Showing papers on "Multiresolution analysis published in 1997"

Multiresolution analysis for surfaces of arbitrary topological type

Abstract: Multiresolution analysis and wavelets provide useful and efficient tools for representing functions at multiple levels of detail. Wavelet representations have been used in a broad range of applications, including image compression, physical simulation, and numerical analysis. In this article, we present a new class of wavelets, based on subdivision surfaces, that radically extends the class of representable functions. Whereas previous two-dimensional methods were restricted to functions difined on R2, the subdivision wavelets developed here may be applied to functions defined on compact surfaces of arbitrary topological type. We envision many applications of this work, including continuous level-of-detail control for graphics rendering, compression of geometric models, and acceleration of global illumination algorithms. Level-of-detail control for spherical domains is illustrated using two examples: shape approximation of a polyhedral model, and color approximation of global terrain data.

A mathematical introduction to wavelets

Abstract: This book presents a mathematical introduction to the theory of orthogonal wavelets and their uses in analysing functions and function spaces, both in one and in several variables. Starting with a detailed and self contained discussion of the general construction of one dimensional wavelets from multiresolution analysis, the book presents in detail the most important wavelets: spline wavelets, Meyer's wavelets and wavelets with compact support. It then moves to the corresponding multivariable theory and gives genuine multivariable examples. Wavelet decompositions in Lp spaces, Hardy spaces and Besov spaces are discussed and wavelet characterisations of those spaces are provided. Also included are some additional topics like periodic wavelets or wavelets not associated with a multiresolution analysis. This will be an invaluable book for those wishing to learn about the mathematical foundations of wavelets.

Multiresolution Flux Decomposition

Abstract: Geophysical variables are orthogonally decomposed by averaging timeseries using different averaging lengths, referred to as a (Haar)multiresolution decomposition. This simple and economic decomposition isassociated with cospectra that formally satisfy Reynolds averaging rules foreach averaging length. The multiresolution decomposition provides a naturalestimate of the random error in estimating a mean turbulent flux. The Fourierand multiresolution decompositions are compared using aircraft data fromBOREAS.

Separation of discontinuous adventitious sounds from vesicular sounds using a wavelet-based filter

Abstract: The separation of pathological discontinuous adventitious sounds (DAS) from vesicular sounds (VS) is of great importance to the analysis of lung sounds, since DAS are related to certain pulmonary pathologies. An automated way of revealing the diagnostic character of DAS by isolating them from VS, based on their nonstationarity, is presented in this paper. The proposed algorithm combines multiresolution analysis with hard thresholding in order to compose a wavelet transform-based stationary-nonstationary filter (WTST-NST). Applying the WTST-NST filter to fine/coarse crackles and squawks, selected from three lung sound databases, the coherent structure of DAS is revealed and they are separated from VS. When compared to other separation tools, the WTST-NST filter performed more accurately, objectively, and with lower computational cost. Due to its simple implementation it can easily be used in clinical medicine.

Multiresolution reproducing kernel particle method for computational fluid dynamics

Abstract: Multiresolution analysis based on the reproducing kernel particle method (RKPM) is developed for computational fluid dynamics. An algorithm incorporating multiple-scale adaptive refinement is introduced. The concept of using a wavelet solution as an error indicator is also presented. A few representative numerical examples are solved to illustrate the performance of this new meshless method.

Multiresolution Schemes for the Numerical Solution of 2-D Conservation Laws I

Abstract: A generalization of Harten's multiresolution algorithms to two-dimensional (2-D) hyperbolic conservation laws is presented. Given a Cartesian grid and a discretized function on it, the method computes the local-scale components of the function by recursive diadic coarsening of the grid. Since the function's regularity can be described in terms of its scale or multiresolution analysis, the numerical solution of conservation laws becomes more efficient by eliminating flux computations wherever the solution is smooth. Instead, in those locations, the divergence of the solution is interpolated from the next coarser grid level. First, the basic 2-D essentially nonoscillatory (ENO) scheme is presented, then the 2-D multiresolution analysis is developed, and finally the subsequent scheme is tested numerically. The computational results confirm that the efficiency of the numerical scheme can be considerably improved in two dimensions as well.

