The Biorthogonality of Multiple Vector-valued Bivariate Wavelet Packets

The notion of a sort of biorthogonal multiple vector-valued bivariate wavelet packets,which are associated with a quantity dilation matrix,is introduced.The biorthogonality property of the multiple vector-valued wavelet packets in higher dimensions is studied by means of Fourier transform and integral transform biorthogonality formulas concerning these wavelet packets are obtained.

A CLASS OF BIDIMENSIONAL NONSEPARABLEWAVELET PACKETS

2-band wavelet packets in L-2 (R-s) were constructed in [3]. In this note, a way to construct bidimensional orthonormal wavelet packets related to the dilation matrix M = ((1)(1) (1)(-1)) is obtained. M-wavelets are used ill quincunx subsampling in two dimensions for image processing. What is more., the approach of this paper can be generalized to construct wavelet packets in L-2 (R-s) related to a general diltion matrix.

Deep neural network based on multi-level wavelet and attention for structured illumination microscopy

Structured illumination microscopy(SIM)is a popular and powerful super-resolution(SR)technique in biomedical research.However,the conventional reconstruction algorithm for SIM heavily relies on the accurate prior knowledge of illumination patterns and signal-to-noise ratio(SNR)of raw images.To obtain high-quality SR images,several raw images need to be captured under high fluorescence level,which further restricts SIM’s temporal resolution and its applications.Deep learning(DL)is a data-driven technology that has been used to expand the limits of optical microscopy.In this study,we propose a deep neural network based on multi-level wavelet and attention mechanism(MWAM)for SIM.Our results show that the MWAM network can extract high-frequency information contained in SIM raw images and accurately integrate it into the output image,resulting in superior SR images compared to those generated using wide-field images as input data.We also demonstrate that the number of SIM raw images can be reduced to three,with one image in each illumination orientation,to achieve the optimal tradeoff between temporal and spatial resolution.Furthermore,our MWAM network exhibits superior reconstruction ability on low-SNR images compared to conventional SIM algorithms.We have also analyzed the adaptability of this network on other biological samples and successfully applied the pretrained model to other SIM systems.

Enhanced Fourier Transform Using Wavelet Packet Decomposition

Many domains, including communication, signal processing, and image processing, use the Fourier Transform as a mathematical tool for signal analysis. Although it can analyze signals with steady and transitory properties, it has limits. The Wavelet Packet Decomposition (WPD) is a novel technique that we suggest in this study as a way to improve the Fourier Transform and get beyond these drawbacks. In this experiment, we specifically considered the utilization of Daubechies level 4 for the wavelet transformation. The choice of Daubechies level 4 was motivated by several reasons. Daubechies wavelets are known for their compact support, orthogonality, and good time-frequency localization. By choosing Daubechies level 4, we aimed to strike a balance between preserving important transient information and avoiding excessive noise or oversmoothing in the transformed signal. Then we compared the outcomes of our suggested approach to the conventional Fourier Transform using a non-stationary signal. The findings demonstrated that the suggested method offered a more accurate representation of non-stationary and transient signals in the frequency domain. Our method precisely showed a 12% reduction in MSE and a 3% rise in PSNR for the standard Fourier transform, as well as a 35% decrease in MSE and an 8% increase in PSNR for voice signals when compared to the traditional wavelet packet decomposition method.

A Generalization of the Mean Size Formula of Wavelet Packets in L_p

The purpose of this paper is to investigate the mean size formula of wavelet packets in Lp for 0 〈 p ≤ ∞. We generalize a mean size formula of wavelet packets given in terms of the p-norm joint spectral radius and we also give some asymptotic formulas for the Lp-norm or quasi-norm on the subdivision trees. All results will be given in the general setting,

