Elsevier

Computers & Graphics

Volume 31, Issue 2, April 2007, Pages 243-251
Computers & Graphics

Technical Section
Ellipsoidal-blob approximation of 3D models and its applications

https://doi.org/10.1016/j.cag.2006.12.004Get rights and content

Abstract

This paper presents a technique for automatically approximating a given mesh model with an ellipsoidal blobby model. Firstly, an ellipsoid decomposition algorithm is introduced to approximate given models by ellipsoids. After that, a blobby implicit surface employing ellipsoidal blobs is modelled to fit the sampling points on the given mesh. Finally, the reconstructed ellipsoidal blobby model is applied in two applications: the geometry data reduction and the target shape controlled cloud animation.

Introduction

The computer graphics, computer-aided design and computer vision literatures are filled with a diverse array of surface representations. The reason for this variety is that there is no single representation that can satisfy the needs of all problems in various applications. Implicit surface is one of the most popular representations which are widely employed in computer graphics applications including geometric modelling, three-dimensional metamorphosis and collision detection. Among variety of implicit surfaces, one important class is the so-called blobby model [1] and its variants—metaball [2] and soft object [3]. However, the method to efficiently and effectively approximate a given mesh model by a blobby model is still a problem under research. For simple objects (e.g., spheres and peanut-like objects, etc.), it is easy to obtain their corresponding blobby representations. For a model with complex shape such as a human body, it is a difficult and tedious work to construct blobs for the model manually. The motivation of the work presented in this paper is to seek an automatic method to approximate given mesh models by a blobby representation.

Here we present an automatic approach to approximate a given mesh model with an implicit surface employing ellipsoids as primitives—named as ellipsoidal-blobs. Our algorithm consists of two steps: in the first step, the given mesh model is sampled into points and then decomposed into a set of ellipsoids; in the second step, the final blobby model is reconstructed and computed from the ellipsoids through numerical optimization. The reconstructed blobby model has many applications. To demonstrate the functionality of an ellipsoidal blobby model, we apply it in the applications of the geometry data reduction and the target shape controlled cloud animation.

Previous related works will be firstly reviewed in the following section. After that, a modified ellipsoid decomposition algorithm will be introduced in Section 3. The mathematical representation and the reconstruction method of ellipsoidal blobby models are then presented in Section 4. In Section 5, two applications will be demonstrated. Lastly, our paper ends with the section of conclusion and discussion.

Section snippets

Previous work

An implicit surface S is usually defined by a continuous scalar function f(x) with xR3. The geometry of S is given by the locus of points at which the function f(x)=0. In [1], [2], [3], the implicit surfaces are defined as the summation of radial symmetric functions, which are generally in the form off(x)=-t+i=1nωifi(x).In this formula, the parameter t is a threshold of isosurface S, n is the number of primitives, ωi is the weight for the ith primitive (with default value 1.0), and the

Ellipsoid decomposition

For a given polygonal mesh, obviously there are many different possible ellipsoid decompositions. In [28], Bischoff and Kobbelt designed an algorithm to find one candidate among this multitude of decompositions, where the computed decomposition is a local optimum with respect to the shape, the orientation and the distribution of ellipsoids. However, some small features on given models are missed in their algorithm. To avoid this, under the framework of [28], a modified scheme is developed.

Reconstruction of ellipsoidal blobby models

With the ellipsoids decomposed from the given mesh model M as input, an ellipsoidal blobby model Ω approximating M is reconstructed from the ellipsoids by taking their centers as the skeletons with associated field functions. The isosurface of Ω approximates the surface of M. After that, the parameters of blobs in Ω are optimized to reduce the approximation error.

Data reduction of geometric model

The first application of the ellipsoidal blobby models is in the area of geometry data reduction—i.e., lossy geometry compression. For a given mesh model, if it has nv vertices and nf triangular faces, the total bytes to recording this model is 12nv+12nf where every vertex has three float numbers for its R3 coordinate, every triangle has three integers for encoding indices of vertices, and both integer and float numbers occupy 4 bytes. On the other side, for the approximation with nb

Conclusion and discussion

This paper developed an automatic scheme for approximating a given polygonal mesh with an ellipsoidal blobby model. The experimental results prove that the reconstructed implicit surfaces can approximate the original model very well. Based on this, we demonstrate two applications of the blobby models: the geometry data reduction and the target shape controlled cloud animation. In summary, our work presented in this paper has the following contributions.

  • A novel implicit ellipsoidal blobby model

Acknowledgements

The authors would like to acknowledge the helpful comments given by the reviewers. This work was supported by the National Natural Science Foundation of China (Grant No. 60573153), Natural Science Foundation of Zhejiang Province (Grant No. R105431), Program for New Century Excellent Talents in University (Grant No. NCET-05-0519) and the 863 program (Grant No. SQ2006AA01Z302275). It was partially supported by the Hong Kong RGC/CERG grant CUHK/412405 and the CUHK project CUHK/2050341.

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