Showing papers on "Multiresolution analysis published in 1993"

Volume data and wavelet transforms

Abstract: The application of 3D orthogonal wavelet transforms to real volume data is discussed. Examples of the wavelet transform and the reconstruction of 1D functions are presented. The application of the 3D wavelet transform to real volume data generated from a series of 115 slices of magnetic resonance (MR) images is described. >

Time-varying system identification and model validation using wavelets

Abstract: Parametric identification of time-varying (TV) systems is possible if each TV coefficient can be expanded onto a finite set of basis sequences. The problem then becomes time invariant with respect to the parameters of the expansion. The authors address the question of selecting this set of basis sequences. They advocate the use of a wavelet basis because of its flexibility in capturing the signal's characteristics at different scales, and discuss how to choose the optimal wavelet basis for a given system trajectory. They also develop statistical tests to keep only the basis sequences that significantly contribute to the description of the system's time-variation. By formulating the problem as a regressor selection problem, they apply an P-test and an AIC based approach for multiresolution analysis of TV systems. The resulting algorithm can estimate TV AR or ARMAX models and determine their orders. They apply this algorithm to both synthetic and real speech data and compare it with the Kalman filtering TV parameter estimator. >

Wavelet approximation methods for pseudodifferential equations II: Matrix compression and fast solution

Abstract: This is the second part of two papers which are concerned with generalized Petrov-Galerkin schemes for elliptic periodic pseudodifferential equations in ℝ n . This setting covers classical Galerkin methods, collocation, and quasi-interpolation. The numerical methods are based on a general framework of multiresolution analysis, i.e. of sequences of nested spaces which are generated by refinable functions. In this part, we analyse compression techniques for the resulting stiffness matrices relative to wavelet-type bases. We will show that, although these stiffness matrices are generally not sparse, the order of the overall computational work which is needed to realize a certain accuracy is of the formO(N(logN) b ), whereN is the number of unknowns andb ≥ 0 is some real number.

A discrete-time multiresolution theory

Abstract: Multiresolution analysis and synthesis for discrete-time signals is described. Concepts of scale and resolution are first reviewed in discrete time. The resulting framework allows one to treat the discrete wavelet transform, octave-band perfect reconstruction filter banks, and pyramid transforms from a unified standpoint. This approach is very close to previous work on multiresolution decomposition of functions of a continuous variable, and the connection between these two approaches is made. It is shown that they share many mathematical properties such as biorthogonality, orthonormality, and regularity. However, the discrete-time formalism is well suited to practical tasks in digital signal processing and does not require the use of functional spaces as an intermediate step. >

Families of multiresolution and wavelet spaces with optimal properties

Abstract: Under suitable conditions, if the scaling functions ϕ1 and ϕ2 generate the multiresolutions V (j)(ϕ1) and V (j)(ϕ2), then their convolution ϕ1*ϕ2also generates a multiresolution V (j)(ϕ1*ϕ2) More over, if p is an appropriate convolution operator from l 2 into itself and if ϕ is a scaling function generating the multiresolution V (j)(ϕ),then p*ϕis a scaling function generating the same multiresolution V (j)(ϕ)=V (j)(p*ϕ). Using these two properties, we group the scaling and wavelet functions into equivalent classes and consider various equivalent basis functions of the associated function spaces We use the n-fold convolution product to construct sequences of multiresolution and wavelet spaces V (j)(ϕn) and W (j)(ϕn) with increasing regularity. We discuss the link between multiresolution analysis and Shannon's sampling theory. We then show that the interpolating and orthogonal pre- and post-filters associated with the multiresolution sequence V (0)(ϕn)asymptotically converge to the ideal lowpass filter of S...

Multiresolution wavelet analysis of evoked potentials

Abstract: Neurological injury, such as from cerebral hypoxia, appears to cause complex changes in the shape of evoked potential (EP) signals. To characterize such changes we analyze EP signals with the aid of scaling functions called wavelets. In particular, we consider multiresolution wavelets that are a family of orthonormal functions. In the time domain, the multiresolution wavelets analyze EP signals at coarse or successively greater levels of temporal detail. In the frequency domain, the multiresolution wavelets resolve the EP signal into independent spectral bands. In an experimental demonstration of the method, somatosensory EP signals recorded during cerebral hypoxia in anesthetized cats are analyzed. Results obtained by multiresolution wavelet analysis are compared with conventional time-domain analysis and Fourier series expansions of the same signals. Multiresolution wavelet analysis appears to be a different, sensitive way to analyze EP signal features and to follow the EP signal trends in neurologic injury. Two characteristics appear to be of diagnostic value: the detail component of the MRW displays an early and a more rapid decline in response to hypoxic injury while the coarse component displays an earlier recovery upon reoxygenation. >

Some elementary properties of multiresolution analyses of L 2 (R n )

Abstract: . A multiresolution analysis is a family of subspaces which are generated by lattice translates of dilates of one function ϕ. Here we record several, hopefully useful, observations concerning such families. For example: the definitions are reviewed and examined, examples are given, functions ϕ whose translates and dilates generate such analyses are characterized, those multiresolution analyses which are invariant under all translations and those whose scaling functions are characteristic functions are completely described.