Generalized sampling theorems in multiresolution subspaces

Abstract: It is well known that under very mild conditions on the scaling function, multiresolution subspaces are reproducing kernel Hilbert spaces (RKHSs). This allows for the development of a sampling theory. In this paper, we extend the existing sampling theory for wavelet subspaces in several directions. We consider periodically nonuniform sampling, sampling of a function and its derivatives, oversampling, multiband sampling, and reconstruction from local averages. All these problems are treated in a unified way using the perfect reconstruction (PR) filter bank theory. We give conditions for stable reconstructions in each of these cases. Sampling theorems developed in the past do not allow the scaling function and the synthesizing function to be both compactly supported, except in trivial cases. This restriction no longer applies for the generalizations we study here, due to the existence of FIR PR banks. In fact, with nonuniform sampling, oversampling, and reconstruction from local averages, we can guarantee compactly supported synthesizing functions. Moreover, local averaging schemes have additional nice properties (robustness to the input noise and compression capabilities). We also show that some of the proposed methods can be used for efficient computation of inner products in multiresolution analysis. After this, we extend the sampling theory to random processes. We require autocorrelation functions to belong to some subspace related to wavelet subspaces. It turns out that we cannot recover random processes themselves (unless they are bandlimited) but only their power spectral density functions. We consider both uniform and nonuniform sampling.

Multiresolution reproducing kernel particle methods

Abstract: Reproducing Kernel Particle Methods (RKPM) with a built-in feature of multiresolution analysis are addressed. Some fundamental concepts such as reproducing conditions, and correction function are constructed to systematize the framework of RKPM. In particular, Fourier analysis, as a tool, is exploited to further elaborate RKPM in the frequency domain. Furthermore, we address error estimation and convergence properties. We present several applications which confirm the widespread applicability of multiresolution RKPM.

Multiresolution statistical analysis of high-resolution digital mammograms

Abstract: A multiresolution statistical method for identifying clinically normal tissue in digitized mammograms is used to construct an algorithm for separating normal regions from potentially abnormal regions; that is, small regions that may contain isolated calcifications. This is the initial phase of the development of a general method for the automatic recognition of normal mammograms. The first step is to decompose the image with a wavelet expansion that yields a sum of independent images, each containing different levels of image detail. When calcifications are present, there is strong empirical evidence that only some of the image components are necessary for the purpose of detecting a deviation from normal. The underlying statistic for each of the selected expansion components can be modeled with a simple parametric probability distribution function. This function serves as an instrument for the development of a statistical test that allows for the recognition of normal tissue regions. The distribution function depends on only one parameter, and this parameter itself has an underlying statistical distribution. The values of this parameter define a summary statistic that can be used to set detection error rates. Once the summary statistic is determined, spatial filters that are matched to resolution are applied independently to each selected expansion image. Regions of the image that correlate with the normal statistical model are discarded and regions in disagreement (suspicious areas) are flagged. These results are combined to produce a detection output image consisting only of suspicious areas. This type of detection output is amenable to further processing that may ultimately lead to a fully automated algorithm for the identification of normal mammograms.

Automatic registration of satellite images

Abstract: Image registration is one of the basic image processing operations in remote sensing. With the increase in the number of images collected every day from different sensors, automated registration of multi-sensor/multi-spectral images has become an important issue. A wide range of registration techniques has been developed for many different types of applications and data. Given the diversity of the data, it is unlikely that a single registration scheme will work satisfactorily for all different applications. A possible solution is to integrate multiple registration algorithms into a rule-based artificial intelligence system, so that appropriate methods for any given set of multisensor data can be automatically selected. The objective of this paper is to present an automatic registration algorithm which has been developed at INPE. It uses a multiresolution analysis procedure based upon the wavelet transform. The procedure is completely automatic and relies on the grey level information content of the images and their local wavelet transform modulus maxima. The algorithm was tested on SPOT and TM images from forest, urban and agricultural areas. In all cases we obtained very encouraging results.

