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Showing papers on "Fractional Fourier transform published in 2015"


Journal ArticleDOI
TL;DR: Two new post-transformations for the short-time Fourier transform that achieve a compact time-frequency representation while allowing for the separation and the reconstruction of the modes are introduced.
Abstract: This paper considers the analysis of multicomponent signals, defined as superpositions of real or complex modulated waves. It introduces two new post-transformations for the short-time Fourier transform that achieve a compact time-frequency representation while allowing for the separation and the reconstruction of the modes. These two new transformations thus benefit from both the synchrosqueezing transform (which allows for reconstruction) and the reassignment method (which achieves a compact time-frequency representation). Numerical experiments on real and synthetic signals demonstrate the efficiency of these new transformations, and illustrate their differences.

345 citations


Journal ArticleDOI
TL;DR: This paper defines a new tensor–tensor product alternative to the t-product and generalizes the transform-based approach to any invertible linear transform, and introduces the algebraic structures induced by each new multiplication in the family, which is that of C⁎-algebras and modules.

184 citations


Journal ArticleDOI
TL;DR: This work shows that using the STFT leads to improved performance over recovery from the oversampled Fourier magnitude with the same number of measurements, and suggests an efficient algorithm for recovery of a sparse input from theSTFT magnitude.
Abstract: We consider the classical 1D phase retrieval problem. In order to overcome the difficulties associated with phase retrieval from measurements of the Fourier magnitude, we treat recovery from the magnitude of the short-time Fourier transform (STFT). We first show that the redundancy offered by the STFT enables unique recovery for arbitrary nonvanishing inputs, under mild conditions. An efficient algorithm for recovery of a sparse input from the STFT magnitude is then suggested, based on an adaptation of the recently proposed GESPAR algorithm. We demonstrate through simulations that using the STFT leads to improved performance over recovery from the oversampled Fourier magnitude with the same number of measurements.

121 citations


Journal ArticleDOI
TL;DR: WFRFT is effective, the proposed methods can be used in practical, and all three proposed methods were superior to eight state‐of‐the‐art algorithms.
Abstract: To classify brain images into pathological or healthy is a key pre-clinical state for patients. Manual classification is tiresome, expensive, time-consuming, and irreproducible. In this study, we aimed to present an automatic computer-aided system for brain-image classification. We used 90 T2-weighted images obtained by magnetic resonance images. First, we used weighted-type fractional Fourier transform WFRFT to extract spectrums from each magnetic resonance image. Second, we used principal component analysis PCA to reduce spectrum features to only 26. Third, those reduced spectral features of different samples were combined and were fed into support vector machine SVM and its two variants: generalized eigenvalue proximal SVM and twin SVM. A 5 × 5-fold cross-validation results showed that this proposed "WFRFT+PCA+generalized eigenvalue proximal SVM" yielded sensitivity of 99.53%, specificity of 92.00%, precision of 99.53%, and accuracy of 99.11%, which are comparable with the proposed "WFRFT+PCA+twin SVM" and better than the proposed "WFRFT+PCA+SVM." Besides, all three proposed methods were superior to eight state-of-the-art algorithms. Thus, WFRFT is effective, and the proposed methods can be used in practical. © 2015 Wiley Periodicals, Inc. Int J Imaging Syst Technol, 25, 317-327, 2015

115 citations


Journal ArticleDOI
Jibin Zheng1, Tao Su1, Wentao Zhu1, Xuehui He1, Qing Huo Liu2 
TL;DR: This coherent detection algorithm can detect high-speed targets without the brute-force searching of unknown motion parameters and achieve a good balance between the computational cost and the antinoise performance.
Abstract: In this paper, by employing the symmetric autocorrelation function and the scaled inverse Fourier transform (SCIFT), a coherent detection algorithm is proposed for high-speed targets. This coherent detection algorithm is simple and can be easily implemented by using complex multiplications, the fast Fourier transform (FFT) and the inverse FFT (IFFT). Compared to the Hough transform and the keystone transform, this coherent detection algorithm can detect high-speed targets without the brute-force searching of unknown motion parameters and achieve a good balance between the computational cost and the antinoise performance. Through simulations and analyses for synthetic models and the real data, we verify the effectiveness of the proposed coherent detection algorithm.

