Discrete Spherical Harmonic Oscillator Transforms on the Cartesian Grids Using Transformation Coefficients

Abstract

The analog harmonic oscillators are well-studied in quantum physics, including their energy states, wavefunctions, orthogonal properties, and eigenfunctions of the Fourier transform. In addition, the continuous solutions in different dimension and coordinate systems are known. Some discrete equivalents of the 1D wavefunctions were also studied. However, in the 3D spherical coordinate system, the discrete equivalents of the 3D wavefunctions are not established. In this paper, we focus on the spherical harmonic oscillator wavefunctions (SHOWs) the spherical harmonic oscillator transforms (SHOTs), and their discrete implementation. The SHOWs can be synthesized by linear combinations of the Hermite Gaussian functions with proper transformation coefficients. We find that computing the coefficients can be speeded up using the fast Fourier transforms or some recursive relations. These coefficients relate the Hermite transforms with the SHOTs. Some applications of the discrete SHOWs and the discrete SHOTs are introduced. First, the SHOWs are exactly the eigenfunctions of the 3D DFT. Also, the SHOTs can be used to derive the spherical harmonic oscillator descriptor (SHOD), which is a rotational invariant descriptor. We find that the SHOD is not only compatible with the existing rotational descriptors for the spherically sampled data but also outperforms the existing rotational descriptors for 3D Cartesian sampled, bandlimited input data. Besides, the SHOTs can be used to decompose 3D signals into spherical components. Hence, 3D signal reconstruction is done using partially chosen spherical component, and 3D data compression for MRI data is demonstrated using SHOTs for medical applications.