Biorthogonal Wavelet Bases Research Articles

Experimental Research on Evaluation of Soil Water Content Using Ground Penetrating Radar and Wavelet Packet-Based Energy Analysis

Soil water content is one of the most important factors affecting the safety and stability of buildings or structures, especially in roadbeds, slopes, earth dams and foundations. Accurate assessments of soil water content can ensure the quality of construction, reduce construction costs and prevent accidents, among other benefits. In this study, ground penetrating radar (GPR) was used to detect and evaluate changes in soil water content. The GPR signal is usually nonstationary and nonlinear; however, traditional Fourier theory is typically suitable for periodic stationary signals, and cannot reflect the law of the frequency and energy of the GPR signal changing with time. Wavelet transform has good time-frequency localization characteristics, and therefore represents a new method for analyzing and processing GPR signals. According to the time-frequency characteristics of GPR signals, in this paper, a new biorthogonal wavelet basis which was highly matched with the GPR waveform was constructed using the lifting framework of wavelet theory. Subsequently, an evaluation method, namely, the wavelet packet-based energy analysis (WPEA) method, was proposed. The method was utilized to calculate the wavelet packet-based energy indexes (WPEI) of the GPR single-channel signals for clay samples with water contents ranging from 10% to 24%. The research results showed that there was a highly correlated linear relationship between the WPEI and the soil water contents, and the relationship between the two was fitted with a linear fitting function. The feasibility of the method was verified by comparing our results with those obtained using classical wavelet bases to perform the wavelet packet transform. The large-area, continuous scanning measurement method of GPR was shown to be suitable for evaluations of soil water contents in roadbeds, slopes, earth dams, and foundations.

Calculation of the permittivity of inhomogeneous media based on the TEDAWT method

In the field of nondestructive inspection, ground-penetrating radar (GPR) is a widely used investigation tool, usually to evaluate quality and safety in structural engineering. However, its development has been limited due to differences in the dielectric constant in inhomogeneous media. In this study, a time-energy density analysis of wavelet transform (TEDAWT) method for GPR signal singularity recognition was proposed. In addition, a biorthogonal wavelet basis with high similarity to GPR signal waveform was innovatively constructed and added into the wavelet toolbox. A model experiment was conducted to validate the feasibility of inverse calculation of the permittivity of wet clay by the TEDAWT method. Results showed that this method offered the advantages of signal denoising, resolution improvement, singularity recognition in GPR signal processing. Subsequently, a basic flow chart was established, which could be used to inversely calculate the permittivity of inhomogeneous media, and be applied to the quantitative identification of the thickness of the reinforcement protective layer in an expressway tunnel. The calculation results showed that the inverse calculation value of reinforcement protection layer thickness was 3.74% lower than the average relative theoretical error value. That improved detection accuracy provided a new method to quantitatively identify inhomogeneous media by GPR.

Homogeneous Dirichlet wavelets on the interval diagonalizing the derivative operator, and application to free-slip divergence-free wavelets

This paper presents a new construction of a homogeneous Dirichlet wavelet basis on the unit interval, linked by a diagonal differentiation-integration relation to a standard biorthogonal wavelet basis. This allows to compute the solution of Poisson equation by renormalizing the wavelet coefficients - as in Fourier domain but using locally supported basis functions with boundary conditions-, which yields a linear complexity O(N) for this problem. Another application concerns the construction of free-slip divergence-free wavelet bases of the hypercube, in general dimension, with an associated decomposition algorithm as simple as in the periodic case.

Intensive interferences processing for GPR signal based on the wavelet transform and F-K filtering

To reduce the shielding effect of strong jamming signals such as direct waves and multiple echoes on the effective signals in ground penetrating radar (GPR) images and effectively improve the GPR image resolution, a method to remove strong interference signals in GPR that combines the wavelet transform and F-K filtering is proposed. First, we choose a biorthogonal wavelet basis that matches the waveform of GPR signal, and an original GPR image is decomposed into subimages of different frequency bands based on the optimal wavelet basis. Second, according to the difference in the distribution of the effective signals and the strong interference of different subsignal images in the wavenumber domain, different bandpass filters are designed to accurately separate the effective signals and strong interference, and the subimages are reconstructed after the interference being removed. The processing of the forward simulated and measured GPR images shows that the combined method of wavelet transform and F-K filtering can produce different F-K filters according to the characteristics of different subsignal images and accurately process the image, suppressing the strong interference and retaining the effective signals to the greatest extent. Compared to the results of the predictive deconvolution method and the conventional F-K method, the cavity reflection in the image processed by the proposed method is more obvious, the coaxial signal is continuous and clear, and the image resolution is improved, which proves the feasibility and effectiveness of this method in separating strong interference waves.

