Showing papers on "Fractional Fourier transform published in 2016"
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
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
TL;DR: In the context of classical optics, this work implements discrete fractional Fourier transforms of exemplary wave functions and experimentally demonstrate the shift theorem and applies this approach in the quantum realm to Fourier transform separable and path-entangled biphoton wave functions.
Abstract: Fourier transforms, integer and fractional, are ubiquitous mathematical tools in basic and applied science. Certainly, since the ordinary Fourier transform is merely a particular case of a continuous set of fractional Fourier domains, every property and application of the ordinary Fourier transform becomes a special case of the fractional Fourier transform. Despite the great practical importance of the discrete Fourier transform, implementation of fractional orders of the corresponding discrete operation has been elusive. Here we report classical and quantum optical realizations of the discrete fractional Fourier transform. In the context of classical optics, we implement discrete fractional Fourier transforms of exemplary wave functions and experimentally demonstrate the shift theorem. Moreover, we apply this approach in the quantum realm to Fourier transform separable and path-entangled biphoton wave functions. The proposed approach is versatile and could find applications in various fields where Fourier transforms are essential tools.
98 citations
TL;DR: This paper investigates the GSE for lowpass and bandpass signals with multiple sampling rates in the fractional Fourier transform (FRFT) domain and derives the periodic nonuniform sampling scheme and the derivative interpolation method by designing different fractional filters and selecting specific sampling rates.
Abstract: The objective of generalized sampling expansion (GSE) is the reconstruction of an unknown, continuously defined function $f\left(t\right)$ from samples of the responses from $M$ linear time-invariant (LTI) systems that are each sampled using the $1/M$ th Nyquist rate. In this paper, we investigate the GSE for lowpass and bandpass signals with multiple sampling rates in the fractional Fourier transform (FRFT) domain. First, we propose an improvement of Papoulis’ GSE, which has multiple sampling rates in the FRFT domain. Based on the proposed GSE, we derive the periodic nonuniform sampling scheme and the derivative interpolation method by designing different fractional filters and selecting specific sampling rates. In addition, the Papoulis GSE and the previous GSE associated with FRFT are shown to be special instances of our results. Second, we address the problem of the GSE of fractional bandpass signals. A new GSE for fractional bandpass signals with equal sampling rates is derived. We show that the restriction of an even number of channels in the GSE for fractional bandpass signals is unnecessary, and perfect signal reconstruction is possible for any arbitrary number of channels. Further, we develop the GSE for a fractional bandpass signal with multiple sampling rates. Lastly, we discuss the application of the proposed method in the context of single-image super-resolution reconstruction based on GSE. Illustrations and simulations are presented to verify the validity and effectiveness of the proposed results.
98 citations
TL;DR: This coherent detection algorithm is an extension of the scaled inverse Fourier transform (SCIFT)-based detection algorithm and can acquire a better antinoise performance and higher peak to sidelobe ratios along the Doppler frequency and the scaled range cell.
Abstract: In this paper, we propose a coherent detection algorithm for high-speed targets by employing the parametric symmetric autocorrelation function and the frequency-domain deramp-keystone transform (FDDKT). This coherent detection algorithm is an extension of the scaled inverse Fourier transform (SCIFT)-based detection algorithm. However, compared to the SCIFT-based detection algorithm, the proposed coherent detection algorithm can acquire a better antinoise performance and higher peak to sidelobe ratios along the Doppler frequency and the scaled range cell. Simulations and analyses for synthetic models and the real radar data are provided to verify the effectiveness of the proposed coherent detection algorithm.
87 citations
TL;DR: By virtue of the fractional Fourier transform (FrFT), an accurate and efficient method for parameter estimation is proposed, which outperforms the existing FrFT domain filter and short-time Fr FT domain filter in multicomponent separation, and that the proposed parameter estimation scheme surpasses the matching pursuit based method in accuracy.