How to Make Wavelets

Abstract: (1993) How To Make Wavelets The American Mathematical Monthly: Vol 100, No 6, pp 539-556

A multiresolution method for distributed parameter estimation

Abstract: A multiresolution method for distributed parameter estimation (or inverse problems) is studied numerically. The identification of the coefficient of an elliptic equation in one dimension is considered as our model problem. First, multiscale bases are used to analyze the degree of ill-posedness of the inverse problem. Second, based on some numerical results, it is shown that the method of scale-by-scale multiresolution yields robust and fast convergence. Finally, it is shown how the method gives a natural regularization approach which is complementary to Tikhonov’s regularization.

On the instability of arbitrary biorthogonal wavelet packets

Abstract: Starting from a multiresolution analysis and the corresponding orthonormal wavelet basis, Coifman and Meyer have constructed wavelet packets, a library from which many different orthonormal bases can be picked. This paper proves that when the same procedure is applied to biorthogonal wavelet bases, not all the resulting wavelet packets lead to Riesz bases for $L^2 (\mathbb{R})$.

Wavelet transforms associated with finite cyclic groups

Abstract: Multiresolution analysis via decomposition on wavelet bases has emerged as an important tool in the analysis of signals and images when these objects are viewed as sequences of complex or real numbers. An important class of multiresolution decompositions are the Laplacian pyramid schemes, in which the resolution is successively halved by recursively low-pass filtering the signal under analysis and decimating it by a factor of two. In general, the principal framework within which multiresolution techniques have been studied and applied is the same as that used in the discrete-time Fourier analysis of sequences of complex numbers. An analogous framework is developed for the multiresolution analysis of finite-length sequences of elements from arbitrary fields. Attention is restricted to sequences of length 2/sup n/, for n a positive integer, so that the resolution may be recursively halved to completion. As in finite-length Fourier analysis, a cyclic group structure of the index set of such sequences is exploited to characterize the transforms of interest for the particular cases of complex and finite fields. >

Wavelets and self-affine tilings

Abstract: Given a self-affine periodic tiling ofR n we construct an associatedr-regular multiresolution analysis and wavelet basis with the same lattice of translations and scaling matrix as the tiling.

Wavelets and Fast Numerical Algorithms

Abstract: Wavelet based algorithms in numerical analysis are similar to other transform methods in that vectors and operators are expanded into a basis and the computations take place in this new system of coordinates. However, due to the recursive definition of wavelets, their controllable localization in both space and wave number (time and frequency) domains, and the vanishing moments property, wavelet based algorithms exhibit new and important properties. For example, the multiresolution structure of the wavelet expansions brings about an efficient organization of transformations on a given scale and of interactions between different neighbouring scales. Moreover, wide classes of operators which naively would require a full (dense) matrix for their numerical description, have sparse representations in wavelet bases. For these operators sparse representations lead to fast numerical algorithms, and thus address a critical numerical issue. We note that wavelet based algorithms provide a systematic generalization of the Fast Multipole Method (FMM) and its descendents. These topics will be the subject of the lecture. Starting from the notion of multiresolution analysis, we will consider the so-called non-standard form (which achieves decoupling among the scales) and the associated fast numerical algorithms. Examples of non-standard forms of several basic operators (e.g. derivatives) will be computed explicitly.

Nonorthogonal wavelet packets

Abstract: The notion of orthonormal wavelet packets introduced by Coifman and Meyer is generalized to the nonorthogonal setting in order to include compactly supported and symmetric basis functions. In particular, dual (or biorthogonal) wavelet packets are investigated and a stability result is established. Algorithms for implementations are also developed.

Iterated filter banks with rational rate changes connection with discrete wavelet transforms

Abstract: Some properties of two-band filter banks with rational rate changes ("rational filter banks") are first reviewed. Focusing then on iterated rational filter banks, compactly supported limit functions are obtained, in the same manner as previously done for dyadic schemes, allowing a characterization of such filter banks. These functions are carefully studied and the properties they share with the dyadic case are highlighted. They are experimentally observed to verify a "shift property" (strictly verified in the dyadic ease) up to an error which can be made arbitrarily small when their regularity increases. In this case, the high-pass outputs of an iterated filter bank can be very close to samples of a discrete wavelet transform with the same rational dilation factor. Straightforward extension of the formalism of multiresolution analysis is also made. Finally, it is shown that if one is ready to put up with the loss of the shift property, rational iterated filter banks can be used in the same manner as if they were dyadic filter banks, with the advantage that rational dilation factors can be chosen closer to 1. >

Continuous wavelet decompositions, multiresolution, and contrast analysis

Abstract: A continuous version of multiresolution analysis is described, starting from usual continuous wavelet decompositions. Scale discretization leads to decomposition into functions of arbitrary bandwidth, satisfying QMF-like conditions. Finally, a nonlinear multiresolution scheme is described, providing multiplicative reconstruction formulas.