A Blind Spectral Unmixing in Wavelet Domain

In this article, a wavelet-based energy minimization framework is developed for joint estimation of endmembers and abundances without assuming pure pixels while considering noisy scenario. Spectrally dense and overlapped hyperspectral data are represented using biorthogonal wavelet bases that yield a compact linear mixing model in the wavelet domain. It acts as the data term and helps to reduce solution space of the unmixed components. Three prior terms are incorporated to better handle the ill-posedness, i.e., the logarithm of determinant volume regularizer enforces minimum endmember simplex, smoothness (spatial) prior to individual abundance maps, and spectral constraint through learning dictionary of abundances. Alternating nonnegative least-squares is employed to optimize the regularized and constrained nonnegative matrix factorization functional in the wavelet domain. We conduct theoretical analysis, discuss convergence and algorithmic details. Experiments are conducted on synthetic and three real benchmark hyperspectral data AVIRIS Cuprite, HYDICE Urban, and AVIRIS Jasper Ridge. The efficacy of the proposed algorithm is evaluated by comparing results with state of the art.

Wavelet–Fourier CORSING techniques for multidimensional advection–diffusion–reaction equations

AbstractWe present and analyze a novel wavelet–Fourier technique for the numerical treatment of multidimensional advection–diffusion–reaction equations based on the COmpRessed SolvING (CORSING) paradigm. Combining the Petrov–Galerkin technique with the compressed sensing approach the proposed method is able to approximate the largest coefficients of the solution with respect to a biorthogonal wavelet basis. Namely, we assemble a compressed discretization based on randomized subsampling of the Fourier test space and we employ sparse recovery techniques to approximate the solution to the partial differential equation (PDE). In this paper we provide the first rigorous recovery error bounds and effective recipes for the implementation of the CORSING technique in the multidimensional setting. Our theoretical analysis relies on new estimates for the local $a$-coherence, which measures interferences between wavelet and Fourier basis functions with respect to the metric induced by the PDE operator. The stability and robustness of the proposed scheme are shown by numerical illustrations in the one-, two- and three-dimensional cases.

Comment on: “Fast 3D inversion of gravity data using solution space priorconditioned Lanczos bidiagonalization”, by Mohammad Rezaie, Ail Moradzadeh and Ali Nejati Kalateh

To reduce the shielding effect of strong jamming signals such as direct waves and multiple echoes on the effective signals in ground penetrating radar (GPR) images and effectively improve the GPR image resolution, a method to remove strong interference signals in GPR that combines the wavelet transform and F-K filtering is proposed. First, we choose a biorthogonal wavelet basis that matches the waveform of GPR signal, and an original GPR image is decomposed into subimages of different frequency bands based on the optimal wavelet basis. Second, according to the difference in the distribution of the effective signals and the strong interference of different subsignal images in the wavenumber domain, different bandpass filters are designed to accurately separate the effective signals and strong interference, and the subimages are reconstructed after the interference being removed. The processing of the forward simulated and measured GPR images shows that the combined method of wavelet transform and F-K filtering can produce different F-K filters according to the characteristics of different subsignal images and accurately process the image, suppressing the strong interference and retaining the effective signals to the greatest extent. Compared to the results of the predictive deconvolution method and the conventional F-K method, the cavity reflection in the image processed by the proposed method is more obvious, the coaxial signal is continuous and clear, and the image resolution is improved, which proves the feasibility and effectiveness of this method in separating strong interference waves.

Signal Reconstruction Using New Biorthogonal Frequency-Modified Kravchenko Wavelets When Representing the Measurement Process as a Convolution Model

Abstract—A new biorthogonal system of wavelet bases has been constructed, which is oriented toward reconstructing the useful signal of a measuring system if the measurement process is represented as a convolution model. New biorthogonal wavelet bases are obtained by using a instrumental function to modify a Kravchenko orthogonal wavelet system with a finite spectrum. The properties of new biorthogonal frequency-modified wavelets are studied, and digital filters that realize fast computational algorithms are constructed. Schemes for multiresolution analysis are proposed, which, during discrete wavelet transform, immediately solve the problem of reconstructing the useful signal, as well as effective noise suppression, which can significantly speed up computations.

Multimedia Image Compression Method Based on Biorthogonal Wavelet and Edge Intelligent Analysis

At present, network image communication is still restricted by channel coding, image and multimedia transmission and other key technologies. Therefore, the transmission process needs to convert the image signal into a digital signal, and then use the relevant band compression technology to reduce these signals to narrow the occupied frequency band, that is, to reduce the amount of information to be transmitted in synchronous transmission. Wavelet transform is a powerful tool for image compression because of its low entropy, multi-resolution, decorrelation and flexible base selection. In this paper, a method of multimedia image compression based on biorthogonal wavelet packet is proposed, which includes the establishment of linear phase biorthogonal wavelet basis, the selection of 3 or 4 levels of wavelet decomposition and reconstruction stage, and the combination of improved band division and preservation strategy. Finally, a compression test is performed based on the selected wavelet basis function and the optimal decomposition and reconstruction layers of the standard test image, which enables to obtain a more ideal compression ratio.

