# 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 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 and the bust by [PERSON], with orders \(20,50\), and 300. For \(N=20\), the reconstructed face of [PERSON] has no features. For \(N=50\), we start seeing faint features within the face; some elements such as the eye are still missing. For \(N=300\) the face is reconstructed reasonable 0. Figure 10 shows the quality of reconstruction of a protein shape from its Zernike moments at different orders. We consider two well studied proteins often used as models in computational structural biology, calmodulin (top row) and a triose phosphate isomerase (bottom row). Calmodulin is a calcium binding protein expressed in all eukaryotic cells. It is a small protein that consists of two small domains separated by a linker region. The flexibility of this linker is key to the ability of calmodulin to bind to a wide range of substrates [67]. We considered the apo form (without ligand) of the human version of protein available in the Protein Data Bank [32] with the code 1 CLL. This conformation is defined as the position of its 1133 atoms. We use the standard model in chemistry of representing a structure as a union of balls, with each ball corresponding to an atom. The surface of this union of ball is represented as its skin surface [68]. It is computed from the boundary of the union of these balls, where the center of a ball is given by the coordinates of the atom, and its radius is set to its van der Waals radius plus a probe radius of \(R=1.4\) A. We generated a quality mesh  
Figure 10: Reconstruction of Calmodulin (**top row**), and TIM (**bottom row**) from the Zernike moments computed from a triangle mesh representing their boundaries. Calmodulin (PDB code 1 CLL) is a human small calcium binding protein while TIM (PDB code 1 TIM) is the chicken triose phosphate isomerase. Both proteins are represented as a union of balls, with each ball corresponding to an atom. The surface of this union of balls, triangulated with the program smesh [65; 66], serves as input to the computation of the Zernike moments of their shapes. 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]. for the skin surface of calmodulin using the program smesh, described in details in [65; 66]. The corresponding triangular mesh includes 203,400 vertices and 406,864 triangles. We computed its Zernike moments using the finite precision algorithm with a tolerance of \(10^{-8}\), up to order 300. We then reconstructed a shape for calmodulin using the same procedure described above, with orders 20, 50, and 300. Those two domains of calmodulin are clearly separated in the absence of a ligand, as illustrated from its original surface. For \(N=20\), the reconstruction shows the position of these two domains. However, it does not include any local features, while some of those features (i.e., the presence of spherical patches) appear for \(N=50\), they are only clearly visible at much higher orders, as illustrated here for \(N=300\). Triose phosphate isomerase (TIM) is an enzyme that catalyzes the reversible interconversion of triose phosphate isomers [69]. We considered the chicken version of the TIM protein available in the Protein Data Bank [32] with the code 1 TIM. This conformation is defined from the positions of its 1870 atoms. We generated a quality mesh for its skin surface using the same procedure described above for calmodulin. The corresponding triangular mesh includes 398,658 vertices and 797,644 triangles. We then computed its Zernike moments using the finite precision algorithm with a tolerance of \(10^{-8}\), up to order 300. We reconstructed a shape for TIM using the same procedure described above, with orders 20, 50, and 300.