Elsevier

Computers & Graphics

Volume 22, Issue 1, 25 February 1998, Pages 13-26
Computers & Graphics

Scene Simplification
Multiresolution representations for surfaces meshes based on the vertex decimation method

https://doi.org/10.1016/S0097-8493(97)00080-0Get rights and content

Abstract

The vertex decimation is a general mesh simplification approach. By successively removing vertices a hierarchy of different levels of detail (LOD) is generated during the simplification process. This paper enhances our 1996 mesh simplification algorithm with error control (R. Klein, G. Liebich and W. Straßer. Mesh reduction with error control. In Visualization 96, ed. R. Yagel, ACM, November 1996) to preserve some information about the normals of the original faces into the resulting simplified data. This enables us to build multiresolution models (MRM) which allow one to control the normal deviation and to extract view dependent-adaptive lighting sensitive approximations of the original meshes. Furthermore, the paper gives a general overview on MRMs generated by vertex removal algorithms. Where necessary material from our previous publications scattered in various conference proceedings and short papers (some of them hard to access) is included and extended with detailed algorithms and proofs. Although formulated for a vertex removal algorithm the results apply to other simplification algorithms and MRMs as well, namely to edge and triangle collapse algorithms.

Introduction

The visualization of large models of real-world objects, like cars, trains, airplanes, etc. is a major challenge in the context of virtual reality. In most scenarios several such models have to be visualized and animated simultaneously. Examples are the optimization of the cabin of a train or the optimization of a driver’s position in a car. In such a setting not only the particular train or car has to be visualized but also other cars, pedestrians, buildings, etc. Furthermore, due to new acquisition techniques and more elaborate modeling and animation techniques the size of the models is still growing. Despite the performance of modern graphics hardware the visualization of such scenarios cannot be realized in real-time without special techniques to reduce the number of triangles that are rendered.

Multiresolution modeling means maintaining objects at different level-of-detail (LOD) or approximation, where the LOD may be different in distant areas of the object. With respect to visualization it means to render the model at any given moment with minimal LOD sufficient to produce an image of reasonable quality. The quality of the resulting image not only depends on the geometric distance between the simplified object and the original model but also on the illumination of the object. The illumination depends on the different light sources and their positions, on the different materials, colors, textures and normals of the object, and last but not least on the shading model and shading technique which is used to render the model. A certain LOD may be sufficient to produce an image of reasonable quality if the object contains only diffuse surfaces that reflect radiation equally in all directions, but the same level of detail may be insufficient if the surface of the object also contains specular components. As an example Fig. 1 shows a part of a car bonnet with different surface properties. The level of detail which is used is shown on the left side. The bonnet is rendered using hardware Gouraud-shading. In contrast to the bonnet on the right side, the middle one contains only low specular components. In this case the chosen level of detail is sufficient while in the other case broad artifacts are visible. Due to the hardware Gouraud-shading in the area of the highlights a higher level of detail is necessary to get a sufficient image quality. A similar problem occurs near silhouettes w.r.t. the direction of light: minimal changes of the geometry may change a vertex from being lit and normally colored to pointing away from the light source and therefore being colored black. This leads to an inhomogeneous illumination in areas where the normals of the object surface are nearly perpendicular to the direction of light and slightly perturbed due to the mesh simplification process (see Fig. 2). Note that the illumination artifacts described here especially occur on smooth surfaces used in CAD-applications. In these applications the illumination plays an important role as a tool to visualize artifacts in the smoothness of the original surfaces. Although mesh simplification is necessary to speed up the visualization, artifacts in the illumination cannot be tolerated and must be avoided. Note that for objects with rough non-specular surfaces, like the brain surface reconstructed from CT-data or the surface of an animal used in an animation, illumination artifacts are less important.

As shown by the examples the illumination artifacts can be avoided if in certain areas like the neighborhood of highlights or along the silhouettes a better approximation of the object, that means a higher level of detail, is used. The simplest way is to use a sufficient high level of detail for the complete object, which can guarantee the desired image quality, but in many cases it does not allow any simplification of the scene geometry. The situation might be improved a little if during the simplification process aside from the geometric distance between original and simplified object deviations between the normals are also taken into consideration.

