Multiresolution Analysis

In this paper, we describe an intelligent signal analysis system employing the wavelet transformation towards solving vehicle engine diagnosis problems. Vehicle engine diagnosis often involves multiple signal analysis. The developed system first partitions a leading signal into small segments representing physical events or stateds based on wavelet mutli-resolution analysis. Second, by applying the segmentation result of the leading signal to the other signals, the detailed properties of each segment, including inter-signal relationships, are extracted to form a feature vector. Finally a fuzzy intelligent system is used to learn diagnostic features from a training set containing feature vectors extracted from signal segments at various vehicle states. The fuzzy system applies its diagnostic knowledge to classify signals as abnormal or normal. The implementation of the system is described and experiment results are presented.

In this paper, we present an efficient coarse-tofine multiresolution framework for multidimensional scaling and demonstrate its performance on a large-scale nonlinear dimensionality reduction and embedding problem in a texture feature extraction step for the unsupervised image segmentation problem. We demonstrate both the efficiency of our multiresolution algorithm and its real interest to learn a nonlinear low-dimensional representation of the texture feature set of an image which can then subsequently be exploited in a simple clustering-based segmentation algorithm. The resulting segmentation procedure has been successfully applied on the Berkeley image database, demonstrating its efficiency compared to the best existing state-ofthe-art segmentation methods recently proposed in the literature.

This paper relies on wavelet multiresolution analysis to capture the dependence structure of currency markets and reveal the complex dynamics across different timescales. It investigates the nature and direction of causal relationships among the most widely traded currencies denoted relative to the United States Dollar (USD), namely Euro (EUR), Great Britain Pound (GBP) and Japanese Yen (JPY). The timescale analysis involves the estimation of linear vis-à-vis nonlinear and spectral causality of wavelet components and aggregate series as well as the detection of short-vs. long-run linkages and cross-scale correlations. Moreover, this study attempts to probe into the micro-foundations of across-scale heterogeneity in the causality pattern on the basis of trader behavior with different time horizons. New stylized properties emerge in the volatility structure and the implications for the flow of information across scales are inferred. The examined period starts from the introduction of the Euro and covers the dot-com bubble, the financial crisis of 2007-2010 and the Eurozone debt crisis. Technically, this paper presents an invariant discrete wavelet transform that deals efficiently with phase shifts, dyadic-length and boundary effects. It also proposes a new entropy-based methodology for the determination of the optimal decomposition level. Overall, there is no indication of a global causal behavior that dominates at all timescales. When the nonlinear effects are accounted for, the evidence of dynamical bidirectional causality implies that the pattern of leads and lags changes over time. These results may prove useful to quantify the process of integration as well as influence the greater predictability of currency markets.

Abstract—Recently, digital image inpainting has attracted strong research interest because of its extensive applications in real life. The terminology ”inpainting ” refers to automatic restoration of image defects such as scratches or blotches as well as removal of unwanted objects as, for instance, subtitles, logos, etc, such that it is undetectable by viewers without the reference to the original image. Many works on this subject have been pub-lished in recent years. This paper introduces a novel unsupervised image completion framework using a modified exemplar-based method in conjunction with a pyramidal representation of an image. A top-down iterative completion is performed gradually with multi-resolution patches and a window-based priority. The proposed approach is verified on different natural images. Also, a comparison with some existing methods coming from literature is carried out and the results show improvement in favor of our approach. Keywords—inpainting, image complet...

The objective of image fusion is to combine information from multiple images of the same scene. The result of image fusion is a new image which is more suitable for human and machine perception or further image-processing tasks such as segmentation, feature extraction and object recognition. Di erent fusion methods have been proposed in literature, including multiresolution analysis. This paper is an image fusion tutorial based on wavelet decomposition, i.e. a multiresolution image fusion approach. We can fuse images with the same or di erent resolution level, i.e. range sensing, visual CCD, infrared, thermal or medical. The tutorial performs a synthesis between the multiscale-decomposition-based image approach (Proc. IEEE 87 (1999) 1315), the ARSIS concept (Photogramm. Eng. Remote Sensing 66 (1) (2000) 49) and a multisensor scheme (Graphical Models Image Process. 57 (3) (1995) 235). Some image fusion examples illustrate the proposed fusion approach. A comparative analysis is carried out against classical existing strategies, including those of multiresolution.

