Christian Wülker

Jürgen Prestin, Christian Wülker

Spherical Gauss-Laguerre (SGL) basis functions, i.e., normalized functions of the type $L_{n-l-1}^{(l + 1/2)} (r^2) r^{l} Y_{lm}(\vartheta,φ)$, $|m| \leq l < n \in \mathbb{N}$, $L_{n-l-1}^{(l + 1/2)}$ being a generalized Laguerre polynomial, $Y_{lm}$ a spherical harmonic, constitute an orthonormal basis of the space $L^{2}$ on $\mathbb{R}^{3}$ with Gaussian weight $\exp(-r^{2})$. These basis functions are used extensively, e.g., in biomolecular dynamic simulations. However, to the present, there is no reliable algorithm available to compute the Fourier coefficients of a function with respect to the SGL basis functions in a fast way. This paper presents such generalized FFTs. We start out from an SGL sampling theorem that permits an exact computation of the SGL Fourier expansion of bandlimited functions. By a separation-of-variables approach and the employment of a fast spherical Fourier transform, we then unveil a general class of fast SGL Fourier transforms. All of these algorithms have an asymptotic complexity of $\mathcal{O}(B^{4})$, $B$ being the respective bandlimit, while the number of sample points on $\mathbb{R}^{3}$ scales with $B^{3}$. This clearly improves the naive bound of $\mathcal{O}(B^{7})$. At the same time, our approach results in fast inverse transforms with the same asymptotic complexity as the forward transforms. We demonstrate the practical suitability of our algorithms in a numerical experiment. Notably, this is one of the first performances of generalized FFTs on a non-compact domain. We conclude with a discussion, including the layout of a true $\mathcal{O}(B^{3} \log^{2} B)$ fast SGL Fourier transform and inverse, and an outlook on future developments.

Jürgen Prestin, Christian Wülker

Spherical Gauss-Laguerre (SGL) basis functions, i.e., normalized functions of the type $L_{n-l-1}^{(l + 1/2)}(r^2) r^{l} Y_{lm}(\vartheta,φ)$, $|m| \leq l < n \in \mathbb{N}$, constitute an orthonormal polynomial basis of the space $L^{2}$ on $\mathbb{R}^{3}$ with radial Gaussian weight $\exp(-r^{2})$. We have recently described reliable fast Fourier transforms for the SGL basis functions. The main application of the SGL basis functions and our fast algorithms is in solving certain three-dimensional rigid matching problems, where the center is prioritized over the periphery. For this purpose, so-called SGL translation matrix elements are required, which describe the spectral behavior of the SGL basis functions under translations. In this paper, we derive a closed-form expression of these translation matrix elements, allowing for a direct computation of these quantities in practice.

Christian Wülker, Gregory S. Chirikjian

Concepts from mathematical crystallography and group theory are used here to quantize the group of rigid-body motions, resulting in a "motion alphabet" with which to express robot motion primitives. From these primitives it is possible to develop a dictionary of physical actions. Equipped with an alphabet of the sort developed here, intelligent actions of robots in the world can be approximated with finite sequences of characters, thereby forming the foundation of a language in which to articulate robot motion. In particular, we use the discrete handedness-preserving symmetries of macromolecular crystals (known in mathematical crystallography as Sohncke space groups) to form a coarse discretization of the space $\rm{SE}(3)$ of rigid-body motions. This discretization is made finer by subdividing using the concept of double-coset decomposition. More specifically, a very efficient, equivolumetric quantization of spatial motion can be defined using the group-theoretic concept of a double-coset decomposition of the form $Γ\backslash \rm{SE}(3) / Δ$, where $Γ$ is a Sohncke space group and $Δ$ is a finite group of rotational symmetries such as those of the icosahedron. The resulting discrete alphabet is based on a very uniform sampling of $\rm{SE}(3)$ and is a tool for describing the continuous trajectories of robots and humans. The general "signals to symbols" problem in artificial intelligence is cast in this framework for robots moving continuously in the world, and we present a coarse-to-fine search scheme here to efficiently solve this decoding problem in practice.