Empirical Wavelet Transforms

This toolbox proposes the original implementation of Empirical Wavelet Transforms

August 2022: new version 5.0!

The 2D Empirical Watershed Wavelet transform and the 2D Empirical Voronoi Wavelet transform have been added, check the documentation! A few minor bugs were also fixed.

New in version 4.0:

The 1D transform can now handle complex signals (i.e the empirical wavelets are themselves complex since they are not necessarily symmetric in the Fourier domain). The construction of the curvelet filters has been revised, simplified in order to guarantee almost perfect reconstruction. All other 2D transforms have been cleaned and simplified when possible. The plotting functions now add some title to each subfigure. In term of organization, almost all functions now contain the acronym 'EWT' in their name (most of the time as a prefix) to avoid any conflict with external functions.

In this toolbox, we implement the Empirical Wavelet Transform for 1D and 2D signals/images. The principle consists in detecting Fourier supports on which Littlewood-Paley like wavelets are build. In 2D, we revisit different well-known transforms: tensor wavelets, Littlewood-Paley wavelets, ridgelets and curvelets.

The toolbox also provides the scripts used to generate the experiments in the papers:

- J.Gilles, "Empirical wavelet transform" to appear in IEEE Trans. Signal Processing, 2013.

Preprint available at ftp://ftp.math.ucla.edu/pub/camreport/cam13-33.pdf

- J.Gilles, G.Tran, S.Osher "2D Empirical transforms. Wavelets, Ridgelets and Curvelets Revisited", SIAM Journal on Imaging Sciences, Vol.7, No.1, 157--186, 2014.

Preprint available at ftp://ftp.math.ucla.edu/pub/camreport/cam13-35.pdf

- J.Gilles, K.Heal, "A parameterless scale-space approach to find meaningful modes in histograms - Application to image and spectrum segmentation", submitted 2014.

Preprint available at ftp://ftp.math.ucla.edu/pub/camreport/cam14-05.pdf

- Y.Huang, V.De Bortoli, F.Zhou, J.Gilles, "Review of wavelet-based unsupervised texture segmentation, advantage of adaptive wavelets", IET Image Processing, Vol.12, No.9, 1626--1638, August 2018.

Preprint available at https://jegilles.sdsu.edu/doc/IET2018.pdf

- Y.Huang, F.Zhou, J.Gilles, "Empirical curvelet based Fully Convolutional Network for supervised texture image segmentation", Neurocomputing, Vol.349, 31--43, July 2019. Preprint available at https://jegilles.sdsu.edu/doc/2019EWTSupervisedTextures.pdf

- J.Gilles, "Continuous empirical wavelets systems", Advances in Data Science and Adaptive Analysis, Vol. 12, No 03n04, 2050006, 2020.

Preprint available at https://jegilles.sdsu.edu/doc/2020-cewt.pdf

- B.Hurat, Z.Alvarado, J.Gilles. "The Empirical Watershed Wavelet", Journal of Imaging, Special Issue "2020 Selected Papers from Journal of Imaging Editorial Board Members", Vol.6, No.12, 140, 2020.

Paper available at https://www.mdpi.com/2313-433X/6/12/140

- J. Gilles. "Empirical Voronoi Wavelets". Submitted.

See the README file and the Documentation folder inside the archive for more instructions