On-the-Fly Adaptive Subdivision Terrain

Transcription

1 On-the-Fly Adaptive Subdivision Terrain Dir Rose, Martin Kada, Thomas Ertl University of Stuttgart, Institute of Computer Siene Visualization and Interative Systems Group Breitwiesenstraße 20-22, Stuttgart, Germany rose Abstrat In this paper we present a method to ahieve interative rendering of smooth terrain based on oarse data. Therefore we use adaptive subdivision surfaes whih are alulated on the fly by using information from the direted edge based mesh of the previously rendered frame and throwing away triangles not used any longer. This way we are able to gain fast, high quality meshes whih are refined depending on the point of view with marginal memory onsumption. Additionally, this tehnique is ompatible to traditional data strutures like progressive meshes. 1 Introdution Although terrain rendering has improved a lot with regard to quality and speed in the reent years, there are still some shortomings when approahing a surfae. Massive use of textures may give a terrain a natural appearane and this works fine when viewed from afar, but when losing up the flat faets of a terrain an be learly spotted in general ases. This may be annoying, e.g. in flight simulators whih show a smooth surfae from above but oarse hills when in low level flight or rolling on the ground. Low resolution an originate from several fats. Rendering speed should not be the bottlenek, sine progressive meshes and level-of-detail overome this problem. The main aspet is memory onsumption, both main memory and hard drive or CD/DVD apaity. It seems impossible nowadays to save the elevation data of the whole world with a resolution of 1m 1m 1m. When disregarding the oeans this still yields about bytes of unompressed, diffiult to manage data. Therefore either the range of interation or the resolution has to be restrited. In many ases the latter is hosen and the typial distane between two sample points is about 100m, and 30m or below for more interesting regions. This results in the mentioned visual defet when approahing the ground. We want to show a way to polish rough edges with almost no supplementary use of memory by adaptively subdividing the terrain data on the fly. This strategy meets today s typial omputer equipment, e.g. onsole platforms, providing a low- to mid-end graphis board, relatively little memory, and nevertheless a quite fast proessor. 2 Related Work In reent years a lot of researh has been onduted in simplifiation of omplex meshes. In a first preproessing step these tehniques [14, 9] an be used to redue high resolution data to a manageable amount of samples. In a seond preproessing pass a progressive mesh struture [3] may be alulated. During interation these progressive meshes or similar algorithms [8, 10, 11, 15] an be used to build up a view-dependent level-of-detail representation of the sene. In our senario we adapt data strutures based on these methods. However, when approahing the ground we want to swith the diretion and refine the mesh. Sine we had to drop most of the detailed smooth information due to lak of memory, we must find a way to regain or reinvent the lost data. For this purpose we use subdivision pathes originally developed as an enhanement to B-spline surfaes. In general these are used to desribe objets, e.g. in CAD, with just a few ontrol points. The possibility to use subdivision algorithms as a method to reonstrut surfaes is shown in [12]. VMV 2001 Stuttgart, Germany, November 21 23, 2001

