The Undecimated Wavelet Decomposition and its Reconstruction

By Jean-Luc Starck, Mohamed-Jalal Fadili, Fionn Murtagh. IEEE Transactions on Image Processing, 2007

514 citations recorded by the source corpus.

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

Subfields: Applied Mathematics, Computer Vision and Pattern Recognition, Media Technology

View Paper: https://www.semanticscholar.org/paper/79b15ea8d009bcaf63f3871c5bd63ea244919def

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Citations in the corpus

An optimized algorithm for multiscale wideband deconvolution of radio astronomical images (2017, methodology): “MORESANE (Dabbech et al. 2015) is a convex optimisation algorithm that uses the isotropic undecimated wavelet transform (IUWT; Starck et al. 2007) to benefit from sparsity in the transformed space.”
Wavelets, Ridgelets, and Curvelets for Poisson Noise Removal (2008, methodology): “The IUWT [33] uses the filter bank where is typically a symmetric low-pass filter such as the -Spline filter.”
Compressed Sensing with Coherent and Redundant Dictionaries (2010, unknown): “4 shows reconstruction results after just one reweighting iteration; the root-mean squared error (RMSE) is significantly reduced, by a factor between 3 and 4.”
Fast Retinal Vessel Detection and Measurement Using Wavelets and Edge Location Refinement (2012, methodology): “It affords a particularly simple implementation that can be readily appreciated without recourse to wavelet theory: at each iteration j, scaling coefficients cj are computed by lowpass filtering, and wavelet coefficients wj by subtraction [38].”
From Sparse Solutions of Systems of Equations to Sparse Modeling of Signals and Images (2009, background): “How can we wisely choose A to perform well on the signals we have in mind? One line of work considered choosing preconstructed dictionaries, such as undecimated wavelets [149], steerable wavelets [145, 37, 136], contourlets [38, 39, 40, 70, 71], curvelets [146, 12], and others [22, 123].”
Hyperspectral Pansharpening: A Review (2015, background): “The huge interest of the community on this topic is evidenced by the existence of sessions dedicated to this topic in the most important remote sensing and Earth observation conferences as well as by the launch of public contests, of which the one sponsored by the data fusion committee of the IEEE…”
A Critical Comparison Among Pansharpening Algorithms (2015, background): “…to several modalities of MRA: decimated wavelet transform (DWT) [17], undecimated wavelet transform (UDWT) [18], " à trous " wavelet transform (ATWT) [19], Laplacian pyramid (LP) [20], and nonseparable transforms, either based on wavelets (e.g., contourlet [21]) or not (e.g., curvelet [22]).”
DEEP U BAND AND R IMAGING OF GOODS-SOUTH: OBSERVATIONS, DATA REDUCTION AND FIRST RESULTS, (2009, methodology): “Using the previous analysis filter h(k, l), one gets the isotropic undecimated wavelet decomposition (Starck et al. 2007).”
Objective comparison of particle tracking methods (2014, methodology): “METHOD 4: PARAMETERS The user parameters of this method (and typical values) are:  Number of iterations of convolution with the Gaussian kernel (10-500).”
Remote Sensing Image Fusion via Sparse Representations Over Learned Dictionaries (2013, unknown): “One approach focuses on the mathematical model, such as undecimated wavelet [23] and discrete cosine transformation [24].”