the dark-shaded triangle in (a) is ollapsed the mesh in (b) is produed. If the dark triangle in (b) is ollapsed the mesh in () is produed whih ontains a vertex of valene 1. 5 = α T 3 os α 1 i= 1 i where α i i = 1 3 are T s interior angles; α(t) ranges from zero for degenerate triangles to one for equilateral triangles. When multiplied by the area of the triangle α(t) assigns a greater weight to large equilateral triangles and a smaller weight to narrow small triangles. Triangles that have high-valene verties have diffiulty in passing the star-shaped requirement for ollapsibility and in general they do not approximate surfaes well. We seek to avoid these situations by adding a term that depends on the potential valene of the vertex to whih the triangle will be ollapsed. The topologial term V(T) penalizes triangles that produe verties of high valene when ollapsed (see Fig. 19). This term is given by 5 5 = VT val 6 m v where m v is hosen to be a maximum-valene normalizing fator and is the point to whih the triangle is ollapsed. The algorithm onsiders verties of valene six to be optimal. The error weight E(T) is based upon preditive error. The triangle T is ollapsed and the error between the modified triangulation and the approximating surfae (see Setion 4.3) is alulated along the edges of the modified triangulation. E(T) is the maximum edge error. When multiplied by the area of a triangle large triangles whose modified stenil triangles fit the surfae poorly reeive large weights and small triangles whose modified stenils fit the surfae well reeive small weights. Thus if a triangle an be ollapsed and the resulting retriangulation of the stenil is a good fit for the loally approximating surfae then the triangle will reeive a low error weight. 5 Fig. illustrates urvature and error weights on a omplex mesh. Fig. a shows a portion of the Crater-Lake data set textured with the urvature weight. Areas with high urvature are white and areas with low urvature are dark. Fig. b shows the same data set textured with the value of the error weight. The sequene of meshes and triangle-ollapse operations provides an anestral hierarhy for any triangle T of a mesh j whih allows T to be assoiated with a set of verties in the original mesh. T is either a triangle in both j and j 1 or T was modified from a triangle T in j 1 through the ollapse of a triangle T C. In the first ase the anestral 5. We note that this weight is not sale invariant. If the data set is resaled the sequene of ollapsed triangles may hange.

11 GIENG ET AL.: CONSTRUCTING HIERARCHIES FOR TRIANGLE MESHES 155 (a) (b) Fig.. Curvature and error weights on a portion of the Crater Lake data set. Fig. 1. The triangle T is the result of triangle T after the ollapse of T C. The anestral points of T are the union of the verties of T and T C. points assoiated with T in j are just the anestral points of T in j 1. In the seond ase the anestral points of T are the union of the anestral points of T and those of T C (see Fig. 1). In this way every triangle T has a set of anestral TABLE 1 SUMMARIES OF THE MESH REDUCTION PROCESS FOR THE FOUR DATA SETS Data Set Level Triangles Perent Fig. Saddle % a % % e % f Mount St. Helens % 3a % % 3e % 3f Torus % 4a % % 4e % 4f Crater Lake % 5a % % 5e % 5g points in the original mesh whih are the points that affet the onstrution of the verties of T. When realulating the weight of the triangle T we an use points from the full anestral hierarhy to determine the approximating surfae. In this way points of the original mesh are used to alulate the weights of the triangles and the new ollapse points reduing error aumulation. 7 RESULTS The algorithm that we have presented allows the representation of large triangular meshes at multiple levels of detail requiring a relatively small number of triangulation levels to be stored. Our algorithm is based on the idea of ollapsing a ertain perentage of triangles in an intermediate mesh in a single step. This priniple leads to signifiant redutions regarding storage requirements. Furthermore it is possible to smoothly traverse the hierarhy upward and downward. We have applied our algorithm to several large triangulated models and have ahieved very enouraging results. Table 1 summarizes the results of the data sets shown in the olor plates. Fig. illustrates a saddle surfae originally ontaining triangles. We have illustrated the mesh at various lev-

12 156 IEEE TRANSACTIONS ON VISUALIZATION AND COMPUTER GRAPHICS VOL. 4 NO. APRIL-JUNE 1998 (a) (b) () (d) Fig.. Saddle surfae. (e) (f) els and the triangles seleted for ollapse at level 1 and at level 1. Due to a user-defined limit on the number of triangles seleted for eah level the algorithm initially selets triangles lose to the saddle point. However as the algorithm proeeds the triangles seleted for ollapse are nearly uniformly distributed over the mesh. Even though the mesh at level 3 has only.4 perent of the original number of triangles it still gives an adequate representation. Fig. 3 is taken from the digital elevation model of Mount St. Helens in the Casade Range in the western United States. This model originally has triangles. Fig. 3b shows the triangles and their stenils hosen for ollapse in the initial level. We note that these are distributed in the nearly flat areas of the data set. At level 1 most of the triangles in the flat areas have been ollapsed and the area weight begins to fore the algorithm to hoose triangles in the highly urved areas of the mesh (Fig. 3d). At level 3 the redued data set has less than.5 perent of the original number of triangles. While most of the detail has disappeared this still is a very good representation of the Mount St. Helens region.