2 So far subdivision approahes were too slow to allow interative frame rates. In the following setions we will present a way to overome this problem, therefore allowing smooth looking surfaes even on low-memory platforms. 3 Subdivision Surfaes d a b There are many forms of subdivision surfaes [17]. These surfaes are generated by onseutively dividing an initial oarse mesh, onverging against a smooth surfae. The subdivision is ontrolled by a subdivision mask or stenil. This mask determines whih oordinates of neighboring points are ombined to alulate the newly inserted point of an edge. Loop and Catmull-Clark surfaes [5] are approximating subdivision surfaes. Similar to B- splines, these surfaes are lose to the ontrol points but do not ontain them. Due to this property we annot use this kind of surfaes beause we want to keep the original oordinates and not alter them. We use an interpolating sheme instead, the butterfly surfae [6], whose initial mask (the solid lines of Figure 1) looks like a butterfly. For better results this mask an be extended by two additional points [7, 18], as shown in Figure 1. The points are multib Figure 1: Butterfly subdivision mask for the white point in the middle plied by different weights, with the parameter allowing a more subtle ontrol of the resulting surfae:! " a $# d % & (1) (' Setting results in the original butterfly mask. The tangent spae an be alulated using the omplete 1- and 2-ring neighborhood of a point as shown in Figure 4 and with this information it is easy to obtain the normal at this point. For the weights please refer to [17]. We have implemented an adaptive algorithm that only subdivides a triangle when this will result in a visible hange. This an be either a hange in height by one pixel or more or it an be a signifiant hange in normal diretion. As a first order approximation for both ases we an use the distane between the yet to be inserted point and the enter point of the orresponding edge. Saled by the ratio of window size to z depth, it is a quality measure in sreen spae dimensions. Setting the error bound to 1 pixel or below will produe an adaptive mesh that looks the same as an infinitely subdivided surfae. During the subdivision proess a red-green triangulation strategy [1, 2] is used to avoid raks between neighboring triangles. Therefore we permit only a differene of one level in subdivision between two neighboring triangles. This proedure also guarantees the easy registration of the subdivision mask at any time. The )* ontinuity at the borders permits a smooth transition to the surrounding original oarse mesh. Subdivision of the border will not be allowed until it is needed due to higher order divisions of neighboring triangles. In this ase the border is subdivided linearly to guarantee that there will not be any raks between the subdivided parts and the original surrounding mesh. 4 On-the-Fly Control The key to ahieve interative frame rates lies in an effiient algorithm for inserting and destroying newly subdivided triangles whih are not needed any more. The main data struture is based on a lean version of direted edges [4]. Here a half-edge is represented by the address of the vertex where the direted edge points to. Thus a triangle is defined by three onseutive half-edges. The neighboring triangle of an edge an be aessed via another pointer that addresses the opposing half-edge. Progressive meshes use the same or a similar data organization in general whih failitates the ombination with our algorithm. If a neighboring triangle is subdivided one level less than the original triangle, then its half-edge points to only one of the juxtaposed half-edge as an be seen in Figure 2. This is one reason why we introdue mid-verties to the original half-edge struture. These midverties belong to every edge and ontain the position whih would arise when subdividing this edge. Additionally we an attah the pointer to the seond neighboring half-edge to it and therefore we 88

3 Figure 2: Half-edge struture with white mid-vertex an save the ostly searh for this seond edge. Another advantage of mid-verties is the ease to alulate the next subdivision level of a triangle. In regions of high urvature we an benefit from these prealulated verties beause the registration of the subdivision mask is more favourable. The alulation of the mid-verties itself is quite simple sine the neighboring triangles do not have the strong fragmentation they may have one step later. In some ases suh a fragmentation may indue a laborious searh for the butterfly mask verties along the half-edges, as shown in Figure 3a. The retrieval is straightforward if the mid-verties have been prealulated on a oarser level (Figure 3b). a) b) Figure 3: a) The way along the direted edges to the outer gray points an be lengthy b) Using prealulated mid-verties may save work The mid-verties are also useful when alulating the tangent vetors sine we need the 1- and 2-ring neighborhood. The mid points help to limit the expansion, sine they are already subdivided one level higher as an be seen in Figure 4. In most ases this prevents the need for virtual verties (see below) and allows a faster alulation of the tangents and aordingly the normals. Finally, we need the mid-verties for the fast quality evaluation in sreen spae. As mentioned in Figure 4: Butterfly tangent mask of the white point using prealulated mid-verties Setion 3 we use this as an overestimated approximation of the expensive Hausdorff distane, whih would be the exat error measure. Another improvement to the original half-edge struture onerns to border verties. Sometimes it an be diffiult or impossible to find all edges belonging to a border vertex by simply moving along neighboring triangles and edges. A separate pointer to the inoming border edge solves this problem and results in a speedup. Butterfly surfaes need the 1-ring of the verties defining the edge to be split (see Figure 1). At borders it is not possible to find a 1-ring that fulfills the requirements for the subdivision mask beause a border vertex does not have a valene of 6 in most ases. Therefore we have to mirror verties from inside along the border to define virtual verties outside so we regain a valene of 6. When the subdivision surfae is embedded in a surrounding progressive mesh, it is possible and advisable to use neighboring verties from the outer mesh sine this will mostly result in a better quality than using virtual verties. A topologial operator manages the reorganization of triangle and edge relationships. When a triangle needs to be subdivided, a dyadi split is performed, whih divides this triangle into four triangles. The middle triangle beomes the new master triangle, whih reuses the memory of the parent triangle. The surrounding three slave triangles are newly reated. The master triangle saves the information about the subdivision depth or the level, respetively. The slave triangles do not need this information beause they will either be of the same depth or they will get their own master tri- 5 89

