Multiresolution Analysis Method Research Articles

Orthogonal and Biorthogonal $\sqrt 3$-Refinement Wavelets for Hexagonal Data Processing

The hexagonal lattice was proposed as an alternative method for image sampling. The hexagonal sampling has certain advantages over the conventionally used square sampling. Hence, the hexagonal lattice has been used in many areas. A hexagonal lattice allows radic3, dyadic and radic7 refinements, which makes it possible to use the multiresolution (multiscale) analysis method to process hexagonally sampled data. The radic3-refinement is the most appealing refinement for multiresolution data processing due to the fact that it has the slowest progression through scale, and hence, it provides more resolution levels from which one can choose. This fact is the main motivation for the study of radic3-refinement surface subdivision, and it is also the main reason for the recommendation to use the radic3-refinement for discrete global grid systems. However, there is little work on compactly supported radic3 -refinement wavelets. In this paper, we study the construction of compactly supported orthogonal and biorthogonal radic3-refinement wavelets. In particular, we present a block structure of orthogonal FIR filter banks with twofold symmetry and construct the associated orthogonal radic3-refinement wavelets. We study the sixfold axial symmetry of perfect reconstruction (biorthogonal) FIR filter banks. In addition, we obtain a block structure of sixfold symmetric radic3-refinement filter banks and construct the associated biorthogonal wavelets.

Power quality disturbances analysis based on EDMRA method

An Energy Difference of Multiresolution Analysis (EDMRA) method for power quality (PQ) disturbances analysis has been proposed in this paper. At each wavelet decomposition level, the squared value of the detail information is calculated as their energy to construct the feature vector for analysis. Following the criteria proposed in this paper, different kinds of power quality disturbances can be detected, localized, and classified effectively. The choice of the decomposition levels of appropriate wavelets are of the critical importance for the EDMRA method, since they will influence quality of the reconstructed signal as well as the computational cost. It is presented in this paper that the Minimum Decomposition Level (MDL) is related to the sampling frequency by the proposed function. The comparison study among different kinds of wavelets for the EDMRA method is presented in details. The EDMRA method is scalable and has robustness characteristics in common design paradigm. It can be realized economically using wavelets with shortest length, such as Harr, Db2, Sym2 or Coif 1. Two types of noise, namely, Gaussian white noise and band limited spectrum noise are considered in this paper to show the effectiveness of the proposed method in noisy environment.

Orthogonal and Biorthogonal FIR Hexagonal Filter Banks With Sixfold Symmetry

Recently, hexagonal image processing has attracted attention. The hexagonal lattice has several advantages in comparison with the rectangular lattice, the conventionally used lattice for image sampling and processing. For example, a hexagonal lattice needs less sampling points; it has better consistent connectivity; it has higher symmetry; and its structure is plausible to human vision systems. The multiresolution analysis method has been used for hexagonal image processing. Since the hexagonal lattice has high degree of symmetry, it is desirable that the hexagonal filter banks designed for multiresolution hexagonal image processing also have high order of symmetry, which is pertinent to the symmetry structure of the hexagonal lattice. The orthogonal or prefect reconstruction (PR) hexagonal filter banks that are available in the literature have only threefold symmetry. In this paper, we investigate the construction of orthogonal and PR finite impulse response (FIR) hexagonal filter banks with sixfold symmetry. We obtain block structures of 7-size refinement (seven-channel two-dimensional) orthogonal and PR FIR hexagonal filter banks with sixfold rotational symmetry. radic7-refinement orthogonal and biorthogonal wavelets based on these block structures are constructed. In this paper, we also consider FIR hexagonal filter banks with axial (line) symmetry, and we present a block structure of FIR hexagonal filter banks with pseudo-sixfold axial symmetry.

Discussion of: Treelets—An adaptive multi-scale basis for sparse unordered data

We would like to congratulate Lee, Nadler and Wasserman on their contribution to clustering and data reduction methods for high $p$ and low $n$ situations. A composite of clustering and traditional principal components analysis, treelets is an innovative method for multi-resolution analysis of unordered data. It is an improvement over traditional PCA and an important contribution to clustering methodology. Their paper [arXiv:0707.0481] presents theory and supporting applications addressing the two main goals of the treelet method: (1) Uncover the underlying structure of the data and (2) Data reduction prior to statistical learning methods. We will organize our discussion into two main parts to address their methodology in terms of each of these two goals. We will present and discuss treelets in terms of a clustering algorithm and an improvement over traditional PCA. We will also discuss the applicability of treelets to more general data, in particular, the application of treelets to microarray data.