13 GIENG ET AL.: CONSTRUCTING HIERARCHIES FOR TRIANGLE MESHES 157 (a) (b) () (d) Fig. 3. Model of Mount St. Helens. (e) (f)

14 158 IEEE TRANSACTIONS ON VISUALIZATION AND COMPUTER GRAPHICS VOL. 4 NO. APRIL-JUNE 1998 (a) (b) () (d) Fig. 4. Triangulation of a torus. (e) (f) The images of Figs. and 3 were generated with weights defined by W5 T A = T 5 T + 5 T + V5 κ α T Here the urvature weight and the area have the maximum influene on the outome of the mesh-redution algorithm. Fig. 4 shows triangulations of a torus initially represented by 5 triangles. Fig. 4d shows the triangles seleted for ollapse in level 1 and Fig. 4d shows the triangles seleted for ollapse at level 5. Notie that the algorithm does not hoose triangles on the equator as the triangles there fit the resulting surfae very well. Eventually the area weight fores the algorithm to onsider these triangles. Fig. 5 shows the triangulation levels of the digital elevation model of the Crater Lake region in the western United States. Fig. 5b shows the triangles hosen for ollapse in the initial level. Notie that the seleted triangles are distributed in the flat areas of the data set. The mesh at level 3 ontains less than.5 perent of the original number of triangles in the data set. The image of Figs. 4 and 5 were generated with weights defined by

15 GIENG ET AL.: CONSTRUCTING HIERARCHIES FOR TRIANGLE MESHES 159 (a) (b) () (d) (e) (f) (g) (h) Fig. 5. Triangulation of the Crater Lake model.

16 16 IEEE TRANSACTIONS ON VISUALIZATION AND COMPUTER GRAPHICS VOL. 4 NO. APRIL-JUNE W T = A T κ T + α T + V T + E T Here the dominant fator is the preditive error weight. The initial preproessing step of the algorithm sets up the hierarhy of meshes in approximately 15 minutes for the large data sets and three to five minutes for the smaller ones. In all meshes the algorithm attempted to selet.5 perent of the triangles in the priority queue at eah level. Sine four triangles disappear with eah ollapse this implies that at most 1 perent of the triangles will be removed at eah level. However due to the stenil overlap the atual numbers removed at eah level will be less than 1 perent. Edge swapping to eliminate yles was not implemented as it was found that the algorithm will selet neighbors of the stenil triangles for ollapse. This eliminates most of the yles as the algorithm progresses. The pitures an be viewed in real time on a Silion Graphis Onyx omputer system. 8 CONCLUSIONS We have introdued an algorithm for the hierarhial representation of very large triangle meshes. The algorithm produes a sequene of meshes 1... Q where eah mesh j is ollapsed to mesh j+1 through a set of simultaneous triangle-ollapse operations. For eah triangle a funtion is generated that approximates the underlying surfae in the area of the triangle and this funtion serves as a basis for assigning weights to eah triangle and for supplying the point to whih triangles are ollapsed. Using this representation allows us to display a large triangle mesh at various levels of detail in real time while preserving the geometry of the original mesh. This work extends previous work on level-of-detail analysis for triangle meshes in several ways. First our algorithm fouses on the triangle as a primitive and the triangle-ollapse operation as the primary redution strategy for the mesh. Seond our algorithm produes a sequene of meshes whih together with the triangle-ollapse operation an be used to produe a ontinuous level-of-detail variation in the model. We have integrated this model into a prototype viewing system that supports interative levelof-detail manipulation of omplex models defined by large triangle meshes. Third whenever we ompute the loation of a new vertex replaing a triangle we onsider the anestral hierarhy reated by the ollapse operations and alulate the weights of triangles using the original surfae data. This ensures that the simplified mesh approximates the original surfae well. ACKNOWLEDGMENTS This work was supported by the U.S. National Siene Foundation under ontrats ASC (Researh Initiation Award) and ASC (CAREER Award) by the U.S. Offie of Naval Researh under ontrat N and the U.S. Army Researh Offie under ontrat ARO MA-RIP. The digital elevation models are ourtesy of the United States Geologial Survey. We would like to thank Eri LaMar for helping with the digital elevation models and all members of the Visualization and Graphis Group of CIPIC for their help. REFERENCES [1] P. Cignoni L. De Floriani C. Montoni E. Puppo and R. Sopigno Multiresolution Modeling and Visualization of Volume Data Based on Simpliial Complexes Pro Symp. Volume Visualization A. Kaufman and W. Krueger eds. pp ACM SIG- GRAPH Ot [] M. Ek T. DeRose T. Duhamp H. Hoppe M. Lounsbery and W. Stuetzle Multiresolution Analysis of Arbitrary Meshes SIG- GRAPH 95 Conf. Pro. R. Cook ed. pp Ann. Conf. Series Addison-Wesley Aug [3] [4] [5] [6] E. Stollnitz T. DeRose and D. Salesin Wavelets for Computer Graphis: Theory and Appliations. San Franiso: Morgan Kaufmann P. Shröder and W. Sweldens Spherial Wavelets: Effiiently Representing Funtions on the Sphere SIGGRAPH 95 Conf. Pro. R. Cook ed. pp Ann. Conf. Series Addison- Wesley Aug W.J. Shroeder J.A. Zarge and W.E. Lorensen Deimation of Triangle Meshes Computer Graphis (SIGGRAPH 9 Pro.) E.E. Catmull ed. vol. 6 pp July 199. K.J. Renze and J.H. Oliver Generalized Unstrutured Deimation IEEE Computer Graphis and Appliations vol. 16 no. 6 pp. 4-3 Nov [7] H. Hoppe Progressive Meshes SIGGRAPH 96 Conf. Pro. H. Rushmeier ed. pp Ann. Conf. Series Addison-Wesley Aug [8] [9] H. Hoppe View-Dependent Refinement of Progressive Meshes SIGGRAPH 97 Conf. Pro. T. Whitted ed. pp Ann. Conf. Series Addison-Wesley Aug J. Popovi and H. Hoppe Progressive Simpliial Complexes SIGGRAPH 97 Conf. Pro. T. Whitted ed pp Ann. Conf. Series Addison-Wesley Aug [1] H. Hoppe T. DeRose T. Duhamp J. MDonald and W. Stuetzle Mesh Optimization Computer Graphis (SIGGRAPH 93 Pro.) J.T. Kajiya ed. vol. 7 pp Aug [11] J.C. Zia and A. Varshney Dynami View-Dependent Simplifiation for Polygonal Models Pro. IEEE Visualization 96 pp Ot [1] M. Garland and P.S. Hekbert Surfae Simplifiation Using Quadrati Error Metris SIGGRAPH 97 Pro. T. Whitted ed. pp Ann. Conf. Series Addison-Wesley Aug [13] B. Hamann A Data Redution Sheme for Triangulated Surfaes Computer Aided Geometri Design vol. 11 pp [14] B. Hamman Curvature Approximation for Triangulated Surfaes Computing vol. 8 (supplement) pp [15] G. Turk Re-Tiling Polygonal Surfaes Computer Graphis (SIG- GRAPH 9 Pro.) E.E. Catmull ed. vol. 6 pp July 199. [16] J. Cohen A. Varshney D. Manoha G. Turk H. Weber P. Agarwal F.P. Brooks Jr. and W. Wright Simplifiation Envelopes SIGGRAPH 96 Conf. Pro. H. Rushmeier ed. pp Ann. Conf. Series Addison-Wesley Aug [17] P. Lindstrom D. Koller W. Ribarsky L.F. Hughes N. Faust and G. Turner Real-Time Continuous Level of Detail Rendering of Height Fields SIGGRAPH 96 Conf. Pro. H. Rushmeier ed. pp Ann. Conf. Series Addison-Wesley Aug [18] T.S. Gieng K.I. Joy B. Hamann G.L. Shussman and I.J. Trotts Smooth Hierarhial Surfae Triangulations Pro. Visualization 97 H. Hagen and R. Yagel eds. pp Ot [19] M. de Berg M. van Kreveld M. Overmars and O. Shwarzkopf Computational Geometry: Algorithms and Appliations. Berlin: Springer-Verlag [] M.P. do Carno Differential Geometry of Curves and Surfaes. Englewood Cliffs N.J.: Prentie Hall 1976.