A further possibility is to take care of the illumination condition already in the simplification process and to simplify the object only in areas where no artifacts are expected. In general this approach leads to high quality images while reducing the number of triangles to be rendered, but in general mesh simplification algorithms are too slow for real-time applications. Nevertheless, there has been some work on real-time simplification algorithms28, 37.

Like Xia et al.[36]and Hoppe[18], in this paper we use a two-stage approach to overcome the problems described above. In an offline simplification process a view-independent multiresolution model MRM is generated. This model contains all the information about the geometric accuracy and about the deviations of the normals of the surfaces in all intermediate levels of detail which are needed to choose the appropriate level according to the viewing parameters, silhouettes and highlights. From this MRM we can extract view-dependent simplified meshes at variable resolution and render them with the desired image quality.

After a discussion of the previous work in Section 2we review in Section 3our simplification algorithm based on vertex removal and show how information about the deviation of normals of the simplified meshes can easily be incorporated. In 4 The MRM based on the Delaunay hierarchy, 5 The explicit MRMwe describe the construction of two different MRMs together with algorithms for the extraction of simplified meshes at variable resolution in more detail. The MRM described in Section 4is well suited for parameterized surface meshes and is based on a constrained Delaunay Triangulation (CDT). In addition to the already published results in two previous short papers23, 27we will give a theoretical foundation as well as a detailed description and analysis of the extraction algorithms based on the CDT. The MRM described in Section 5is an extension of an MRM first presented by Puppo for terrain visualization[30]to free-form surfaces embedded in the three-dimensional space. In Section 6we discuss how the normal information can be used to avoid the extraction of most of the back-facing triangles. Furthermore, we discuss illumination-controlled refinement based on the normal information. We end with some conclusions in Section 7.

Section snippets

Previous work

In the literature a variety of techniques for the generation of multiresolution models has been proposed. To obtain a hierarchy of increasingly coarse approximations of the input object, mesh simplification algorithms are applied. Various techniques have been published that aim to reduce surface complexity2, 5, 7, 14, 17, 19, 20, 25, 29, 32, 33, 34, 35. Most of these techniques simplify triangular meshes either by merging elements or by resampling vertices using different error criteria to

Mesh simplification

Our simplification algorithm is a vertex decimation algorithm first described by Schroeder et al.[34]. It can be used for the simplification and construction of an MRM of arbitrary surface meshes. A simple modification of the algorithm allows one to build up a special MRM for parameterized surfaces, see Section 3.4.

The MRM based on the Delaunay hierarchy

From the coarsest CDT Σm and the sequence of removed vertices (vm+1, vm+2,…,vM) transforming Σm into the triangulation ΣM at full resolution all intermediate unique Delaunay triangulation Σm, Σm+1,…,ΣM can be recomputed through a simple point insertion. Therefore, the multiresolution representation of a surface patch requires a coarse CDT Σm of the domain in the XY-plane, the sequence of points transforming Σm into the triangulation ΣM at full resolution and the corresponding simplified

The explicit MRM

In the following we describe the data-structure of the multiresolution model which can be used for parameterized as well as for non-parameterized surfaces. In each vertex removal step two sets of triangles are involved: the triangles incident to the removed vertex and the new triangles retriangulating the resulting hole. These two sets form a fragment. The set of removed triangles is called ceiling, the set of triangles of the retriangulation is called floor. Each simplification step and its

Applications using the two-stage approach

After computing an MRM, either using the CDT or storing the fragments explicitly, the extraction of variable levels of detail from the MRM with guaranteed approximation error in real-time is feasible. In the following we show how in addition to the view-dependent geometric approximation a view-dependent illumination can also be realized. To achieve this goal in the condition c(Δ) described in Section 3, which decides if the area around a triangle should be refined further, we must take care of

Conclusion and future work

The proposed multiresolution models both support several different levels of detail to co-exist across different regions of the object. The different levels of detail in different regions seamlessly merge with one another without introducing any cracks and discontinuities. The algorithms to refine and to coarsen a simplified mesh allow one to exploit the frame-to-frame coherence between successive frames during the visualization process. Therefore, variable levels of detail can be extracted

Acknowledgements

First of all I would to thank A. Schilling for many fruitful discussions and hints and J. Krämmer for all the programming efforts which were necessary to gain the described results. Thanks also to the anonymous reviewers for their thorough reviews and valuable hints.

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