The wavelet transform is an efficient technique for multi-resolution analysis of non-stationary and fast transient signals. For this reason, wavelet transform has been widely applied in signal analysis through processing, encoding, denoising and encrypting. The objective of this paper is to represent the development process of VHSIC (Very High Speed Integrated

Recently there has been significant development in the use of wavelet methods in various Data Mining processes. This article presents general overview of their applications in Data Mining. It first presents a high-level data-mining framework in which the overall process is divided into smaller components. It reviews a p plications of wavelets for each component. It discusses the impact of wavelets on Data Mining research and outlines potential future research directions and applications.

The paper deals with the problem of pointing out and mark the various kind of defects (mainly scars, wrinkles and fire brandings) of tanned calf leathers used in many industrial areas (e.g. household couches and car seats lining ones). In the first experiments here presented the authors try to use the same inspection method previously successfully applied to quality control

In recent years, the wavelet transform on the basis of multiresolution analysis has been considered as a concrete algorithm of an integral wavelet transform which performs a spectral analysis in time-frequency domain. On the other hand, authors have proposed a new analyzing method which updates the spectrum by using an input data only. The new method proposes analysis approach 1(AA1) which is a fundamental analysis method, and analysis approach 2(AA2) which is an analysis method that achieves high-speed processing. Decomposition and reconstruction of a signal based on AA1 was already proposed by the authors. This paper shows decomposition and reconstruction of a signal based on AA2. The decomposition wave in AA2 can not be obtained by the sama method as in the case of AA1 but by a mathematical processing. By the processing, the decomposition wave in AA2 is identical to the decomposition wave in AA1. Therefore, the decomposition and the reconstruction of the signal in AA2 result in the decomposition and the reconstruction of the signal in AA1. As an example of a numerical analysis, we show decomposition and reconstruction of Japanease vocal sound /i/.

This paper proposes an edge-directed interpolation algorithm for natural images. The basic idea is to first estimate local covariance coefficients from a low-resolution image and then use these covariance estimates to adapt the interpolation at a higher resolution based on the geometric duality between the low-resolution covariance and the high-resolution covariance. The edge-directed property of covariance-based adaptation attributes to its capability of tuning the interpolation coefficients to match an arbitrarily oriented step edge. A hybrid approach of switching between bilinear interpolation and covariance-based adaptive interpolation is proposed to reduce the overall computational complexity. Two important applications of the new interpolation algorithm are studied: resolution enhancement of grayscale images and reconstruction of color images from CCD samples. Simulation results demonstrate that our new interpolation algorithm substantially improves the subjective quality of the interpolated images over conventional linear interpolation.

In this paper, a novel multi-resolution filter for on-line decomposition and analysis of partial discharge (PD) signals based on discrete wavelet transform (DWT) is developed. By applying this filter, as soon as a new sample of the original signal is drawn, the new elements of detail and approximate components at any desired level is derived, and the successive procedure of signal processing in basic DWT is eliminated. Using this filter, the PD or any non-stationary (transient) signal is decomposed to different frequency band components, simultaneously. Therefore, the decomposition components of PD signals in all frequency bandwidths are simultaneously available. The frequency characteristic of this filter is verified, then in order to make the proposed filter more suitable for on-line applications, its order is reduced without losing the accuracy of the results. Results, for simulation and field test data, have shown the appropriateness of the proposed filter for on-line PD detection and measurement.

