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Showing papers on "Non-uniform discrete Fourier transform published in 2016"


Journal ArticleDOI
TL;DR: This work first develops conditions under, under which the short-time Fourier transform magnitude is an almost surely unique signal representation, then considers a semidefinite relaxation-based algorithm (STliFT) and provides recovery guarantees.
Abstract: The problem of recovering a signal from its Fourier magnitude is of paramount importance in various fields of engineering and applied physics. Due to the absence of Fourier phase information, some form of additional information is required in order to be able to uniquely, efficiently, and robustly identify the underlying signal. Inspired by practical methods in optical imaging, we consider the problem of signal reconstruction from the short-time Fourier transform (STFT) magnitude. We first develop conditions under, which the STFT magnitude is an almost surely unique signal representation. We then consider a semidefinite relaxation-based algorithm (STliFT) and provide recovery guarantees. Numerical simulations complement our theoretical analysis and provide directions for future work.

118 citations


Journal ArticleDOI
TL;DR: Demodulated band transform is ideally suited to efficient estimation of both stationary and non-stationary spectral and cross-spectral statistics with minimal susceptibility to spectral leakage.

104 citations


Journal ArticleDOI
TL;DR: In this article, a Fourier-Bessel-based algorithm for principal component analysis (PCA) was proposed for a large set of 2D images, and for each image, the set of its uniform rotations in the plane and their reflections.
Abstract: Cryo-electron microscopy nowadays often requires the analysis of hundreds of thousands of 2-D images as large as a few hundred pixels in each direction. Here, we introduce an algorithm that efficiently and accurately performs principal component analysis (PCA) for a large set of 2-D images, and, for each image, the set of its uniform rotations in the plane and their reflections. For a dataset consisting of $n$ images of size $L \times L$ pixels, the computational complexity of our algorithm is $O(nL^3 + L^4)$ , while existing algorithms take $O(nL^4)$ . The new algorithm computes the expansion coefficients of the images in a Fourier–Bessel basis efficiently using the nonuniform fast Fourier transform. We compare the accuracy and efficiency of the new algorithm with traditional PCA and existing algorithms for steerable PCA.

62 citations


Journal ArticleDOI
TL;DR: The proposed bi-directional algorithm is used to obtain accurate spectral-domain noise statistics for 2-soliton signals using numerical simulation and addresses the significant problem of rounding errors inherent in previously known techniques.
Abstract: The nonlinear Fourier transform represents a signal in terms of its continuous spectrum, discrete eigenvalues, and the corresponding discrete spectral amplitudes. This paper presents a new bi-directional algorithm for computing the discrete spectral amplitudes, which addresses the significant problem of rounding errors inherent in previously known techniques. We use the proposed method to obtain accurate spectral-domain noise statistics for 2-soliton signals using numerical simulation.

47 citations


Journal ArticleDOI
TL;DR: Synthetic data and field data examples show that the efficiency can be improved more than two times and the performance is slightly better in the frequency-space domain compared with the POCS method directly performed in the time- space domain, which demonstrates the validity of the proposed method.
Abstract: Sampling irregularity in observed seismic data may cause a significant complexity increase in subsequent processing. Seismic data interpolation helps in removing this sampling irregularity, for which purpose complex-valued curvelet transform is used, but it is time-consuming because of the huge size of observed data. In order to improve efficiency as well as keep interpolation accuracy, I first extract principal frequency components using forward Fourier transform. The size of the principal frequency-space domain data is at least halved compared with that of the original time-space domain data because the complex-valued components of the representation of a real-valued signal (i.e., a complex-valued signal with zero as its imaginary component) exhibit conjugate symmetry in the frequency domain. Then, the projection onto convex projection (POCS) method is used to interpolate frequency-space data based on complex-valued curvelet transform. Finally, interpolated seismic data in the time-space domain can be obtained using inverse Fourier transform. Synthetic data and field data examples show that the efficiency can be improved more than two times and the performance is slightly better in the frequency-space domain compared with the POCS method directly performed in the time-space domain, which demonstrates the validity of the proposed method.