Abstract: Because of relative motion between transmitter and receiver, the wideband underwater acoustic multipath channel can be more accurately described by a multi-scale multi-lag (MSML) model. The signal components received from different paths can be differentiated in terms of Doppler scales, time delays, and amplitudes. Estimation of these parameters is essential for many underwater applications. In this paper, by virtue of the fractional Fourier transform (FrFT), an accurate and efficient method for parameter estimation is proposed. The algorithm proceeds in an iterative manner, returning parameter estimates of the most dominant signal component successively. With the linear frequency modulation signal employed as the probe signal, the estimation of the Doppler scale factor of each dominant component can be converted into a process of searching for the optimal fractional angle/order of the received signal's FrFT. At each iteration, a sub-iteration is contained to adjust the optimal fractional order. The pulse compression techniques for LFM signal, in both time domain and FrFT domain, are utilized to obtain precise parameter estimates. Once the parameters of a multipath are estimated, the corresponding component will be separated from the received signal—through eliminating the resampled, delayed and attenuated version of the original probe signal. The sparsity of the underwater acoustic channel is considered to reduce the amount of calculation. Simulation results confrm that the proposed algorithm outperforms the existing FrFT domain filter and short-time FrFT domain filter in multicomponent separation, and that the proposed parameter estimation scheme surpasses the matching pursuit based method in accuracy.
75 citations
TL;DR: The proposed computer-aided diagnosis system is effective in detecting abnormal breasts and is better than both the proposed “weighted-type fractional Fourier transform+principal component analysis+k-nearest neighbors” and other five state-of-the-art approaches in terms of sensitivity, specificity, and accuracy.
Abstract: Abnormal breast can be diagnosed using the digital mammography. Traditional manual interpretation method cannot yield high accuracy. In this study, we proposed a novel computer-aided diagnosis system for detecting abnormal breasts. Our dataset contains 200 mammogram images with size of 1024 × 1024. First, we segmented the region of interest from mammogram images. Second, the fractional Fourier transform was employed to obtain the unified time–frequency spectrum. Third, spectrum coefficients were reduced by principal component analysis. Finally, both support vector machine and k-nearest neighbors were used and compared. The proposed “weighted-type fractional Fourier transform+principal component analysis+support vector machine” achieved sensitivity of 92.22% ± 4.16%, specificity of 92.10% ± 2.75%, and accuracy of 92.16% ± 3.60%. It is better than both the proposed “weighted-type fractional Fourier transform+principal component analysis+k-nearest neighbors” and other five state-of-the-art approaches in term...
72 citations
TL;DR: A novel approach based on compressive sensing and chaos is proposed for simultaneously compressing, fusing and encrypting multi-modal images that reduces data volume but also simplifies keys, which improves the efficiency of transmitting data and distributing keys.
66 citations
TL;DR: In this paper, a nonlinear, invertible, low-level image processing transform based on combining the well-known Radon transform for image data, and the 1D cumulative distribution transform proposed earlier is described.
Abstract: Invertible image representation methods (transforms) are routinely employed as low-level image processing operations based on which feature extraction and recognition algorithms are developed. Most transforms in current use (e.g., Fourier, wavelet, and so on) are linear transforms and, by themselves, are unable to substantially simplify the representation of image classes for classification. Here, we describe a nonlinear, invertible, low-level image processing transform based on combining the well-known Radon transform for image data, and the 1D cumulative distribution transform proposed earlier. We describe a few of the properties of this new transform, and with both theoretical and experimental results show that it can often render certain problems linearly separable in a transform space.
64 citations
TL;DR: The fractional Fourier transform makes it possible to represent the SAR signal in a rotated joint time-frequency plane and performs optimal processing and analysis of these residual chirp signals, and the along-track defocus can be compensated for and the target's azimuthal speed estimated.
Abstract: This paper studies the effects of stationary-based processing of moving ship signatures in synthetic aperture radar (SAR) imagery and introduces a methodology to estimate and compensate for them. SAR imaging of moving targets usually results in residual chirps in the azimuthal SLC processed signal. The fractional Fourier transform (FrFT) makes it possible to represent the SAR signal in a rotated joint time-frequency plane and performs optimal processing and analysis of these residual chirp signals. The along-track defocus can thus be compensated for and the target's azimuthal speed estimated. The impact of higher order motion terms (e.g., acceleration) has been also considered. Experiments were conducted on a large number of ship signatures extracted from Radarsat-2 Multi Look Fine and Ultra Fine SAR images. An intercomparison with a standard Doppler Sublook Decomposition Method (SDM) is carried out, as well as a complete performance analysis with AIS data as ground truth.
57 citations
TL;DR: In this paper, the high-order sparse Radon transform (HOSRT) method is introduced to protect the amplitude variation with offset information during the multiple subtraction procedures, and a fast nonlinear filter is adopted in the adaptive subtraction step to avoid the orthogonality assumption.