DISCUSSION OF: TREELETS-AN ADAPTIVE MULTI-SCALE BASIS FOR SPARSE UNORDERED DATA.

We would like to congratulate Lee, Nadler and Wasserman on their contribution to clustering and data reduction methods for high p and low n situations. A composite of clustering and traditional principal components analysis, treelets is an innovative method for multi-resolution analysis of unordered data. It is an improvement over traditional PCA and an important contribution to clustering methodology. Their paper presents theory and supporting applications addressing the two main goals of the treelet method: (1) Uncover the underlying structure of the data and (2) Data reduction prior to statistical learning methods. We will organize our discussion into two main parts to address their methodology in terms of each of these two goals. We will present and discuss treelets in terms of a clustering algorithm and an improvement over traditional PCA. We will also discuss the applicability of treelets to more general data, in particular, the application of treelets to microarray data.

A WAVELET BASED SIGNIFICANCE TEST FOR PERIODICITIES IN INDIAN MONSOON RAINFALL

This paper is a sequel to a recent study of the authors' that uses a combination of multiresolution analysis (MRA) and classical Fourier spectral methods, to identify 17 peaks in the power spectral density of the Homogeneous Indian Monsoon (HIM) rainfall time series constructed in Ref. 1. Here we propose a new procedure for testing the statistical significance of these peaks. In this procedure, using MRA the stationary component of the rainfall time series is first identified. Then (partially) reconstructed time series are derived over each scale band in the stationary component. For each of these time series an appropriately colored reference spectrum is derived. The significance of the detected peaks is then determined using a χ2test against the reference spectra, which together represent a noise process spectrally close to rainfall. It is concluded that HIM rainfall exhibits 10 statistically significant periodicities at a confidence level of 99.9%.

Comparison of wavelet-SVM and wavelet-adaptive network based fuzzy inference system for texture classification

Texture image is a low-level image feature. Nowadays, there are many different applications involving texture analysis, including endoscope, ultrasonographic images recognition, document classification, radar image classification, and texture-based image retrieval. The texture images analysis and classification becomes an important topic. The greatest hardness of texture analysis and classification in the past was the deficiency of enough methods to characterize. Multi-resolution analysis methods such as wavelet and wavelet packet decompositions are more superior to other classic statistical methods. In this paper, a comparison of wavelet-support vector machine (W-SVM) and wavelet-adaptive network based fuzzy inference system (W-ANFIS) approaches for texture image classification is presented. Both W-SVM and W-ANFIS methods are used for classification of the 22 texture images obtained from Brodatz image album. There, 50 64 x 64 image regions were randomly selected (overlapping or nonoverlapping) for each of these 22 images. 25 of these image regions and other 25 of these image regions are used for training and testing of the W-SVM and W-ANFIS methods, respectively.

Matching Pursuit-Based Region-of-Interest Image Coding

Matching pursuit (MP) is a multiresolution signal analysis method and can be used to render a selected region of an image with a specific quality. A novel, scalable, and progressive MP-based region-of-interest image-coding scheme is presented. The method is capable of providing a trade off between rate, distortion, and complexity. The method also provides an interactive way of information refinement for regions of an image with higher receiver's priority. By selecting a proper subset of the huge initial MP dictionary, using the method described in this paper, the complexity burden of MP analysis can be adapted to the computational power of the image encoder.

Using the MRTD based on Daubechies scaling functions to solve the problem of electromagnetic scattering

A new method of multiresolution time domain analysis is proposed, which is based on compactly supported Daubechies scaling functions in the electromagnetic scattering of three dimensional objects. The highly linear dispersional properties are investigated. Using the interfaces of MRTD/FDTD technique, the connecting and absorption boundaries are dealt with. Numerical experiments show that, compared with conventional FDTD, this method can reduce cells without sacrificing solution accuracy, and uses less CPU time. Finally, the characteristics of PBG structure are obtained using this scheme.