Multiresolution analysis frames with applications

Abstract: A generalization of multiresolution analysis (MRA) to arbitrary affine frames, and the constructions of affine frames based on MRAs are considered and characterized. The authors define a frame multiresolution analysis (FMRA) and provide a necessary and sufficient condition for constructing frames of translates. Conditions for constructing FMRAs and associated frames are derived. The results are part of the theoretical background for multiscale signal processing problems including filter bank design, where redundancy and robustness play a role. >

Efficient data fusion using wavelet transform: the case of SPOT satellite images

Abstract: A way of increasing the spatial resolution of SPOT multispectral images (XS) using the corresponding panchromatic image (P) is presented here. Existing methods for merging P and XS are analyzed, before presenting a new method which aims at simulating 10 m resolution multispectral images that contain the same spectral information at the XS images. This method, called ARSIS after its French name 'Amelioration de la Resolution Spatiale par Injection de Structures', is based upon multiresolution analysis and wavelet transform. Different versions have been implemented, which differ on the model that describes the similarity of the spatial variability on P and XS. ARSIS can also be applied to other sensors, featuring different spectral bands and spatial resolutions.

Continuous refinement equations and subdivision

Abstract: This paper is concerned with the study of a general class of functional equations covering as special cases the relation which defines theup-function as well as equations which arise in multiresolution analysis for wavelet construction. We discuss various basic properties of solutions to these functional equations such as regularity, polynomial containment within the space spanned by their integer shifts and their computability by subdivision algorithms.

Wavelet analysis in recruitment of loudness compensation

Abstract: Wavelet-based multiband dynamic range compression is developed to compensate for a common hearing impairment known as recruitment of loudness. The algorithm combines standard compression with intensity-level dependent gain calculation. Complexity and performance are similar to traditional techniques. Further, the methods established can be applied to the more adaptive wavelet packets and local cosine bases which model the speech signal more closely. >

Choosing discrete orthogonal wavelets for signal analysis and approximation

Abstract: The discrete wavelet transform decomposes a discrete time signal into an approximation sequence and a detailed sequence at each level of resolution. The approximation at any resolution is the projection of the signal onto the orthogonal space spanned by the translates of an analyzing scaling function. The choice of scaling function can have a large impact on the error in the approximation at a given resolution. A systematic method for generating scaling functions is developed. This method ensures that a scaling function will be found that is close to the optimum. The resulting scaling functions can be used by themselves or serve as starting point for further optimization. >

Nonorthogonal multiresolution analysis using wavelets

Abstract: . Orthogonal multiresolution analysis theory and algorithms are well understood by now. However, for many applications such as image compression or edge detection, the orthogonal framework is not satisfactory since certain natural constraints cannot be achieved. For instance, it is not possible to use finite impulse response linear phase filters. To remove these drawbacks, we introduce here nonorthogonal multiresolution analyses which may be considered as a generalization of the orthogonal setting and provide more freedom for the design of multiresolution analyses.

Multiscale statistical signal processing: identification of a multiscale AR process from a sample of an ordinary signal

Abstract: The theory of stochastic processes on homogeneous trees aims at contributing to the theory of multiresolution stochastic modeling and associated techniques of multiscale statistical signal processing. The author solves the problem of identifying such a multiscale process indexed by the nodes of a tree from the observation of this process on one single level of resolution. In particular he considers multiscale autoregressive processes, which are evolving by descending on a "hanging" homogeneous tree. >

Wavelet packets applied to mammograms

Abstract: A multiresolution analysis based on wavelets is particularly effective for signals with marked discontinuities, such as microcalcifications produce in mammograms. The projection methods that generate the resumes and details can yield an entire family of orthonormal bases, or wavelet packets, which can be superior to wavelets for certain classes of signals. Results of applying wavelet packets to snippets from several mammograms are presented as a method of data compression, allowing reduction of the digitized data before applying pattern recognition techniques.© (1993) COPYRIGHT SPIE--The International Society for Optical Engineering. Downloading of the abstract is permitted for personal use only.

Wavelets transform in artificial vision inspection of threading

Abstract: We present an application of multiresolution analysis with orthonormal wavelets of 1D signal to quality control by artificial vision. The purpose of the control is to check the thread of a polyethylene bottle stopper. Using a section picture of the stopper provided by a linear CCD camera, we calculate the wavelet coefficients of the first three levels of resolution. The energy densities of these coefficients calculated on a given area, provide three discriminant parameters which permit to distinguish correctly between the two classes (defectless, defective) according to a classifying method which can run without supervision after a period of training. >

Multiresolution 2-dimensional edge analysis using wavelets

Abstract: The authors show that wavelet transforms can be used in an empirical way to improve edge-detected pictures. Specifically, the first derivative of a Gaussian distribution function is used as the wavelet for detecting edges. The wavelet transforms are calculated using integer scales instead of dyadic scales. This empirical approach is justified by comparing the method with an exact reconstruction process that uses wavelet transforms of the original image. The approach was demonstrated in an experiment that used a magnetic resonance image of a brain. >