37 citations


Journal ArticleDOI
TL;DR: The proposed DLCT is based on the well-known CM-CC-CM decomposition and has perfect reversibility, which doesn't hold in many existing DLCTs, and somewhat outperforms the CDDHFs-based method in the approximation accuracy.
Abstract: In this paper, a discrete LCT (DLCT) irrelevant to the sampling periods and without oversampling operation is developed. This DLCT is based on the well-known CM-CC-CM decomposition, that is, implemented by two discrete chirp multiplications (CMs) and one discrete chirp convolution (CC). This decomposition doesn’t use any scaling operation which will change the sampling period or cause the interpolation error. Compared with previous works, DLCT calculated by direct summation and DLCT based on center discrete dilated Hermite functions (CDDHFs), the proposed method implemented by FFTs has much lower computational complexity. The relation between the proposed DLCT and the continuous LCT is also derived to approximate the samples of the continuous LCT. Simulation results show that the proposed method somewhat outperforms the CDDHFs-based method in the approximation accuracy. Besides, the proposed method has approximate additivity property with error as small as the CDDHFs-based method. Most importantly, the proposed method has perfect reversibility, which doesn’t hold in many existing DLCTs. With this property, it is unnecessary to develop the inverse DLCT additionally because it can be replaced by the forward DLCT.

30 citations


Journal ArticleDOI
TL;DR: Experimental results demonstrate that the auto-focus measure proposed in this paper exhibits good unimodal performance, high accuracy, and very few local peaks, and is more resistant to Gaussian and impulse noise.

28 citations


Journal ArticleDOI
TL;DR: The high rejection to distortion in the electrical network, frequency adaptability, flexibility, and good performance in power quality monitor application render the proposed method a promising alternative for signal processing from the mains.
Abstract: This paper presents a three-phase harmonic and sequence components measurement method based on modulated sliding discrete Fourier transform (mSDFT) and a variable sampling period technique. The proposal allows measuring the harmonic components of a three-phase signal and computes the corresponding imbalance by estimating the instantaneous symmetrical components. In addition, an adaptive variable sampling period is used to obtain a sampling frequency multiple of the main frequency. By doing so, DFT typical errors, known as spectral leakage and picket-fence effect, are mitigated in steady state. The proposal is tested with different disturbances by simulation and experimental results. Some results obtained with a power quality monitor implemented with the proposed system are also presented. The high rejection to distortion in the electrical network, frequency adaptability, flexibility, and good performance in power quality monitor application render the proposed method a promising alternative for signal processing from the mains.

27 citations


Journal ArticleDOI
TL;DR: In this article, a non-iterative method for the construction of the Short-Time Fourier Transform (STFT) phase from the magnitude is presented, which is based on the direct relationship between the partial derivatives of the phase and the logarithm of the magnitude of the un-sampled STFT with respect to the Gaussian window.
Abstract: A non-iterative method for the construction of the Short-Time Fourier Transform (STFT) phase from the magnitude is presented. The method is based on the direct relationship between the partial derivatives of the phase and the logarithm of the magnitude of the un-sampled STFT with respect to the Gaussian window. Although the theory holds in the continuous setting only, the experiments show that the algorithm performs well even in the discretized setting (Discrete Gabor transform) with low redundancy using the sampled Gaussian window, the truncated Gaussian window and even other compactly supported windows like the Hann window. Due to the non-iterative nature, the algorithm is very fast and it is suitable for long audio signals. Moreover, solutions of iterative phase reconstruction algorithms can be improved considerably by initializing them with the phase estimate provided by the present algorithm. We present an extensive comparison with the state-of-the-art algorithms in a reproducible manner.