Abstract: The Radon transform is widely used for multiple elimination. Since the Radon transform is not an orthogonal transform, it cannot preserve the amplitude of primary reflections well. The prediction and adaptive subtraction method is another widely used approach for multiple attenuation, which demands that the primaries are orthogonal with the multiples. However, the orthogonality assumption is not true for non-stationary field seismic data. In this paper, the high-order sparse Radon transform (HOSRT) method is introduced to protect the amplitude variation with offset information during the multiple subtraction procedures. The HOSRT incorporates the high-resolution Radon transform with the orthogonal polynomial transform. Because the Radon transform contains the trajectory information of seismic events and the orthogonal polynomial transform contains the amplitude variation information of seismic events, their combination constructs an overcomplete transform and obtains the benefits of both the high-resolution property of the Radon transform and the amplitude preservation of the orthogonal polynomial transform. A fast nonlinear filter is adopted in the adaptive subtraction step in order to avoid the orthogonality assumption that is used in traditional adaptive subtraction methods. The application of the proposed approach to synthetic and field data examples shows that the proposed method can improve the separation performance by preserving more useful energy.
01 Jun 2016
TL;DR: A novel feature extraction scheme for automatic classification of musical instruments using Fractional Fourier Transform (FrFT)-based Mel Frequency Cepstral Coefficient (MFCC) features that shows significant improvement in classification accuracy and robustness against Additive White Gaussian Noise compared to other conventional features.
Abstract: This paper presents a novel feature extraction scheme for automatic classification of musical instruments using Fractional Fourier Transform (FrFT)-based Mel Frequency Cepstral Coefficient (MFCC) features. The classifier model for the proposed system has been built using Counter Propagation Neural Network (CPNN). The discriminating capability of the proposed features have been maximized for between-class instruments and minimized for within-class instruments compared to other conventional features. Also, the proposed features show significant improvement in classification accuracy and robustness against Additive White Gaussian Noise (AWGN) compared to other conventional features. McGill University Master Sample (MUMS) sound database has been used to test the performance of the system.
TL;DR: In this paper, the Taylor-Fourier transform (DTFT) is used to identify low-frequency electromechanical modes in power systems, based on the time-frequency analysis of nonlinear signals that arise after a large disturbance.
Abstract: The digital Taylor-Fourier transform (DTFT) is used to identify low-frequency electromechanical modes in power systems. The identification process is based on the time-frequency analysis of nonlinear signals that arise after a large disturbance. The DTFT creates a signal decomposition, from which mono-component signals are extracted by spectral analysis using a filter bank. This analysis is accomplished through sliding-window data, which is updated each sample, yielding estimates of the reconstructed signal and providing information of its instantaneous damping and frequency. Results demonstrate the applicability of the proposition.
TL;DR: A new method based on fractional Fourier transform based on single-hidden-layer feed-forward neural network trained by the Levenberg–Marquardt algorithm to detect hearing loss more efficiently and accurately.
Abstract: In order to detect hearing loss more efficiently and accurately, this study proposed a new method based on fractional Fourier transform (FRFT). Three-dimensional volumetric magnetic resonance images were obtained from 15 patients with left-sided hearing loss (LHL), 20 healthy controls (HC), and 14 patients with right-sided hearing loss (RHL). Twenty-five FRFT spectrums were reduced by principal component analysis with thresholds of 90%, 95%, and 98%, respectively. The classifier is the single-hidden-layer feed-forward neural network (SFN) trained by the Levenberg–Marquardt algorithm. The results showed that the accuracies of all three classes are higher than 95%. In all, our method is promising and may raise interest from other researchers.
TL;DR: In this article, an integral transform J[Φ(τ)] = 1/μ ∫ 0∞Φ (τ)e−μτ dτ is proposed for the first time.
Abstract: In this paper, an new integral transform J[Φ(τ)] =1/μ ∫0∞Φ(τ)e−μτ dτ is
proposed for the first time. The integral transform is used to solve the
differential equation arising in heat-transfer problem.
TL;DR: In this paper, 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 QFT is provided.
Abstract: 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.
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TL;DR: It is shown that sampling in the SAFT domain is equivalent to orthogonal projection of functions onto a subspace of bandlimited basis associated with the SAFt domain, which leads to least-squares optimal sampling theorem.
Abstract: The Special Affine Fourier Transformation or the SAFT generalizes a number of well known unitary transformations as well as signal processing and optics related mathematical operations. Shift-invariant spaces also play an important role in sampling theory, multiresolution analysis, and many other areas of signal and image processing. Shannon's sampling theorem, which is at the heart of modern digital communications, is a special case of sampling in shift-invariant spaces. Furthermore, it is well known that the Poisson summation formula is equivalent to the sampling theorem and that the Zak transform is closely connected to the sampling theorem and the Poisson summation formula. These results have been known to hold in the Fourier transform domain for decades and were recently shown to hold in the Fractional Fourier transform domain by A. Bhandari and A. Zayed.
