The Undecimated Wavelet Decomposition and its Reconstruction

This paper describes the undecimated wavelet transform and its reconstruction. In the first part, we show the relation between two well known undecimated wavelet transforms, the standard undecimated wavelet transform and the isotropic undecimated wavelet transform. Then we present new filter banks specially designed for undecimated wavelet decompositions which have some useful properties such as being robust to ringing artifacts which appear generally in wavelet-based denoising methods. A range of examples illustrates the results

[1]  A. Antoniadis,et al.  Wavelet shrinkage for natural exponential families with quadratic variance functions , 2001 .

[2]  A. Cohen.  Numerical Analysis of Wavelet Methods , 2003 .

[3]  Fionn Murtagh,et al.  Image Processing and Data Analysis - The Multiscale Approach , 1998 .

[4]  Jean-Luc Starck,et al.  Astronomical image and data analysis , 2002 .

[5]  Helmut Bölcskei,et al.  Frame-theoretic analysis of oversampled filter banks , 1998, IEEE Trans. Signal Process..

[6]  Albert Bijaoui,et al.  DeQuant: a flexible multiresolution restoration framework , 2004, Signal Process..

[7]  Thomas Brox,et al.  On the Equivalence of Soft Wavelet Shrinkage, Total Variation Diffusion, Total Variation Regularization, and SIDEs , 2004, SIAM J. Numer. Anal..

[8]  P. Dutilleux.  An Implementation of the “algorithme à trous” to Compute the Wavelet Transform , 1989 .

[9]  François Malgouyres,et al.  Minimizing the total variation under a general convex constraint for image restoration , 2002, IEEE Trans. Image Process..

[10]  Michael Unser,et al.  Texture classification and segmentation using wavelet frames , 1995, IEEE Trans. Image Process..

[11]  Robert D. Nowak,et al.  Multiscale Modeling and Estimation of Poisson Processes with Application to Photon-Limited Imaging , 1999, IEEE Trans. Inf. Theory.

[12]  Robert D. Nowak,et al.  Wavelet-domain filtering for photon imaging systems , 1999, IEEE Trans. Image Process..

[13]  Ingrid Daubechies,et al.  Ten Lectures on Wavelets , 1992 .

[14]  Jacques Froment,et al.  Reconstruction of Wavelet Coefficients Using Total Variation Minimization , 2002, SIAM J. Sci. Comput..

[15]  Mark J. Shensa,et al.  The discrete wavelet transform: wedding the a trous and Mallat algorithms , 1992, IEEE Trans. Signal Process..

[16]  Ph. Tchamitchian,et al.  Wavelets: Time-Frequency Methods and Phase Space , 1992 .

[17]  E. Kolaczyk.  WAVELET SHRINKAGE ESTIMATION OF CERTAIN POISSON INTENSITY SIGNALS USING CORRECTED THRESHOLDS , 1999 .

[18]  Emmanuel J. Candès,et al.  New multiscale transforms, minimum total variation synthesis: applications to edge-preserving image reconstruction , 2002, Signal Process..

[19]  E. Kolaczyk,et al.  Nonparametric Estimation of Intensity Maps Using Haar Wavelets and Poisson Noise Characteristics , 2000 .

[20]  Martin Vetterli,et al.  Oversampled filter banks , 1998, IEEE Trans. Signal Process..

[21]  Richard Kronland-Martinet,et al.  A real-time algorithm for signal analysis with the help of the wavelet transform , 1989 .

[22]  Michel Barlaud,et al.  Image coding using wavelet transform , 1992, IEEE Trans. Image Process..

[23]  Jean-Christophe Olivo-Marin,et al.  Tracking fluroescent spots in biological video microscopy , 2003, SPIE BiOS.

[24]  Bo Zhang,et al.  Tracking of multiple fluorescent biological objects in three dimensional video microscopy , 2003, Proceedings 2003 International Conference on Image Processing (Cat. No.03CH37429).

[25]  G. Battle.  A block spin construction of ondelettes. Part I: Lemarié functions , 1987 .

[26]  Jean-Luc Starck,et al.  Very high quality image restoration by combining wavelets and curvelets , 2001, SPIE Optics + Photonics.

[27]  Martin Vetterli,et al.  Discrete-time wavelet extrema representation: design and consistent reconstruction , 1995, IEEE Trans. Signal Process..

[28]  Edward H. Adelson,et al.  The Laplacian Pyramid as a Compact Image Code , 1983, IEEE Trans. Commun..

[29]  Michael Elad,et al.  Why Simple Shrinkage Is Still Relevant for Redundant Representations? , 2006, IEEE Transactions on Information Theory.

[30]  Jean-Pierre Antoine,et al.  Two-dimensional wavelet analysis in image processing , 1994 .

[31]  G. Nason,et al.  A Haar-Fisz Algorithm for Poisson Intensity Estimation , 2004 .

[32]  D. Donoho,et al.  Redundant Multiscale Transforms and Their Application for Morphological Component Separation , 2004 .

[33]  Robert W. Heath,et al.  Designing structured tight frames via an alternating projection method , 2005, IEEE Transactions on Information Theory.

[34]  Michael Elad,et al.  Submitted to Ieee Transactions on Image Processing Image Decomposition via the Combination of Sparse Representations and a Variational Approach , 2022 .

[35]  P. Laguna,et al.  Signal Processing , 2002, Yearbook of Medical Informatics.

[36]  D. Donoho,et al.  Simultaneous cartoon and texture image inpainting using morphological component analysis (MCA) , 2005 .

[37]  Fionn Murtagh,et al.  Multiresolution Support Applied to Image Filtering and Restoration , 1995, CVGIP Graph. Model. Image Process..

[38]  Vivek K. Goyal,et al.  Quantized Overcomplete Expansions in IRN: Analysis, Synthesis, and Algorithms , 1998, IEEE Trans. Inf. Theory.

[39]  S. Mallat.  A wavelet tour of signal processing , 1998 .