Showing papers on "Fractional Fourier transform published in 2018"

A fast and efficient approach to color-image encryption based on compressive sensing and fractional Fourier transform

Abstract: This paper introduces a novel method of fast and efficient measurement matrices and random phase masks for color image encryption, in which Kronecker product (KP) is combined with chaotic map. The encryption scheme is based on two-dimension (2D) compressive sensing (CS) and fraction Fourier transform (FrFT). In this algorithm, the KP is employed to extend low dimension seed matrices to obtain high dimension measurement matrices and random phase masks. The low dimension seed matrices are generated by controlling chaotic map. The original image is simultaneously encrypted and compressed by the 2D CS, then re-encrypted with FrFT. The proposed encryption scheme fulfills high speed, low complexity and high security. Numerical simulation results demonstrate the excellent performance and security of the proposed scheme.

An efficient computational approach for time-fractional Rosenau–Hyman equation

Abstract: In this work, we concentrate on the analysis of the time-fractional Rosenau–Hyman equation occurring in the formation of patterns in liquid drops via q-homotopy analysis transform technique and reduced differential transform approach. The q-homotopy analysis transform algorithm can provide rapid convergent series by choosing the appropriate values of auxiliary parameters ħ and n at large domain. The reduced differential transform technique gives wider applicability due to reduction in computations and makes the calculation much simpler and easier. The proposed techniques are realistic and free from any assumption and perturbation for solving the time-fractional Rosenau–Hyman equation.

Approximate Fast Graph Fourier Transforms via Multilayer Sparse Approximations

Abstract: The fast Fourier transform is an algorithm of paramount importance in signal processing as it allows to apply the Fourier transform in $\mathcal {O}(n \log n)$ instead of $\mathcal {O}(n^2)$ arithmetic operations. Graph signal processing is a recent research domain that generalizes classical signal processing tools, such as the Fourier transform, to situations where the signal domain is given by any arbitrary graph instead of a regular grid. Today, there is no method to rapidly apply graph Fourier transforms. In this paper, we propose a method to obtain approximate graph Fourier transforms that can be applied rapidly and stored efficiently. It is based on a greedy approximate diagonalization of the graph Laplacian matrix, carried out using a modified version of the famous Jacobi eigenvalues algorithm. The method is described and analyzed in detail, and then applied to both synthetic and real graphs, showing its potential.

New Convolutions for Quadratic-Phase Fourier Integral Operators and their Applications

Abstract: We obtain new convolutions for quadratic-phase Fourier integral operators (which include, as subcases, e.g., the fractional Fourier transform and the linear canonical transform). The structure of these convolutions is based on properties of the mentioned integral operators and takes profit of weight-functions associated with some amplitude and Gaussian functions. Therefore, the fundamental properties of that quadratic-phase Fourier integral operators are also studied (including a Riemann–Lebesgue type lemma, invertibility results, a Plancherel type theorem and a Parseval type identity). As applications, we obtain new Young type inequalities, the asymptotic behaviour of some oscillatory integrals, and the solvability of convolution integral equations.

On the Estimation of LFM Signal Parameters: Analytical Formulation

Abstract: We propose analytical formulations, approximations, upper and lower bounds for the angle sweep of maximum magnitude of fractional Fourier transform of mono- and multicomponent linear frequency modulated (LFM) signals. We employ a successive coarse-to-fine grid-search algorithm to estimate the chirp rates of multicomponent nonseparable LFM signals. Extensive numerical simulations show the validity of analytical formulations and performance of the proposed estimator. Obtained analytical results may also find themselves other application areas, where nonstationary signals are of interest.