Multiresolution cross-analysis of rainfall rates and karstic spring runoffs. In order to quantify the quality of the rainfall/discharge relationship across time-scales, we propose the use of both orthogonal wavelet multiresolution analysis and cross-correlation analysis. By using the two techniques together, it is possible to show, scale-by-scale, the influence of the input to the system (rainfall) on the response (discharge) of the aquifer and also to relate these results to the internal structure of the aquifer and to the degree of organisation of the karst drainage. An application of this method to three Pyrenean karsts is also shown. To cite this article: D. Labat et al., C. R. Geoscience 334 (2002) 551-556.  2002 Académie des sciences / Éditions scientifiques et médicales Elsevier SAS karstic systems / rainfall / discharge / multiresolution wavelet / cross-correlation analyses * Correspondance et tirés à part.

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 defined on ‫ޒ‬ 2 , 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-ofdetail control for spherical domains is illustrated using two examples: shape approximation of a polyhedral model, and color approximation of global terrain data.

This paper proposes an edge-directed interpolation algorithm for natural images. The basic idea is to first estimate local covariance coefficients from a low-resolution image and then use these covariance estimates to adapt the interpolation at a higher resolution based on the geometric duality between the low-resolution covariance and the high-resolution covariance. The edge-directed property of covariance-based adaptation attributes to its capability of tuning the interpolation coefficients to match an arbitrarily oriented step edge. A hybrid approach of switching between bilinear interpolation and covariance-based adaptive interpolation is proposed to reduce the overall computational complexity. Two important applications of the new interpolation algorithm are studied: resolution enhancement of grayscale images and reconstruction of color images from CCD samples. Simulation results demonstrate that our new interpolation algorithm substantially improves the subjective quality of the interpolated images over conventional linear interpolation.

Wavelet analysis, although used extensively in disciplines such as signal processing, engineering, medical sciences, physics and astronomy, has not fully entered the economics discipline yet. In this survey article, wavelet analysis is introduced in an intuitive manner, and the existing economics and finance literature that utilizes wavelets is surveyed and explored. Extensive examples of exploratory wavelet analysis are given, most using Canadian, US and Finnish industrial production data. Finally, potential and possible future applications for wavelet analysis in economics are discussed.

Human visual system makes an extensive use of visual attention in order to select the most relevant information and speed-up the vision process. Inspired by visual attention, several computer models have been developped and many computer vision applications rely today on such models. However, the actual algorithms are not suitable to omnidirectional images, which contain a significant amount of geometrical distorsion. In this paper, we present a novel computational approach that performs in spherical geometry and thus is suitable for omnidirectional images. Following one of the actual models of visual attention, the spherical saliency map is obtained by fusing together intensity, chromatic, and orientation spherical cue conspicuity maps that are themselves obtained through multiscale analysis on the sphere. Finally, the consecutive maxima in the spherical saliency map represent the spots of attention on the sphere. In the experimental part, the proposed method is then compared to the standard one using a synthetic image. Also, we provide examples of spots detection in real omnidirectional scenes which show its advantages. Finally, an experiment illustrates the homogeneity of the detected visual attention in omnidirectional images.

This paper presents a comprehensive physical characterization and modeling of the three-phase common-mode (CM) inductors along with the equivalent circuits that are relevant for their design. Modeling issues that are treated sparsely in previous literature are explained in this paper, and novel insightful aspects are presented. The calculation of the leakage inductance is reviewed, along with the magnetic core saturation issues, and a new expression for the leakage flux path is derived. The influence of the core material characteristics on the performance of the component is discussed, and a new method for the selection of the material for the minimized volume CM inductors is proposed in order to simplify the design procedure. Experimental results which validate the model are presented.

