Showing papers on "Fractional Fourier transform published in 2019"

Hyperspectral Anomaly Detection by Fractional Fourier Entropy

Abstract: Anomaly detection is an important task in hyperspectral remote sensing. Most widely used detectors, such as Reed–Xiaoli (RX), have been developed only using original spectral signatures, which may lack the capability of signal enhancement and noise suppression. In this article, an effective alternative approach, fractional Fourier entropy (FrFE)-based hyperspectral anomaly detection method, is proposed. First, fractional Fourier transform (FrFT) is employed as preprocessing, which obtains features in an intermediate domain between the original reflectance spectrum and its Fourier transform with complementary strengths by space-frequency representations. It is desirable for noise removal so as to enhance the discrimination between anomalies and background. Furthermore, an FrFE-based step is developed to automatically determine an optimal fractional transform order. With a more flexible constraint, i.e., Shannon entropy uncertainty principle on FrFT, the proposed method can significantly distinguish signal from background and noise. Finally, the proposed FrFE-based anomaly detection method is implemented in the optimal fractional domain. Experimental results obtained on real hyperspectral datasets demonstrate that the proposed method is quite competitive.

Medical image encryption using fractional discrete cosine transform with chaotic function.

Abstract: In this advanced era, where we have high-speed connectivity, it is very imperative to insulate medical data from forgery and fraud. With the regular increment in the number of internet users, it is challenging to transmit the beefy medical data. This (medical data) is always reused for different diagnosis purposes, so the information of the medical images need to be protected. This paper introduces a new scheme to ensure the safety of the medical data, which includes the use of a chaotic map on the fractional discrete cosine transform (FrDCT) coefficients of the medical data/images. The imperative FrDCT provides a high degree of freedom for the encryption of the medical images. The algorithm consists of two significant steps, i.e., application of FrDCT on an image and after that chaotic map on FrDCT coefficients. The proposed algorithm discusses the benefits of FrDCT over fractional Fourier transform (FRFT) concerning fractional order α. The key sensitivity and space of the proposed algorithm for different medical images inspire us to make a platform for other researchers to work in this area. Experiments are conducted to study different parameters and challenges. The proposed method has been compared with state-of-the-art techniques. The results suggest that our technique outperforms many other state-of-the-art techniques. Graphical Abstract Overview of the proposed algorithm.

Double chaotic image encryption algorithm based on optimal sequence solution and fractional transform

Abstract: In order to solve the shortcomings of the traditional chaotic encryption and problems of low security, the double chaotic image encryption algorithm based on fractional Fourier transform is proposed. In this algorithm, the optimization algorithm is obtained with the help of Henon mapping and fractional Fourier transforms, then the ciphertext image obtained by the optimization algorithm is taken as input, and according to the sequence of the optimal solution, the logistic chaos is synthesized to obtain the final ciphertext image. This algorithm combines chaotic systems and Fourier transforms, which allows the plaintext to be well hidden, and spatial and frequency domain scrambling is achieved. After experiments, the results show that the improved encryption algorithm achieves better encryption effect. It not only has strong sensitivity and large key space, but also can resist attacks effectively. It has certain application value in image information security.

Separation of Human Micro-Doppler Signals Based on Short-Time Fractional Fourier Transform

Abstract: As an important signature associated with human movement, human micro-Doppler (m-D) signature can provide the basis for activity classification. In particular, the m-D signal of limbs can provide a highly distinctive feature for the activity with reduced limbs movement, which can be used to detect people who are carrying weapon or injured. Fully exploiting the elaborate m-D features that correspond to the motion of limbs can improve the classification accuracy of such activities. Therefore, it is significant to separate the limb-swing micro-Doppler signature from the torso signature and process them separately. In this paper, a novel separation method is proposed, which uses the short-time fractional Fourier transform (STFrFT) with different orders and window lengths to sparsely characterize the echoes from limbs and torso, respectively. Then STFrFT based sparse representation is combined with the morphological component analysis (MCA) to realize the m-D signal separation. Simulation and experimental results verify the effectiveness of the proposed algorithm, where the real data of different activities are utilized to demonstrate its adaptability.

Image Encryption Based on Block-wise Fractional Fourier Transform with Wavelet Transform

Abstract: In the era of multimedia, images excessively present a backbone of communication. When images are transferred over an unsecured network, then the confidentiality of information is a big cha...