25 citations


Journal ArticleDOI
TL;DR: In this paper, a discrete Fourier transform method (DFTM) was proposed to discriminate between the signal of neutrons and gamma rays in organic scintillation detectors, which is based on the transformation of signals into the frequency domain using the sine and cosine Fourier transforms in combination with the DFT.
Abstract: A discrete Fourier transform method (DFTM) for discrimination between the signal of neutrons and gamma rays in organic scintillation detectors is presented. The method is based on the transformation of signals into the frequency domain using the sine and cosine Fourier transforms in combination with the discrete Fourier transform. The method is largely benefited from considerable differences that usually is available between the zero-frequency components of sine and cosine and the norm of the amplitude of the DFT for neutrons and gamma-ray signals. Moreover, working in frequency domain naturally results in considerable suppression of the unwanted effects of various noise sources that is expected to be effective in time domain methods. The proposed method could also be assumed as a generalized nonlinear weighting method that could result in a new class of pulse shape discrimination methods, beyond definition of the DFT. A comparison to the traditional charge integration method (CIM), as well as the frequency gradient analysis method (FGAM) and the wavelet packet transform method (WPTM) has been presented to demonstrate the applicability and efficiency of the method for real-world applications. The method, in general, shows better discrimination Figure of Merits (FoMs) at both the low-light outputs and in average over the studied energy domain. A noise analysis has been performed for all of the abovementioned methods. It reveals that the frequency domain methods (FGAM and DFTM) are less sensitive to the noise effects.

23 citations


Journal ArticleDOI
TL;DR: In this paper, a new method based on the shearlet transform is presented for phase extraction in fringe projection profilometry (FPP) from a single fringe pattern, which is more effective and accurate than the Fourier transform method and wavelet transform method.

01 Jan 2016
TL;DR: The the fourier transform and its application is universally compatible with any devices to read and is available in the book collection an online access to it is set as public so you can download it instantly.
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Posted Content
TL;DR: This study presents time-frequency analysis by the Fourier transform which maps the time-domain signal into the frequency-domain and defines the concept of 'frequen-taneous time' which is frequency derivative of phase.
Abstract: The nonstationary nature of signals and nonlinear systems require the time-frequency representation. In time-domain signal, frequency information is derived from the phase of the Gabor's analytic signal which is practically obtained by the inverse Fourier transform. This study presents time-frequency analysis by the Fourier transform which maps the time-domain signal into the frequency-domain. In this study, we derive the time information from the phase of the frequency-domain signal and obtain the time-frequency representation. In order to obtain the time information in Fourier domain, we define the concept of 'frequen-taneous time' which is frequency derivative of phase. This is very similar to the group delay, which is also defined as frequency derivative of phase and it provide physical meaning only when it is positive. The frequen-taneous time is always positive or negative depending upon whether signal is defined for only positive or negative times, respectively. If a signal is defined for both positive and negative times, then we divide the signal into two parts, signal for positive times and signal for negative times. The proposed frequentaneous time and Fourier transform based time-frequency distribution contains only those frequencies which are present in the Fourier spectrum. Simulations and numerical results , on many simulated as well as read data, demonstrate the efficacy of the proposed method for the time-frequency analysis of a signal.

01 Jan 2016
TL;DR: Thank you very much for downloading the nonuniform discrete fourier transform and its applications in signal processing and maybe you have knowledge that, people have search hundreds of times for their favorite books like this, but end up in malicious downloads.
Abstract: Thank you very much for downloading the nonuniform discrete fourier transform and its applications in signal processing. Maybe you have knowledge that, people have search hundreds times for their favorite books like this the nonuniform discrete fourier transform and its applications in signal processing, but end up in malicious downloads. Rather than enjoying a good book with a cup of coffee in the afternoon, instead they cope with some infectious virus inside their computer.

Proceedings ArticleDOI
01 Oct 2016
TL;DR: With AWFT, this work is able to extract meaningful intrinsic localized orientation-sensitive structures on surfaces, and use them in applications such as shape segmentation, salient point detection, feature point description, and matching.
Abstract: We propose Anisotropic Windowed Fourier Transform (AWFT), a framework for localized space-frequency analysis of deformable 3D shapes. With AWFT, we are able to extract meaningful intrinsic localized orientation-sensitive structures on surfaces, and use them in applications such as shape segmentation, salient point detection, feature point description, and matching. Our method outperforms previous approaches in the considered applications.