4 angle defining their level of subdivision if they are subdivided further. If the differene of levels between neighboring triangles is greater than one, the oarser triangle needs to be subdivided too. In these ases the stored depth an be used for a fast omparison. When refining it is reasonable to first split all oarse triangles and subsequently the more and more refined ones. This leads to the following algorithm: Finally the surfae an be rendered. Although we used a red-green triangulation strategy, we allow differenes in subdivision depth of level 1. This may result in gaps where different levels meet. But fortunately it is easy to fill these gaps with a final just in time subdivision whih does not need to be stringently dyadi in analogy to a green split. Basially there are three ases whih need a path to avoid holes shown in Figure 5. for all triangles T (inreasing level) if needed then subdivide(t) proedure subdivide(triangle T) for all neighboring triangles N of T if subd_level(n) < subd_level(t) subdivide(n) dyadi_split(t) However, before the refinement proess is performed, the mesh is investigated in areas where it an be simplified again if the new point of view permits it. Simplifiation is only possible when all triangles in the neighborhood have the same or smaller subdivision level beause otherwise the ondition that only a differene of 1 level is allowed would be broken. If the triangles meet the ondition, the information of the master triangle will be used to rebuild the old, one level oarser triangle and to undo the dyadi split. It makes sense to start with the finest triangles asending to the oarsest triangles, resulting in the pseudoode: for all mastertriangles TMid (dereasing level) if quality is too good then get_slaves T1,T2,T3 of TMid undivide(tmid,t1,t2,t3) undivide(triangle TMid,T1,T2,T3) for all neighboring triangles N of TMid,T1,T2,T3 if subd_level(n) > subd_level(tmid) return undo_dyadi_split(t) Figure 5: Temporary split to avoid raks The texture oordinates are alulated during the rendering proess by a simple orthographi projetion of the texture along the height axis onto the fine mesh. This is valid sine terrain textures are gained mostly from pitures taken from vertially above by satellites or planes, whih also orresponds to an approximately linear mapping funtion. Additionally textures may be used for hardware aelerated lighting alulations of small details [16]. 5 Results The frame rates of Table 1 were obtained on an Athlon K7C 1.33GHz with a GeforeII-MX graphis board by rotating the heightfield whih an be seen on the olor plates in the appendix around various axes and averaging the times. When using uniform subdivision, the number of triangles is the same, for a viewport size of both and pixels. The frame rates for higher resolutions are limited by the fill rate of the graphis board. The appearane does not improve very muh at subdivision level 5 or higher, good results an be ahieved at level 3 or 4. Aordingly, the adaptive method with an error bound of 2 or 1 pixels yields the same quality but 90