102 citations


Journal ArticleDOI
TL;DR: In this article, it is shown that by applying the symmetric form f = f 1 + i f 2 + f 3 j + I f 4 j of quaternion from Hitzer and the novel module or L p -norm of the Quaternion Fourier transform f ˆ, the uncertainty principle of Heisenberg-Weyl's inequality and uncertainty principle associated with the two-sided quaternions transform cannot both be highly concentrated.

92 citations


Journal ArticleDOI
TL;DR: A novel double-image encryption–compression scheme proposed by combining compressive sensing with discrete fractional random transform based on two circular matrices, demonstrating the validity and security of the scheme.

79 citations


Journal ArticleDOI
TL;DR: The proposed color image encryption method using Color Blend and Chaos Permutation operations in the reality-preserving multiple-parameter fractional Fourier transform (RPMPFRFT) domain is feasible, secure, sensitive to keys and robust to noise attack and data loss.

79 citations


Journal ArticleDOI
TL;DR: This letter proposes an optimized transform for the prediction residual, based on a generalized version of previously developed Graph Fourier Transform (GFT), which outperforms combinations of previous intra-prediction and ADST coding by 2.5 dB in PSNR on average.
Abstract: Intra-prediction is employed in block-based image coding to reduce energy in the prediction residual before transform coding. Conventional intra-prediction schemes copy directly from known pixels across block boundaries as prediction. In this letter, we first cluster differences between neighboring pixel pairs. Then, for each pixel pair, we add the cluster mean to the known pixel for prediction of the neighboring unknown pixel. The cluster indices are transmitted per block, allowing the decoder to mimic the same intra-prediction. We then propose an optimized transform for the prediction residual, based on a generalized version of previously developed Graph Fourier Transform (GFT). Experimental results show that our generalized intra-prediction plus transform coding outperforms combinations of previous intra-prediction and ADST coding by 2.5 dB in PSNR on average.

78 citations


Journal ArticleDOI
TL;DR: Experiments with simulated and real radar data sets indicate that the proposed RLCAF can achieve higher integration gain and detection probability of a marine target in a low signal-to-clutter ratio environment.
Abstract: Robust and effective detection of a marine target is a challenging task due to the complex sea environment and target's motion. A long-time coherent integration technique is one of the most useful methods for the improvement of radar detection ability, whereas it would easily run into the across range unit (ARU) and Doppler frequency migration (DFM) effects resulting distributed energy in the time and frequency domain. In this paper, the micro-Doppler (m-D) signature of a marine target is employed for detection and modeled as a quadratic frequency-modulated signal. Furthermore, a novel long-time coherent integration method, i.e., Radon-linear canonical ambiguity function (RLCAF), is proposed to detect and estimate the m-D signal without the ARU and DFM effects. The observation values of a micromotion target are first extracted by searching along the moving trajectory. Then these values are carried out with the long-time instantaneous autocorrelation function for reduction of the signal order, and well matched and accumulated in the RLCAF domain using extra three degrees of freedom. It can be verified that the proposed RLCAF can be regarded as a generalization of the popular ambiguity function, fractional Fourier transform, fractional ambiguity function, and Radon-linear canonical transform. Experiments with simulated and real radar data sets indicate that the RLCAF can achieve higher integration gain and detection probability of a marine target in a low signal-to-clutter ratio environment.