Proceedings ArticleDOI
12 Jul 2016
TL;DR: This work introduces the periodic nonlinear Fourier transform (PNFT) and proposes a proof-of-concept communication system based on it by using a simple waveform with known nonlinear spectrum (NS).
Abstract: In this work we introduce the periodic nonlinear Fourier transform (PNFT) and propose a proof-of-concept communication system based on it by using a simple waveform with known nonlinear spectrum (NS). We study the performance (addressing the bit-error-rate (BER), as a function of the propagation distance) of the transmission system based on the use of the PNFT processing method and show the benefits of the latter approach. By analysing our simulation results for the system with lumped amplification, we demonstrate the decent potential of the new processing method.

01 Jan 2016
TL;DR: The the fourier transform and its application is universally compatible with any devices to read and is available in the book collection an online access to it is set as public so you can download it instantly.
Abstract: Thank you very much for downloading the fourier transform and its application. As you may know, people have search hundreds times for their chosen books like this the fourier transform and its application, but end up in infectious downloads. Rather than enjoying a good book with a cup of coffee in the afternoon, instead they are facing with some malicious bugs inside their desktop computer. the fourier transform and its application is available in our book collection an online access to it is set as public so you can download it instantly. Our book servers spans in multiple countries, allowing you to get the most less latency time to download any of our books like this one. Merely said, the the fourier transform and its application is universally compatible with any devices to read.

Journal ArticleDOI
TL;DR: This paper considers using a new interval-valued inversion process within this new framework to upsample a signal, i.e. reconstruct a high resolution signal with a low resolution signal, in a semi-blind context.

Proceedings ArticleDOI
19 Aug 2016
TL;DR: This work develops a hexagonal FFT in ASA coordinates that uses only the standard Fourier transform, allowing the user to implement the hexagonally sampled FFT using standard FFT routines.
Abstract: The discrete Fourier transform is an important tool for processing digital images. Efficient algorithms for computing the Fourier transform are known as fast Fourier transforms (FFTs). One of the most common of these is the Cooley-Tukey radix-2 decimation algorithm that efficiently transforms one-dimensional data into its frequency domain representation. The orthogonality of rectangular sampling allows the separability of the Fourier kernel which enables the use of the Cooley-Tukey algorithm on two-dimensional digital images that have been sampled rectangularly. Hexagonal sampling provides many benefits over rectangular sampling, but it does not result in the orthogonal rows and columns that can be transformed independently as is done with rectangular samples. Use of the Array Set Addressing (ASA) coordinate system for hexagonally sampled images has been shown to provide a separable Fourier kernel, leading to an efficient FFT, however its implementation is composed of nonstandard transforms that require custom routines to evaluate. This work develops a hexagonal FFT in ASA coordinates that uses only the standard Fourier transform, allowing the user to implement the hexagonal FFT using standard FFT routines.

Journal ArticleDOI
TL;DR: In this article, a spectral analysis technique for multilevel modulation was proposed, which separates the multileve PWM waveform into a spectral image of the reference, and sideband basis functions which are then expanded using a one-dimensional Fourier series.
Abstract: This paper presents a novel spectral analysis technique for multilevel modulation. Conventionally, such analyses use a double Fourier series technique, but this approach can become intractable when complex reference waveforms (e.g., multilevel space vector offsets) and regular sampling processes are considered. In contrast, the strategy proposed in this paper separates the multilevel pulse width modulation (PWM) waveform into a spectral image of the reference, and sideband basis functions which are then expanded using a one-dimensional Fourier series. The coefficients of this Fourier series are defined by a one-dimensional Fourier integral that is simpler in form compared to the corresponding double integral associated with the double Fourier series. This analysis technique naturally incorporates regular sampling, and a discrete formulation is developed that enables complex PWM reference waveforms, including centered space vector offsets, to be solved. Results of this analysis are validated against previously published multilevel inverter double Fourier series results and matching switched simulations.

