# Exponential shapelets: basis functions for data analysis of isolated features

## TL;DR

Exponential shapelets are a new set of basis functions derived from quantum mechanics that efficiently model isolated features in data, outperforming traditional shapelets in data compression and analysis.

## Contribution

The paper introduces exponential shapelets, a novel basis function set based on hydrogen atom eigenfunctions, extending shapelet methods to 2D and non-circular features with improved data compression.

## Key findings

- Exponential shapelets outperform Gauss-Hermite/Gauss-Laguerre shapelets in data compression.
- They inherit mathematical properties under Fourier transform, facilitating convolution operations.
- Applications demonstrated in astronomy, physics, and geodesy.

## Abstract

We introduce one- and two-dimensional `exponential shapelets': orthonormal basis functions that efficiently model isolated features in data. They are built from eigenfunctions of the quantum mechanical hydrogen atom, and inherit mathematics with elegant properties under Fourier transform, and hence (de)convolution. For a wide variety of data, exponential shapelets compress information better than Gauss-Hermite/Gauss-Laguerre (`shapelet') decomposition, and generalise previous attempts that were limited to 1D or circularly symmetric basis functions. We discuss example applications in astronomy, fundamental physics and space geodesy.