ThreshLab: Matlab algorithms for wavelet noise reduction

Version 6.1.3, released June 2024

Maarten Jansen

Key words: Wavelet, multiscale, sparse, nonequidistant, threshold, inverse problem

General Description

Main features

• Wavelet transforms: fast and undecimated, (bi)orthogonal, 1D, 2D, first generation, i.e., classical filterbanks (Mallat's algorithm) on equispaced data, but also second generation, using the lifting scheme on nonequispaced data.
  Special attention was paid to fast implementation through Matlab blass routines and polyphase implementation of filter banks.
• Multiscale local polynomial transforms (MLPT): 1D, 2D
• Noise reduction by wavelet thresholding, including threshold assessment by minimum prediction error, minimum SURE, minimum GCV, EBayesThresh.
• Hidden Markov Field modelling of wavelet coefficients
• Variable selection in sparse high-dimensional models including LARS/lasso algorithms, mirror correction (Jansen, 2013, Biometrika), best subtree selection

History of added features

• Version 1.2.1 added: CVS - Conditional Variance Stabilisation for Poisson intensities, including a Bayesian model within this framework. It also includes routines for Fisz-wavelet transforms (Fryzlewicz, Nason). Some of the test routines require that you download software from Theofanis Sapatinas' collection of Poisson Wavelet Denoising Software.
  See the related paper on Poisson intensity estimation:
  M. Jansen: Multiscale Poisson data smoothing, JRSSb, 68(1), pages 27-48, 2006.
• Version 2.1.1 added: Continuous unbalanced Haar transform for change point detection.
• Version 3.1.1 (Nov. 2009) added:
  • Iterative soft- and hard-thresholding with variable thresholds for inverse problems
  • Lasso, hard-thresholding, mirror correction
    See these papers:
    M. Jansen. Generalized Cross Validation in variable selection with and without shrinkage. Journal of Statistical Planning and Inference. To Appear.
    M. Jansen. Information criteria for variable selection under sparsity. Biometrika, 101(1), pages 37-55, 2014.
• Version 3.1.2 (Mar. 2010) added: EBayesThresh
• Version 3.1.3 (Mar. 2010) added: minor updates
• Version 3.1.4 (Sep. 2010) added: minor updates and renamed PiefLab (the old name of this package) into ThreshLab (version Pi)
• Version 3.1.5 (Mar. 2011) added: minor updates
• Version 3.1.6 (Feb. 2012) added: minor updates
• Version 3.2.1 (Mar. 2012) added: multiscale kernel and local polynomial transforms
  See this paper:
  M. Jansen. Multiscale local polynomial smoothing in a lifted pyramid for non-equispaced data. IEEE Transactions on Signal Processing, 61(3), pages 545-555, 2013.
• Version 3.2.2 (Sep 2012) added: minor updates
• Version 3.2.3 (Dec 2012) added: minor updates
• Version 3.2.4 (Nov 2013) added: minor updates
• Version 4.1.1 (Dec 2014) added: Second generation spline wavelets
• Version 4.1.2 (Feb 2015) added: new version of finescaleBsplinecoefs.m + minor updates
• Version 4.1.3 (June 2015) added: minor updates and numerical issues (e.g. in cumgauss.m)
• Version 4.1.4 (July 2015) added: minor updates; renamed main routines for Multiscale Local Polynomial Decomposition (old names are now in folder ThreshLab/OldNames)
• Version 4.2.1 (Sep 2015) added: 2D scattered data Multiscale Local Polynomial Decomposition (FMLPT2D and IMLPT2D)
• Version 4.2.2 (Apr 2016) added: various small updates
• Version 4.3.1 (Jan 2017) added: Best subtree selection, along with numerous small updates
• Version 4.3.2 (March 2017) added: various small updates (Slight simplification in parameter handling FMLPT1D; exact calculation of B-spline moments in Bsplinemoments.m,...)
• Version 4.3.3 (Nov 2018) added: minor updates
• Version 4.3.4 (Nov 2018) added: minor updates (fixed bugs in FMLPT1D1step.m and IMLPT1D1step.m)
• Version 4.4.1 (Dec 2018): New (equivalent) implementations of FMLPT1D.m and IMLPT1D.m (simpler code, no longer calling FMLPT1D1step.m and IMLPT1D1step.m; removed all code for adaptive MLPT); New implementation (not equivalent) of splitfrombandwidth.m; several minor updates
• Version 4.4.2 (Jan 2019) added: minor updates (extended output parameters FMLPT1D.m)
• Version 4.4.3 (Apr 2019) added: minor maintenance (corrections in illustratefinescalecoefs and derivBsplinebasis)
• Version 4.4.4 (Sep 2019) added: minor maintenance (adding Gasser-Müller estimation in kernelestimation.m, correction coiflet.m, adding integralkernel.m)
• Version 4.5.1 (Oct 2019) added: refinementfromliftingsteps.m in ThreshLab2/Irregular/1D/
  minor maintenance (allpowercoefsBsplines.m, Bsplinebasis.m, Bsplinebasisbyrefinement.m, refinementfromliftingsteps.m)
• Version 4.5.2 (Dec 2019) added:
  DDbasisbyrefinement.m, grammatrixBsplines.m, updatesemiorth.m in ThreshLab2/Irregular/1D/;
  major extension in Bsplinebasisbyrefinement (end points handling) and in FWT_2Gspline.m (semi-orthogonal B-spline wavelet transforms);
  Revision of drawaxes for proper handling of legend;
  minor maintenance (CDF_LDL.m, CDFspline.m, constructMLPTtree.m, finescalehist.m,weightedLSpol.m, refinescalingfunctionsMLPT.m
• Version 4.6.1 (April 2020) added:
  Second generation wavelet packet transforms (FWT_2Gpacket.m and IWT_2Gpacket.m in ThreshLab2/Irregular/1D/)
  Revision of second generation nondecimated wavelet transforms (filenames starting with RWT_2G and IRT_2G in ThreshLab2/Irregular/1D/)
  minor maintenance
• Version 4.7.1 (August 2021) added
  Routines in tree-strctured selection
  minor maintenance
• Version 5.1.1 (February 2022) added
  routines and functions for the book Wavelets from a Statistical Perspective
  minor maintenance
• Version 5.2.1 (July 2022) added
  routines for sparse parameter selection in trees and graphs
  minor maintenance
• Version 5.3.1 (May 2023) added
  routines for broken basis wavelet transforms;
  • see preprint Constructing wavelets by welding segments of smooth functions
  • Get started with ThreshLab/ThreshLab2/Irregular/BrokenbasisWT/illustrateFWT_2Gbroken.m
  
