芦野 隆一

This paper deals with the linear canonical wavelet transform. It is a non-trivial generalization of the ordinary wavelet transform in the framework of the linear canonical transform. We first present a direct relationship between the linear canonical wavelet transform and ordinary wavelet transform. Based on the relation, we provide an alternative proof of the orthogonality relation for the linear canonical wavelet transform. Some of its essential properties are also studied in detail. Finally, we explicitly derive several versions of inequalities associated with the linear canonical wavelet transform.

In our previous work, we established some basic poverties of the linear canonical transform and obtained alternative form of convolution and correlation theorems. In this paper, we study essential properties of the linear canonical transform (LCT). The properties are modifications of the classical Fourier transform properties. They are very need in applying LCT in signal processing. In addition, we formulate an inequality associated with the LCT, which is different from the uncertainty principle in literature.

Several essential properties of the linear canonical transform (LCT) are provided. Some results related to the sampling theorem in the LCT domain are investigated. Generalized wave and heat equations on the real line are introduced, and their solutions are constructed using the sampling formulae. Some examples are presented to illustrate our results.

The convolution theorem for the linear canonical transformation is introduced. Correlation theorems related to the linear canonical transformation are established by using the relation between convolution and correlation definitions in the linear canonical transform domains.

The continuous fractional wavelet transform (CFrWT) is a nontrivial generalization of the classical wavelet transform (WT) in the fractional Fourier transform (FrFT) domain. Firstly, the Riemann Lebesgue lemma for the FrFT is derived, and secondly, the CFrWT in terms of the FrFT is introduced. Based on the CFrWT, a different proof of the inner product relation and the inversion formula of the CFrWT are provided. Thereafter, a logarithmic uncertainty relation for the CFrWT is investigated and the convolution theorem related to the CFrWT is established using the convolution of the FrFT. The CFrWT on a generalized Sobolev space is introduced and its important properties are presented.

The continuous quaternion wavelet transform (CQWT) is a generalization of the classical continuous wavelet transform within the context of quaternion algebra. First of all, we show that the directional quaternion Fourier transform (QFT) uncertainty principle can be obtained using the component-wise QFT uncertainty principle. Based on this method, the directional QFT uncertainty principle using representation of polar coordinate form is easily derived. We derive a variation on uncertainty principle related to the QFT. We state that the CQWT of a quaternion function can be written in terms of the QFT and obtain a variation on uncertainty principle related to the CQWT. Finally, we apply the extended uncertainty principles and properties of the CQWT to establish logarithmic uncertainty principles related to generalized transform.

Observed signals are usually recorded as linear mixtures of original sources. Our purpose is to separate observed signals into original sources. To analyse observed signals, it is important to use several wavelet functions having different characteristics and compare their continuous wavelet transforms. The notion of the continuous multiwavelet transform and its essentials are introduced. An application to blind image separation is presented.

DOI: 10.1007/978-3-319-48812-7_75

DOI: 10.1007/978-3-319-48812-7_73

https://doi.org/10.11540/jsiamt.27.2_216

Based on the relationship between the Fourier transform (FT) and linear canonical transform (LCT), a logarithmic uncertainty principle and Hausdorff-Young inequality in the LCT domains are derived. In order to construct the windowed linear canonical transform (WLCT), Gabor filters associated with the LCT is introduced. Using the basic connection between the classical windowed Fourier transform (WFT) and the WLCT, a new proof of inversion formula for the WLCT is provided. This relation allows us to derive Lieb's uncertainty principle associated with the WLCT. Some useful properties of the WLCT such as bounded, shift, modulation, switching, orthogonality relation, and characterization of range are also investigated in detail. By the Heisenberg uncertainty principle for the LCT and the orthogonality relation property for the WLCT, the Heisenberg uncertainty principle for the WLCT is established. This uncertainty principle gives information how a complex function and its WLCT relate. Lastly, the logarithmic uncertainty principle associated with the WLCT is obtained.

Article ID 5874930, 11 pages, 2016. <br /> doi:10.1155/2016/5874930<br /> Open access journal<br /> http://www.hindawi.com/journals/aaa/2016/5874930/

The correspondence between quaternion convolution and quaternion product associated with the hypercomplex Fourier transforms is studied. Some useful properties of relationship between quaternion convolution and the hypercomplex Fourier transform are obtained.

We provide a short and simple proof of an uncertainty principle associated with the quaternion linear canonical transform (QLCT) by considering the fundamental relationship between the QLCT and the quaternion Fourier transform (QFT). We show how this relation allows us to derive the inverse transform and Parseval and Plancherel formulas associated with the QLCT. Some other properties of the QLCT are also studied.

The quaternion linear canonical transform (QLCT) can be thought as a generalization of the linear canonical transform (LCT) to quaternion algebra. The relationship between the QLCT and the quaternion Fourier transform (QFT) is derived. Based on this fact and properties of the QLCT, a logarithmic uncertainty principle associated with the quaternion linear canonical transform is established.

