Low-complexity wavelet filter design for image compression

Image compression algorithms based on the wavelet transform are an increasingly attractive and flexible alternative to other algorithms based on block orthogonal transforms. While the design of orthogonal wavelet filters has been studied in significant depth, the design of nonorthogonal wavelet filters, such as linear-phase (LP) filters, has not yet reached that point. Of particular interest are wavelet transforms with low complexity at the encoder. In this article, we present known and new parameterizations of the two families of LP perfect reconstruction (PR) filters. The first family is that of all PR LP filters with finite impulse response (FIR), with equal complexity at the encoder and decoder. The second family is one of LP PR filters, which are FIR at the encoder and infinite impulse response (IIR) at the decoder, i.e., with controllable encoder complexity. These parameterizations are used to optimize the subband/wavelet transform coding gain, as defined for nonorthogonal wavelet transforms. Optimal LP wavelet filters are given for low levels of encoder complexity, as well as their corresponding integer approximations, to allow for applications limited to using integer arithmetic. These optimal LP filters yield larger coding gains than orthogonal filters with an equivalent complexity. The parameterizations described in this article can be used for the optimization of any other appropriate objective function

First-arrival Travel-Time Tomography using Second Generation Wavelets

International audienceWavelet decomposition of the slowness model has been proposed as a multiscale strategy for seismic first-arrival time tomography. We propose the introduction of so-called second generation wavelets which could be used for any mesh structure and does not need a number of samples as the power of two in each direction. Moreover, one can handle easily boundary effects. A linearized procedure for inverting delayed travel-times considering either slowness coefficients or wavelet coefficients. The ray tracing is solved at each iteration through an eikonal solver while the linear system to be solved at each iteration goes through an iterative solver as LSQR algorithm. We develop wavelet decomposition over constant patches (Haar wavelet) or over linear patches (Battle-Lemarie wavelet) of coefficients at different scales. This decomposition is introduced in the linear system to be solved and wavelet coefficients are considered as unknowns to be inverted. Synthetic examples show that the inversion behaves in a better way as wavelet decomposition seems to act as a preconditioner of the linear system. Local discretisation is possible but requires additional implementation as artefacs once built inside the model description never disappear because of the linearized approach. A binary mask operator is designed for each scale grid and could be applied locally leading to quite different spatial resolution depending on the analysis we could perform of the expected resolution at a given position of the medium. We show that indeed it is possible to design this binary operator and we apply it to synthetic examples as a crosswell experiment inside the Marmousi model. An application to a surface-surface experiment has been performed and the waveled decomposition shows that indeed we may recover detailed features nearby the free surface while preventing imprints of ray coverage at greater depths giving us smooth features at that depths. In spite of the increase demand of computer resources, the wavelet decomposition seems to be a rather promising alternative for controlling the resolution variation of seismic first-arrival tomography

A Panorama on Multiscale Geometric Representations, Intertwining Spatial, Directional and Frequency Selectivity

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.Comment: 65 pages, 33 figures, 303 reference

Representation and design of wavelets using unitary circuits

The representation of discrete, compact wavelet transformations (WTs) as circuits of local unitary gates is discussed. We employ a similar formalism as used in the multiscale representation of quantum many-body wave functions using unitary circuits, further cementing the relation established in the literature between classical and quantum multiscale methods. An algorithm for constructing the circuit representation of known orthogonal, dyadic, discrete WTs is presented, and the explicit representation for Daubechies wavelets, coiflets, and symlets is provided. Furthermore, we demonstrate the usefulness of the circuit formalism in designing WTs, including various classes of symmetric wavelets and multiwavelets, boundary wavelets, and biorthogonal wavelets

Representation and design of wavelets using unitary circuits

The representation of discrete, compact wavelet transformations (WTs) as circuits of local unitary gates is discussed. We employ a similar formalism as used in the multiscale representation of quantum many-body wave functions using unitary circuits, further cementing the relation established in the literature between classical and quantum multiscale methods. An algorithm for constructing the circuit representation of known orthogonal, dyadic, discrete WTs is presented, and the explicit representation for Daubechies wavelets, coiflets, and symlets is provided. Furthermore, we demonstrate the usefulness of the circuit formalism in designing WTs, including various classes of symmetric wavelets and multiwavelets, boundary wavelets, and biorthogonal wavelets