Estimation of spatial derivatives and identification of continuous spatio-temporal dynamical systems

A new approach for the estimation of spatial derivatives and the identification of a class of continuous spatio-temporal dynamical systems from experimental data is presented in this study. The proposed identification approach is a combination of implicit Adams integration and an orthogonal forward regression algorithm (OFR), in which the operators are expanded using polynomials as basis functions. The noisy experimental data are de-noised by using biorthogonal spline wavelet filters and the spatial derivatives are estimated using a multiresolution analysis method. Finally, a bootstrap method is applied to refine the identified parameters from the OFR algorithm. The resulting identified models of the spatio-temporal evolution form a system of partial differential equations. Examples are provided to demonstrate the efficiency of the proposed method.

Recent advances in wavelet analyses: Part 1. A review of concepts

This contribution provides a review of the most recent wavelet applications in the field of earth sciences and is devoted to introducing and illustrating new wavelet analysis methods in the field of hydrology. Wavelet analysis remains unknown in the field of hydrology even though it clearly overcomes the well-known limits of the classical Fourier analysis. New wavelet-based tools are proposed to hydrologists in order to make wavelet analysis more attractive. First, a multiresolution continuous wavelet analysis method is shown to significantly improve the determination of the temporal-scale structure of a given signal. Second, the concept of wavelet entropy in both continuous and multiresolution frameworks is introduced allowing for an estimation of the temporal evolution of a given hydrological or climatologic signal's complexity. New insights in the scale-dependence of the relationship are exposed by introducing wavelet cross-correlation and wavelet coherence. Continuous wavelet cross-correlation provides a time–scale distribution of the correlation between two signals, whereas continuous wavelet coherence provides a qualitative estimator of the temporal evolution of the degree of linearity of the relationship between two signals on a given scale. These methods are applied to four large river runoffs and two global climatic indexes in a companion paper.

Stochastic texture analysis for monitoring stochastic processes in industry

Several continuous manufacturing processes use stochastic texture images for quality control and monitoring. Large amounts of pictorial data are acquired, providing both important information about the materials produced and about the manufacturing processes involved. However, it is often difficult to measure objectively the similarity among such images, or to discriminate between texture images of materials with distinct properties. The degree of discrimination required by industrial processes sometimes goes beyond the limits of human visual perception. This work presents a new method for multi-resolution stochastic texture analysis, interpretation and discrimination based on the wavelet transform. A multi-resolution distance measure for stochastic textures is proposed, and applications of our method in the non-woven textiles industry are reported. The conclusions include ideas for future work.

Application of Spatial Multiresolution Analysis and Synthesis to Piezoelectric Inhomogeneous Drive Systems

The method of multiresolution analysis and synthesis is applied on a space wave-number basis to investigate the frequency characteristics of a piezoelectric transducer with a spatially inhomogeneous `piezoelectric drive ability', with which conventional methods such as the electrical equivalent circuit method cannot deal easily and systematically. The inhomogeneous distribution of the piezoelectric drive ability is analyzed into some spatial functions with different resolutions, and then the synthesis of these components is performed on a circuit structure in the framework of `complex series dynamics'. The circuit structure reflects the electrical and mechanical boundary conditions of the transducer, and the frequency characteristics can be calculated easily and systematically using a recursive procedure up to the desired resolutions.

WAVELET APPROACH FOR PERFORMANCE MONITORING AND DIAGNOSIS OF A HYDRAULIC PUMP

Hydraulic pump performance monitoring methods that can detect failures by using the outlet pressure signals of the pump are investigated. Two faults diagnosis methods, namely conventional spectral analysis method based on FFT and wavelet based multi-resolution analysis method, are introduced and their efficiency and reliability are discussed. The performance of both diagnoses methods were evaluated based on simulation results and experimental results. Validation results obtained from using both methods in analyzing the same sets of data indicated that the wavelet transform based fault diagnosis method showed a more sensitive and robust detecting results for all three tested pump faults than that obtained from a spectrum analyses approach.