Trainable wavelet-like transform for feature extraction to audio classification

A method of learning optimal orthonormal filters for feature extraction from 1-D signal based on learning wavelet-like transform was proposed. Filters had been learned by using backpropagation simultaneously with neural network, which was used as a classifier. Orthonormality of filters during the learning process was provided by several quadratic regularization terms that follow from the orthogonality of the scaling functions. The proposed method was evaluated on the environmental sound classification task. We used the trainable wavelet-like transform and wavelet transform with different bases as feature extraction methods with fixed architecture of the neural network. The proposed method obtained the best results. The spectrum characteristics of learned filters of wavelet-like transform were compared with the corresponding characteristics of reverse biorthogonal wavelet basis rbior1.5 that obtained the closest accuracy results.

Construction and application of a new biorthogonal wavelet basis for a quantitative analysis of GPR signals

Due to interference from coherent reflected waves and clutter, it is difficult for ground-penetrating radar (GPR) signals to enable exact quantitative analyses of tunnel void defects. According to GPR pulse features, a new wavelet basis (named the GPR3.3 wavelet) was constructed using an improved biorthogonal wavelet construction method. In addition, based on the wavelet transform modulus maximum method, a new quantitative analysis method for GPR signals that is based on the biorthogonal wavelet basis (the QAGBW method) is proposed. The QAGBW method based on the GPR3.3 wavelet was applied in a quantitative analysis of analog and measured signals. The results show that the GPR3.3 wavelet, which can identify the singularity of a GPR pulse, is an optimal wavelet basis for quantitative analysis. Furthermore, the QAGBW method based on the GPR3.3 wavelet has greater precision than the regularized deconvolution method for the quantitative analysis of the depth and vertical size of voids because it restrains the interference of coherent reflected waves and clutter.

Serum free cortisol measurement by LC-MSMS

Due to interference from coherent reflected waves and clutter, it is difficult for ground-penetrating radar (GPR) signals to enable exact quantitative analyses of tunnel void defects. According to GPR pulse features, a new wavelet basis (named the GPR3.3 wavelet) was constructed using an improved biorthogonal wavelet construction method. In addition, based on the wavelet transform modulus maximum method, a new quantitative analysis method for GPR signals that is based on the biorthogonal wavelet basis (the QAGBW method) is proposed. The QAGBW method based on the GPR3.3 wavelet was applied in a quantitative analysis of analog and measured signals. The results show that the GPR3.3 wavelet, which can identify the singularity of a GPR pulse, is an optimal wavelet basis for quantitative analysis. Furthermore, the QAGBW method based on the GPR3.3 wavelet has greater precision than the regularized deconvolution method for the quantitative analysis of the depth and vertical size of voids because it restrains the interference of coherent reflected waves and clutter.

A wavelet video coding algorithm with balanced significance probability tree based on energy weighting

This work presents a 3-D wavelet video coding algorithm. By analyzing the contribution of each biorthogonal wavelet basis to reconstructed signal’s energy, we weight each wavelet subband according to its basis energy. Based on distribution of weighted coefficients, we further discuss a 3-D wavelet tree structure named balanced significance probability tree, which places the coefficients with similar probabilities of being significant on the same layer. It is implemented by using hybrid spatial orientation tree and temporal-domain block tree. Subsequently, a novel 3-D wavelet video coding algorithm is proposed based on the energy-weighted balanced significance probability tree. Experimental results illustrate that our algorithm always achieves good reconstruction quality for different classes of video sequences. Compared with asymmetric 3-D orientation tree, the average peak signal-to-noise ratio (PSNR) gain of our algorithm are 1.24dB, 2.54dB and 2.57dB for luminance (Y) and chrominance (U,V) components, respectively. Compared with temporal-spatial orientation tree algorithm, our algorithm gains 0.38dB, 2.92dB and 2.39dB higher PSNR separately for Y, U, and V components. In addition, the proposed algorithm requires lower computation cost than those of the above two algorithms.

Construction of Dual Multiple Knot B-Spline Wavelets on Interval

The paper is concerned with the construction of biorthogonal multiple knot B-spline(MKBS) scaling functions and multiple knot B-spline wavelet (MKBSW) basis functions. The resulting bases are on the unit interval with the desired number of vanishing wavelet moments. Moreover, in order to be the same the general structure of MKBS and MKBSW basis functions, a new basis of MKBS is introduced. This arises a significant decrease in the computational costs.