This paper primarily concerns the problem of isolated handwritten numeral recognition of major Indian scripts. The principal contributions presented here are 1) pioneering development of two databases for handwritten numerals of the two most popular Indian scripts, 2) a multistage cascaded recognition scheme using wavelet-based multiresolution representations and multilayer perceptron (MLP) classifiers, and 3) application of 2 for the recognition of mixed handwritten numerals of three Indian scripts-Devanagari, Bangla, and English. The present databases include, respectively, 22,556 and 23,392 handwritten isolated numeral samples of Devanagari and Bangla collected from real-life situations, and these can be made available free of cost to researchers of other academic institutions. In the proposed scheme, a numeral is subjected to three MLP classifiers corresponding to three coarse-to-fine resolution levels in a cascaded manner. If rejection occurs even at the highest resolution, another MLP is used as the final attempt to recognize the input numeral by combining the outputs of three classifiers of the previous stages. This scheme has been extended to the situation when the script of a document is not known a priori or the numerals written on a document belong to different scripts. Handwritten numerals in mixed scripts are frequently found in Indian postal mail and tabular form documents.

This paper presents a wavelet norm entropy-based effective feature extraction method for power quality (PQ) disturbance classification problem. The disturbance classification schema is performed with waveletneural network (WNN). It performs a feature extraction and a classification algorithm composed of a wavelet feature extractor based on norm entropy and a classifier based on a multi-layer perceptron. The PQ signals used in this study are seven types. The performance of this classifier is evaluated by using total 2800 PQ disturbance signals which are generated the based model. The classification performance of different wavelet family for the proposed algorithm is tested. Sensitivity of WNN under different noise conditions which are different levels of noises with the signal to noise ratio is investigated. The rate of average correct classification is about 92.5% for the different PQ disturbance signals under noise conditions.

The objective of image fusion is to combine information from multiple images of the same scene. The result of image fusion is a new image which is more suitable for human and machine perception or further image-processing tasks such as segmentation, feature extraction and object recognition. Di erent fusion methods have been proposed in literature, including multiresolution analysis. This paper is an image fusion tutorial based on wavelet decomposition, i.e. a multiresolution image fusion approach. We can fuse images with the same or di erent resolution level, i.e. range sensing, visual CCD, infrared, thermal or medical. The tutorial performs a synthesis between the multiscale-decomposition-based image approach (Proc. IEEE 87 (1999) 1315), the ARSIS concept (Photogramm. Eng. Remote Sensing 66 (1) (2000) 49) and a multisensor scheme (Graphical Models Image Process. 57 (3) (1995) 235). Some image fusion examples illustrate the proposed fusion approach. A comparative analysis is carried out against classical existing strategies, including those of multiresolution.

This paper relies on wavelet multiresolution analysis to capture the dependence structure of currency markets and reveal the complex dynamics across different timescales. It investigates the nature and direction of causal relationships among the most widely traded currencies denoted relative to the United States Dollar (USD), namely Euro (EUR), Great Britain Pound (GBP) and Japanese Yen (JPY). The timescale analysis

Detection and diagnosis of faults in a large-scale distributed system is a formidable task. Interest in monitoring and using traces of user requests for fault detection has been on the rise recently. In this paper we propose novel fault detection methods based on abnormal trace detection. One essential problem is how to represent the large amount of training trace data compactly as an oracle. Our key contribution is the novel use of varied-length n-grams and automata to characterize normal traces. A new trace is compared against the learned automata to determine whether it is abnormal. We develop algorithms to automatically extract n-grams and construct multi-resolution automata from training data. Further both deterministic and multihypothesis algorithms are proposed for detection. We inspect the trace constraints of real application software and verify the existence of long n-grams. Our approach is tested in a real system with injected faults and achieves good results in experiments.

Wavelets are relatively new mathematical tools that have proven to be quite useful for analyzing time series and spatial data. We provide a basic introduction to wavelet analysis that concentrates on their interpretation in the context of analyzing time series.

