Multidimensional Wavelet Research Articles

Layer-based sparse representation of multiview images

This article presents a novel method to obtain a sparse representation of multiview images. The method is based on the fact that multiview data is composed of epipolar-plane image lines which are highly redundant. We extend this principle to obtain the layer-based representation, which partitions a multiview image dataset into redundant regions (which we call layers) each related to a constant depth in the observed scene. The layers are extracted using a general segmentation framework which takes into account the camera setup and occlusion constraints. To obtain a sparse representation, the extracted layers are further decomposed using a multidimensional discrete wavelet transform (DWT), first across the view domain followed by a two-dimensional (2D) DWT applied to the image dimensions. We modify the viewpoint DWT to take into account occlusions and scene depth variations. Simulation results based on nonlinear approximation show that the sparsity of our representation is superior to the multi-dimensional DWT without disparity compensation. In addition we demonstrate that the constant depth model of the representation can be used to synthesise novel viewpoints for immersive viewing applications and also de-noise multiview images.

Wavelets in functional data analysis: Estimation of multidimensional curves and their derivatives

A wavelet-based method is proposed to obtain accurate estimates of curves in more than one dimension and of their derivatives. By means of simulation studies, this novel method is compared to another locally-adaptive estimation technique for multidimensional functional data, based on free-knot regression splines. This comparison shows that the proposed method is particularly attractive when the curves to be estimated present strongly localized features. The multidimensional wavelet estimation method is thus applied to multi-lead electrocardiogram records, where strongly localized features are indeed expected.

The Traits of Two-Directional Orthogonal Multivariate Small-Wave Packages with Finite Support

In this paper, we introduce the notion of vector-valued multiresolution analysis and two-directional vector-valued multivariate wavelet packages associated with an integer-valued dilation matrix. A novel method for constructing multi-dimensional vector-valued wavelet packet is presented. Their characteristics are researched by means of operator theory, time-frequency analysis method and matrix theory. Three orthogonality formulas concerning the wavelet packets are establ- lished. Orthogonality decomposition relation formulas of the space are derived by constru- cting a series of subspaces of wavelet packets.

Wavelets with Crystal Symmetry Shifts

A generalization of multi-dimensional wavelet theory is introduced in which the usual lattice of translational shifts is replaced by a discrete subgroup of the group of affine, area preserving, transformations of Euclidean space. The dilation matrix must now be compatible with the group of shifts. An existence theorem for a multiwavelet in the presence of a multiresolution analysis is established and examples are given to illustrate the theory with two dimensional crystal symmetry groups as shifts.

The Characterization of Biorthogonal Multidimensional Vector Wavelet Wraps with Nine-Scale Dilation Factor

Wavelet analysis is nowadays a widely used tool in applied mathematics. The advantages of wavelet wraps and their promising features in various application have attracted a lot of interest and effort in recent years. In this paper, we develop the notion of a sort of multivariate vector wavelet wraps. A new approcah for designing the multidimensional vector wavelet wraps is formul- ated. Their characters are investigated by virtue of iterative method, time-frequency analysis metho- d and matrix theory. There biorthogonality formulas regarding the wavelet wraps are provided. Biorthogonality decomposition relation formulas of the space L2(Rn)r are obtained by constructing a series of subspaces of the vector-valued wavelet wraps. Moreover, several Riesz bases of space L2(Rn)r are constructed from the wavelet wraps.

Wavelet SVM in Reproducing Kernel Hilbert Space for hyperspectral remote sensing image classification

Combining Support Vector Machine (SVM) with wavelet analysis, we constructed wavelet SVM (WSVM) classifier based on wavelet kernel functions in Reproducing Kernel Hilbert Space (RKHS). In conventional kernel theory, SVM is faced with the bottleneck of kernel parameter selection which further results in time-consuming and low classification accuracy. The wavelet kernel in RKHS is a kind of multidimensional wavelet function that can approximate arbitrary nonlinear functions. Implications on semiparametric estimation are proposed in this paper. Airborne Operational Modular Imaging Spectrometer II (OMIS II) hyperspectral remote sensing image with 64 bands and Reflective Optics System Imaging Spectrometer (ROSIS) data with 115 bands were used to experiment the performance and accuracy of the proposed WSVM classifier. The experimental results indicate that the WSVM classifier can obtain the highest accuracy when using the Coiflet Kernel function in wavelet transform. In contrast with some traditional classifiers, including Spectral Angle Mapping (SAM) and Minimum Distance Classification (MDC), and SVM classifier using Radial Basis Function kernel, the proposed wavelet SVM classifier using the wavelet kernel function in Reproducing Kernel Hilbert Space is capable of improving classification accuracy obviously.