  minor maintenance
• Version 5.3.2 (March 2024) added
  corrections in average interpolating prediction
• Version 6.1.1 (April 2024) major revision 1D 2nd generation wavelets on nonequispaced knots, introducing
  • FWT2G1D.m and IWT2G1D.m for
  • Deslauriers-Dubuc, average interpolation, B-spline, broken basis refinement;
  • extendible to other and generalised refinements and offset methods.
• Version 6.1.2 (May 2024) added
  Inverse 2D tensor product 2nd generation wavelet transform
  minor maintenance
• Version 6.1.3 (JUne 2024) added
  Maintainance and extensions to second generation transforms (FWT2G1D and FWT2G2Dtensprod)

Installation

• Users without access to Matlab may download and install first GNU Octave (a free alternative)
  ThreshLab should run on octave as well. If you experience serious troubles, you are welcome to report them! (see contact info at the bottom of this page)
• Download the tar-file to your favorite directory.
• unwrap the tar-file. In Unix/Linux this is done by: tar xvf threshlab.tar . This creates a directory ThreshLab with subdirectories.
• cd ThreshLab and edit the startup.m file to adapt it to your specific path. This startup file sets the Matlab path, thereby giving access to all .m-files in the subdirectories.
• starting Matlab in ThreshLab should now make all .m-files available
• Alternatively, you can work with this zip file

Getting started

• You can download several test images
• Type `help ThreshLab' (mind the capitals!) after having started Matlab in the ThreshLab-directory. The answer will be a list of .m-files and subdirectories. Browse through this list by typing `help ' (omitting the extension `.m').
• The subdirectory ThreshLab/Tests has several test procedures (mainly with images)
• The command `gcvdemo' starts a very simple demonstration of GCV threshold estimation. See ThreshLab/Tests/GCVdemo.

Copyright

• Research in or financed by weapon or tobacco industry is excluded without any exception from the use of this software.
• Any commercial use without explicit permission is strictly forbidden.
• All research papers presenting results which are (partly) based on this software should include in their bibliography one or more references to the corresponding papers. Check this list of papers.
• All redistributions of this software, even if modified or compiled or translated into a different programming language or binary code, should mention the origin of the software and include these copyright instructions.

publications