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If you cannot find anything more, look for something else (Bridget Fountain)
Si vous ne trouvez plus rien, cherchez autre chose (Brigitte Fontaine)
Web www.laurent-duval.eu lcd.siva.free.fr

Companion page to: A panorama on Multiscale Geometric Representations, intertwining spatial, directional and frequency selectivity (or "On 2D wavelets")


The richness of natural images makes the quest for optimal representations in image processing and computer vision challenging. The latter observation has not prevented the design of image representations, which trade off between efficiency and complexity, while achieving accurate rendering of smooth regions as well as reproducing faithful contours and textures. The most recent ones, proposed in the past decade, share an hybrid heritage highlighting the multiscale and oriented nature of edges and patterns in images. This paper presents a panorama of the aforementioned literature on decompositions in multiscale, multi-orientation bases or dictionaries. They typically exhibit redundancy to improve sparsity in the transformed domain and sometimes its invariance with respect to simple geometric deformations (translation, rotation). Oriented multiscale dictionaries extend traditional wavelet processing and may offer rotation invariance. Highly redundant dictionaries require specific algorithms to simplify the search for an efficient (sparse) representation. We also discuss the extension of multiscale geometric decompositions to non-Euclidean domains such as the sphere or arbitrary meshed surfaces. The etymology of panorama suggests an overview, based on a choice of partially overlapping "pictures". We hope that this paper will contribute to the appreciation and apprehension of a stream of current research directions in image understanding.

Laurent Jacques, Laurent Duval, Caroline Chaux and Gabriel Peyré, Signal Processing, Volume 91, Issue 12, December 2011, Special issue on Advances in Multirate Filter Bank Structures and Multiscale Representations, DOI:10.1016/j.sigpro.2011.04.025

See also: WITS: Where Is The Starlet? (wavelet names in *let)

Full bibliographical reference list on 2D wavelets

Accepted version updated to: [arxiv:1101.5320] or [hal-00588381]

All full-resolution 2D wavelet representations (curvelets, contourlets, bandelets, ridgelets, dual-tree, Morlet, gaussian and laplacian pyramids) in a single zip file: Jacques_L_2011_j-sp_panorama-figures.zip

Two-dimensional wavelet and multiscale toolboxes

  • PyrTools: matlabPyrTools - Matlab source code for multi-scale image processing. Includes tools for building and manipulating Laplacian pyramids, QMF/Wavelets, and steerable pyramids (base page: http://www.cns.nyu.edu/~eero/software.php)
  • Shearlab (using the power of traditional multiscale analysis to handle the geometry of multidimensional phenomena...): programs for shearlet transforms (base page: http://www.shearlet.org/)
  • Curvelab: Matlab and C++ programs for curvelet transforms (base page: http://www.curvelet.org/)
  • YAWTb toolbox: Yet Another Wavelet Toolbox, with continuous and discrete wavelet transform, easy visualization, modular programming, use of external routines written in C/C++ for efficiency.
  • BeamLab toolbox: a collection of Matlab functions [...] to implement a variety of computational algorithms related to beamlet, curvelet, ridgelet analysis.
  • Contourlet toolbox: home of the standard contourlets and Nonsubsampled Contourlet Toolbox
  • Waveatom.org: Matlab and C++ programs for orthonormal basis or a tight frame of directional wave packets, well suited for representing oscillatory patterns in images (base page: http://www.waveatom.org/)
  • LISQ toolbox: A toolbox for the lifting scheme on 2D quincunx grids by Paul De Zeeuw
  • Codes for M-band dual-tree wavelets: a matlab toolbox

Sample Image representations

The exemple image is taken from the Gedenktafel für Frigyes Riesz und Alfréd Haar in Szeged/Commemorative plaque for Frigyes Riesz and Alfréd Haar in Szeged (Hungary)

Picture courtesy of Professor Károly Szatmáry

Szeged University Memorial plaque in honor of A. Haar and F. Riesz
A szegedi matematikai iskola világhírű megalapítói
(The worldwide famous founders of the mathematical school in Szeged)

Szeged University Memorial plaque: Gaussian pyramid
Szeged University Memorial plaque: Gaussian pyramid

Szeged University Memorial plaque: Laplacian pyramid
Szeged University Memorial plaque: Laplacian pyramid

Additional links