Advances in Wavelet Algorithms and Applications

Wickerhauser, Mladen Victor

doi:10.1142/9789812776679_0016

Cited by 9 publications

(11 citation statements)

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“…There are, several works on theoretical estimates [14] or on numerical work with wavelets [68] but we are not aware of rigorous computer-assisted estimates.…”

Section: Choice Of Basismentioning

confidence: 99%

A Framework for the Numerical Computation and A Posteriori Verification of Invariant Objects of Evolution Equations

Figueras¹,

Gameiro²,

Lessard³

et al. 2017

SIAM J. Appl. Dyn. Syst.

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We develop a theoretical framework for computer-assisted proofs of the existence of invariant objects in semilinear PDEs. The invariant objects considered in this paper are equilibrium points, traveling waves, periodic orbits and invariant manifolds attached to fixed points or periodic orbits. The core of the study is writing down the invariance condition as a zero of an operator. These operators are in general not continuous, so one needs to smooth them by means of preconditioners before classical fixed point theorems can be applied. We develop in detail all the aspects of how to work with these objects: how to precondition the equations, how to work with the nonlinear terms, which function spaces can be useful, and how to work with them in a computationally rigorous way. In two companion papers, we present two different implementations of the tools developed in this paper to study periodic orbits.

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“…There are, several works on theoretical estimates [14] or on numerical work with wavelets [68] but we are not aware of rigorous computer-assisted estimates.…”

Section: Choice Of Basismentioning

confidence: 99%

A Framework for the Numerical Computation and A Posteriori Verification of Invariant Objects of Evolution Equations

Figueras¹,

Gameiro²,

Lessard³

et al. 2017

SIAM J. Appl. Dyn. Syst.

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“…The system finely divides lower-frequency bands and roughly divides higher-frequency bands. The wavelet packet transform (WPT) [15] is a solution to improve Figure 4 shows that the WPT recursively uses the previous level's outputs as new inputs. The output of each channel is called the packet.…”

Section: Discrete Wavelet Transformmentioning

confidence: 99%

Development of Quasi-Shift-Invariant Complex Discrete Wavelet Transform

Kobayashi¹,

Nakano

2017

Journal of Signal Processing

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The downsampling of a discrete wavelet transform (DWT) has a side effect of the lack of shiftinvariance. There are two main solutions for this effect: one is the stationary wavelet transform (SWT), which does not apply downsampling. The other is the complex DWT (CDWT), which uses dual multiresolution analysis (MRA). We choose the CDWT as a target of research. It is well known that wavelet functions become Hilbert transform pairs if the low-pass filters (LPFs) on the reconstruction side have half-sample shifts. In this paper, we propose a quasi-shift-invariant (QSI) CDWT for bi-/orthogonal wavelets as a new CDWT. We report three new works (W1-W3) on it: (W1) we generalized the condition of Hilbert transform pairs and employed a complex wavelet function as a conjugate analytical signal. (W2) We defined a structure that achieves shift-invariance. The structure requires half-sample delays between the inputs of real and imaginary parts. (W3) We proposed an implementation of the QSI-CDWT and confirmed that our method has higher shift-invariance than the conventional CDWT. However, two problems (P1, P2) remain unsolved: (P1) our method requires more resources, such as memory and calculation time, than the conventional CDWT. (P2) Our theory cannot make all packets shift-invariant in a classical wavelet packet transform tree.

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“…Different families of discrete wavelet functions have been tried. They Developing automatic recognition system of drill wear in standard laminated chipboard drilling process include Daubechies, symlets, coiflets, biorthogonal and Meyer functions of different orders [17,18].…”

Section: Feature Generationmentioning

confidence: 99%

Developing automatic recognition system of drill wear in standard laminated chipboard drilling process

Kurek

Kruk

Osowski

et al. 2016

Bulletin of the Polish Academy of Sciences Technical Sciences

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Abstract. The paper presents an automatic approach to recognition of the drill condition in a standard laminated chipboard drilling process. The state of the drill is classified into two classes: "useful" (sharp enough) and "useless" (worn out). The case "useless" indicates symptoms of excessive drill wear, unsatisfactory from the point of view of furniture processing quality. On the other hand the "useful" state identifies tools which are still able to drill holes acceptable due to the required processing quality. The main problem in this task is to choose an appropriate set of diagnostic features (variables), based on which the recognition of drill state ("useful" versus "useless") can be made. The features have been generated based on 5 registered signals: feed force, cutting torque, noise, vibration and acoustic emission. Different statistical parameters describing these signals and also their Fourier and wavelet representations have been used for defining the features. Sequential feature selection is applied to detect the most class discriminative set of features. The final step of recognition is done by using three types of classifiers, including support vector machine, ensemble of decision trees and random forest. Six standard drills of 12 mm diameter with tungsten carbide tips were used in experiments. The results have confirmed good quality of the proposed diagnostic system. Key words: diagnostic expert systems, neural networks, wavelet packets, wear monitoring.Developing automatic recognition system of drill wear in standard laminated chipboard drilling process [5,7].The key issue in diagnostics of cutting tool state is the selection of sensors. Usually force sensors, electrical power, acoustic emission, vibration and acoustic pressure are applied in TCM [4,8]. The best results in metal working are achieved using force sensors.The disadvantage of this kind of measurement in normal production is that the force or torque transducers are relatively expensive and there are difficulties in mounting these sensors to the cutting tool or work piece. Therefore vibration sensors -easy to install, but at the same time less accurate because of background noise -are most commonly used in wood industry [5]. Vibration measurement is easy, since an accelerometer can be mounted close to the spindle bearing and no modifications of machine tools are needed [9].Quite similar to vibration is the sound signal, which can also be used in assessment of drill state. Mechanical vibration of the cutting tool, machine holder and drill are partly transferred to airborne vibration. It means that part of the information contained in vibration can also be obtained from sound measurement. The acquisition of the sound can be done easily using a microphone. However, the sound pressure sensors (microphones), are even more susceptible to background [7,8]. The sound measurement in an extended range of frequencies from 20 kHz to 80 kHz is usually applied [4]. This range is called the acoustic emission. The effectiveness of acoustic emission...

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Advances in Wavelet Algorithms and Applications

Cited by 9 publications

References 0 publications

A Framework for the Numerical Computation and A Posteriori Verification of Invariant Objects of Evolution Equations

A Framework for the Numerical Computation and A Posteriori Verification of Invariant Objects of Evolution Equations

Development of Quasi-Shift-Invariant Complex Discrete Wavelet Transform

Developing automatic recognition system of drill wear in standard laminated chipboard drilling process

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