# Stable Evaluation of 3D Zernike Moments for Surface Meshes
###### Abstract  
The 3D Zernike polynomials form an orthonormal basis of the unit ball. The associated 3D Zernike moments have been successfully applied for 3D shape recognition; they are popular in structural biology for comparing protein structures and properties. Many algorithms have been proposed for computing those moments, starting from a voxel-based representation or from a surface based geometric mesh of the shape. As the order of the 3D Zernike moments increases, however, those algorithms suffer from decrease in computational efficiency and more importantly from numerical accuracy. In this paper, new algorithms are proposed to compute the 3D Zernike moments of a homogeneous shape defined by an unstructured triangulation of its surface that remove those numerical inaccuracies. These algorithms rely on the analytical integration of the moments on tetrahedra defined by the surface triangles and a central point and on a set of novel recurrent relationships between the corresponding integrals. The mathematical basis and implementation details of the algorithms are presented and their numerical stability is evaluated. shape signatures; Zernike polynomials; Zernike moments +
Footnote †: journal:  
0000-0002-4000-0000-0000-00000-0000-0000-00  
where the integration is performed over the volume \(V\) of the shape \(S\) considered, \(N=i+j+k\) is the order of the moment, and \(f(x,y,z)\) is a vector field over \(S\) that may represent a potential, a grey scale for image processing, or an indicator function such that \(f(x,y,z)=1\) inside the shape and 0 otherwise. These geometric moments and their invariant extensions have been used extensively in pattern recognition (for review, see for example [10]). They are usually easy to compute, though at a high computational costs. Spherical harmonics [11; 12] and their rotational invariants [13] form another class of moment-based descriptors for analyzing star-shaped objects that are topologically equivalent to a sphere. They are becoming increasingly popular in cellular bio-imaging for example, as it is usually safe to assume that cells observed through a microscope are topologically closed and equivalent to a sphere; this prior makes it possible to 0 their shape mathematically, thereby simplifying the definition of their surface and provide a shape description (see for example [3] for representing cell organelles and more recently [14] for studies of cell dynamics). In many cases however, the hypothesis that the object is star-shaped does not hold. The Zernike moments, first introduced in two dimensions by Dutch Nobel price [PERSON] in 1934 [15], circumvent this problem through the introduction of a radial term. 2D Zernike moments have been used in cellular biology as an alternative to Quantitative Phase Imaging [16] for classifying breast cancer tumors based on mammograms [17], for characterizing subcellular structures [18], and for measuring changes in cancer cell shapes [19]. They have proved to be superior to geometric moments in 2D image retrieval (see for example [20]). After they were generalized to 3D by [PERSON] [21], they have been applied in many domains, such as tools for shape retrieval in computer graphics [22; 23], terrain matching and building reconstruction [24; 25], as well as in astronomy [26]. Most current applications, however, are related to biology, coincident with the applications of geometric moments. In biochemistry, for example, they have been proposed as a tool for protein shape comparisons [27; 28] and alignments [29], to the point that they have become a standard tool [30; 31] associated with the Protein Structure Database (PDB) [32]. Zernike moments are also used to analyze and search the protein models [33] generated by AlphaFold, the deep learning method that is currently used to predict the structure of proteins with high accuracy [34; 35].