3D (pseudo) Zernike moments: Fast computation via symmetry properties of spherical harmonics and recursive radial polynomials

Based on pseudo-Zernike radial polynomials and spherical harmonics, we introduce a new form of three-dimensional (3D) moments that we call 3D pseudo-Zernike moments (3DPZMs). Then, using recursive generation of; Zernike radial polynomials, pseudo-Zernike radial polynomials, associated Legendre functions, and introducing a novel method to define 3D points-of-symmetry of spherical harmonics multiplied by the 3D object, we present an algorithm for the fast computation of three-dimensional (3D) Zernike moments (3DZMs) and 3DPZMs. The methods that we propose may play an important role in 3D object analysis and recognition. Asymptotic computational complexity and simulation tests have shown that the proposed symmetry-based algorithm is much faster than the direct (non-symmetry). 3DPZMs not only outperform 3DZMs, but they generate, for the same moment order, twice as much as the number of invariants that 3DZMs generate.