Fast and Refined Processing of Radar Maneuvering Target Based on Hierarchical Detection via Sparse Fractional Representation

Abstract: Reliable and fast detection of maneuvering target in complex background is important for both civilian and military applications. It is rather difficult due to the complex motion resulting in energy spread in time and frequency domain. Also, high detection performance and computational efficiency are difficult to balance in case of more pulses. In this paper, we propose a fast and refined processing method of radar maneuvering target based on hierarchical detection, utilizing the advantages of moving target detection (MTD), and the proposed sparse fractional representation. The method adopts two-stage threshold processing. The first stage is the coarse detection processing screening out the rangebins with possible moving targets. The second stage is called the refined processing, which uses robust sparse fractional Fourier transform (RSFRFT) or robust sparse fractional ambiguity function (RSFRAF) dealing with high-order motions, i.e., accelerated or jerk motion. And the second stage is carried out only within the rangebins after the first stage. Therefore, the amount of calculation can be greatly reduced while ensuring high detection performance. Finally, real radar experiment of UAV target detection is carried out for verification of the proposed method, which shows better performance than the traditional MTD method, and the FRFT-FRAF hierarchical coherent integration detection with less computational burden.

Hilbert transform, spectral filters and option pricing

Abstract: We show how spectral filters can improve the convergence of numerical schemes which use discrete Hilbert transforms based on a sinc function expansion, and thus ultimately on the fast Fourier transform. This is relevant, for example, for the computation of fluctuation identities, which give the distribution of the maximum or the minimum of a random path, or the joint distribution at maturity with the extrema staying below or above barriers. We use as examples the methods by Feng and Linetsky (Math Finance 18(3):337–384, 2008) and Fusai et al. (Eur J Oper Res 251(4):124–134, 2016) to price discretely monitored barrier options where the underlying asset price is modelled by an exponential Levy process. Both methods show exponential convergence with respect to the number of grid points in most cases, but are limited to polynomial convergence under certain conditions. We relate these rates of convergence to the Gibbs phenomenon for Fourier transforms and achieve improved results with spectral filtering.

STLFM signal based adaptive synchronization for underwater acoustic communications

Abstract: In underwater acoustic communications (UAC), signal synchronization plays a key role in the performance. It is usually performed using a known preamble transmitted prior to the data. However, the underwater acoustic (UWA) channel is characterized as time-varying and frequency-varying, which makes the preamble fluctuated as well as the transmitted data. Thus, it contains uncertainty to set a constant threshold for synchronization by using information (e.g., Doppler shift) extracted from the preamble. In this paper, we propose an adaptive scheme for UAC synchronization. The scheme uses the symmetrical triangular linear frequency modulation (STLFM) signal to design a fractional Fourier transform (FrFT)-based detection algorithm. It establishes the frame synchronization by detecting the deviation of the two energy peaks which usually emerge in their “optimal” FrFT domain in pairs. Instead of detecting the absolute peaks, the proposed method performs an initial synchronization and a precise correction based on the relative positional relationship and amplitude attenuation of the two peaks, which makes full use of the two peaks of the STLFM signal in the FRFT domain. The effectiveness of the scheme has been verified by simulations and field works. The results suggest that it is able to peak the time-varying signal amplitude for each frame in UWA channels. Besides, the proposed scheme performs better accuracy and stability in the frame synchronization compared to the traditional LFM method, which is shown as three times less detection error and five-to-ten times dropping of mean square error.

Fractional Fourier Analysis Using the Möbius Inversion Formula

Abstract: Compared with the Fourier analysis, the fractional Fourier analysis is more suitable to process linear frequency modulation type non-stationary functions. To the best of our knowledge, the theoretical framework of the fractional Fourier analysis has not well established yet, especially for the fractional Fourier series (FrFS) and the discrete fractional Fourier transform (DFrFT) algorithms. To tackle with these problems, the efficient FrFS and DFrFT algorithms based on the Mobius function are proposed. First, the existence and applicability of the FrFS are analyzed basing on the Mobius inversion formula. Second, two kinds of fast algorithms for the infinite/finite FrFS are proposed. Then, based on the amplitude scaling relationship between the FrFS and samples of the FrFT, two efficient DFrFT algorithms are obtained, which are noted as the arithmetic discrete fractional Fourier transform (ADFrFT)-I and the ADFrFT-II. Importantly, the multiplication complexity of the two proposed ADFrFT algorithms is reduced to O(M), which is less than that of the state-of-the-art DFrFT algorithms. The parallel butterfly structure of the ADFrFT-II algorithm is suitable for the very large scale integration implementation. Finally, the simulations justify the efficiency of the ADFrFT algorithms in filtering and parameter evaluation of radar and optical signals.