Wavelet Packets Analysis Based on IIR Two-channel Bi-orthogonal Filter Banks

A framework of IIR two-channel bi-orthogonal filter banks is presented. The design of the analysis/synthesis systems reduces to the design of a single transfer function. The wavelet bases corresponding to the IIR maximally flat filters are generated. Use the wavelet bases in wavelet packets analysis, take information entropy function as information cost function, adjust the best bases dynamically, and the section of signal's spectramfits its characteristics well. A lot of audio signals are processed. Compared to that of the wavelet transform and the wavelet packet transform using the Daubechies' wavelet which has the same order with our IIR wavelet, the properties of our system are better

Wavelets and Wavelet Packets Related to a Class of Dilation Matrices

The construction of wavelets generated from an orthogonal multiresolution analysis can be reduced to the unitary extension of a matrix,which is not easy in most cases.Jia and Micchelli gave a solution to the problem in the case where the dilation matrix is 21 and the dimension does not exceed 3.In this paper,by the method of unitary extension of a matrix,we obtain the construction of wavelets and wavelet packets related to a class of dilation matrices.

Research on Sleeve Grouting Density Detection Based on the Impact Echo Method

Grouting defects are an inherent challenge in construction practices,exerting a considerable impact on the operational structural integrity of connections.This investigation employed the impact-echo technique for the detection of grouting anomalies within connections,enhancing its precision through the integration of wavelet packet energy principles for damage identification purposes.A series of grouting completeness assessments were meticulously conducted,taking into account variables such as the divergent material properties of the sleeves and the configuration of adjacent reinforcement.The findings revealed that:(i)the energy distribution for the highstrength concrete cohort predominantly occupied the frequency bands 42,44,45,and 47,whereas for other groups,it was concentrated within the 37 to 40 frequency band;(ii)the delineation of empty sleeves was effectively discernible by examining the wavelet packet energy ratios across the spectrum of frequencies,albeit distinguishing between sleeves with 50%and full grouting density proved challenging;and(iii)the wavelet packet energy analysis yielded variable detection outcomes contingent on the material attributes of the sleeves,demonstrating heightened sensitivity when applied to ultrahigh-performance concrete matrices and GFRP-reinforced steel bars.

Fuzzy Cluster Neural Network Based on Wavelet Transform and Its Vibration Application

This paper advances a new approach based on wavelet and wavelet packet transforms in tandem with a fuzzy cluster neural network,abbreviated WPFCNN.Wavelets and wavelet packets decompose a vibration signal into different bands at different levels and provides multiresolution or multiscale views of a signal which is stationary or nonstationary. Fuzzy mathematics processes uncertain problems in engineering and converts the attributes extracted by wavelet packets to fuzzy membership degree.To achieve self-organizing classification,the MAXNET neural network is employed.WPFCNN integrates the advantages of wavelet packets and fuzzy cluster with MAXNET.The approach is adopted to process and classify vibration signal of a NH_3 compressor in a petrochemical plant.The results indicate that it is a useful and effective intelligence classification in the field of condition monitoring and fault diagnosis.

Mean size formula of wavelet subdivision tree on Heisenberg group

The purpose of this paper is to investigate the mean size formula of wavelet packets （wavelet subdivision tree） on Heisenberg group. The formula is given in terms of the p-norm joint spectral radius. The vector refinement equations on Heisenberg group and the subdivision tree on the Heisenberg group are discussed. The mean size formula of wavelet packets can be used to describe the asymptotic behavior of norm of the subdivision tree.

Structural health monitoring of long-span suspension bridges using wavelet packet analysis

During the service life of civil engineering structures such as long-span bridges, local damage at key positions may continually accumulate, and may finally result in their sudden failure. One core issue of global vibration-based health monitoring methods is to seek some damage indices that are sensitive to structural damage, This paper proposes an online structural health monitoring method for long-span suspension bridges using wavelet packet transform （WPT）. The WPT- based method is based on the energy variations of structural ambient vibration responses decomposed using wavelet packet analysis. The main feature of this method is that the proposed wavelet packet energy spectrum （WPES） has the ability to detect structural damage from ambient vibration tests of a long-span suspension bridge. As an example application, the WPES-based health monitoring system is used on the Runyang Suspension Bridge under daily environmental conditions. The analysis reveals that changes in environmental temperature have a long-term influence on the WPES, while the effect of traffic loadings on the measured WPES of the bridge presents instantaneous changes because of the nonstationary properties of the loadings. The condition indication indices VD reflect the influences of environmental temperature on the dynamic properties of the Runyang Suspension Bridge. The field tests demonstrate that the proposed WPES-based condition indication index VD is a good candidate index for health monitoring of long-span suspension bridges under ambient excitations.

