The discrete wavelet transform: wedding the a trous and Mallat algorithms

By Mark J. Shensa. 1992

Two separately motivated implementations of the wavelet transform are brought together. It is observed that these algorithms are both special cases of a single filter bank structure, the discrete wavelet transform, the behavior of which is governed by the choice of filters. In fact, the a trous algorithm is more properly viewed as a nonorthonormal multiresolution algorithm for which the discrete wavelet transform is exact. Moreover, it is shown that the commonly used Lagrange a trous filters are in one-to-one correspondence with the convolutional squares of the Daubechies filters for orthonormal wavelets of compact support. A systematic framework for the discrete wavelet transform is provided, and conditions are derived under which it computes the continuous wavelet transform exactly. Suitable filter constraints for finite energy and boundedness of the discrete transform are also derived. Relevant signal processing parameters are examined, and it is observed that orthonormality is balanced by restrictions on resolution. >

Subfields: Numerical Methods & Scientific Computing

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Citations in the corpus

Axial-DeepLab: Stand-Alone Axial-Attention for Panoptic Segmentation (2020, methodology): “Some employ atrous convolutions [34,77,67,13], larger kernel [70], or image pyramids [98,85], either designed by hand or searched by algorithms [103,12,60].”
DeepLab: Semantic Image Segmentation with Deep Convolutional Nets, Atrous Convolution, and Fully Connected CRFs (2016, methodology): “The second method, originally proposed by [82] and used in [3], [16] is to subsample the input feature map by a factor equal to the atrous convolution rate r, deinterlacing it to produce r(2) reduced resolution maps, one for each of the r r possible shifts.”
Multi-Scale Context Aggregation by Dilated Convolutions (2015, neutral): “It plays a key role in the algorithme `a trous , an algorithm for wavelet decomposition (Holschneider et al., 1987; Shensa, 1992).”
The redundant discrete wavelet transform and additive noise (2005, methodology): “(49) We note that an alternative implementation of the RDWT was independently proposed by Shensa [17] and Beylkin [18].”
Time-series modeling with undecimated fully convolutional neural networks (2015, methodology): “ signal can lead to very large changes to the 3 representation coefﬁcients. In order to address this issue, we consider using an undecimated FCN that is motivated by the undecimated wavelet transform [12] which is shift-invariant. 3.2 Undecimated fully convolutional neural networks The undecimated w”
DeepGCNs: Can GCNs Go As Deep As CNNs? (2019, methodology): “Dilated wavelet convolution is an algorithm originating from the wavelet processing domain [12, 33].”
Gated Residual Networks with Dilated Convolutions for Supervised Speech Separation (2018, unknown): “Dilated convolutions were originally developed in the algorithme à trous, an algorithm for wavelet decomposition [13].”
Water Bodies' Mapping from Sentinel-2 Imagery with Modified Normalized Difference Water Index at 10-m Spatial Resolution Produced by Sharpening the SWIR Band (2016, methodology): “In this paper, four popular pan-sharpening algorithms, including PCA [44], IHS [45], High Pass Filter (HPF) [46] and À Trous Wavelet Transform (ATWT) [47] were applied to downscale the Sentinel-2 SWIR band.”
Wavelets and Subband Coding (2013, methodology): “The computation of the wavelet series coefficients using filter banks was studied by Mallat [181, 179] and Shensa [261], among others.”
The Nonsubsampled Contourlet Transform: Theory, Design, and Applications (2006, methodology): “Thus, most state-of-the-art wavelet denoising algorithms (see for example [6]–[8]) use an expansion with less shift sensitivity than the standard maximally decimated wavelet decomposition—the most common being the nonsubsampled wavelet transform (NSWT) computed with the à trous algorithm [9].”
Image fusion—the ARSIS concept and some successful implementation schemes (2003, methodology): “A theoretical analysis of the “à trous” method is given in Shensa (1992).”