A multiresolution methodology for signal-level fusion and data assimilation with applications to remote sensing

Abstract: This paper covers the design of multiscale stochastic models that can be used to fuse measurements of a random field or random process provided at multiple resolutions. Such sensor fusion problems arise in a variety of contexts, including many problems in remote sensing and geophysics. An example, which is used in this paper as a vehicle to illustrate our methodology, is the estimation of variations in hydraulic conductivity as required for the characterization of groundwater flow. Such a problem is typical in that the phenomenon to be estimated cannot be measured at fine scales throughout the region of interest, but instead must be inferred from a combination of measurements of very different types, including point measurements of hydraulic conductivity at irregular collections of points and indirect measurements that provide only coarse and nonlocal information about the conductivity field. Fusion of such disparate and irregular measurement sets is a challenging problem, especially when one includes the objective of producing, in addition to estimates, statistics characterizing the errors in those estimates. In this paper, we show how modeling a random field at multiple resolutions allows for the natural fusion (or assimilation) of measurements that provide information of different types and at different resolutions. The key to our approach is to take advantage of the fast multiscale estimation algorithms that efficiently produce both estimates and error variances even for very large problems. The major innovation required in our case, however, is to extend the modeling of random fields within this framework to accommodate multiresolution measurements. In particular to take advantage of the fast algorithms that the models admit, we must be able to model each nonlocal measurement as the measurement of a single variable of the multiresolution model at some appropriate resolution and scale. We describe how this can be done and illustrate its effectiveness for an ill-posed inverse problem in groundwater hydrology.

Wavelet analysis and nonlinear dynamics in a nonextensive setting

Abstract: We undertake the study of signals originated in time-dependent nonlinear systems by recourse to a wavelet-based multiresolution analysis as adapted to a nonextensive (Tsallis) scenario. Diverse applications are discussed. It is shown that the Tsallis environment provides one with more detailed information than the conventional Shannon one.

Detection of epileptic events in electroencephalograms using wavelet analysis

Abstract: This study deals with the problem of identification of epileptic events in electroencephalograms using multiresolution wavelet analysis The following problems are analyzed: time localization and characterization of epileptiform events, and computational efficiency of the method The algorithm presented is based on a polynomial spline wavelet transform The multiresolution representation obtained from this wavelet transform and the corresponding digital filters derived allows time localization of epileptiform activity The proposed detector is based on the multiresolution energy function Electroencephalogram records from epileptic patients were analyzed, and results obtained are shown Some comparisons with other methods are given

A thresholding multiresolution block matching algorithm

Abstract: In this paper, we present a thresholding multiresolution block matching algorithm. Preventing blocks that satisfy a predefined accuracy criterion from further processing saves computation. In three experiments which have quite different motion complexities, the proposed algorithm outperforms the fastest existing multiresolution block matching algorithm. Specifically, it reduces the processing time ranging from 14% to 20%, while maintaining almost the same quality of the reconstructed image.

Multiresolution Reproducing Kernel Particle Methods in Acoustics

Abstract: In the analysis of complex phenomena of acoustic systems, the computational modeling requires special attention in order to give a realistic representation of the physics. As a powerful tool, the finite element method has been widely used in the study of complex systems. In order to capture the important physical phenomena, p-finite elements and/or hp-finite elements are employed. The Reproducing Kernel Particle Methods (RKPM) are emerging as an effective alternative due to the absence of a mesh and the ability to analyze a specific frequency range. In this study, a wavelet particle method based on the multiresolution analysis encountered in signal processing has been developed. The interpolation functions consist of spline functions with a built-in window which permits translation as well as dilation. A variation in the size of the window implies a geometrical refinement and allows the filtering of the desired frequency range. An adaptivity similar to hp-finite element method is obtained through the choice of an optimal dilation parameter. The analysis of the wave equation shows the effectiveness of this approach. The frequency/wave number relationship of the continuum case can be closely simulated by using the reproducing kernel particle methods. A similar methodology is also developed for the Timoshenko beam.