Multiresolution Analysis for Reconstruction of Conductivity Profiles in Eddy Current Nondestructive Evaluation Using Probe Impedance Data

This paper presents a wavelet-based multiresolution analysis method for solving the inverse problem in eddy current testing (ECT). Using the probe impedance signals as the measurement data, the reconstruction of conductivity profiles can be performed by a minimization scheme. The method allows to identify the regions where the perturbation may be localized and to retrieve the unknown parameters only in those regions. Some numerical simulation results show the feasibility of the technique.

Multi-resolution dyadic wavelet denoising approach for extraction of visual evoked potentials in the brain

More powerful techniques need to be developed to extract small and weak visual evoked potentials (VEPs) from the spontaneous cerebral electric activity EEG. The authors present a wavelet decomposition algorithm suitable for identification and detection of very weak VEPs. The cross-correlation analysis between Daubechies wavelet (i.e. dbN) functions /spl psi//sub N/(t), for N=4, 5 ,...,10 and a representative noiseless VEP signal is performed to choose the proper wavelet function, say that with maximum correlation coefficient (highest resemblance) with respect to the representative VEP signal sequence. In this way, the specific choice of the best wavelet prototype function is no longer arbitrary for the application of obtaining pattern reversal VEPs. Extensive clinical experiments have demonstrated that the multiresolution wavelet analysis method can identify and estimate the peak latency of VEP signal well, with only a much reduced trial of ensemble averaging (EA) required. The major advantages of the wavelet transform are that it can 'zoom-in' to time discontinuities, and that orthonormal bases, localised in time and frequency, can be constructed. With this zoom-in property of the wavelet analysis, the irregularities or abnormalities of signals can easily be detected. Also the characteristics of EP signals can be captured by means of wavelet analysis, which can be further used for the detection and recognition of the abnormalities in the brain.

Global elevation vibration and seasonal changes derived by the analysis of GPS height

We use the spectral analysis and the multi-resolution wavelet analysis methods to study GPS time series in height component generated from continuously operating stations, which are globally distributed, and separate stationary signals from the non-stationary, then set up discrete models of stationary height signals by using AR model methods. Comparisons made in this paper show that the correlative length of GPS time series in height component varies from 2 days to 31 days, rectifying the integral deficiency from zero in random walk process. After analyzing, both annual and biannual vibrations of integrated expanding and contracting movements in the Earth are detected. The biannual maximum value is found in March—April and October—November while the annual maximum value is in September—November. Meantime, we observe that the movements follow different laws in the Northern and Southern Hemispheres. The Southern Hemisphere is expanding compared with the Northern Hemisphere.

Comparison between statistical and multiresolution wavelets analysis methods for speech classifications

This paper considers the use of MultiResolution Wavelets Analysis (MRWA) to classify speech for multifunctional control applications. While the classification problem is the focus of this paper, th...

A Coordinate-Invariant Approach to Multiresolution Motion Analysis

Multiresolution motion analysis has gained considerable research interest as a unified framework to facilitate a variety of motion editing tasks. Within this framework, motion data are represented as a collection of coefficients that form a coarse-to-fine hierarchy. The coefficients at the coarsest level describe the global pattern of a motion signal, while those at fine levels provide details at successively finer resolutions. Due to the inherent nonlinearity of the orientation space, the challenge is to generalize multiresolution representations for motion data that contain orientations as well as positions. Our goal is to develop a multiresolution analysis method that guarantees coordinate-invariance without singularity. To do so, we employ two novel ideas: hierarchical displacement mapping and motion filtering. Hierarchical displacement mapping provides an elegant formulation to describe positions and orientations in a coherent manner. Motion filtering enables us to separate motion details level-by-level to build a multiresolution representation in a coordinate-invariant way. Our representation facilitates multiresolution motion editing through level-wise coefficient manipulation that uniformly addresses issues raised by motion modification, blending, and stitching.

On-line disturbance classification using nearest neighbor rule

Abstract This paper presents an automated on-line disturbance classification technique for different power quality problems. This technique is based on wavelet multi-resolution analysis and nearest neighbors pattern recognition method. The wavelet-multi-resolution transform is introduced as a powerful tool for feature extraction. It has the ability to extract discriminative, translation invariant features with small dimensionality in order to classify different disturbances. The nearest neighbor pattern recognition technique is then implemented to classify different disturbances and evaluate the efficiency of the extracted features.