A New Perspective on the Two-Dimensional Fractional Fourier Transform and Its Relationship with the Wigner Distribution

Abstract: The fractional Fourier transform $$F_{\theta }(w)$$ with an angle $$\theta $$ of a function f(t) is a generalization of the standard Fourier transform and reduces to it when $$\theta =\pi /2. $$ It has many applications in signal processing and optics because of its close relations with a number of time-frequency representations. It is known that the Wigner distribution of the fractional Fourier transform $$F_{\theta }(w)$$ may be obtained from the Wigner distribution of f by a two-dimensional rotation with the angle $$\theta $$ in the $$t-w$$ plane The fractional Fourier transform has been extended to higher dimensions by taking the tensor product of one-dimensional transforms; hence, resulting in a transform in several but separable variables. It has been shown that the Wigner distribution of the two-dimensional fractional Fourier transform $$F_{\theta ,\phi }(v,w)$$ may be obtained from the Wigner distribution of f(x, y) by a simple four-dimensional rotation with the angle $$\theta $$ in the $$x-y$$ plane and the angle $$\phi $$ in the $$v-w$$ plane. The aim of this paper is two-fold: (1) To introduce a new definition of the two-dimensional fractional Fourier transform that is not a tensor product of two copies of one-dimensional transforms. The new transform, which is more general than the one that exists in the literature, uses a relatively new family of Hermite functions, known as Hermite functions of two complex variables. (2) To give an explicit matrix representation of a four-dimensional rotation that verifies that the Wigner distribution of the new two-dimensional fractional Fourier transform $$F_{\theta ,\phi }(v,w)$$ may be obtained from the Wigner distribution of f(x, y) by a four-dimensional rotation. The matrix representation is more general than the one for the tensor product case and it corresponds to a four-dimensional rotation with two planes of rotations, one with the angle $$(\theta +\phi )/2$$ and the other with the angle $$(\theta -\phi )/2$$ .

An asymmetric hybrid cryptosystem using equal modulus and random decomposition in hybrid transform domain

Abstract: In this paper, an asymmetric hybrid cryptosystem with coherent superposition, equal modulus and random decomposition in hybrid transform domain is proposed. To further strengthen the security of the cryptosystem, a hyperchaotic system is used as a pixel-swapping procedure. The hybrid transform is generated by utilizing fractional Fourier transform of various orders and Walsh transform. The hyperchaotic framework’s parameters and starting conditions alongside the fractional orders of the fractional Fourier transform extend the key-space and consequently give extra strength to the proposed cryptosystem. The designed cryptosystem has an extended key-space to avoid any brute-force attack and is non-linear in nature. The scheme is validated on gray-scale images. Computer based simulations have been performed to verify the validity and the performance of the proposed cryptosystem against different types of attacks. Results demonstrate that the proposed cryptosystem not only offers higher protection against noise attacks but is also invulnerable to special attack.

An asymmetric hybrid cryptosystem using hyperchaotic system and random decomposition in hybrid multi resolution wavelet domain

Abstract: In this paper, an asymmetric hybrid cryptosystem utilizing four-dimensional (4D) hyperchaotic framework by means of coherent superposition and random decomposition in hybrid multi-resolution wavelet domain is put forward. The 4D hyperchaotic framework is utilized for creating permutation keystream for a pixel swapping procedure. The hybrid multi-resolution wavelet is formed by combining Walsh transform and fractional Fourier transform of various orders. The 4D hyperchaotic framework’s parameters and preliminary conditions alongside the fractional orders extend the key-space and consequently give additional strength to the proposed cryptosystem. The proposed cryptosystem has an extended key-space to avoid any brute-force attack and is nonlinear in nature. The scheme is validated on greyscale images. Computer-based simulations have been executed to validate the robustness of the proposed scheme against different types of attacks. Results demonstrate that the proposed cryptosystem along with offering higher protection against noise and occlusion attacks is also unassailable to special attack.

Uncertainty Principles for the Offset Linear Canonical Transform

Abstract: The offset linear canonical transform (OLCT) provides a more general framework for a number of well-known linear integral transforms in signal processing and optics, such as Fourier transform, fractional Fourier transform, linear canonical transform. In this paper, to characterize simultaneous localization of a signal and its OLCT, we extend some different uncertainty principles (UPs), including Nazarov’s UP, Hardy’s UP, Beurling’s UP, logarithmic UP and entropic UP, which have already been well studied in the Fourier transform domain over the last few decades, to the OLCT domain in a broader sense.

