42,057 research outputs found
Novel Fourier Quadrature Transforms and Analytic Signal Representations for Nonlinear and Non-stationary Time Series Analysis
The Hilbert transform (HT) and associated Gabor analytic signal (GAS)
representation are well-known and widely used mathematical formulations for
modeling and analysis of signals in various applications. In this study, like
the HT, to obtain quadrature component of a signal, we propose the novel
discrete Fourier cosine quadrature transforms (FCQTs) and discrete Fourier sine
quadrature transforms (FSQTs), designated as Fourier quadrature transforms
(FQTs). Using these FQTs, we propose sixteen Fourier-Singh analytic signal
(FSAS) representations with following properties: (1) real part of eight FSAS
representations is the original signal and imaginary part is the FCQT of the
real part, (2) imaginary part of eight FSAS representations is the original
signal and real part is the FSQT of the real part, (3) like the GAS, Fourier
spectrum of the all FSAS representations has only positive frequencies, however
unlike the GAS, the real and imaginary parts of the proposed FSAS
representations are not orthogonal to each other. The Fourier decomposition
method (FDM) is an adaptive data analysis approach to decompose a signal into a
set of small number of Fourier intrinsic band functions which are AM-FM
components. This study also proposes a new formulation of the FDM using the
discrete cosine transform (DCT) with the GAS and FSAS representations, and
demonstrate its efficacy for improved time-frequency-energy representation and
analysis of nonlinear and non-stationary time series.Comment: 22 pages, 13 figure
Construction of Hilbert Transform Pairs of Wavelet Bases and Gabor-like Transforms
We propose a novel method for constructing Hilbert transform (HT) pairs of
wavelet bases based on a fundamental approximation-theoretic characterization
of scaling functions--the B-spline factorization theorem. In particular,
starting from well-localized scaling functions, we construct HT pairs of
biorthogonal wavelet bases of L^2(R) by relating the corresponding wavelet
filters via a discrete form of the continuous HT filter. As a concrete
application of this methodology, we identify HT pairs of spline wavelets of a
specific flavor, which are then combined to realize a family of complex
wavelets that resemble the optimally-localized Gabor function for sufficiently
large orders.
Analytic wavelets, derived from the complexification of HT wavelet pairs,
exhibit a one-sided spectrum. Based on the tensor-product of such analytic
wavelets, and, in effect, by appropriately combining four separable
biorthogonal wavelet bases of L^2(R^2), we then discuss a methodology for
constructing 2D directional-selective complex wavelets. In particular,
analogous to the HT correspondence between the components of the 1D
counterpart, we relate the real and imaginary components of these complex
wavelets using a multi-dimensional extension of the HT--the directional HT.
Next, we construct a family of complex spline wavelets that resemble the
directional Gabor functions proposed by Daugman. Finally, we present an
efficient FFT-based filterbank algorithm for implementing the associated
complex wavelet transform.Comment: 36 pages, 8 figure
Instantaneous frequency and amplitude of complex signals based on quaternion Fourier transform
The ideas of instantaneous amplitude and phase are well understood for
signals with real-valued samples, based on the analytic signal which is a
complex signal with one-sided Fourier transform. We extend these ideas to
signals with complex-valued samples, using a quaternion-valued equivalent of
the analytic signal obtained from a one-sided quaternion Fourier transform
which we refer to as the hypercomplex representation of the complex signal. We
present the necessary properties of the quaternion Fourier transform,
particularly its symmetries in the frequency domain and formulae for
convolution and the quaternion Fourier transform of the Hilbert transform. The
hypercomplex representation may be interpreted as an ordered pair of complex
signals or as a quaternion signal. We discuss its derivation and properties and
show that its quaternion Fourier transform is one-sided. It is shown how to
derive from the hypercomplex representation a complex envelope and a phase.
A classical result in the case of real signals is that an amplitude modulated
signal may be analysed into its envelope and carrier using the analytic signal
provided that the modulating signal has frequency content not overlapping with
that of the carrier. We show that this idea extends to the complex case,
provided that the complex signal modulates an orthonormal complex exponential.
Orthonormal complex modulation can be represented mathematically by a polar
representation of quaternions previously derived by the authors. As in the
classical case, there is a restriction of non-overlapping frequency content
between the modulating complex signal and the orthonormal complex exponential.
We show that, under these conditions, modulation in the time domain is
equivalent to a frequency shift in the quaternion Fourier domain. Examples are
presented to demonstrate these concepts
Wavelet-based denoising for 3D OCT images
Optical coherence tomography produces high resolution medical images based on spatial and temporal coherence of the optical waves backscattered from the scanned tissue. However, the same coherence introduces speckle noise as well; this degrades the quality of acquired images.
In this paper we propose a technique for noise reduction of 3D OCT images, where the 3D volume is considered as a sequence of 2D images, i.e., 2D slices in depth-lateral projection plane. In the proposed method we first perform recursive temporal filtering through the estimated motion trajectory between the 2D slices using noise-robust motion estimation/compensation scheme previously proposed for video denoising. The temporal filtering scheme reduces the noise level and adapts the motion compensation on it. Subsequently, we apply a spatial filter for speckle reduction in order to remove the remainder of noise in the 2D slices. In this scheme the spatial (2D) speckle-nature of noise in OCT is modeled and used for spatially adaptive denoising. Both the temporal and the spatial filter are wavelet-based techniques, where for the temporal filter two resolution scales are used and for the spatial one four resolution scales.
The evaluation of the proposed denoising approach is done on demodulated 3D OCT images on different sources and of different resolution. For optimizing the parameters for best denoising performance fantom OCT images were used. The denoising performance of the proposed method was measured in terms of SNR, edge sharpness preservation and contrast-to-noise ratio. A comparison was made to the state-of-the-art methods for noise reduction in 2D OCT images, where the proposed approach showed to be advantageous in terms of both objective and subjective quality measures
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