O Eletrocardiograma (ECG) é um exame de extrema importância para avaliar a saúde
do coração de um indivíduo. A análise de seu traçado possibilita diagnosticar uma série
de doenças e anomalias. Para isso, sistemas computacionais precisam segmentar os
sinais de ECG, cujo primeiro passo é a detecção dos complexos QRS (ondas Q, R e S).
Muitos métodos têm sido propostos para tal finalidade. Os primeiros utilizavam técnicas
de filtragem e diferenciação, mas recentemente alguns pesquisadores têm concebido
metodologias baseadas na análise de multirresolução da transformada Wavelet. Seguindo
caminho semelhante, neste trabalho é proposta uma metodologia baseada na decomposição
em valores singulares em multirresolução (MRSVD), para solução do problema de
detecção dos complexos QRS de um sinal de ECG, que pode ser dividida em quatro etapas
essenciais: pré-processamento, inspeção, decisão e correção. Em quase todas elas há implementações
inovadoras: na primeira utiliza-se, de forma inédita, a MRSVD; na segunda
etapa exploram-se características fisiológicas do músculo cardíaco para certificar-se de que
um determinado segmento compreende ou não uma onda R autêntica; na última, são corrigidas
as detecções baseadas nas amplitudes das ondas R. A avaliação da qualidade de
detecção, para os experimentos realizados, mostra que o algoritmo proposto atingiu taxa
de previsibilidade de 99, 20%, taxa de sensibilidade de 99, 70% e erro de apenas 1, 10%,
contra 97, 10%, 99, 00% e 3, 70%, respectivamente, obtidas por outro método.

The main purpose of the thesis is the systematic study of quadrature mirror filters (QMF ) and their parametrization in a simple way, adapted to building implementable wavelet bases with a variety of properties. Using the framework given by Multiresolution Analysis, we will focus on bases for L 2 (R) and realizable infinite impulse response QMF with real coefficients, which allow implementations (decomposition/reconstruction of a function in the basis) based on very efficient algorithms. However, the structural representations that we will describe are valid for more general functions and can be applied not only to build new bases but also to recover all previous constructions of wavelet bases for L 2 (R) with either rational or non-rational filters. In particular we will give a different presentation of such constructions as those of Meyer, Daubechies, Battle-Lemarié, and Strömberg. This different way of looking at them will give sometimes new properties, sometimes new descriptions for these known bases.

Abstract: We present a MR image segmentation algorithm based on the conventional Expectation Maximization (EM) algorithm and the multiresolution analysis of images. Although the EM algorithm was used in MRI brain segmentation, as well as, image segmentation in general, it fails to utilize the strong spatial correlation between neighboring pixels. The multiresolution-based image segmentation techniques, which have emerged as a powerful method for producing high-quality segmentation of images, are combined here ...

Partial discharge (PD) measurement is a promising method for condition assessment of insulation systems in high voltage equipments. Conventional PD measuring systems have some difficulties in the measurements especially in online conditions and noisy environments. There is a growing tendency to use new measuring systems for separation of PD pulses from different sources and rejection of external noises. In this investigation a new ultra wideband PD measuring system and relevant methodology are discussed. The proposed system uses an efficient detection method for extracting PD pulses embedded in noisy signal, meanwhile, a wavelet multiresolution analysis is applied to reduce background and communication noise level. In this paper some new and modified features for pulse classification procedure have been proposed. The proposed system, methodology and features have been implemented on a hydrogenerator and also a power transformer. Finally performance of the system is discussed.

A prevision of solar activity minimum and maximum epochs for the next 1000 years was obtained using a sum of sine waves derived from spectral analysis of the time series of 10 years average sunspot number for the period 1016-2006. The predicted 1000 year solar activity evidence the possibility of forthcoming four minima and one maximum. The 10 years average sunspot number activity in each anomalous period was used in linear equations to obtain estimates of the solar irradiance, solar wind velocity, and the southward component of the interplanetary magnetic field.