FEATURE EXTRACTION OF VIBRATION SIGNALS BASED ON WAVELET PACKET TRANSFORM

A method is proposed for the analysis of vibration signals from components ofrotating machines, based on the wavelet packet transformation (WPT) and the underlying physicalconcepts of modulation mechanism. The method provides a finer analysis and better time-frequencylocalization capabilities than any other analysis methods. Both details and approximations are splitinto finer components and result in better-localized frequency ranges corresponding to each node ofa wavelet packet tree. For the punpose of feature extraction, a hard threshold is given and theenergy of the coefficients above the threshold is used, as a criterion for the selection of the bestvector. The feature extraction of a vibration signal is accomplished by computing thereconstruction signal and its spectrum. When applied to a rolling bear vibration signal featureextraction, the proposed method can lead to be very effective.

Classification using wavelet packet decomposition and support vector machine for digital modulations

To make the modulation classification system more suitable for signals in a wide range of signal to noise rate （SNR）, a feature extraction method based on signal wavelet packet transform modulus maxima matrix （WPTMMM） and a novel support vector machine fuzzy network （SVMFN） classifier is presented. The WPTMMM feature extraction method has less computational complexity, more stability, and has the preferable advantage of robust with the time parallel moving and white noise. Further, the SVMFN uses a new definition of fuzzy density that incorporates accuracy and uncertainty of the classifiers to improve recognition reliability to classify nine digital modulation types （i.e. 2ASK, 2FSK, 2PSK, 4ASK, 4FSK, 4PSK, 16QAM, MSK, and OQPSK）. Computer simulation shows that the proposed scheme has the advantages of high accuracy and reliability （success rates are over 98% when SNR is not lower than 0dB）, and it adapts to engineering applications.

Damage Detection Methods for Offshore Platforms Based on Wavelet Packet Transform

The wavelet packet transform is used for the damage detection of offshore platforms. When some damage occurs, the dynamic response parameters of the structure will shift subtly. However, in some cases, the dynamic parameters, such as dynamic response, are not sensitive, and it is very difficult to predict the existence of damage. The present paper aims to describe how to find small damage by the use of wavelet packet transform. As the wavelet packet transform can be used to quickly find the singularity of the response signal on different scales, the acceleration signal of a damaged offshore platform in the time domain is transformed through the wavelet packet. Experimental results show that the Daubechies 4 wavelet transform can be used to detect damage.

Radar Emitter Signal Recognition Using Wavelet Packet Transform and Support Vector Machines

This paper presents a novel method for radar emitter signal recognition. First, wavelet packet transform （WPT） is introduced to extract features from radar emitter signals. Then, rough set theory is used to select the optimal feature subset with good discriminability from original feature set, and support vector machines （SVMs） are employed to design classifiers. A large number of experimental results show that the proposed method achieves very high recognition rates for 9 radar emitter signals in a wide range of signal-to-noise rates, and proves a feasible and valid method.

SORTING RADAR SIGNAL BASED ON WAVELET CHARACTERISTICS OF WIGNER-VILLE DISTRIBUTION

Common sorting method have low sorting rates and is sensitive to the Signal-to-Noise Ratio(SNR),wavelet characteristics of Wigner-Ville distribution are applied to sort unknown complicated radar signal,high sorting accuracy can be got.The Wigner-Ville distribution of received signal is calculated,then it is predigested to two-dimensional characteristics.Using wavelet transformation to extract characteristics from two-dimensional of Wigner-Ville distribution,the best characteristics are selected to be used as sorting parameters.Experiment results demonstrated that the characteristics of eight typical radar emitter signals extracted by this method showed good performance of noise-resistance and clustering at large-scale SNR.