The peak latencies of auditory brainstem response (ABR) are useful to support of human auditory brain functional diagnosis. For example is useful to estimate audiogram in the human hearing test. In the previous study we proposed a method of analysis of the peak latencies of the ABR using stationary wavelet transform (SWT). Furthermore, in this paper we observe the ABR peak latencies near around human auditory threshold in which depicts in the audiogram by using SWT. At the same time we discuss about the optimum wavelet function to analyze the waveform of the human audiometry threshold depicted in the audiogram by using SWT.

It is well-known that an orthonormal scaling function generates an orthonormal wavelet function in the theory of multiresolution analysis. We consider two families of unitary operators. One is a family of extensions of the Hilbert transform called fractional Hilbert transforms. The other is a new family of operators which are a kind of modified translation operators. A fractional Hilbert transform of a given orthonormal wavelet (rasp. scaling) function is also an orthonormal wavelet (resp. scaling) function, although a fractional Hilbert transform of a scaling function has bad localization in many cases. We show that a modified translation of a scaling function is also a scaling function, and it generates a fractional Hilbert transform of the corresponding wavelet function. We also show a good localization property of the modified translation operators. The modified translation operators act on the Meyer scaling functions as the ordinary translation operators. We give a class of scaling functions, on which the modified translation operators act as the ordinary translation operators.

Generalized convolution and correlation theorems for the Wigner-Ville distribution (transform) associated with linear canonical transform (WVD-LCT) are established. The proposed theorems are modified forms of the convolution and correlation theorems of the linear canonical transform and classical Wigner-Ville distribution.

We show that the fractional Hilbert transform preserves the biorthogonal wavelet property. For a naturally generated biorthogonal wavelet pair constructed from a given biorthogonal scaling pair, we propose a design to construct a better biorthogonal scaling pair for the naturally generated biorthogonal wavelet pair using the fractional Hilbert transform. We also propose a calculation method for low-pass and high-pass filters.

N-tree discrete wavelet transforms based on the fractional Hilbert transforms of biorthogonal wavelets are proposed. Application of N-tree discrete wavelet transforms to separating mixtures of shifted images is considered. Some experimental results demonstrate the validity of the proposed method.

The relationship between the slow component of auditory brainstem response (ABR) and the number of averaging is investigated using the multi-resolution discrete stationary wavelet analysis. A new model to analyze the phase shifts of the spontaneous electroencephalogram (EEG) is presented.

The continuous reduced biquaternion wavelet transform (CRBWT) is a generalization of the classical wavelet transform to the reduced biquaternion setting. Some important properties of the CRBWT such as inner product, norm relation, and inversion formula are discussed. The convolution and correlation theorems for the CRBWT are established.

The simplest spatio-temporal mixing model of blind source separation for images is discussed. Shift parameters are estimated by total correlation functions of continuous wavelet transforms. An image separation algorithm using an annular sector multiwavelet is proposed.

A two-dimensional (2D) quaternion Fourier transform (QFT) defined with the kernel e(-i+j+k/root 3w.x) is proposed. Some fundamental properties, such as convolution, Plancherel and vector differential theorems, are established. The heat equation in quaternion algebra is presented as an example of the application of the QFT to partial differential equations. The wavelet transform is extended to quaternion algebra using the kernel of the QFT.

The purpose of blind source separation is to separate the original sources from the sensor array, without knowing the transmission channel characteristics. Besides methods based on independent component analysis, several methods based on time-frequency analysis have been proposed. In this paper, a new method of multistage separation is proposed, which improves our formerly proposed methods using the time-scale information matrix based on the continuous multiwavelet transform.

We briefly introduce the quaternion linear canonical transform (QLCT), which is a generalization of the quaternion Fourier transform (QFT) to the linear canonical transform (LCT) domain. We show that the QLCT can be reduced to the quaternion Fourier transform (QFT). We then derive the inverse transform of the QLCT using the properties of the QFT. We finally provide an example which describes the relationship between the QFT and the QLCT.

It is available to construct a discrete wavelet transform corresponding to an analytic wavelet transform. It is necessary to design good biorthogonal scaling functions which generate the Hilbert transforms of biorthogonal wavelets. In this paper, a design method of good scaling functions is proposed. We calculate the low-pass filter coefficients, which correspond to these scaling functions, directly from the original biorthogonal low-pass filter coefficients. In numerical experiments, it is shown that DWTs and IDWTs with twenty taps filter coefficients reconstruct signals with adequate precision.

In clinical applications, the rapid detection of the auditory brainstem response (ABR) peak characteristics is required. The input stimulus intensity influences the peak latency of the ABR waveform. In the sound stimulus intensity - latency curve (called I-L curve), the fifth peak (which is called wave V) of ABR waveform is used to diagnose conductive hearing loss or sensorineural hearing loss. In our previous study, we proved that the optimal decomposition level of the fast ABR is the level of D5 (frequency band is from 781 Hz to 1562 Hz), and explained that we used the reconstructed waveform of the ABR by inverse DWT (IDWT). Further more we proposed automated detection of the peak latency of wave V near-threshold according to input stimulus intensities using the template of normal I-L range. In this paper, we examine the following two points. Firstly, we apply the MRA to recorded ABR waveforms that are different number of averagings and we observe the reconstructed waveform at each resolution level. Secondly, we apply the MRA to recorded ABR waveforms that are obtained at the different sound stimulus intensity and we observe the reconstructed waveform at each resolution level.