A rational subdivision scheme using cosine-modulated wavelets

This paper proposes a rational subdivision scheme using cosine-modulated wavelets. Subdivision schemes constructed from iterated filter banks can be used to generate wavelets and limit functions for multiresolution analysis. The proposed subdivision scheme is based on a kind of nonuniform filter banks called recombination nonuniform filterbanks (RN FB). It is shown that if the component FBs in a RNFB are wavelet FBs, then the necessary condition for convergence to limit functions in the subdivision scheme is also satisfied. Therefore, the design of different rational subdivision schemes is considerably simplified. An efficient RNFB, called RN cosine modulated FBs (CMFB), constructed from uniform CMFBs and cosinemodulated transmultiplexers (TMUX) are further investigated. Using a design technique for designing RN CMFB and cosine modulated wavelets (CMW) previously reported by the authors, very smooth limit functions can be generated from the rational subdivision scheme. A design example is given to illustrate the proposed method.published_or_final_versio

Using iterated rational filter banks within the ARSIS method for producing 10 m Landsat multispectral images

International audienceThe ARSIS concept makes use of wavelet transforms and multiresolution analysis to improve the spatial resolutions of images from a set made of a low resolution image in the same spectral band and a high resolution image in an other spectral band. The paper deals with the use of rational filter banks in ARSIS. Indeed, first this concept was operationally applied with dyadic wavelet transforms that limit the merging of images with a ratio between spatial resolutions equals to a power of two. Provided some conditions, rational filter banks can be seen as a good approximation of rational wavelet transforms and, thus, enable a more general merging of images with ARSIS. The advantages of those rational filter banks compared to other methods are discussed and illustrated by an example of fusion of a 10 m SPOT Panchromatic image and a 30 m Landsat Thematic Mapper (TM) multispectral image into a synthetic 10 m multispectral image called hereafter TM-HR

A New Design Algorithm for Two-Band Orthonormal Rational Filter Banks and Orthonormal Rational Wavelets

In this paper, we present a new algorithm for the design of orthonormal two-band rational filter banks. Owing to the connection between iterated rational filter banks and rational wavelets, this is also a design algorithm for orthonormal rational wavelets. It is basically a simple iterative procedure, which explains its exponential convergence and adaptability under various linear constraints (e.g., regularity). Although the filters obtained from this algorithm are suboptimally designed, they show excellent frequency selectivity. After an in-depth account of the algorithm, we discuss the properties of the rational wavelets generated by some designed filters. In particular, we stress the possibility to design "almost" shift error-free wavelets, which allows the implementation of a rational wavelet transform

Wavelets and Subband Coding

First published in 1995, Wavelets and Subband Coding offered a unified view of the exciting field of wavelets and their discrete-time cousins, filter banks, or subband coding. The book developed the theory in both continuous and discrete time, and presented important applications. During the past decade, it filled a useful need in explaining a new view of signal processing based on flexible time-frequency analysis and its applications. Since 2007, the authors now retain the copyright and allow open access to the book

Quantitative Fourier Analysis of Approximation Techniques: Part II—Wavelets

In a previous paper, we proposed a general Fourier method which provides an accurate prediction of the approximation error, irrespective of the scaling properties of the approximating functions. Here, we apply our results when these functions satisfy the usual two-scale relation encountered in dyadic multiresolution analysis. As a consequence of this additional constraint, the quantities introduced in our previous paper can be computed explicitly as a function of the refinement filter. This is in particular true for the asymptotic expansion of the approximation error for biorthonormal wavelets, as the scale tends to zero. One of the contributions of this paper is the computation of sharp, asymptotically optimal upper bounds for the least-squares approximation error. Another contribution is the application of these results to B-splines and Daubechies scaling functions, which yields explicit asymptotic developments and upper bounds. Thanks to these explicit expressions, we can quantify the improvement that can be obtained by using B-splines instead of Daubechies wavelets. In other words, we can use a coarser spline sampling and achieve the same reconstruction accuracy as Daubechies: Specifically, we show that this sampling gain converges to pi as the order tends to infinity

Wavelet Filter Banks in Perceptual Audio Coding

This thesis studies the application of the wavelet filter bank (WFB) in perceptual audio coding by providing brief overviews of perceptual coding, psychoacoustics, wavelet theory, and existing wavelet coding algorithms. Furthermore, it describes the poor frequency localization property of the WFB and explores one filter design method, in particular, for improving channel separation between the wavelet bands. A wavelet audio coder has also been developed by the author to test the new filters. Preliminary tests indicate that the new filters provide some improvement over other wavelet filters when coding audio signals that are stationary-like and contain only a few harmonic components, and similar results for other types of audio signals that contain many spectral and temporal components. It has been found that the WFB provides a flexible decomposition scheme through the choice of the tree structure and basis filter, but at the cost of poor localization properties. This flexibility can be a benefit in the context of audio coding but the poor localization properties represent a drawback. Determining ways to fully utilize this flexibility, while minimizing the effects of poor time-frequency localization, is an area that is still very much open for research

Fusion of high spatial and spectral resolution images: the ARSIS concept and its implementation

International audienceIn various applications of remote sensing, when high spatial resolution is required in addition with classification results, sensor fusion is a solution. From a set of images with different spatial and spectral resolutions, the aim is to synthesize images with the highest spatial resolution available in the set and with an appropriate spectral content. Several sensor fusion methods exist; most of them improve the spatial resolution but with a poor quality of the spectral content of the resulting image. Based on a multiresolution modeling of the information, the ARSIS concept (from its French name "Amélioration de la Résolution Spatiale par Injection de Structures") was designed in the aim of improving the spatial resolution together with a high-quality in the spectral content of the synthesized images. The general case of application of this concept is described. A quantitative comparison of all presented methods is achieved for a SPOT image. Another example of the fusion of SPOT XS (20 m) and KVR-1000 (2 m) images is given. Practical information for the implementation of the wavelet transform, the multiresolution analysis, and the ARSIS concept by practitioners is given with particular relevance to SPOT and Landsat imagery

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