Journal ArticleDOI
TL;DR: This paper defines the time spread and the fractional frequency spread for discrete signals and derives an uncertainty relation between these two spreads, which are extended to the linear canonical transform, which is a generalized form of the FRFT.
Abstract: The fractional Fourier transform (FRFT), which generalizes the classical Fourier transform, has gained much popularity in recent years because of its applications in many areas, including optics, radar, and signal processing. There are relations between duration in time and bandwidth in fractional frequency for analog signals, which are called the uncertainty principles of the FRFT. However, these relations are only suitable for analog signals and have not been investigated in discrete signals. In practice, an analog signal is usually represented by its discrete samples. The purpose of this paper is to propose an equivalent uncertainty principle for the FRFT in discrete signals. First, we define the time spread and the fractional frequency spread for discrete signals. Then, we derive an uncertainty relation between these two spreads. The derived results are also extended to the linear canonical transform, which is a generalized form of the FRFT.

Proceedings ArticleDOI
20 Mar 2016
TL;DR: This paper proposes to leverage the recently introduced Flexible Approximate MUltilayer Sparse Transforms (FAST) in order to compute approximate FFTs on graphs, showing good potential.
Abstract: Signal processing on graphs is a recent research domain that seeks to extend classical signal processing tools such as the Fourier transform to irregular domains given by a graph. In such a graph setting, a way to rapidly apply the Fourier transform, i.e. a Fast Fourier Transform (FFT), is lacking. In this paper, we propose to leverage the recently introduced Flexible Approximate MUlti-layer Sparse Transforms (FAST) in order to compute approximate FFTs on graphs. The approach is first described, then validated on several types of classical graphs and finally used for fast filtering, showing good potential.

Journal ArticleDOI
TL;DR: Compared with three other popular methods, product high-order match phase transform, TC-dechirp Clean and modified discrete chirp Fourier transform, the proposed SCDCT-based method is more computationally efficient and has better estimation performance in low signal-to-noise ratio (SNR) circumstance.

Journal ArticleDOI
TL;DR: A second method is proposed based on expanding the DFT of the interferogram and the spectrum by a Haar or box function and it recovered the spectrum and got rid of the fictitious spectral components and spectral harmonic overlap.
Abstract: We analyze the Fourier transform spectrometer based on a symmetric/asymmetric Fabry-Perot interferometer. In this spectrometer, the interferogram is obtained by recording the intensity as a function of the interferometer length. Then, we recover the spectrum by applying the discrete Fourier transform (DFT) directly on the interferogram. This technique results in spectral harmonic overlap and fictitious wavenumber components outside the original spectral range. For this purpose, in this work, we propose a second method to recover the spectrum. This method is based on expanding the DFT of the interferogram and the spectrum by a Haar or box function. By this second method, we recovered the spectrum and got rid of the fictitious spectral components and spectral harmonic overlap.

Journal ArticleDOI
TL;DR: A new decomposition called CM-CC-CM-CC decomposition is proposed, which decomposes the 2D NsLCT into two 2D CMs and two2D chirp convolutions, which have a perfect reversibility property, meaning that one can reconstruct the input signal/image losslessly from the output.
Abstract: As a generalization of the 2D Fourier transform (2D FT) and 2D fractional Fourier transform, the 2D nonseparable linear canonical transform (2D NsLCT) is useful in optics and signal and image processing. To reduce the digital implementation complexity of the 2D NsLCT, some previous works decomposed the 2D NsLCT into several low-complexity operations, including 2D FT, 2D chirp multiplication (2D CM), and 2D affine transformations. However, 2D affine transformations will introduce interpolation error. In this paper, we propose a new decomposition called CM-CC-CM-CC decomposition, which decomposes the 2D NsLCT into two 2D CMs and two 2D chirp convolutions. No 2D affine transforms are involved. Simulation results show that the proposed methods have higher accuracy, lower computational complexity, and smaller error in the additivity property compared with the previous works. Plus, the proposed methods have a perfect reversibility property, meaning that one can reconstruct the input signal/image losslessly from the output.