A multiresolution approach to discrimination in SAR imagery

Abstract: We develop and test a new algorithm for discriminating man-made objects from natural clutter in synthetic-aperture radar (SAR) imagery. This algorithm exploits characteristic variations in speckle pattern as image resolution is varied from course to fine. We model these variations as an autoregression in scale, and then use the autoregressive model to define a multiresolution log-likelihood ratio discriminant. We incorporate this discriminant into the existing Lincoln Laboratory SAR system for automatic target recognition (ATR), and test the augmented system by applying it to millimeter-wave SAR imagery having 0.3 m resolution and representing 56 square kilometers of terrain. At a probability of detection of 0.95, the addition of the multiresolution discriminant reduces the number of natural-clutter false alarms by a factor of six.

Spline Pyramids for Inter-Modal Image Registration Using Mutual Information

Abstract: We propose a new optimizer for multiresolution image registration. It is adapted to a criterion known as mutual information and is well suited to inter-modality. Our iteration strategy is inspired by the Marquardt-Levenberg algorithm, even though the underlying problem is not least-squares. We develop a framework based on a continuous polynomial spline representation of images. Together with the use of Parzen histogram estimates, it allows for closedform expressions of the gradient and Hessian of the criterion. Tremendous simplifications result from the choice of Parzen windows satisfying the partition of unity, also based on B-splines. We use this framework to compute an image pyramid and to set our optimizer in a multiresolution context. We perform several experiments and show that it is particularly well adapted to a coarse-to-fine optimization strategy. We compare our approach to the popular Powell algorithm and conclude that our proposed optimizer is faster, at no cost in robustness or precision.

Haar wavelets over triangular domains with applications to multiresolution models for flow over a sphere

Abstract: Some new piecewise constant wavelets defined over nested triangulated domains are presented and applied to the problem of multiresolution analysis of flow over a spherical domain. These new, nearly orthogonal wavelets have advantages over the existing weaker biorthogonal wavelets. In the planar case of uniform areas, the wavelets converge to one of two fully orthogonal Haar wavelets. These new, fully orthogonal wavelets are proven to be the only possible wavelets of this type.

Multiresolution analysis in extraction of reference lines from documents with gray level background

Abstract: Based on wavelets, a theoretical method has been developed to process multi-gray level documents. In this method, two-dimensional multiresolution analysis, a wavelet decomposition algorithm, and compactly supported orthonormal wavelets are used to transform a document image into sub-images. According to these sub-images, the reference lines of a multi-gray level document can be extracted, and knowledge about the geometric structure of the document can be acquired. Particularly, this approach is more efficient to process form documents with gray level background. Experiments indicate that this new method can be applied to process documents with promising results.

Multiresolution detection of coherent radar targets

Abstract: We develop and investigate several novel multiresolution algorithms for detecting coherent radar targets embedded in clutter. These multiresolution detectors exploit the fact that prominent target scatterers interfere in a characteristic manner as resolution is changed, while multiresolution clutter signatures are random. We show, both on simulated and collected synthetic aperture radar data, that these multiresolution algorithms yield significant detection improvements over single-pixel, single-resolution constant false alarm rate (CFAR) methods that use only the finest available resolution.

Wavelet‐based density estimation and application to process monitoring

Abstract: An application of wavelets and multiresolution analysis to density estimation and process monitoring is presented. Wavelet-based density-estimation techniques are developed as an alternative and superior method to other common density-estimation techniques. Also shown is the effectiveness of wavelet estimators when the observations are dependent. The resulting density estimators are then used in defining a normal operating region for the process under study so that any abnormal behavior by the process can be monitored. Results of applying these techniques to a typical multivariate chemical process are also presented.

