Atangana-Baleanu Regularized Wavelet Compression For Astronomical Time-Series

Astronomical light curves are noisy and irregular, so compression must reduce size without erasing weak transients. We propose a fractional wavelet compression method where wavelet coefficients are regularized via an Atangana Baleanu Caputo derivative with a nonsingular Mittag Leffler kernel. The induced long memory smoothing suppresses noise while preserving coherent transits, flares and oscillations. We give the coefficient level formulation, an efficient implementation, and comparisons with classical discrete wavelet thresholding, showing competitive compression with improved retention of low-amplitude events.

💡 Research Summary

This preprint presents a novel data compression framework designed specifically for the challenges of astronomical time-series analysis, particularly light curves from missions like TESS. The core problem is that these datasets are massive, noisy, irregularly sampled, and contain faint but scientifically critical transient events (e.g., exoplanet transits, stellar flares). Traditional compression using the Discrete Wavelet Transform (DWT) followed by coefficient thresholding is effective but often suffers from two drawbacks: the removal of low-amplitude astrophysical signals and the introduction of ringing artifacts around sharp features, due to its non-adaptive, point-wise processing of coefficients.

The authors’ key innovation is to integrate concepts from fractional calculus into the wavelet compression pipeline. Instead of applying a simple hard or soft threshold to each wavelet coefficient independently, they propose to regularize the temporal evolution of coefficients within each wavelet scale using the Atangana-Baleanu fractional derivative in the Caputo sense (ABC derivative). The ABC derivative is chosen for its non-singular Mittag-Leffler kernel, which provides numerical stability and models a smooth transition between exponential and power-law memory decay. This mathematical operation imposes a “long-memory” regularization on the sequence of coefficients. In practice, it means that the value of a coefficient at a given time is influenced by a weighted sum of its predecessors, where the weights are defined by the fractional order and the Mittag-Leffler kernel. Coefficients that are part of a temporally coherent feature (like a transit) reinforce each other and are preserved, while isolated noise coefficients are suppressed. This process inherently considers the time-domain correlations within the signal, which global thresholding ignores.

The paper is structured to first establish the background on wavelet compression and its challenges in astronomy. It then provides a primer on fractional calculus, highlighting the properties and advantages of the ABC operator. The proposed AB-regularized wavelet compression method is formulated at the coefficient level. The authors also discuss an efficient numerical implementation suitable for platforms like MATLAB or Python, handling the convolution with the Mittag-Leffler kernel.

The experimental validation involves tests on both synthetic time-series (combining sinusoidal signals, impulses, and noise) and real light curves from the TESS archive. Performance is compared against classical DWT with soft thresholding. Metrics include compression ratio, signal-to-noise ratio (SNR) improvement, and fidelity in preserving low-amplitude peaks. The results demonstrate that the AB-regularized method achieves competitive compression ratios while offering superior retention of weak transient signals and reduced visual artifacts compared to the standard approach. The method effectively balances noise suppression with feature preservation by leveraging the built-in memory effect of the fractional operator.

In conclusion, the study successfully demonstrates that fractional calculus, specifically the ABC derivative, can be a powerful tool for enhancing wavelet-based compression of non-stationary, long-memory time-series data like astronomical light curves. The work opens avenues for future research, such as developing adaptive algorithms to choose the optimal fractional order (α) for different data segments or extending the framework to multi-band (multi-channel) astronomical data.