Optimal model for 4-band biorthogonal wavelets bases for fast calculation

A class of 4-band symmetric biorthogonal wavelet bases has been constructed, in which any wavelet system the high-pass filters can be determined by exchanging position and changing the sign of the two low-pass filters. Thus, the least restrictive conditions are needed for forming a wavelet so that the free degrees can be reversed for application requirement. Some concrete examples with high vanishing moments are also given, the properties of the transformation matrix are studied and the optimal model is constructed. These wavelets can process the boundary conveniently, and they lead to highly efficient computations in applications.

Multivariate intensity estimation via hyperbolic wavelet selection

We propose a new statistical procedure that can overcome the curse of dimensionality without structural assumptions on the function to estimate. It relies on a least-squares type penalized criterion and a new collection of models built from hyperbolic biorthogonal wavelet bases. We study its properties in a unifying intensity estimation framework, where an oracle-type inequality and adaptation to mixed smoothness are shown to hold. We also show how to implement the estimator with an algorithm whose complexity is manageable.

Compactly Supported Biorthogonal Wavelet Bases on the Body Centered Cubic Lattice

AbstractIn this work, we present a family of compact, biorthogonal wavelet filter banks that are applicable to the Body Centered Cubic (BCC) lattice. While the BCC lattice has been shown to have superior approximation properties for volumetric data when compared to the Cartesian Cubic (CC) lattice, there has been little work in the way of designing wavelet filter banks that respect the geometry of the BCC lattice. Since wavelets have applications in signal de‐noising, compression, and sparse signal reconstruction, these filter banks are an important tool that addresses some of the scalability concerns presented by the BCC lattice. We use these filters in the context of volumetric data compression and reconstruction and qualitatively evaluate our results by rendering images of isosurfaces from compressed data.

A parametrization technique to design joint time–frequency optimized discrete-time biorthogonal wavelet bases

The accurate and efficient representation of a signal in terms of elementary atoms has been a challenge in many signal processing applications including harmonic analysis. The wavelet bases have been proved to be very efficient and flexible atoms. Towards the goal of obtaining optimal wavelet bases, we present a simple and efficient parametrization technique for constructing linear phase biorthogonal discrete-time wavelet bases that have joint time–frequency localization (JTFL) close to the lower bound of 0.25. In this paper, we first develop a parametrization technique to design biorthogonal filter banks (FBs). Then an optimization method is formulated to design jointly time–frequency localized discrete wavelet bases employing the designed FBs. Finally, the performance of the optimal wavelet bases is evaluated in image coding application. The proposed parametrization method presents a general and yet a very simple framework to construct a linear phase biorthogonal FB of desired order, with the prescribed number of vanishing moments (VMs) and free parameters. Several examples are presented to demonstrate the effectiveness and flexibility of the technique to design different classes of FB with various degrees of freedom. The performance of the designed FBs is compared with the other popular biorthogonal wavelet FBs.

Design of Optimized Wavelet Packet Algorithm to Improve Perception of Sensorineural Hearing Impaired

A novel optimized wavelet packet algorithm is proposed to improve the perception of sensorineural hearing-impaired people. In this work, we have developed optimized wavelet packet along with, biorthogonal wavelet basis functions using MATLAB Code. Here, we have created eight bands based on auditory filters of quasi octave bandwidth. Evaluation was carried out by conducting listening tests on seven subjects with bilateral mild to severe sensorineural hearing loss. The speech material used for the listening test consisted of a set of fifteen nonsense syllables in VCV context. The test results show that the proposed algorithm improves the recognition score, speech quality and transmission of overall feature specifically over the unprocessed signal. The response time also reduces significantly.

Besov regularity for operator equations on patchwise smooth manifolds

We study regularity properties of solutions to operator equations on patchwise smooth manifolds $$\partial \Omega $$?Ω, e.g., boundaries of polyhedral domains $$\Omega \subset \mathbb {R}^3$$Ω?R3. Using suitable biorthogonal wavelet bases $$\Psi $$?, we introduce a new class of Besov-type spaces $$B_{\Psi ,q}^\alpha (L_p(\partial \Omega ))$$B?,q?(Lp(?Ω)) of functions $$u:\partial \Omega \rightarrow \mathbb {C}$$u:?Ω?C. Special attention is paid on the rate of convergence for best n-term wavelet approximation to functions in these scales since this determines the performance of adaptive numerical schemes. We show embeddings of (weighted) Sobolev spaces on $$\partial \Omega $$?Ω into $$B_{\Psi ,\tau }^\alpha (L_\tau (\partial \Omega )), 1/\tau =\alpha /2 + 1/2$$B?,??(L?(?Ω)),1/?=?/2+1/2, which lead us to regularity assertions for the equations under consideration. Finally, we apply our results to a boundary integral equation of the second kind which arises from the double-layer ansatz for Dirichlet problems for Laplace's equation in $$\Omega $$Ω.