Journal ArticleDOI
TL;DR: A procedure for the selection of optimal parameters with controlled accuracies is proposed to reach the best performance of ENUF method, a good alternative to the standard Ewald method with the same computational precision but a dramatically higher computational efficiency.
Abstract: We present new algorithms to improve the performance of ENUF method (F. Hedman, A. Laaksonen, Chem. Phys. Lett. 425, 2006, 142) which is essentially Ewald summation using Non-Uniform FFT (NFFT) technique. A NearDistance algorithm is developed to extensively reduce the neighbor list size in real-space computation. In reciprocal-space computation, a new algorithm is developed for NFFT for the evaluations of electrostatic interaction energies and forces. Both real-space and reciprocal-space computations are further accelerated by using graphical processing units (GPU) with CUDA technology. Especially, the use of CUNFFT (NFFT based on CUDA) very much reduces the reciprocal-space computation. In order to reach the best performance of this method, we propose a procedure for the selection of optimal parameters with controlled accuracies. With the choice of suitable parameters, we show that our method is a good alternative to the standard Ewald method with the same computational precision but a dramatically higher computational efficiency.

Proceedings ArticleDOI
01 Jun 2016
TL;DR: In this paper, the authors proposed a new method combining the Short Time Fourier Transform (STFT) and Zoom-FRFT, to estimate the MLFM signal parameters, which can improve both the parameters estimation precision and the computation cost significantly.
Abstract: Traditional parameters estimation methods for the Multi-component Linear Frequency Modulation (MLFM) signal based on Fractional Fourier Transform (FRFT) could not achieve the satisfactory precision and the lesser computation cost. In this paper, we propose a new method combining the Short Time Fourier Transform (STFT) and Zoom-FRFT, to estimate the MLFM signal parameters. Firstly, the coarse estimation can be achieved from the straight line detection of short time Fourier spectrum. Then, we could calculate the analysis range of transform order and Fractional domain spectrum by the FRFT domain spectrum distribution characteristics of interference signal. Finally, we can estimate the optimal order and precise peak position by the optimum seeking method in the Zoom-FRFT domain. The simulation results show that, this method can improve both the parameters estimation precision and the computation cost significantly, and can flexibly choose the window width and the zoom times.