Cluster expansion and the configurational theory of alloys

A rigorous mathematical foundation for the cluster-expansion method is presented. It is shown that the cluster basis developed by Sanchez et al. [Physica A 128, 334 (1984)] is a multidimensional discrete Fourier transform while the general formalism of Sanchez [Phys. Rev. B 48, 14013 (1993)] corresponds to a multidimensional discrete wavelet transform. For functions that depend nonlinearly on the concentration, it is shown that the cluster basis corresponding to a multidimensional discrete Fourier transform does not converge, as it is usually assumed, to a finite cluster expansion or to an Ising-type model representation of the energy of formation of alloys. The multidimensional wavelet transform, based on a variable basis cluster expansion, is shown to provide a satisfactory solution to the deficiencies of the discrete Fourier-transform approach. Several examples aimed at illustrating the main findings and conclusions of this work are given.

The Biorthogonality Property of the Multidimensional Wavelet Packets with Multi-Scale Factor

In this article, we introduce a sort of vector-valued wavelet packets with multi-scale dilation of space , which are generalizations of multivariaale wavelet packets. A method for designing a sort of biorthogonal vector-valued wavelet packets in higher dimensions is presented and their biorthogonality property is characterized by virtue of time-frequency analysis method, matrix theory, and operator theory. Three biorthogonality formulas regarding these wavelet packets are established. Furtherore, it is shown how to obtain new Riesz bases of space from these wavelet packets.

The Properties of a Sort of Multidimensional Wavelet Packets According to an Integer-Valued Dilation Matrix

Wavelet analysis has become a popular subject in scientific research in the past twenty years. In this work, we develop the concept of a class of vector-valued multivarition matrix. A new method for constructing multidimensional vector-valued wavelet packets is formulated. Their characteristics are researched by means of operator theory, time-frequency analysis method and matrix theory. There orthogonality formulas regarding the wavelet packets are provided. Orthogonality decomposition relation formulas of the space L2(Rs)r are obtained by constructing a series of subspaces of the vector-valued wavelet packet. Furthermore, several orthonormal wavelet packet bases of space L2(Rs)r are constructed from the wavelet packets.

Adaptive Nonseparable Wavelet Transform via Lifting and its Application to Content-Based Image Retrieval

We present in this paper a novel way to adapt a multidimensional wavelet filter bank, based on the nonseparable lifting scheme framework, to any specific problem. It allows the design of filter banks with a desired number of degrees of freedom, while controlling the number of vanishing moments of the primal wavelet ((~)N moments) and of the dual wavelet ( N moments). The prediction and update filters, in the lifting scheme based filter banks, are defined as Neville filters of order (~)N and N, respectively. However, in order to introduce some degrees of freedom in the design, these filters are not defined as the simplest Neville filters. The proposed method is convenient: the same algorithm is used whatever the dimensionality of the signal, and whatever the lattice used. The method is applied to content-based image retrieval (CBIR): an image signature is derived from this new adaptive nonseparable wavelet transform. The method is evaluated on four image databases and compared to a similar CBIR system, based on an adaptive separable wavelet transform. The mean precision at five of the nonseparable wavelet based system is notably higher on three out of the four databases, and comparable on the other one. The proposed method also compares favorably with the dual-tree complex wavelet transform, an overcomplete nonseparable wavelet transform.