A Novel Doppler Rate Estimator Based on Fractional Fourier Transform for High-Dynamic GNSS Signal

Abstract: When the fractional Fourier transform (FRFT) is introduced into the weak and high-dynamic global navigation satellite system (GNSS) signal acquisition, the 2-D search cell will be transferred to a 3-D one with respect to the code chip, the Doppler shift, and the Doppler rate. The proper determinations of the code bin and Doppler shift bin in the acquisition process have already been covered in the previous researches. The aim of this paper is to provide an exhaustive analysis of the approach to specify an optimal FRFT order bin, in terms of the Doppler shift rate. The lower and upper bound of FRFT order ranges is determined by the incoming signal dynamics. Then, we propose a precise model to yield an optimal FRFT order bin. Besides, a novel and fast Doppler estimator based on the non-linear least square (NLS) method is presented to improve the performance of the digital FRFT implementation. Finally, an alternate search procedure is proposed to reduce the singular estimations of the NLS method. The simulating examples demonstrate the performance of the proposed algorithms. It has been verified that the computation efficiency and the estimation accuracy have been significantly improved by proposed techniques.

A lossless image compression algorithm using wavelets and fractional Fourier transform

Abstract: The necessity of data transfer at a high speed, in fast-growing information technology, depends on compression algorithms. Maintaining quality of data reconstructed at high compression rate is a very difficult part of the data compression technique. In this paper, a new lossless image compression algorithm is proposed, which uses both wavelet and fractional transforms for image compression. Even though wavelets are the best choice for feature extraction from the source image at different frequency resolutions, the low-frequency sub-bands of wavelet decomposition are the untouched part in compression method in most of the existing methods. On the other hand, fractional Fourier transform is a convenient form of generalized Fourier transform that helps in the compact lossless coding of the source image with optimal fractional orders. Hence, we have used discrete fractional Fourier transform to compress those sensitive sub-bands of the wavelet transform, carefully. In this method, an image is split into low- and high-frequency sub-bands by using Daubechies wavelet filter and level 1 quantization is applied for both low-frequency and high-frequency sub-bands. The low-frequency sub-bands are compressed by using fractional Fourier transform with optimal fractional orders, and at the same time, high-frequency sub-bands are compressed by eliminating zeroes and storing only nonzero blocks and its position. The compressed wavelet coefficients are further compressed by the application of level 2 quantization and stored as a reduced array. This reduced array is encoded by using arithmetic encoder followed by run-length coding. The experimental results of the proposed algorithm with a different set of test images are compared with some of the existing image compression algorithms. The results show that the proposed method has significant improvement in image reconstruction quality.

Towards a Dictionary for the Bargmann Transform

Abstract: There is a canonical unitary transformation from $L^2(\R)$ onto the Fock space $F^2$, called the Bargmann transform. The purpose of this article is to translate some important results and operators from the context of $L^2(\R)$ to that of $F^2$. Examples include the Fourier transform, the Hilbert transform, Gabor frames, pseudo-differential operators, and the uncertainty principle.

Sliding 2D Discrete Fractional Fourier Transform

Abstract: The two-dimensional discrete fractional Fourier transform (2D DFrFT) has been shown to be a powerful tool for 2D signal processing. However, the existing discrete algorithms aren't the optimal for real-time applications, where the input signals are stream data arriving in a sequential manner. In this letter, a new sliding algorithm is proposed to solve this problem, termed as the 2D sliding DFrFT (2D SDFrFT). The proposed 2D SDFrFT algorithm directly computes the 2D DFrFT in current window using the results of previous window, which greatly reduces the computations. During the derivation, we find that the (m + δ, n)th DFrFT bin in previous window is needed for computing the (m, n)th DFrFT bin in current window, where the increment δ isn't always an integer. Further, a method is proposed to convert the increment δ to a certain integer by determining appropriate sampling interval. The theoretical analysis demonstrates that when compute the new 2D DFrFT in a shifted window in sliding process, our proposed algorithm has the lowest computational cost among existing 2D DFrFT algorithms.

Asymmetric hybrid encryption scheme based on modified equal modulus decomposition in hybrid multi-resolution wavelet domain

Abstract: In this paper, an asymmetric hybrid encryption scheme, using coherent superposition and modified equal modulus decomposition in a hybrid multi-resolution wavelet, is proposed. The hybrid multi-resolution wavelet is generated using fractional Fourier transform of multiple orders and Walsh transform. The fractional orders of the fractional Fourier transform increase the key space and hence provide additional strength to the cryptosystem. The designed scheme has a large key space to avoid brute-force attack and is non-linear in nature. The scheme is validated on grey-scale images. Computer-based simulations have been performed to verify the validity and performance of the proposed scheme against various attacks. Scheme's robustness to the special attack is also checked. Results show that single equal modulus decomposition with fractional Fourier transform and hybrid transform are vulnerable to the special attack, whereas the proposed scheme endures the special attack.