Separation of closely spaced modes by combining complex envelope displacement analysis with method of generating intrinsic mode functions through filtering algorithm based on wavelet packet decomposition

One of the important issues in the system identification and the spectrum analysis is the frequency resolution, i.e., the capability of distinguishing between two or more closely spaced frequency components. In the modal identification by the empirical mode decomposition （EMD） method, because of the separating capability of the method, it is still a challenge to consistently and reliably identify the parameters of structures of which modes are not well separated. A new method is introduced to generate the intrin- sic mode functions （IMFs） through the filtering algorithm based on the wavelet packet decomposition （GIFWPD）. In this paper, it is demonstrated that the CIFWPD method alone has a good capability of separating close modes, even under the severe condition beyond the critical frequency ratio limit which makes it impossible to separate two closely spaced harmonics by the EMD method. However, the GIFWPD-only based method is impelled to use a very fine sampling frequency with consequent prohibitive computational costs. Therefore, in order to decrease the computational load by reducing the amount of samples and improve the effectiveness of separation by increasing the frequency ratio, the present paper uses a combination of the complex envelope displacement analysis （CEDA） and the GIFWPD method. For the validation, two examples from the previous works are taken to show the results obtained by the GIFWPD-only based method and by combining the CEDA with the GIFWPD method.

Application of the wavelet packet and its energy spectrum to identify nugget splash during the aluminum alloys spot welding

Nugget splash during aluminum alloys spot welding has a detrimental effect on weld nugget integrity, strength and durability of the welded joints. This investigation is performed to identify nugget splash from voltage signals because these are easily accessible on-line. In the present work, we propose a novel method based on the wavelet packet transform and its energy spectrum for pattern recognition of splash signal. The result demonstrates that this novel method is more accuracy and a useful way of monitoring the spot welding quality.

HYBRID WAVELET PACKET-TEAGER ENERGY OPERATOR ANALYSIS AND ITS APPLICATION FOR GEARBOX FAULT DIAGNOSIS

Based on wavelet packet decomposition （WPD） algorithm and Teager energy operator （TEO）, a novel gearbox fault detection and diagnosis method is proposed. Its process is expatiated after the principles of WPD and TEO modulation are introduced respectively. The preprocessed sigaaal is interpolated with the cubic spline function, then expanded over the selected basis wavelets. Grouping its wavelet packet components of the signal based on the minimum entropy criterion, the interpolated signal can be decomposed into its dominant components with nearly distinct fault frequency contents. To extract the demodulation information of each dominant component, TEO is used. The performance of the proposed method is assessed by means of several tests on vibration signals collected from the gearbox mounted on a heavy truck. It is proved that hybrid WPD-TEO method is effective and robust for detecting and diagnosing localized gearbox faults.

Ignition Pattern Analysis for Automotive Engine Trouble Diagnosis Using Wavelet Packet Transform and Support Vector Machines

Engine spark ignition is an important source for diagnosis of engine faults.Based on the waveform of the ignition pattern,a mechanic can guess what may be the potential malfunctioning parts of an engine with his/her experience and handbooks.However,this manual diagnostic method is imprecise because many spark ignition patterns are very similar.Therefore,a diagnosis needs many trials to identify the malfunctioning parts.Meanwhile the mechanic needs to disassemble and assemble the engine parts for verification.To tackle this problem,an intelligent diagnosis system was established based on ignition patterns.First,the captured patterns were normalized and compressed.Then wavelet packet transform（WPT） was employed to extract the representative features of the ignition patterns.Finally,a classification system was constructed by using multi-class support vector machines（SVM） and the extracted features.The classification system can intelligently classify the most likely engine fault so as to reduce the number of diagnosis trials.Experimental results show that SVM produces higher diagnosis accuracy than the traditional multilayer feedforward neural network.This is the first trial on the combination of WPT and SVM to analyze ignition patterns and diagnose automotive engines.