Nonlinear dynamics of accelerator via wavelet approach

Abstract: In this paper we present the applications of methods from wavelet analysis to polynomial approximations for a number of accelerator physics problems. In the general case we have the solution as a multiresolution expansion in the base of compactly supported wavelet basis. The solution is parametrized by the solutions of two reduced algebraical problems, one is nonlinear and the second is some linear problem, which is obtained from one of the next wavelet constructions: Fast Wavelet Transform, Stationary Subdivision Schemes, the method of Connection Coefficients. According to the orbit method and by using construction from the geometric quantization theory we construct the symplectic and Poisson structures associated with generalized wavelets by using metaplectic structure. We consider wavelet approach to the calculations of Melnikov functions in the theory of homoclinic chaos in perturbed Hamiltonian systems and for parametrization of Arnold-Weinstein curves in Floer variational approach.

Computation of optical flow using basis functions

Abstract: The issues governing the computation of optical flow in image sequences are addressed. The trade-off between accuracy versus computation cost is shown to be dependent on the redundancy of the image representation. This dependency is highlighted by reformulating Horn's (1986) algorithm, making explicit use of the approximations to the continuous basis functions underlying the discrete representation. The computation cost of estimating optical flow, for a fixed error tolerance, is shown to be a minimum for images resampled at twice the Nyquist rate. The issues of derivative calculation and multiresolution representation are also briefly discussed in terms of basis functions and information encoding. A multiresolution basis function formulation of Horn's algorithm is shown to lead to large improvements in dealing with high frequencies and large displacements.

Automatic registration of satellite imagery

Abstract: Image registration is one of the basic image processing operations in remote sensing. With the increase in the number of images collected every day from different sensors, automated registration of multi-sensor/multi-spectral images has become an important issue. A wide range of registration techniques has been developed for many different types of applications and data. The objective of this paper is to present an automatic registration algorithm which uses a multiresolution analysis procedure based upon the wavelet transform. The procedure is completely automatic and relies on the grey level information content of the images and their local wavelet transform modulus maxima. The registration algorithm is very simple and easy to apply because it needs basically one parameter. We have obtained very encouraging results on test data sets from the TM and SPOT sensor images of forest, urban and agricultural areas.

Approach to clustering large visual databases using wavelet transform

Abstract: Many applications demand the capability of retrieval based on image content. A classification mechanism is needed to categorize images based on feature similarity. An effective classification of the images can support efficient retrieval of images. In this paper, we investigate a feature-based approach to image clustering and retrieval. Four different texture-based feature sets of images are extracted using Haar and Daubechies wavelet transforms. Using multi- resolution property of wavelets, we extract the features at different levels. The experimental results of our clustering approach on air photo images are reported.© (1997) COPYRIGHT SPIE--The International Society for Optical Engineering. Downloading of the abstract is permitted for personal use only.

An introduction to discrete finite frames

Abstract: The frame concept was first introduced by Duffin and Schaeffer (1952), and it is widely used today to describe the behavior of vectors for signal representation. The Gabor (1946) expansion and wavelet transform are two special well-known cases. The goal of this article is to describe the frame theory and introduce a simple tutorial method to find discrete finite frame operators and their frame bounds. An easily implementable method for finding the discrete finite frame and subframe operators has been presented by Kaiser (1994). We introduce the method of Kaiser to compute the discrete finite frame operator. Using subframe operators, the biorthogonal basis and projection vectors in a subspace can be easily calculated. Gabor and wavelet analysis are two popular tools for signal processing, and they can reveal time-frequency distribution for a nonstationary signal. Both schemes can be regarded as signal decompositions onto a set of basis functions, and their basis functions are derived from a single prototype function through simple operations. Therefore, the basis functions used in Gabor and wavelet analysis can be regarded as special frames. For completeness we also make some simple introductions on the results of special frames such as discrete Gabor and wavelet analysis.

Wavelets for biomedical signal processing

Abstract: In this paper, we will discuss the general idea of the wavelet representation, in its continuous and discrete versions, as well as in terms of a multiresolution approximation. In addition, the general expression for the affine class, and the relationship between the affine and Cohen's classes are presented. Also, the shift-scale invariant class is defined. This class basically combines the properties of both classes. Finally a recent development, namely, the use of unitary transformations in both Cohen's and the affine classes, with the consequent generation of even more specific tools for signal analysis will be discussed.