76 citations


Journal ArticleDOI
17 Dec 2015-Entropy
TL;DR: The proposed “FRFE + WTT + TSVM” method is superior to 20 state-of-the-art methods and introduced an advanced classifier: twin support vector machine (TSVM).
Abstract: Aim: To detect pathological brain conditions early is a core procedure for patients so as to have enough time for treatment. Traditional manual detection is either cumbersome, or expensive, or time-consuming. We aim to offer a system that can automatically identify pathological brain images in this paper. Method: We propose a novel image feature, viz., Fractional Fourier Entropy (FRFE), which is based on the combination of Fractional Fourier Transform (FRFT) and Shannon entropy. Afterwards, the Welch’s t-test (WTT) and Mahalanobis distance (MD) were harnessed to select distinguishing features. Finally, we introduced an advanced classifier: twin support vector machine (TSVM). Results: A 10 × K-fold stratified cross validation test showed that this proposed “FRFE + WTT + TSVM” yielded an accuracy of 100.00%, 100.00%, and 99.57% on datasets that contained 66, 160, and 255 brain images, respectively. Conclusions: The proposed “FRFE + WTT + TSVM” method is superior to 20 state-of-the-art methods.

Journal ArticleDOI
TL;DR: A new discrete fractional transform defined by the fractional order, periodicity and vector parameters is presented, which is named as the discrete multiple-parameter fractional angular transform and a double-image encryption scheme is proposed, which has an obvious advantage that no phase keys are used in the encryption and decryption process.

Journal ArticleDOI
Zhichao Zhang1
TL;DR: The simulation results indicate that the new integral transform achieves better detection performance than the WDL and AFL, and many useful and novel properties of WAL are derived.

Journal ArticleDOI
TL;DR: The results show that the proposed scheme offers a good sensitivity to image content alterations and is robust to the common content-preserving operations, and especially to large angle rotation operations.

Journal ArticleDOI
Xuan Rao1, Haihong Tao1, Jia Su1, Jian Xie1, Xiangyang Zhang1 
TL;DR: A novel coherent integration algorithm, improved axis rotation fractional Fourier transform (IAR-FRFT), is proposed to detect the weak targets with a constant radial acceleration to eliminate the linear range migration and alleviate the nonlinear range migration via anImproved axis rotation transform.
Abstract: A novel coherent integration algorithm, improved axis rotation fractional Fourier transform (IAR-FRFT), is proposed to detect the weak targets with a constant radial acceleration. IAR-FRFT could eliminate the linear range migration and alleviate the nonlinear range migration via an improved axis rotation transform and realize coherent integration via fractional Fourier transform. Numerical experiments verify the performance of IAR-FRFT on four aspects: coherent integration time, coherent integration gain, computational complexity, and multitarget detection.

Journal ArticleDOI
TL;DR: A method for accurate and efficient parameter estimation and decomposition of sinusoidally frequency modulated signals is presented and theory is illustrated on signals with one and more components, including noise and disturbances, as well as time-frequency patterns that deviate from sinusoidal form.
Abstract: A method for accurate and efficient parameter estimation and decomposition of sinusoidally frequency modulated signals is presented. These kinds of signals are of special interest in radars and communications. The proposed method is based on the inverse Radon transform property to transform a two-dimensional sinusoidal pattern into a single point in a two-dimensional plane. Since the signal is well concentrated (sparse) in the inverse Radon transform domain, its reconstruction can be performed from a reduced set of observations (back-projections). Theory is illustrated on signals with one and more components, including noise and disturbances, as well as time-frequency patterns that deviate from sinusoidal form.

Journal ArticleDOI
TL;DR: A multispectral double-image-based cryptosystem that exploits only a tiny number of random white noise samples for proper decryption and provides an additional layer of security to the conventional DRPE system is demonstrated.
Abstract: We demonstrate a multispectral double-image-based cryptosystem that exploits only a tiny number of random white noise samples for proper decryption Primarily, one of the two downsampled images is converted into the phase function after being shuffled by Arnold transform (AT), while the other image is modulated as an amplitude-based image after AT Consecutively, a full double-image encryption can be achieved by employing classical double random phase encryption (DRPE) technique in the fractional Fourier transform domain with corresponding fractional orders In this study, the encrypted complex data is randomly sampled via compressive sensing (CS) framework by which only 25% of the sparse white noise samples are being reserved to realize decryption with zero or small errors As a consequence, together with correct phase keys, fractional orders and proper inverse AT operators, lpminimization must be utilized to decrypt the original information Thus, in addition to the perfect image reconstruction, the proposed cryptosystem provides an additional layer of security to the conventional DRPE system Both the mathematical and numerical simulations were carried out to verify the feasibility as well as the robustness of the proposed system The simulation results are presented in order to demonstrate the effectiveness of the proposed system To the best of our knowledge, this is the first report on integrating CS with encrypted complex samples for information security