17 GIENG ET AL.: CONSTRUCTING HIERARCHIES FOR TRIANGLE MESHES 161 Tran Gieng is a graduate student in the Multiresolution Modeling Group at the California Institute of Tehnology. He graduated in 1997 with a BS in omputer siene and engineering from the University of California at Davis. His researh interests are in the areas of mathematis and omputer siene. He is a member of the ACM. Bernd Hamann reeived a BS in omputer siene a BS in mathematis and an MS in omputer siene from the Tehnial University of Braunshweig Germany. He reeived his PhD in omputer siene from Arizona State University in He is an assoiate professor of omputer siene and odiretor of the Center for Image Proessing and Integrated Computing (CIPIC) at the University of California Davis and an adjunt professor of omputer siene at Mississippi State University. His primary interests are visualization omputer-aided geometri design (CAGD) and omputer graphis. Dr. Hamann is a member of the ACM the IEEE and SIAM. Kenneth I. Joy reeived a BA and MA in mathematis from the University of California at Los Angeles and a PhD in mathematis from the University of Colorado in He is an assoiate professor of omputer siene at the University of California at Davis. He ame to UC Davis in 198 in the Department of Mathematis and was a founding member of the Computer Siene Department in His primary interests are visualization geometri modeling and omputer graphis. Dr. Joy is a member of the ACM the IEEE and SIAM. Greg L. Shussman is a graduate student in omputer siene at the University of California Davis. His researh interests are omputer graphis and sientifi visualization. He is a member of the ACM and the IEEE. Issa J. Trotts is pursuing a BS in omputer siene and a BS in mathematis at the University of California Davis. His researh interests inlude omputer graphis visualization and omputer-aided geometri design (CAGD). He is a member of the ACM.