The main goal of this article is to show that these results also hold true in the SAFT domain. We provide a short, self-contained proof of Shannon's theorem for functions bandlimited in the SAFT domain and then show that sampling in the SAFT domain is equivalent to orthogonal projection of functions onto a subspace of bandlimited basis associated with the SAFT domain. This interpretation of sampling leads to least-squares optimal sampling theorem. Furthermore, we show that this approximation procedure is linked with convolution and semi-discrete convolution operators that are associated with the SAFT domain. We conclude the article with an application of fractional delay filtering of SAFT bandlimited functions.
TL;DR: In this paper, a new convolution structure for the special affine Fourier transform (SAFT) is introduced, which preserves the convolution theorem for the FT, which states that the FT of the convolutions of two functions is the product of their Fourier transforms.
TL;DR: This work proposes a novel image watermarking method using LCWT and QR decomposition that is not only feasible, but also robust to some geometry attacks and image processing attacks.
Abstract: Inspired by the fact that wavelet transform can be written as a classical convolution form, a new linear canonical wavelet transform (LCWT) based on generalised convolution theorem associated with linear canonical transform (LCT) is proposed recently. The LCWT not only inherits the advantages of multi-resolution analysis of wavelet transform (WT), but also has the capability of image representations in the LCT domain. Based on these good properties, the authors propose a novel image watermarking method using LCWT and QR decomposition. Compared with the existing image watermarking methods based on discrete WT or QR, this novel image watermarking method provides more flexibility in the image watermarking. Peak signal-to-noise ratio, normalised cross and structural similarity index measure are used to verify the advantages of the proposed method in simulation experiments. The experiment results show that the proposed method is not only feasible, but also robust to some geometry attacks and image processing attacks.
23 Feb 2016
TL;DR: In this paper, a novel Fractional Fourier Transform (FrFT) based multiplexing scheme is presented as a joint radar-communication technique, which is used to embed data into chirp sub-carriers with different time-frequency rates.
Abstract: The increasing demand of spectrum resources and the need to keep the size, weight and power consumption of modern radar as low as possible, has led to the development of solutions like joint radar-communication systems. In this paper a novel Fractional Fourier Transform (FrFT) based multiplexing scheme is presented as a joint radar-communication technique. The FrFT is used to embed data into chirp sub-carriers with different time-frequency rates. Some optimisation procedures are also proposed, with the objective of improving the bandwidth occupancy and the bit rate and/or Bit Error Ratio (BER). The generated waveform is demonstrated to be robust to distortions introduced by the channel, leading to low BER, while keeping good radar characteristics compared to a widely used Linear Frequency Modulated (LFM) pulse with same duration and bandwidth.
TL;DR: Compared with many methods, extensive experimental results validate that the proposed method can obtain the better-edge characteristic, less blur and less aliasing of the SISR reconstruction.
Abstract: In this stydy, the authors present a single image super-resolution (SISR) reconstruction based on high-order derivative interpolation (HDI) in the fractional Fourier transform (FRFT) domain. First, the HDI formula is derived using a simple technique, which is based on the relationship between the fractional band-limited signal and the traditional band-limited signal. This interpolation formula contains the derivative information of the image and the FRFT domain filter functions (FDFF). Moreover, the advantages of the FDFF are also analysed. Second, the new SISR reconstruction is presented via the HDI. The main advantage is that the presented method involves the derivatives of an image in the resizing process. Moreover, the authors take advantage of the FDFF to resize the image. Furthermore, three evaluation criteria and some simulations are presented to validate the effectiveness of the proposed method. Last, the proposed method is applied to colour image processing. For a colour image case, the RGB colour space is chosen for super-resolution reconstruction. In addition to peak signal-to-noise ratio, the authors have also used the correlation to assess the quality of the reconstruction. Compared with many methods, extensive experimental results validate that the proposed method can obtain the better-edge characteristic, less blur and less aliasing.
TL;DR: The extended fractional Fourier transform is introduced to address the single lens imaging with a fast algorithm for the transform by convolution combined with parallel iterative phase retrieval algorithm to reconstruct the complex amplitude of the object.