Quaternion discrete fractional random transform for color image adaptive watermarking

Abstract: As for the fractional transforms, to date only fractional Fourier transform (FrFT) has been applicable to color images in a holistic manner by using the quaternion algebra, yet, discrete fractional random transform (DFRNT) with the useful intrinsic randomness is still to be explored. This paper first defines quaternion DFRNT (QDFRNT) which generalizes DFRNT to efficiently process quaternion signals, and then applies QDFRNT to color image adaptive watermarking. For the QDFRNT, this paper also derives the relationship between QDFRNT of a quaternion signal and DFRNT of four components for this signal to efficiently compute QDFRNT. For the color image adaptive watermarking based on QDFRNT and SVM, in order to efficiently utilize the color information in the adaptive process, this paper also exploits the human vision system’s (HVS) masking properties of texture, edge and color tone directly from the color host image to adaptively adjust the watermark strength for each block. In addition, the constraints in watermark embedding are discussed to preserve the watermarking energy. Experimental results show that: (a) the proposed efficient computation method takes only half the computational time of the direct method; (b) the comparison of five color models (RGB, YUV, YIQ, CIEL*a*b* and YCbCr) shows that the proposed QDFRNT-based watermarking scheme using YCbCr color model has the overall best performance and can achieve a good balance between invisibility and robustness to the Checkmark attacks; (c) The proposed scheme is superior to the existing schemes respectively using DCT, DFRNT, discrete quaternion Fourier transform (DQFT), discrete fractional quaternion Fourier transform (DFrQFT), and quaternion radial moments (QRMs). Moreover, the fractional order and the random matrix of QDFRNT enhance the security of the proposed scheme.

A cryptosystem based on deterministic phase masks and fractional Fourier transform deploying singular value decomposition

Abstract: In this paper, an asymmetric cryptosystem has been proposed to enhance the security of DRPE. The traditional DRPE scheme is thus tweaked by using fractional Fourier transform (FrFT), a class of structured phase masks called as deterministic phase masks (DMKs) and deploying singular value decomposition (SVD). In specific, we propose to organise the encryption procedure by using two DMKs and FrFT, additionally deploying SVD. On the decryption front, the input image is recovered by utilising the inverse singular value decomposition (ISVD) and an angular portion of the deterministic phase masks. The use of FrFT for encryption and decryption would enhance the robustness of DRPE scheme. Deployment of SVD on our asymmetric cryptosystem provides three components for cipher image is yet another added feature that hardens the security of DRPE scheme. DMKs are formed by the deviation from conventional rectangular function and limited range values which delivers key components with reduced size, better performance and lower complexity. The capability study of defined method, includes analysis on SVD, histogram and correlation coefficient. Our system is subject to an occlusion attack and noise attack to evaluate its performance and reliability. Computational analysis outputs and security investigation are offered in aspect to determine the security potential of proposed system. Comparative results are shown for values of mean-square-error and peak-signal-to-noise ratio of DRPE schemes.

Multiple-parameter fractional quaternion Fourier transform and its application in colour image encryption

Abstract: In this study, by using the quaternion algebra, multiple-parameter fractional quaternion Fourier transform (MPFrQFT) is proposed to generalise the conventional multiple-parameter fractional Fourier transform (MPFrFT) to quaternion signal processing in a holistic manner. First, the new transform MPFrQFT and its inverse transform are defined. An efficient discrete implementation method of MPFrQFT is then proposed, in which the relationship between MPFrQFT and MPFrFT of four components is utilised for a quaternion signal. Finally, a new colour image encryption algorithm based on the proposed MPFrQFT and the double random phase encoding technique is proposed to evaluate the performance of the proposed MPFrQFT. Experimental results demonstrate that: (i) the computational time of the proposed implementation method is almost a half of the direct method's time; (ii) the proposed MPFrQFT-based encryption algorithm has an overall better performance than eight compared algorithms in security test and robustness test: it is more secure than the compared frequency-based algorithms due to the larger key space and the more sensitive key `transform orders'; it is also more robust than the compared spatial-domain algorithms.

Two-dimensional fractional Fourier transform and some of its properties

Abstract: The fractional Fourier transform (FrFT), which is a generalization of the Fourier transform, has become the focus of many research papers in recent years because of its applications in electrical e...