Synthesis Lectures on Computational Electromagnetics will publish 50-to 100-page publications on topics that include advanced and state-of-the-art methods for modeling complex and practical electromagnetic boundary value problems. Each lecture develops, in a unified manner, the method based on Maxwell's equations along with the boundary conditions and other auxiliary relations, extends underlying concepts needed for sequential material, and progresses to more advanced techniques and modeling. Computer software, when appropriate and available, is included for computation, visualization and design. The authors selected to write the lectures are leading experts on the subject that have extensive background in the theory, numerical techniques, modeling, computations and software development. The series is designed to: • Develop computational methods to solve complex and practical electromagnetic boundary-value problems of the 21st century. • Meet the demands of a new era in information delivery for engineers, scientists, technologists and engineering managers in the fields of wireless communication, radiation, propagation, communication, navigation, radar, RF systems, remote sensing, and biotechnology who require a better understanding and application of the analytical, numerical and computational methods for electromagnetics.

Diffraction and holography are fertile areas for application of signal theory and processing. Recent work on 3DTV displays has posed particularly challenging signal processing problems. Various procedures to compute Rayleigh-Sommerfeld, Fresnel and Fraunhofer diffraction exist in the literature. Diffraction between parallel planes and tilted planes can be efficiently computed. Discretization and quantization of diffraction fields yield interesting theoretical and practical results, and allow efficient schemes compared to commonly used Nyquist sampling. The literature on computer-generated holography provides a good resource for holographic 3DTV related issues. Fast algorithms to compute Fourier, Walsh-Hadamard, fractional Fourier, linear canonical, Fresnel, and wavelet transforms, as well as optimization-based techniques such as best orthogonal basis, matching pursuit, basis pursuit etc., are especially relevant signal processing techniques for wave propagation, diffraction, holography, and related problems. Atomic decompositions, multiresolution techniques, Gabor functions, and Wigner distributions are among the signal processing techniques which have or may be applied to problems in optics. Research aimed at solving such problems at the intersection of wave optics and signal processing promises not only to facilitate the development of 3DTV systems, but also to contribute to fundamental advances in optics and signal processing theory.

We present a notion of frame multiresolution analysis on the Heisenberg group, abbreviated by FMRA, and study its properties. Using the irreducible representations of this group, we shall define a sinc-type function which is our starting point for obtaining the scaling function. Further, we shall give a concrete example of a wavelet FMRA on the Heisenberg group which is analogous to the Shannon MRA on R.

Given a sample autocovariance sequence of finite length for some observed random process, the spectrum estimation problem involves the extension of this sequence for the required Fourier transformation. The maximum entropy approach which is based on the optimal use of information contents, leads to a dual sequence of reflection coefficients with reciprocal spectrum of the process. The estimation of the maximum entropy spectrum implies results identical to those using autoregressive modeling in one dimension under appropriate white noise assumptions. In cases of a non-white noise component, the approach is generalized to an autoregressive-moving-average model. Recent developments in multiresolution analysis with spectral domain decompositions also offer possibilities of subband spectrum estimation for specific applications. Using a simulated data sequence with two close frequencies, the estimated spectrum from a two-level decomposition with autoregressive modeling shows better resolution than with conventional processing. Geodetic and geophysical applications are briefly indicated.

This paper proposes an edge-directed interpolation algorithm for natural images. The basic idea is to first estimate local covariance coefficients from a low-resolution image and then use these covariance estimates to adapt the interpolation at a higher resolution based on the geometric duality between the low-resolution covariance and the high-resolution covariance. The edge-directed property of covariance-based adaptation attributes to its capability of tuning the interpolation coefficients to match an arbitrarily oriented step edge. A hybrid approach of switching between bilinear interpolation and covariance-based adaptive interpolation is proposed to reduce the overall computational complexity. Two important applications of the new interpolation algorithm are studied: resolution enhancement of grayscale images and reconstruction of color images from CCD samples. Simulation results demonstrate that our new interpolation algorithm substantially improves the subjective quality of the interpolated images over conventional linear interpolation.

