Spatiospectral concentration in the Cartesian plane

Frederik J Simons and Dong V. Wang

Geosciences Department
Princeton University
Princeton NJ 08544, USA

Abstract

We pose and solve the analogue of Slepian's time-frequency concentration problem in the two-dimensional plane, for applications in the natural sciences. We determine an orthogonal family of strictly bandlimited functions that are optimally concentrated within a closed region of the plane, or, alternatively, of strictly spacelimited functions that are optimally concentrated in the Fourier domain. The Cartesian Slepian functions can be found by solving a Fredholm integral equation whose associated eigenvalues are a measure of the spatiospectral concentration. Both the spatial and spectral regions of concentration can, in principle, have arbitrary geometry. However, for practical applications of signal representation or spectral analysis such as exist in geophysics or astronomy, in physical space irregular shapes, and in spectral space symmetric domains will usually be preferred. When the concentration domains are circularly symmetric in both spaces, the Slepian functions are also eigenfunctions of a Sturm-Liouville operator, leading to special algorithms for this case, as is well known. Much like their one-dimensional and spherical counterparts with which we discuss them in a common framework, a basis of functions that are simultaneously spatially and spectrally localized on arbitrary Cartesian domains will be of great utility in many scientific disciplines.

Figures

1. Figure 01 Radial Slepian eigenfunctions on the disk in the spatial domain
   
2. Figure 02 Radial Slepian eigenfunctions on the disk in the spectral domain
   
3. Figure 03 Eigenvalues of the concentration problem to the disk
   
4. Figure 04 Slepian eigenfunctions on the disk in the spatial domain
   
5. Figure 05 Slepian eigenfunctions on the Colorado Plateaus in the spatial domain
   Bonus 05 Slepian eigenfunctions on the Columbia Plateau
   
6. Figure 06 Cumulative energy of the Slepian eigenfunctions on the Colorado Plateaus
   Bonus 06 Cumulative energy of the Slepian eigenfunctions on the Columbia Plateau
   
7. Figure 07 Directionally sensitive Slepian eigenfunctions on the Colorado Plateaus
   

Frederik Simons