# Stable Evaluation of 3D Zernike Moments for Surface Meshes
## 6 Numerical Results
### Reconstructing a Shape from Its 3D Zernike Moments: Importance of the Order \(N\)  
We consider three different experiments for the reconstruction of a shape from its Zernike moments based on generic shapes, Figure 9, on shapes of protein structures, Figure 10, and on shapes of a human organ, the brain, Figure 11.  
Figure 9 shows the quality of reconstruction of a shape from its Zernike moments at different orders for standard 3D models from the computer graphics community. The first row corresponds to a model of a gargoyle, available at the repository AIMS@SHAPE  
Figure 8: Summation of the series in Equation (33) for \(n\) up to \(N\) with \(N=20,50\) and 200 for a scaled sphere with radius of \(r_{0}=0.75\). The series converges to a straight line at value 1 between the origin and \(r_{0}\), and vanishes beyond. Note the presence of overshoots, similar to oscillations in truncated Fourier expansions. Those overshoots concentrate as \(N\) increases. Figure 7: Accuracy of the computation of Zernike moments for a cube with 12 facets as a function of the order \(n\). The reference values were computed symbolically using a symbolic algebra system. The approximate algorithm is indistinguishable from the exact one. (accessed on 23 September 2022) ([[http://visionair.ge.imati.cnr.it/](http://visionair.ge.imati.cnr.it/)]([http://visionair.ge.imati.cnr.it/](http://visionair.ge.imati.cnr.it/))). The original mesh includes 50,002 vertices and 100,000 triangles. We computed the Zernike moments of this shape using the finite precision algorithm with a tolerance of \(10^{-8}\), up to order 300. We then reconstructed the shape by computing a field over the unit ball, where each node in the field is assigned a value corresponding to Equation (33), with different orders of \(N\) and then identifying the surface of the reconstructed shape with the 0.5 isosurface of this field. The isosurface is built using the regularized marching tetrahedra algorithm [50]. As expected, the quality of the reconstruction depends on the maximum order \(N\) considered. For \(N=20\), the reconstructed surface only shows the global envelope with no details. For \(N=50\), the shape of the gargoyle is better defined; however, fine details are still missing. For example, we do not see any texture in the wings. \(N=300\) gives us a more accurate representation of the shape, with fine details within the wings and in the mane of the gargoyle. The second row corresponds to a statue of the bust of [PERSON], sculpted by [PERSON]. It is available as part of the package MeshLab [64]. The mesh includes 27,861 vertices and 49,954 triangles. We computed its Zernike moments using the finite precision algorithm with a tolerance of \(10^{-8}\), up to order 300. We then reconstructed its shape using the same procedure described above for the gargoyle, with orders \(20,50\), and \(300\). For \(N=20\), the reconstructed bust is incomplete, with no details on its surface. For \(N=50\), we start seeing faint features within the face, but the bottom of the bust is still  
Figure 9: Reconstruction of a gargoyle (**top row**), a statue (**middle row**), and of the head of an ogre from the Zernike moments computed from a triangle mesh representing their boundaries. We compare the reconstructions generated from three different maximum order \(N\), \(N=20\), \(N=50\), and \(N=300\). The original shape is shown on the left. All figures were generated with MeshLab [64]. incomplete. \(N=300\) gives us a much more accurate representation of the model, especially in the face. The third row corresponds to the smiling version of [PERSON], introduced by [PERSON] et al. It is available at the 3D model repository of Keenan Crane (accessed on 23 September 2022) ([[https://www.cs.cmu.edu/~kmcrane/Projects/ModelRepository/](https://www.cs.cmu.edu/~kmcrane/Projects/ModelRepository/)]([https://www.cs.cmu.edu/~kmcrane/Projects/ModelRepository/](https://www.cs.cmu.edu/~kmcrane/Projects/ModelRepository/))). The mesh includes 27,861 vertices and 49,954 triangles.