Journal ArticleDOI
TL;DR: In this paper, a positive-sequence phase-angle estimation method based on discrete Fourier transform for the synchronization of three-phase power-electronic converters under distorted and variable-frequency conditions is proposed.
Abstract: This paper proposes a positive-sequence phase-angle estimation method based on discrete Fourier transform for the synchronization of three-phase power-electronic converters under distorted and variable-frequency conditions. The proposed method is designed based on a fixed sampling rate and, thus, it can simply be employed for control applications. First, analytical analysis is presented to determine the errors associated with the phasor estimation using standard discrete Fourier transform in a variable-frequency environment. Then, a robust phase-angle estimation technique is proposed, which is based on a combination of estimated positive and negative sequences, tracked frequency, and two proposed compensation coefficients. The proposed method has one cycle transient response and is immune to harmonics, noises, voltage imbalances, and grid frequency variations. An effective approximation technique is proposed to simplify the computation of the compensation coefficients. The effectiveness of the proposed method is verified through a comprehensive set of simulations in Matlab software. Simulation results show the robust and accurate performance of the proposed method in various abnormal operating conditions.

Journal ArticleDOI
TL;DR: An image encryption technique to simultaneously encrypt double or multiple images into one encrypted image using computational integral imaging (CII) and fractional Fourier transform (FrFT) is proposed.

Proceedings ArticleDOI
14 Jun 2015
TL;DR: This work shows that two specific masks or five specific masks are sufficient for a convex relaxation of the phase retrieval problem to provably recover almost all signals (up to global phase), a significant improvement over the existing results.
Abstract: Signal recovery from the magnitude of the Fourier transform, or equivalently, from the autocorrelation, is a classical problem known as phase retrieval. Due to the absence of phase information, some form of additional information is required in order to be able to uniquely identify the underlying signal. In this work, we consider the problem of phase retrieval using masks. Due to our interest in developing robust algorithms with theoretical guarantees, we explore a convex optimization-based framework. In this work, we show that two specific masks (each mask provides 2n Fourier magnitude measurements) or five specific masks (each mask provides n Fourier magnitude measurements) are sufficient for a convex relaxation of the phase retrieval problem to provably recover almost all signals (up to global phase). We also show that the recovery is stable in the presence of measurement noise. This is a significant improvement over the existing results, which require O(log2 n) random masks (each mask provides n Fourier magnitude measurements) in order to guarantee unique recovery (up to global phase). Numerical experiments complement our theoretical analysis and show interesting trends, which we hope to explain in a future publication.

Journal ArticleDOI
TL;DR: A function for calculating Euclidean distance transform in large binary images of dimension three or higher in Matlab that significantly outperforms the Matlab’s standard distance transform function “bwdist” both in terms of the computation time and the possible data sizes.
Abstract: In this note, we introduce a function for calculating Euclidean distance transform in large binary images of dimension three or higher in Matlab. This function uses transparent and fast line-scan algorithm that can be efficiently implemented on vector processing architectures such as Matlab and significantly outperforms the Matlab’s standard distance transform function “bwdist” both in terms of the computation time and the possible data sizes. The described function also can be used to calculate the distance transform of the data with anisotropic voxel aspect ratios. These advantages make this function especially useful for high-performance scientific and engineering applications that require distance transform calculations for large multidimensional and/or anisotropic datasets in Matlab. The described function is publicly available from the Matlab Central website under the name “bwdistsc”, “Euclidean Distance Transform for Variable Data Aspect Ratio”.