Abstract: A moveable lens is used for determining amplitude and phase on the object plane. The extended fractional Fourier transform is introduced to address the single lens imaging. We put forward a fast algorithm for the transform by convolution. Combined with parallel iterative phase retrieval algorithm, it is applied to reconstruct the complex amplitude of the object. Compared with inline holography, the implementation of our method is simple and easy. Without the oversampling operation, the computational load is less. Also the proposed method has a superiority of accuracy over the direct focusing measurement for the imaging of small size objects.
TL;DR: A new generalized fractional Fourier transform is presented, which can overcome the problem of multi-decryption-key hinders the application of this algorithm and enlarge the key space.
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.
TL;DR: This new approach is based on reusing the calculations of the STFT at consecutive time instants, which leads to significant savings in hardware components with respect to fast Fourier transform based STFTs.
Abstract: This brief presents the feedforward short-time Fourier transform (STFT). This new approach is based on reusing the calculations of the STFT at consecutive time instants. This leads to significant savings in hardware components with respect to fast Fourier transform based STFTs. Furthermore, the feedforward STFT does not have the accumulative error of iterative STFT approaches. As a result, the proposed feedforward STFT presents an excellent tradeoff between hardware utilization and performance.
TL;DR: Wang et al. as discussed by the authors proposed to use the statistical property features of transformed signals by the fractional Fourier transform in the optimal fractional order domain as fault features, such as range, mean, standard deviation, skewness, kurtosis, entropy, median, third central moment, and centroid.
Abstract: Feature extraction plays an important role in the field of fault diagnosis of analog circuits. How to effectively extract fault features is crucial to diagnostic accuracy. The components tolerance and circuit nonlinearities of analog circuits can cause some part overlapping of primal signal among different component faults in time domain and frequency domain. Currently, the existing method aims at wavelet features, statistical property features, conventional frequency features and conventional time-domain features. There is no decoupling ability for the feature extraction methods mentioned above. To solve the problem, a new fault features extraction method is proposed. The diagnostic results are compared with those from other methods. Firstly, it is proposed to use the statistical property features of transformed signals by the fractional Fourier transform in the optimal fractional order domain as fault features, such as range, mean, standard deviation, skewness, kurtosis, entropy, median, the third central moment, and centroid. And then, KPCA is used to reduce the dimensionality of candidate features so as to obtain the optimal features. Next, normalization is applied to rescale input features. Finally, extracted features are trained by SVM to diagnose faulty components in analog circuits. The simulation results show that compared with traditional methods, the proposed method is quite efficient to improve diagnostic accuracy.
TL;DR: In this paper, the authors proposed a new uncertainty principle for the two-sided quaternion Fourier transform (QFT), which describes that the spread of a quaternionsvalued function and its two sides QFT are inversely proportional.
Abstract: This paper proposes a new uncertainty principle for the two-sided quaternion Fourier transform. This uncertainty principle describes that the spread of a quaternion-valued function and its two-sided quaternion Fourier transform (QFT) are inversely proportional. We obtain a tighter lower bound about the product of the spread of quaternion signal in the QFT domain. As a consequence, we show that the quaternionic Gabor filters minimize the uncertainty.
TL;DR: Experimental results including peak- to-peak signal-to-noise ratio between the original and reconstructed image are shown to analyze the validity of this technique and demonstrated the proposed method to be secure, fast, complex and robust.
TL;DR: Computer simulation results and security analysis are presented to show the robustness of the proposed opto-digital image encryption technique and the great importance of the new non-linear preprocessing introduced to enhance the security of the cryptosystem and overcome the problem of linearity encountered in the existing permutation-based opto, digital image encryption schemes.
TL;DR: In this paper, the Fourier scattering transform is combined with time-frequency (Gabor) representations to construct a feature extractor which combines Mallat's scattering transform framework with timefrequency representations.
Abstract: In this paper we address the problem of constructing a feature extractor which combines Mallat's scattering transform framework with time-frequency (Gabor) representations. To do this, we introduce a class of frames, called uniform covering frames, which includes a variety of semi-discrete Gabor systems. Incorporating a uniform covering frame with a neural network structure yields the Fourier scattering transform $\mathcal{S}_\mathcal{F}$ and the truncated Fourier scattering transform. We prove that $\mathcal{S}_\mathcal{F}$ propagates energy along frequency decreasing paths and its energy decays exponentially as a function of the depth. These quantitative estimates are fundamental in showing that $\mathcal{S}_\mathcal{F}$ satisfies the typical scattering transform properties, and in controlling the information loss due to width and depth truncation. We introduce the fast Fourier scattering transform algorithm, and illustrate the algorithm's performance. The time-frequency covering techniques developed in this paper are flexible and give insight into the analysis of scattering transforms.