Autonomous Underwater Vehicles (AUVs) usually operate in ocean environments characterized by complex spatial variability which can jeopardize their missions. To avoid this, planning safety routes with minimum energy cost is of primary importance. This work revisits the benefits, in terms of travelling time, of path planning in marine environments showing spatial variability. By means of a path planner presented in a previous paper, this work focuses on the application to a real environment of such techniques. Extensive computations have been carried out to calculate optimal paths on realistic ocean environments, based on autonomous underwater glider properties as the mobile platform.

This paper suggests the use of wavelet multiresolution analysis (WMRA) as a reliable tool for digital signal processing in structural health monitoring (SHM) systems. A damage occurrence detection algorithm using WMRA augmented with artificial neural networks (ANN) is described. The suggested algorithm allows intelligent monitoring of structures in real time. The probability of damage occurrence is determined by evaluating the wavelet norm index (WNI) representing the energy of a signal describing the change in the system dynamics due to damage. An example application of the proposed algorithm is presented using a finite element simulated structural dynamics of a prestressed concrete bridge. The new algorithm showed very promising results.

We apply a methodology developed by García-Fernández and Palacios-González (2008a,b) to the measurement of income polarization using Israeli data over the past decade. During this period the Israeli economy experienced sharp economic fluctuations, from rapid growth to a severe recession and back to rapid growth during four and a half years. During and at the end of the recession a deep and permanent cut in social benefits was carried out, accompanied by small scale proactive labor market policy. The largely export-led growth period thereafter was mainly concentrated in hi-tech industries, to the advantage of the high-skilled labor force. Such a development is expected to have a destabilizing effect on social stability as captured in polarization measures. Nevertheless, the values of the suggested measure indicate that polarization fluctuated around a horizontal trend during the period considered.

In this paper, split step wavelet Galerkin method (SSWGM)is proposed for solving underwater wave propagation in range-independent fluid/solid media. Parabolic equation model is applied for transforming elliptic wave to parabolic wave equation that enable us to use marching approaches in numerical algorithms. Wavelet Galerkin method is used to discretize the depth operators by using 1-periodic Daubechies scaling basis as shape functions. This discretization leads to circulant and bandlimit system which can be solved by fast Fourier transformations and this improves the accuracy and cost of computation. The numerical solution of SSWGM is applied for deep and shallow water environment involving water column over bottom.

Abstract}In this work extensions to principal component analysis (PCA) for wastewater treatment (WWT) process monitoring are discussed. Conventional PCA has some limitations when used for WWT monitoring. Firstly, PCA assumes that data are stationary, which is normally not the case in WWT monitoring. Secondly, PCA is most suitable for monitoring data that display events in one time-scale. However, in WWT operation, disturbances and events occur in different time-scales. These two limitations make conventional PCA unsuitable for WWT monitoring. The first limitation can be overcome by use of adaptive PCA. In adaptive PCA, the PCA model is continuously updated using an exponential memory function. Variable mean, variance and co-variance are thus adapted to the changing conditions. The second problem can be solved by time-scale decomposition of data prior to analysis. The time-scale decomposition methodology involves wavelets and multiresolution analysis (MRA) in combination with PCA. MRA provides a tool for investigation and monitoring of process measurement at different timescales by decomposing measurement data into separate frequency bands. Time-scale decomposition increases the sensitivity of the monitoring, which makes it possible to detect small but significant events in data displaying large variations. Moreover, time-scale information is sometimes important in the interpretation of a disturbance to determine its physical cause. Also, by decomposing data, the problem of changing process conditions is partly solved. All the presented methods are illustrated with examples using real WWT process data. #

In this paper, we propose a new wavelet-based reconstruction method suited to three-dimensional (3-D) cone-beam (CB) tomography. It is derived from the Feldkamp algorithm and is valid for the same geometrical conditions. The demonstration is done in the framework of nonseparable wavelets and requires ideally radial wavelets. The proposed inversion formula yields to a filtered backprojection algorithm but the filtering step is implemented using quincunx wavelet filters. The proposed algorithm reconstructs slice by slice both the wavelet and approximation coefficients of the 3-D image directly from the CB projection data.