Journal ArticleDOI
Xuepan Zhang1, Guisheng Liao1, Shengqi Zhu1, Yongchan Gao1, Jingwei Xu1 
TL;DR: An efficient Radon transform (RT) and an efficient fractional Fourier transform (FRFT) to estimate the radial velocity and azimuth velocity and the symmetry property can be used for clutter canceling.
Abstract: Efficient motion parameter estimation is a key challenge for moving-target imaging and localization in the synthetic aperture radar ground moving-target indication system. However, the existing methods suffer from ambiguities, complex realization, or heavy computation load of O(MN). To solve these problems, we propose an efficient Radon transform (RT) and an efficient fractional Fourier transform (FRFT) to estimate the radial velocity and azimuth velocity. By exploiting the geometry information, we model a geometry relationship between the motion parameters and two transform angles of RT or FRFT. The matched motion parameters can be estimated by the geometry relationship of the mismatched results, and the computation complexity is reduced from O(MN) to O(2N) effectively. Additionally, the symmetry property can be used for clutter canceling. Simulated and experimental results demonstrate the validity of the proposed methods. Compared with conventional motion parameter estimation methods, the proposed methods are much more efficient in acquiring accurate estimation results.

Posted Content
TL;DR: Improved upper bounds are given for computing the Fourier transform for the general linear groups over finite fields, the classical Weyl groups, and homogeneous spaces of finite groups, while also recovering the best known algorithms for the symmetric group.
Abstract: We present a general diagrammatic approach to the construction of efficient algorithms for computing the Fourier transform of a function on a finite group. By extending work which connects Bratteli diagrams to the construction of Fast Fourier Transform algorithms %\cite{sovi}, we make explicit use of the path algebra connection to the construction of Gel'fand-Tsetlin bases and work in the setting of quivers. We relate this framework to the construction of a {\em configuration space} derived from a Bratteli diagram. In this setting the complexity of an algorithm for computing a Fourier transform reduces to the calculation of the dimension of the associated configuration space. Our methods give improved upper bounds for computing the Fourier transform for the general linear groups over finite fields, the classical Weyl groups, and homogeneous spaces of finite groups, while also recovering the best known algorithms for the symmetric group and compact Lie groups.

Proceedings ArticleDOI
TL;DR: In this paper, the authors combined the 1D nonstationary seislet transform with empirical-mode decomposition (EMD) in the f-x domain, and the resulting representation showed remarkable sparsity.
Abstract: The seislet transform uses a prediction operator that is connected to the local slope or frequency of seismic events. We have combined the 1D nonstationary seislet transform with empirical-mode decomposition (EMD) in the f-x domain. We used EMD to decompose data into smoothly variable frequency components for the following 1D seislet transform. The resultant representation showed remarkable sparsity. We developed a detailed algorithm and used a field example to demonstrate the application of the new seislet transform for sparsity-promoting seismic data processing.

Journal ArticleDOI
TL;DR: The theory of a DHT that is shown to arise from a discretization scheme based on the theory of Fourier-Bessel expansions is proposed and evaluated and can be used to approximate the continuous forward and inverse Hankel transform in the same manner that the discrete Fourier transform is known to be able to approximation the continuous Fouriertransform.
Abstract: Previous definitions of a discrete Hankel transform (DHT) have focused on methods to approximate the continuous Hankel integral transform. In this paper, we propose and evaluate the theory of a DHT that is shown to arise from a discretization scheme based on the theory of Fourier-Bessel expansions. The proposed transform also possesses requisite orthogonality properties which lead to invertibility of the transform. The standard set of shift, modulation, multiplication, and convolution rules are derived. In addition to the theory of the actual manipulated quantities which stand in their own right, this DHT can be used to approximate the continuous forward and inverse Hankel transform in the same manner that the discrete Fourier transform is known to be able to approximate the continuous Fourier transform.

Journal ArticleDOI
TL;DR: This paper proposes reality-preserving fractional versions of the discrete Fourier, Hartley, generalized Fouriers, and generalized Hartley transforms, which have random outputs and many parameters and thus are very flexible.
Abstract: Real transforms require less complexity for computations and less memory for storages than complex transforms. However, discrete fractional Fourier and Hartley transforms are complex transforms. In this paper, we propose reality-preserving fractional versions of the discrete Fourier, Hartley, generalized Fourier, and generalized Hartley transforms. All of the proposed real discrete fractional transforms have as many as $O(N^{2})$ parameters and thus are very flexible. The proposed real discrete fractional transforms have random eigenvectors and they have only two distinct eigenvalues 1 and $-$ 1. Properties and relationships of the proposed real discrete fractional transforms are investigated. Besides, for the real conventional discrete Hartley and generalized discrete Hartley transforms, we propose their alternative reality-preserving fractionalizations based on diagonal-like matrices to further increase their flexibility. The proposed real transforms have all of the required good properties to be discrete fractional transforms. Finally, since the proposed new transforms have random outputs and many parameters, they are all suitable for data security applications such as image encryption and watermarking.

Journal ArticleDOI
TL;DR: A new fractional chirp scaling algorithm (FrCSA) for highly squinted missile-borne SAR is proposed from high resolution point of view and results indicate that the FrCSA offers better focusing capabilities, greater peak sidelobe ratio (PSLR), and integrated sidelobe ratios (ISLR) by appropriately choosing the rotation angles for the range and azimuth fractional Fourier transform (FrFT).
Abstract: Since synthetic aperture technology was employed in radar signal processing, a lot of imaging algorithms have been developed for highly squinted synthetic aperture radar (SAR). However, the high-resolution imaging for highly squinted SAR data is still a difficult issue due to the large range migration and strong space-variant characteristic. To accommodate for this problem, a new fractional chirp scaling algorithm (FrCSA) for highly squinted missile-borne SAR is proposed from high resolution point of view in this paper. The performance of the FrCSA is compared to the classical CSA based on fast Fourier transform (FFT). Simulation and real data results indicate that the FrCSA offers better focusing capabilities, greater peak sidelobe ratio (PSLR), and integrated sidelobe ratio (ISLR) by appropriately choosing the rotation angles for the range and azimuth fractional Fourier transform (FrFT).

Journal ArticleDOI
TL;DR: A range compression method using the fractional Fourier transform (FrFT) based on minimum entropy criterion is presented to obtain high-resolution one-dimensional (1-D) range profile and a new method of imaging time selection based on frequency smooth degree (FSD) is proposed.
Abstract: The existing bistatic inverse synthetic aperture radar (Bi-ISAR) imaging methods usually uses a “stop-and-go” assumption where the target can be considered not in motion (stop condition) during the fast-time and in motion (go condition) during the slow time. However, for the high-speed target, this assumption is violated; furthermore, the conventional compression via Fourier transform is also invalid due to the quadratic phase term induced by the high-speed motion. In this case, a range compression method using the fractional Fourier transform (FrFT) based on minimum entropy criterion is presented to obtain high-resolution one-dimensional (1-D) range profile. Moreover, to achieve azimuth focusing for the complex-motion target, a new method of imaging time selection based on frequency smooth degree (FSD) is proposed. Simulated and real data are provided to verify the effectiveness of the proposed method.

Journal ArticleDOI
TL;DR: The numerical simulations have demonstrated the validity and high security level of the image cryptosystem based on the novel transform which is similar to fractional Fourier transform and gyrator transform to some extent.

Journal ArticleDOI
TL;DR: The advantages between legitimate partners are extended via developing novel security codes on top of the proposed cross-layer DFRFT security communication model, aiming to achieve an error-free legitimate channel while preventing the eavesdropper from any useful information.
Abstract: Discrete fractional Fourier transform (DFRFT) is a generalization of discrete Fourier transform. There are a number of DFRFT proposals, which are useful for various signal processing applications. This paper investigates practical solutions toward the construction of unconditionally secure communication systems based on DFRFT via cross-layer approach. By introducing a distort signal parameter, the sender randomly flip-flops between the distort signal parameter and the general signal parameter to confuse the attacker. The advantages of the legitimate partners are guaranteed. We extend the advantages between legitimate partners via developing novel security codes on top of the proposed cross-layer DFRFT security communication model, aiming to achieve an error-free legitimate channel while preventing the eavesdropper from any useful information. Thus, a cross-layer strong mobile communication secure model is built.