Efficient Spherical Harmonics Representation of 3D Objects M. MOUSA R. CHAINE S. AKKOUCHE E. GALIN LIRIS, Claude Bernard University Lyon1 - France {mmousa, rchaine, sakkouch, egalin}@liris.cnrs.fr Abstract In this paper, we present a new and efficient spherical harmonics decomposition for spherical functions defining 3D triangulated objects. Such spherical functions are in- trinsically associated to star-shaped objects. However, our results can be extended to any triangular object or oriented point set surface after segmentation into star-shaped sur- face patches and recomposition of the results in the implicit framework. There is thus no restriction about the genus number of the object. We demonstrate that the evaluation of the spherical harmonics coefficients can be performed by a Monte Carlo integration over the edges, which makes the computation more accurate and faster than previous tech- niques, and provides a better control over the precision error in contrast to the voxel-based methods. We present several applications of our research, including fast surface reconstruction from point clouds, local surface smoothing and interactive geometric texture transfer. 1. Introduction In computer graphics community, the spherical harmon- ics have gained much of interest due to their contribution in many fields such as global illumination [5], shape descrip- tors [11, 12], shape reconstruction of star-shaped point set [22], frequency-based representations and filtering [25, 16]. The spherical harmonics decomposition is naturally applied to spherical domains (star-shaped objects) and may be ex- tended to non spherical domains using a prior spherical pa- rameterization. This can be used in particular to zero genus 3D objects described by triangular meshes [25]. Current methods for computing the spherical harmonics transform often sample these spherical functions over a reg- ular grid on the sphere or a 3D grid surrounding the ob- ject (voxelization) and apply a discrete spherical harmonics transform algorithm [6, 14, 20] to this regular grid to com- pute the harmonics coefficients. Spatial grid-based methods [4, 11, 19] are prone to numerical errors associated with the size of the cells. It is very difficult to predict the size of the Figure 1. The model of Thai statue at different levels of details represented by our spheri- cal harmonics representation using a band- width of 24, 64 and 480. The initial data set of the model is given as an oriented set of 3000k points without any information about the connectivity. voxel which satisfies a given error tolerance. In addition, the complexity of the method increases tremendously as the voxel size decreases. In this paper, we present a new and efficient algorithm for computing the spherical harmonics decomposition of spherical functions measuring the radial extent of the points of 3D objects represented as meshes or oriented point sets. The main contribution of our approach is as follows: Efficient spherical harmonics The evaluation of the spherical harmonics coefficients is distributed over the edges thanks to the Curl theorem that permits to reduce a