Noise reduction for enhanced component identification in multi-dimensional biomolecular NMR studies

The objective of the research presented in this paper is to shed light into the benefits of multi-dimensional wavelet-based methodology applied to NMR biomolecular data analysis. Specifically, the emphasis is on noise reduction for enhanced component identification in multi-dimensional mixture regression. The contributions of this research are multi-fold. First, the wavelet-based noise reduction method applies to multi-dimensional data whereas most of the existing work focuses on one- or two-dimensional data only. The proposed wavelet-based methodology is founded on rigorous analysis of the dependence between wavelet coefficients, an important aspect of multi-dimensional wavelet de-noising. The wavelet de-noising rule is based on Stein’s unbiased risk estimator (SURE) where the smoothness thresholds vary with the resolution level and orientation of the wavelet transform and selected by controlling the False Discovery Rate of the significant wavelet coefficients. Second, this paper highlights the application of the wavelet methodology to multi-dimensional NMR data analysis for protein structure determination. The noise reduction method is general and applicable to multi-dimensional data arising in many other research fields, prominently in biology science. Our empirical investigation shows that reducing the noise using the method in this paper results in more detectable true components and fewer false positives without altering the shape of the significant components.

Distributed detection and estimation based on the turbo approach.

We develop a centralized information fusion architecture from basic principles of information theory and Bayesian statistics. It is well known that any clustering, quantizing, or thresholding of data causes loss of information unless a sufficient statistic is computed in the processing. For the case of wideband active ranging systems, the coherent output of an optimum beamformer and a matched filter is a sufficient statistic that can be transmitted to the fusion center. For unknown target velocity, range, and bearing, the wideband space‐time matched filter output can be interpreted as a multidimensional wavelet transform or a delay‐scale‐bearing map. In this paper, an iterative, Bayesian, joint estimation‐detection approach is used for computation of sufficient statistics and multisensor information fusion. An approach borrowed from sequential Bayesian processing is used to compute prior densities for joint Bayesian estimation‐detection. In this approach a posteriori densities become priors after a coordinate transformation that transforms the outputs of each sensor to a common reference frame for all sensors. In this paper, receiver operating characteristics and Cramer–Rao lower bounds are given for several undersea signal processing cases of interest.

Classification of impulse fault response patterns in transformers using cascaded wavelet network

Accurate identification of winding faults that may develop during impulse testing of transformer is of great importance for ensuring reliability and quality of power supply. In the present work a wavelet network based methodology has been proposed for feature extraction as well as classification of those features for impulse fault pattern identification. The classification involves two networks, viz. a single dimensional wavelet network (SDWN) for feature extraction and a multi dimensional wavelet network (MDWN) for analysing the extracted features of the optimally trained SDWN to classify the fault type and location along the transformer winding. Results show that the proposed methodology can perform efficiently for classification of various types of impulse faults in transformers.

Sufficient Conditions for Irregular Wavelet Frames

Finding verifiable conditions for wavelet systems to be wavelet frames is among the core problems in wavelet analysis. In this paper, we give some simple and sufficient conditions that ensure a multidimensional irregular wavelet system to be a frame or a weighted frame. Quantitative results are provided, and explicit frame bounds are given.

Forecasting volatility based on wavelet support vector machine

One of the challenging problems in forecasting the conditional volatility of stock market returns is that general kernel functions in support vector machine (SVM) cannot capture the cluster feature of volatility accurately. While wavelet function yields features that describe of the volatility time series both at various locations and at varying time granularities, so this paper construct a multidimensional wavelet kernel function and prove it meeting the mercer condition to address this problem. The applicability and validity of wavelet support vector machine (WSVM) for volatility forecasting are confirmed through computer simulations and experiments on real-world stock data.

Image/video denoising based on a mixture of Laplace distributions with local parameters in multidimensional complex wavelet domain

Noise reduction in the wavelet domain can be expressed as an estimation problem in a Bayesian framework. So, the proposed distribution for the noise-free wavelet coefficients plays a key role in the performance of wavelet-based image/video denoising. This paper presents a new image/video denoising algorithm based on the modeling of wavelet coefficients in each subband with a mixture of Laplace probability density functions (pdfs) that uses local parameters for the mixture model. The mixture model is able to capture the heavy-tailed nature of wavelet coefficients and the local parameters model the intrascale dependency between the coefficients. Therefore, this relatively new pdf potentially can fit better the statistical properties of the wavelet coefficients than several other existing models. Within this framework, we describe a novel method for image/video denoising based on designing a maximum a posteriori (MAP) estimator, which relies on the mixture distributions for each wavelet coefficient in each subband. We employ two different versions of expectation maximization (EM) algorithm to find the parameters of mixture model and compare our new method with other image denoising techniques that are based on (1) non mixture pdfs that are not local, (2) non mixture pdfs with local variances, (3) mixture pdfs without local parameters and (4) methods that consider both heavy-tailed and locality properties. The simulation results show that our proposed technique is among the best reported in the literature both visually and in terms of peak signal-to-noise ratio (PSNR). In addition, we use the proposed algorithm for video denoising in multidimensional complex wavelet domain. Because 3-D complex wavelet transform provides a motion-based multiscale decomposition for video, we can see that our algorithm for video denoising, has very good performance without explicitly using motion estimation.

