On the Representation of Functions with Gaussian Wave Packets

1. Introduction. The main purpose of this paper is to develop algorithms to obtain sparse representations of functions of several variables using Gaussian wave packets. Numerically, we will primarily focus on the two-dimensional case; our main interest is the representation of waves related to the wave equation. Sparse representations are useful, in some cases essential, for large-scale problems that arise in, for example, seismic imaging [8, 23, 22, 33]. To analyze functions whose features vary at different resolution, one may use wavelet transforms, which employ pairs of translation and dilation operators [12, 24]. One can also consider a time– frequency analysis [19] using a combination of translation and modulation in the short–time Fourier transform. On the other hand, the use of Gaussian wave packets allows us to incorporate all three of the operations translation, dilation and modulation in the one-dimensional case. A classic one-dimensional example is music; it is often used as a motivation for the usage of both wavelets and time-frequency analysis. It also serves to motivate the three-parameter transform that we propose as musical notes contain (at least) three characteristic features: the tone, the time it is played and its duration. In higher dimensions Gaussian wave packets can also be used to include rotation invariance and anisotropic dilation parameters. A simple way to represent functions in two dimensions, either by the wavelet or time-frequency methods, is to employ tensor products of the chosen one-dimensional functions. This has the disadvantage of representing horizontal and vertical features/waves well but not those that are not aligned with the coordinate axes. Therefore, a lot of work has been targeted into obtaining more rotation-invariant representations. Proposed methods include two-dimensional wavelets [3], steerable pyramids [27], brushlets [2], curvelets [9], shearlets [21] and beamlets [33]. The anisotropic dilation parameter has been shown to play a crucial theoretical role for solutions of wave equations, cf. [28]. From a practical perspective, it is useful when working with waves that have (locally) largely varying curvature. For waves with small curvature, a locally plane wave approximation works well, and it is preferable to work with representations where the extent in the non-oscillatory direction is much larger than in the oscillatory direction, whereas for waves with large curvature the extent in the two directions should be similar. The one-dimensional continuous wavelet — or — continuous short–time Fourier transform are redundant in the sense that they map a function of one …