Temporal Change Detection in SAR Images Using Log Cumulants and Stacked Autoencoder

Abstract: This letter proposes a change detection algorithm for damage assessment caused by fires in Ireland using Sentinel 1 data. The novelty, in this letter, is a feature extraction within tunable $Q$ discrete wavelet transform (TQWT) using higher order log cumulants of fractional Fourier transform (FrFT), which were fed into a stacked autoencoder (SAE) to distinguish changed and unchanged areas. The extracted features were used to train the SAE layerwise using an unsupervised learning algorithm. After training the decoding layer was replaced by a logistic regression layer to perform supervised fine-tuning and classification. The proposed algorithm was compared with the algorithm that used log cumulants of FrFT within the oriented dual-tree wavelet transform using support vector machine (SVM) classifier. The experimental results showed that the proposed combination of algorithms decreased the overall error (OE) for real synthetic aperture radar images by 6%, when TQWT was used instead of oriented dual-tree wavelet transform and OE was decreased by another 5% when SAE was used instead of the SVM classifier.

Sampling of Signals Bandlimited to a Disc in the Linear Canonical Transform Domain for IEEE Signal Processing Letters

Abstract: We derive sampling formula for signals that are bandlimited to a disc of radius $R$ in the linear canonical transform (LCT) domain. By bandlimitedness in a disc $D$ in the LCT domain, we mean that the LCT $F(\omega)$ of a signal $f(t)$ vanishes outside the disc $D.$ We first express the signal in polar coordinates and then obtain the sampling formula. The samples of the angle $\theta$ are taken at $2N+1$ uniformly distributed points on the unit circle while the samples of the radial distance $r$ are taken at the zeros of the Bessel function. As a special case, we obtain sampling formula for signals that are bandlimited to a disc in the fractional Fourier transform domain.

An Acquisition Algorithm Based on FRFT for Weak GNSS Signals in A Dynamic Environment

Abstract: To improve the acquisition sensitivity of the weak and dynamic global navigation satellite systems signal, the fractional Fourier transform (FRFT) is introduced to deal with the acquisition process. The acceleration can be estimated using the FRFT approach, and the gain of coherent integration as well as the detection probability could be significantly improved. Partially matched filter technique is used to compare with FRFT. The digital computation complexity and the mean acquisition time are provided. Theoretical analysis for the detection probability has been proposed, and simulations are conducted to verify the high performance of the technique.

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.

Optical excitation fractional Fourier transform (FrFT) based enhanced thermal-wave radar imaging (TWRI).

Abstract: In this paper, we demonstrated a novel thermal-wave radar imaging approach with use of a dual-directional (down then up) chirp (or linear frequency modulation, LFM) modulated laser as an external excitation source and signal processing by Fractional Fourier transform (FrFT), which can enhance the defect detectability and extend the depth-resolution dynamic range. The thermal-wave signal was reconstructed by use of dimensionless normalization scaling (DNS) method, and furthermore, it explored the centralized feature of energy spectral density in FrFT domain. The amplitude and phase angle at the peak energy density in FrFT domain were extracted to form the corresponding image and used for the defect detection and identification. The experiments were carried over a carbon fiber reinforced polymer (CFRP) specimen with the artificial flat bottom holes (FBHs) to validate the defect detection capability using FrFT based enhanced TWRI compared to the FFT based TWRI or conventional lock-in thermography (LIT) by taking the defect signal to noise ratio (SNR) into account.

Symmetric Cryptosystem Based on Chaos Structured Phase Masks and Equal Modulus Decomposition Using Fractional Fourier Transform

Abstract: Chaotic structured phase masks (CSPM), equal modulus decomposition (EMD) and fractional Fourier transform are potentially proposed to design the effective symmetric cryptosystem. The encryption and decryption process of our proposed system is completely established by using double random phase encoding (DRPE) in fractional Fourier domain. Frequently, random phase mask (RPM) are used routinely as secret key in most of the DRPE schemes. Nevertheless, RPM are not optimally robust against many attacks. Henceforth, this method utilises chaotic structured phase masks in the place of random phase masks (RPM). CSPM are assembled with the help of logistic map, Fresnel zone plates (FZP) and radial Hilbert mask (RHM) functions. To design an effectual trap door one-way function, equal modulus decomposition (EMD) is performed for encryption and decryption procedure of our cryptosystem. Various asymmetric cryptosystem was designed for EMD; but constructing EMD effectively in symmetric cryptosystem based on chaos structured phase masks and fractional Fourier transform is considered as a novel work and it is employed. As a result, the proposed symmetric cryptosystem attains high robust and withstand many attacks. