3-Dimensional Object Modeling with Mesh Simplification Based Resolution Adjustment

Transcription

1 3-Dimensional Object Modeling with Mesh Simplification Based Resolution Adjustment Özgür ULUCAY Sarp ERTÜRK University of Kocaeli Electronics & Communication Engineering Department Izmit, Kocaeli / Turkey oulucay@yahoo.com, sertur@kou.edu.tr Abstract A 3-Dimensional (3-D) object modeling technique with mesh simplification and refinement based resolution adjustment is proposed in this paper. Polygonal models which are widely utilized in the modeling of 3-D objects are taken as basis, making use of the polygonal structure and vertex coordinates for the display of 3-D models. The amount of polygons and vertices of a model is proportional to the resolution as well as data quantity. In other words the resolution and data increases with the number of polygons. In this paper, it is proposed to utilize resolution, and hence data amount, adjustable 3-D modeling so that model resolution and transmitted data amount can be regulated according to access constraints. 1. Introduction The process of representing object shapes is commonly referred to as modeling. 3-Dimensional (3-D) modeling of object shapes is commonly utilized in several applications such as computer graphics, computer aided design, computer games, object recognition or tracking and model based video coding. While there are multiple 3-D modeling techniques, each has its own advantages and drawbacks depending on the application area. However, the polygonal shape modeling with triangular facets has become the de facto standard for representation of 3-D shapes on computers. Polygonal models are commonly preferred mainly because of their simplicity in representing complex shapes and ease of handling. A polygonal model consists of three kinds of mesh elements, namely vertices, edges and faces (usually triangles). These elements are used to describe the mesh geometry and mesh connectivity. A common approach is to keep a list of vertex coordinates to defined the geometry, and a second list of polygon connectivity information (for face-based structures) or edge connectivity information (for edge-based structures). One way of obtaining the 3-D model polygonal of an object is to utilize 3-D scanners. However, scanned 3-D object models can consist of millions of vertices, and although a higher sampling rate means improved shape resolution, the high data amount can be impractical for some applications requiring real-time rendering, manipulation, storage or transmission. The approaches for reducing the data amount involved in the representation of 3-D meshes can be classified into three categories: Simplification algorithms that reduce the number of elements in a mesh, algorithms that compactly encode the relations of mesh elements, and algorithms that encode the mesh geometry. Simplification algorithms are essential for reducing the data amount if the original 3-D model is sampled at high resolution and a lower resolution is sufficient for the application in mind. As this is commonly the case if 3D models are acquired by laser scanners, a lot of research has been carried out in the field of mesh simplification. One approach to accomplishing the simplification is to carry out the simplification iteratively, and store the removed information, so that the original model can be progressively reconstructed if desired. This approach is particularly appealing if users access the model from a remote location, and hence transmission of the model data is required. In this case it is usually not sensible to transmit the huge data amount of the original high resolution model, and it is rather practical to transmit a low resolution version first and then improve the resolution through additional enhancement layers transmitted if the user desires a higher resolution. Various mesh simplification algorithms have been proposed in the literature. Vertex clustering by merging cells to cells or clusters and representing each cell by one vertex has been proposed in [1]. While the proposed technique is very simple and

2 quick, it is not suitable for non-uniform meshes and can also fail to preserve topological properties. Face clustering has been proposed in [2], by merging coplanar faces to superfaces, with the utilization of error metrics to keep the topology, however it is difficult to triangulate the patches and there is a possibility of creating holes in the model. Progressive meshes (PM) are a well-known method for mesh simplification based on the edge collapse or edge contraction operation, where a single vertex is removed at each time [3, 4]. While the method is time-consuming and hence not appropriate for realtime mesh simplification applications, it provides simple and effective means for offline stabilization. Recently a multiphase approach consisting of an initial out-of-core uniform vertex clustering, followed by an in-core iterative edge contraction has been proposed in [5]. Figure 1 shows a Bunny model consisting of vertices and polygons, from three different perspectives. Figure 2 displays a 3-D Human Body model composed of vertices and polygons. In order to introduce flexibility the 3-D model display system has been implemented to support arbitrary polygon shapes, yet mainly triangular mesh models are utilized due to their processing simplicity and mesh uniformity. The implemented display system support interactive alteration of color and lighting, as well as supporting translation, rotation and zooming possibilities. In order to introduce a lighting effect, so that every facet is rendered according to a light source located in the 3-D space, the normal of each triangle is calculated using the Newell method [6] to provide brightness values that vary according to the surface normal. Both models displayed in Figure 1 and 2 have a large mount of vertices and polygons that makes transmission of model data and real-time interactive display and animation prohibitively difficult. Mesh simplification of large polygonal models is an important issue in Computer Graphics, and there are mainly two reasons that make simplification an essential component. The first reason is that sophisticated 3D scanners can capture real objects at high resolution, producing very fine meshes with sometimes millions of faces. These models are usually highly oversampled, and it is possible to at least remove redundant polygons before having the model enter the graphics production pipeline [7]. The second reason for requiring mesh simplification algorithms is that detailed geometric models can be quite large even after redundancy is eliminated. The large amount of huge polygonal representations can easily overwhelm most graphics programs, rendering their use in applications impractical. Furthermore if texture mapping is to be utilized, a lower resolution model can be employed as the texture will hide errors anyway. The strategy to overcome such limitations is commonly based on multiresolution models, that allow processing geometry at multiple levels of detail (LoD). Simplification algorithms constitute the main component in the creation of the multiresolution representations. A mesh simplification algorithm takes as input a polygonal surface description and outputs a simpler mesh representing the same surface. It aims to reduce the number of polygons so as to speed up rendering while assuring a good approximation of the original model. If the simplification algorithm continues simplification beyond redundancy removal, there will be a difference between the original model and the simplified version. However, if the difference is small enough the loss in model accuracy can be traded against a reduction in the number of vertices and polygons. Figure 1. The Bunny model composed of vertices and polygons.

3 Figure 2. The 3-D Human Body model composed of vertices and polygons. This paper utilizes triangle collapse based mesh simplification by merging three vertices into a single vertex. Hence, the approach is equivalent to two edge collapse operations carried out in a single turn. This easy iterative simplification procedure enables the construction of multiple models at different levels-of-detail (LoD) providing a multiresolution modeling scheme. The data lost at each simplification step is saved as enhancement layer to reconstruct the higher resolution model from the lower resolution case by refinement. 2. The Proposed Approach The proposed simplification algorithm accomplishes iterative mesh simplification so that 3-D models with varying level of detail are obtained at each step to provide a multiresolution framework. The triangle collapse approach is employed as mesh simplification procedure. While carrying out the simplification, it has been taken care that the simplification is not carried out in a random fashion, but triangles that have less geometric contribution are collapsed first. The initial step of the algorithm is the computation of the surface normals for all triangles in the original model. Then for each triangle, the maximum angle between the normal of that triangle and the normals of neighbor triangles is evaluated. The maximum angle is a direct measure of geometric contribution, a small angle states that the corresponding triangle has little geometric contribution and can be collapsed during mesh simplification. Once the maximum surface normal angle of each triangle is computed, it is possible to order the triangles from minimum angle to maximum angle and sequentially carry out the simplification starting with lower angle polygons. Two approaches are possible, triangles can be collapsed until a certain simplification percentage is achieved or an angle threshold can be set and all triangles with surface normal angles below that threshold can be collapsed. The second approach is adapted in this paper. At each level polygons with surface normal angles below the threshold are ordered from lowest angle upwards. Triangles with the smallest angle are consecutively collapsed. In order to carry out the triangle collapse, the centre coordinate of the three vertices of the triangle is computed and added to the vertex list, while the three vertices of the collapsed triangle are removed from the list. The corresponding triangle and its three first order neighbor triangles (neighbors that have two vertices in common with the triangle to be collapsed) are destroyed and removed from the polygon list. The second order neighbor polygons (polygons with one vertex in common with the triangle to be collapsed) are modified by replacing the vertex of the collapsed triangle with the new vertex obtained as the centre location. Note that modified polygons are marked as changed for the current simplification level so that they don t enter the simplification process ate the same level again, to avoid confusion and maintain uniformity to some extent. This procedure is repeated until all triangles with surface normal angles below the threshold are

4 processed. The main structure of the simplification algorithm is given below. Algorithm : Simplification of 3-D models For i1 = ( 1 : number of polygons) Do Compute Surface Normals; Point each polygon as unprocessed; For i2 = ( 1 : number of polygons )Do Calculate the max. angle between neighbors For i3 = ( 1 : number of polygons) Do If ( Angle < Threshold) Put polygon into simplification queue For j = ( 1 : # of polygons in queue) Do If (unprocessed) For i4 = ( 1 : number of polygons) Do Select the polygon that has smallest angle value 1 st vertex = (1 st vertex+2 nd vertex+3 rd vertex)/3 Destroy polygon Destroy 2 nd vertex and 3 rd vertex For i5 = ( 1 : number of polygons ) Do Look at the vertices of all polygons If (polygon has any of the 3 vertices) vertex = 1 st vertex value For i6 = (1 : number of polygons) Do Find 1 st order neighbors! Destroy 1 st order neighbors Find 2 nd order neighbors! Mark 2 nd order neighbors as processed Figure 3 demonstrates the mesh simplification process on an example mesh for various simplification levels, i.e. various levels of detail. It can be seen that the simplification is carried out fairly uniformly. Figure 4 shows the multiresolution models obtained by the mesh simplification for various levels of detail together with the number of vertices and polygons as well as the uncompressed file size. It can be seen that by iterative simplification, a multiresolution modeling scheme with changing level of detail can be obtained. The 3-D model resolution reduces together with file size after each simplification step. Note that it is possible to further reduce the file size through coding, which is however beyond the scope of this paper. In order to construct an entire multiresolution scheme in which it is not only possible to go from the highest resolution to the lowest resolution by mesh simplification, but which also support refinement from the lowest resolution upwards, it is proposed to save the mesh structures collapsed at each simplification level. This data will serve as enhancement layer, to allow refinement of lower resolution models into higher resolution ones. However it is not sufficient to keep the collapsed vertex and triangle information, but it is also required to keep the modification information which depicts how the mesh structure of the lower resolution level should be changed to incorporate the additional mesh elements of the enhancement layer. Original Mesh Level 1 Level 2 Level 3 Level 4 Level 5 Figure 3. Mesh simplification example for various levels of detail.

5 a b c d e f g a) Original model : 7664 Vertices and Polygons : 413 KB b) Simplified model at a rate of 33% : 5930 Vertices and Polygons : 318 KB c) Simplified model at a rate of 41% : 4588 Vertices and 9172 Polygons : 245 KB d) Simplified model at a rate of 53% : 3644 Vertices and 7284 Polygons : 194 KB e) Simplified model at a rate of 62% : 2954 Vertices and 5900 Polygons : 156KB f) Simplified model at a rate of 69% : 2440 Vertices and 4870 Polygons : 128 KB g) Simplified model at a rate of 74% : 2038 Vertices and 4064 Polygons : 106 KB Figure 4. Simplification results for the 3-D Right Hand model. The support for refinement enables that users can initially download the low resolution model to have a first view of the object, without having to download the bulk data for an initial look. After viewing the low-resolution model, the user can refine the model if desired. Naturally, the user will be required to download more data in total if refinement from low resolution to high resolution is carried out, as the enhancement layers not only contain the dropped mesh element information but also the information of which polygons of the low-resolution models should be changed to accommodate the new ones. However refinement will be only carried out if the user requests a higher resolution model and has sufficient download speed, and as in many cases the lower resolution model will be sufficient the total amount of transmitted data will be reduced. Another advantage of a multiresolution scheme is that the user gets resolution updates with each enhancement layer received. It is also possible to give the user only two options, to download the lowest resolution model and the highest resolution model, however in this case the user will have to wait until the entire data of the highest resolution model is downloaded before it can be viewed. Figure 5 shows the refinement procedure and demonstrates the amount of data involved during the process for the Right Hand model. At the lowest resolution ( model g ) the data amount required to represent the model is 106 Kbytes. The enhancement layer comprising the information required to build the model with one higher resolution level ( model f ) takes 56 Kbytes, so that a total of 162 Kbytes of data is required if the model is refined from the lower resolution (compare to the necessary 128 Kbytes to represent model f unaided). The enhancement layer to construct model e by refining model f is 69 Kbytes, so that a total of 231 Kbytes is required for constructing model e from the lowest resolution model. Continuing in this way it is seen that the total amount of data required to construct the highest resolution model ( model a ) from the lowest resolution model by refinement is 783 Kbytes, almost twice as much compared to the 413 Kbytes required to represent the highest resolution model by itself. Although the difference will reduce if compression through coding is used, it is still an important weight that has to be accepted for having a multiresolution scheme.

6 g f e d c b a Model_g Model_f Model_e Model_d Model_c Model_b Model_a 106 KBytes 162 KBytes 231 Kbytes 319 Kbytes 433 Kbytes 589 Kbytes 783 Kbytes Addition_gf Addition_fe Addition_ed Addition_dc Addition_cb Addition_ba 56 Kbytes 69 Kbytes 88 Kbytes 114 Kbytes 156 Kbytes 194 KByes Figure 5. Refinement process of the Right Hand model 4. Conclusions A 3-Dimensional (3-D) object modeling technique with mesh simplification based resolution adjustment is proposed in this paper. High resolution models are iteratively simplified using triangle collapse, to obtain a multiresolution representation scheme with various levels of detail as well as varying data amount. The structure information dropped during each simplification step is saved in an enhancement layer that is used to reconstruct higher resolution models from lower resolution ones through refining. Because the simplification is carried out according to the angle between neighbor surface normals, polygons with less geometrical contribution are collapsed first. Figure 6 shows for example the models obtained by simplifying the Torso model and it is seen that the nose which has the highest resolution is preserved by smaller polygons, as these are the last to be taken into the simplification process. 5. References [1] J. Rossignac and P. Borrel: Multi-resolution 3D approximations for rendering, Modeling in Computer Graphics, pp Springer-Verlag, June-July [2] A. D. Kalvin and R. H. Taylor: Superfaces: Polygonal mesh simplification with bounded error. IEEE Computer Graphics & Applications, 16(3), pp64 77, May [3] H. Hoppe: Progressive meshes, Proceedings of SIGGRAPH 96, pp , August 1996, New Orleans, USA. [4] H. Hoppe, T. DeRose, T. Duchamp, J. McDonald and W. Stuetzle: Mesh Optimization, Proceedings of SIGGRAPH 93, pp [5] M. Garland and Eric Shaffer: A Multiphase Approach to Efficient Surface Simplification, Proceedings of IEEE VIS 2002, pages , October [6] T. McReynolds and D. Blythe: Advanced Graphics Programming Techniques Using OpenGL, Silicon Graphics, Siggraph 98 Course Notes, ACM SIGGRAPH, April [7] P. Heckbert and M. Garland (ed.): Surface Simplification, Chapter Survey of Polygonal Surface Simplification Algorithms, Course notes of Siggraph 97, ACM SIGGRAPH, July Acknowledgement This work was supported by DPT with project number DPT2004K

7 a ) vertices, polygons, orijinal b) vertices, polygons, Simplified Rate of 41% c) vertices, polygons, Simplified Rate of 72% d) 7644 vertices, polygons, Simplified Rate of 86% e) 3918 vertices, 7788 polygons, Simplified Rate of 93% f) 2176 vertices, 4264 polygons, Simplified Rate of 96% Figure 6. Simplified meshes of the Torso model