5 viewport x viewport y u n i f o r m a d a p t i v e level of subdiv. number of triangles frames per seond error in pixels number of triangles frames per seond Table 1: Frame rates of terrain data starting with 8 8 faets is 2.5 up to 10 times faster than the uniform division. Here the lower values in higher resolutions are aused by fill rate limits as well. Additionally there are more triangles to be rendered due to the dependeny of the error limit on the viewport dimensions. 6 Conlusions and Future Work In this paper we have presented a fast and effiient implementation of the butterfly subdivision algorithm. By using a modified direted edges data struture and by introduing mid-verties, whih are easier and faster to alulate on a oarser level, it is possible to ahieve highly interative frame rates. The ompatibility with onventional tehniques like progressive meshes alleviates the handling and allows the use of both methods at the same time. The field of appliation may be an improvement of traditional terrain rendering or LOD-rendering of CAD models desribed as subdivision surfaes. In the future we would like to study if an implementation via hardware aelerated vertex programming is possible. This would result in an enormous speed burst on ontemporary PC graphis hardware. 7 Aknowledgement The implementation and ideas are based on the diploma thesis [13] of one of the authors and we would like to thank Rüdiger Westermann for valuable hints during this work. Referenes [1] Randolph E. Bank, Andrew H. Sherman, and Alan Weiser, Refinement algorithms and data strutures for regular loal mesh refinement, Sientifi Computing, R. Stepleman, ed., North Holland, Amsterdam, pp. 3 17, [2] Jürgen Bey, Tetrahedral Grid Refinement, Computing, 55, pp , [3] Swen Campagna, Polygonreduktion zur effizienten Speiherung, Übertragung und Darstellung komplexer polygonaler Modelle, PhD-thesis, Herbert Utz Verlag, [4] Swen Campagna, Leif Kobbelt, and Hans- Peter Seidel, Direted Edges A Salable Representation for Triangle Meshes, Tehnial Report 2/1998, Lehrstuhl für Graphishe Datenverarbeitung, Universität Erlangen- Nürnberg, [5] Edwin Catmull and James Clark, Reursively Generated B-Spline Surfaes on Arbitrary Topologial Meshes, Computer Aided Design 10, 6, pp , [6] Nira Dyn, David Levin, and John A. Gregory, A Butterfly Subdivision Sheme for Surfae Interpolation with Tension Control, ACM Transations on Graphis 9, 2, pp , [7] Nira Dyn, Sigalit Hed, and David Levin, Subdivision Shemes for Surfae Interpolation, Workshop in Computational Geometry, pp , [8] Mihael Garland and Paul S. Hekbert, Fast Polygonal Approximation of Terrains and Height Fields, Tehnial Report CMU- CS , Shool of Computer Siene, Carnegie Mellon University, Pittsburgh, [9] Paul S. Hekbert and Mihael Garland, Survey of Polygonal Surfae Simplifiation Algorithms, SIGGRAPH 97, [10] Hugues Hoppe, View-Dependent Refinement of Progressive Meshes, SIGGRAPH 97 Pro- 91

6 eedings, pp , [11] Hugues Hoppe, Smooth view-dependent level-of-detail ontrol and its appliation to terrain rendering, IEEE Visualization 98 Proeedings, pp , [12] Hugues Hoppe, Tony DeRose, Tom Duhamp, Mark Halstead, Hubert Jin, John MDonald, Jean Shweitzer, Werner Stuetzle, Pieewise Smooth Surfae Reonstrution, SIGGRAPH 94 Proeedings, pp , [13] Martin Kada, Unterteilungsverfahren zum beshleunigten Rendering von Oberflähen, Diplomarbeit, Institut für Informatik, Universität Stuttgart, [14] Reinhard Klein, Gunther Liebih, and Wolfgang Straßer, Mesh Redution with Error Control, IEEE Visualization 96 Proeedings, pp , [15] Stefan Roettger, Wolfgang Heidrih, Philipp Slusallek, and Hans-Peter Seidel, Real-Time Generation of Continuous Levels of Detail for Height Fields, Proeedings of WSCG 98, pp , [16] Dir Rose and Thomas Ertl, Rendering Details on Simplified Meshes by Texture Based Shading, Workshop on Vision, Modelling, and Visualization VMV 00, pp infix, [17] Denis Zorin, Stationary Subdivision and Multiresolution Surfae Representations, Ph.D. Thesis, California Institute of Tehnology, [18] Denis Zorin, Peter Shröder, and Wim Sweldens, Interpolating Subdivision for Meshes with Arbitrary Topology, SIG- GRAPH 96 Proeedings, pp ,

7 ,, /. -,, Color Plate On-the-Fly Adaptive Subdivision Terrain a) oarse terrain 8 8 faets ) subdivided 8 8 terrain (adaptive, pixel) b) subdivided 8 8 terrain (level 3) d) omparison: terrain faets 93