Journal ArticleDOI
TL;DR: A fast algorithm is described for the 2-D left-side QDFT which is based on the concept of the tensor representation when the color or four-componnrnt quaternion image is described by a set of 1-D quaternions signals and the 1- D left- side QDFTs over these signals determine values of the2-Dleft-sideQDFT at corresponding subset of frequency-points.
Abstract: We describe a fast algorithm for the 2-D left-side QDFT which is based on the concept of the tensor representation when the color or four-componnrnt quaternion image is described by a set of 1-D quaternion signals and the 1-D left-side QDFTs over these signals determine values of the 2-D left-side QDFT at corresponding subset of frequency-points. The efficiency of the tensor algorithm for calculating the fast left-side 2-D QDFT is described and compared with the existent methods.  The proposed algorithm of the 2r×2r-point 2-D QDFT uses 18N2 less multiplications than the well-known methods: • column-row method • method of symplectic decomposition.  The proposed algorithm is simple to apply and design, which makes it very practical in color image processing in the frequency domain.  The method of quaternion image tensor representation is uique in a sense that it can be used for both left-sida and right-side 2-D QDFTs. 3 Inroduction – Quanterions in Imaging  The quaternion can be considered 4-dimensional generation of a complex number with one real part and three imaginary parts. Any quaternion may be represented in a hyper-complex form Q = a + bi + cj + dk = a + (bi + cj + dk), where a, b, c, and d are real numbers and i, j, and k are three imaginary units with the following multiplication laws: ij = −ji = k, jk = −kj = i, ki = −ik = −j, i2 = j2 = k2 = ijk = −1.  The commutativity does not hold in quaternion algebra, i.e., Q1Q2≠Q2Q1.  A unit pure quaternion is μ=iμi+jμj+kμk such that |μ| = 1, μ 2 = −1 For instance, the number μ=(i+j+k)/√3, μ=(i+j)/√2, and μ=(i-k)/√2  The exponential number is defined as exp(μx) = cos(x) + μ sin(x) = cos(x) + iμi sin(x) +jμj sin(x) +kμk sin(x) 4 RGB Model for Color Images 5  A discrete color image fn,m in the RGB color space can be transformed into imaginary part of quaternion numbers form by encoding the red, green, and blue components of the RGB value as a pure quaternion (with zero real part): fn,m = 0 + (rn,mi + gn,mj + bn,mk) Figure 1: RBG color cube in quaternion space.  The advantage of using quaternion based operations to manipulate color information in an image is that we do not have to process each color channel independently, but rather, treat each color triple as a whole unit. Calculation of the left-side 1-D QDFT  Let fn =(an,bn,cn,dn)=an +ibn +jcn +kdn be the quaternion signal of length N. The left-side 1-D quaternion DFT ( LS QDFT) is defined as 6 If we denote the N-point LS 1-D DFTs of the parts an, bn, cn, and dn of the quaternion signal fn by Ap, Bp, Cp, and Dp, respectively, we can calculate of the LS 1-D QDFT as If the real part is zero, an =0, and fn =(0,bn,cn,dn)=an +ibn +jcn +kdn , the number of operations of multiplication and addition can be estimated as Multiplications and Additions for the left-side 1-D QDFT  In the general case of the quaternion signal fn, the number of operations of multiplication and addition for LS 1-D QDFT can be estimated as 7 The number of operations for the left-side 1-D QDFT can be estimated as Here, we consider that for the fast N-point discrete paired transform-based FFT, the estimation for multiplications and additions are and two 1-D DFTs with real inputs can be calculated by one DFT with complex input, (1) Number of multiplications: Special case 8 The number of operations of multiplication and addition equal or 8N operations of real multiplication less than in (1). The direct and inverse left-side 2-D QDFTs  Given color-in-quaternion image fn,m =an,m +ibn,m +jcn,m +kdn,m , we consider the concept of the left-side 2-D QDFT in the following form: 9 1. Column-row algorithm: The calculation of the separable 2-D N×N-point QDFT by formula 2. The calculation the LS 2-D QDFT by the symplectic decomposition of the color image requires 2N N-point LS 1-D QDFTs. Each of the 1-D QDFT requires two N-point LS 1-D DFTs. Therefore, the column-row method uses 4N N-point LS 1-D DFTs and multiplications or The inverse left-side 2-D QDFT is: Example: N×N-point left-side 2-D QDFT 10 Figure 2. (a) The color image of size 1223×1223 and (b) the 2-D left-side quaternion discrete Fourier transform of the color-inqiuaternion image (in absolute scale and cyclically shifted to the middle). Tensor Representation of the regular 2-D DFT Let fn,m be the gray-scale image of size N×N.  The tensor representation of the image fn,m is the 2D-frequency-and-1D-time representation when the image is described by a set of 1-D splitting-signals each of length N 11 The components of the signals are the ray-sums of the image along the parallel lines Each splitting-signals defines 2-D DFT at N frequency-points of the set on the cartesian lattice Example: Tensor Representation of the 2-D DFT 1-D splitting-signal of the tensor represntation of the image 512×512 12 Figure 3. (a) The Miki-Anoush-Mini image, (b) splitting-signal for the frequency-point (4,1), (c) magnitude of the shifted to the middle 1-D DFT of the signal, and (d) the 2D DFT of the image with the frequency-points of the set T4,1. Tensor Representation of the left-side 2-D QDFT Let fn,m =an,m +ibn,m +jcn,m +kdn,m be the quaternion image of size N×N, (an,m =0). In the tensor representation, the quaternion image is represented by a set of 1-D quaternion splitting-signals each of length N and generated by a set of frequencies (p,s), 13 The components of the signals are defined as Here, the subsets Property of the TT: Example: Tensor Representation of the 2-D LS QDFT The splitting-signal of the tensor represntation of the color image 1223×1223: 14 Figure 5. The 123-point left-side DFT of the (1,4) quaternion splitting-signal; (a) the real part and (b) the i-component of the signal. Figure 4. Color image and (a,b,c) components of the splitting-signal generated by (1,4). Example: Tensor Representation of the 2-D LS QDFT 15 Figure 7. (a) The real part and (b) the imaginary part of the left-side 2-D QDFT of the 2-D color-in-quaternion `girl Anoush\" image. Figure 6. (a) The 1-D left-side QDFT the quaternion splitting-signal f1,4,t (in absolute scale), and (b) the location of 1223 frequency-points of the set T1,4 on the Cartesian grid, wherein this 1-D LS QDFT equals the 2-D LS QDFT of the quaternion image. μ=(i+2j+k)/√6 Tensor Transform: Direction Quaternion Image Components Color image can be reconstructed by its 1-D quaternion splitting-signals or direction color image components defined by 16 Statement 1: The discrete quaternion image of size N×N, where N is prime, can be composed from its (N+1) quaternion direction images or splitting-signals as Color-or-Quaternion Image is The Sum of Direction Image Components 17 Figure 8: (a) The color image and direction images generated by (p,s) equal (b) (1,1), (c) (1,2), and (d) (1,4). The Paired Image Representation: Splitting-Signals and Direction Quaternion Image Components The tensor transform, or representation is redndant for the case N×N, where N is a power of 2. Therefore the tensor transform is modified and new1-D quaternion splitting-signals or direction color image components are calculated by 18 Statement 2: The discrete quaternion image of size N×N, where N=2r, r>1, can be composed from its (3N−2) quaternion direction images as Here JʹN,N is a set of generators (p,s). Such representation of the quaternion image is called the paired transform; it is unitary and therefore not redundant.

