2D Empirical Transforms. Wavelets, Ridgelets and Curvelets revisited

TL;DR

This paper extends the empirical wavelet transform approach to 2D signals, creating adaptive versions of classical transforms like wavelets, ridgelets, and curvelets, with promising applications in image analysis.

Contribution

It introduces empirical counterparts for well-known 2D transforms, demonstrating their potential for adaptive image analysis and processing.

Findings

Empirical 2D transforms can be constructed for classical wavelet-based methods.

Adaptive frames show promising properties for image analysis.

The approach generalizes the empirical wavelet transform to 2D signals.

Abstract

A recently developed new approach, called ``Empirical Wavelet Transform'', aims to build 1D adaptive wavelet frames accordingly to the analyzed signal. In this paper, we present several extensions of this approach to 2D signals (images). We revisit some well-known transforms (tensor wavelets, Littlewood-Paley wavelets, ridgelets and curvelets) and show that it is possible to build their empirical counterpart. We prove that such constructions lead to different adaptive frames which show some promising properties for image analysis and processing.