Modelling of Wastewater Treatment Plant for Monitoring and Control Purposes by State – Space Wavelet Networks

Most of industrial processes are nonlinear, not stationary, and dynamical with at least few different time scales in their internal dynamics and hardly measured states. A biological wastewater treatment plant falls into this category. The paper considers modelling such processes for monitorning and control purposes by using State - SpaceWavelet Neural Networks (SSWN). The modelling method is illustrated based on bioreactors of the wastewater treatment plant. The learning algorithms and basis function (multidimensional wavelets) are also proposed. The simulation results based on real data record are presented.

Lossless Information Fusion for Active Ranging and Detection Systems

The authors develop a centralized information fusion architecture from basic principles of information theory and Bayesian statistics. It is well known that any clustering, quantizing, or thresholding of data causes loss of information unless a sufficient statistic is computed in the processing. For the case of wideband active ranging systems, the coherent output of an optimum beamformer and a matched filter is a sufficient statistic that can be transmitted to the fusion center. For unknown target velocity, range, and bearing, the wideband space-time matched filter output can be interpreted as a multidimensional wavelet transform or a delay-scale-bearing map. In this paper, a Bayesian, joint estimation-detection approach is used for computation of sufficient statistics and multisensor information fusion. An approach borrowed from sequential Bayesian processing is used to compute prior densities for joint Bayesian estimation-detection. In this approach, a posteriori densities become priors after a coordinate transformation that transforms the outputs of each sensor to a common reference frame for all sensors. Reproducing prior densities are used to simplify Bayesian computation

Trends in Remote Spectroscopic Sensing and Imaging - Experimental Techniques and Chemometric Concepts

For studying environmental pollution locally performed analyses are non-representative as the pollutants are distributed heterogeneously. Thus, comprehensive pollution information can only be obtained by means of spatially probing the entire area of interest. Optical spectroscopy is a powerful method because active and passive sensing is feasible over large distances. Recent experimental techniques like light detection and ranging (lidar), differential optical absorption spectroscopy (DOAS), open-path FTIR spectroscopy and chemical imaging are discussed. However, sophisticated experimental techniques alone are not sufficient to establish a successful analytical method - innovative statistical data evaluation (chemometric) techniques are also needed for reliable sensor calibration and data analyses. All spectroscopic open-path techniques are inherently hampered by unpredictable and changing measurement conditions as well as by incomplete chemical knowledge during calibration. Thus, new methods are presented, which calibration models and prevent artifacts from impacting quantification of pollutants. Novel chemical imaging techniques perform spatially resolved spectroscopy and enable multicomponent analyses at unprecedented spatial and temporal resolution. However, spectroscopic imaging produces humongous data sets and thus imposes new challenges as data storage and evaluation is a computational burden and extremely time consuming. Recent data compression algorithms utilize multidimensional wavelet transformations for shrinking data sets considerably prior to chemometric evaluation and archiving. Keywords: Remote pollution sensing, Optical spectroscopy, Chemical imaging, Optimized chemometrics, Robust quantification

A WAVELET NEURAL NETWORK RAMWORK FOR SPEAKER IDNTIFCATION

This paper introduces a new model-free identification methodology to detect and identify speakers and recognize them. The basic module of the methodology is a novel multi-dimensional wavelet neural network. The WNN approach include: a universal approximator; the time frequency localization: property of wavelets leads to reduced networks at a given level of performance; The construct used as the feature mode classifier. Wavelet transform has been successfully applied to the processing of non- stationary speech signal and the feature vector that obtained becomes the input to the wavelet neural network which is trained off-line to map features to used for the classification procedure. An example is employed to illustrate the robustness and effectiveness of the proposed scheme