Journal ArticleDOI
TL;DR: A new linear predictive data extrapolation approach is proposed that involves partitioning the spectrum into multiple spectral subbands and using a different autoregressive (AR) process to model each subband to address the detection and estimation problem of multiple sinusoids in a discrete data sequence.
Abstract: In this letter, we propose a new linear predictive (LP) data extrapolation approach. It involves partitioning the spectrum into multiple spectral subbands and using a different autoregressive (AR) process to model each subband. The new extrapolation approach is then combined with the classical discrete Fourier transform (DFT) to produce a new hybrid LP-DFT spectral estimator to address the detection and estimation problem of multiple sinusoids in a discrete data sequence. Simulation results demonstrate the superiority of the proposed hybrid technique over an existing popular hybrid LP-DFT technique, where a single AR process is used to model the entire spectrum of the data sequence.

Journal ArticleDOI
TL;DR: The effect and the utility of flexible sensor location to make optimal use of a limited number of sensor points are investigated and the nonuniform fast Fourier transform is used to handle this issue and shows its feasibility in three-dimensional experiments with real and synthetic data.
Abstract: To obtain the initial pressure from the collected data on a planar sensor arrangement in photoacoustic tomography, there exists an exact analytic frequency-domain reconstruction formula. An efficient realization of this formula needs to cope with the evaluation of the data’s Fourier transform on a nonequispaced mesh. We use the nonuniform fast Fourier transform to handle this issue and show its feasibility in three-dimensional experiments with real and synthetic data. This is done in comparison to the standard approach that uses linear, polynomial, or nearest neighbor interpolation. Moreover, we investigate the effect and the utility of flexible sensor location to make optimal use of a limited number of sensor points. The computational realization is accomplished by the use of a multidimensional nonuniform fast Fourier algorithm, where nonuniform data sampling is performed both in frequency and spatial domain. Examples with synthetic and real data show that both approaches improve image quality.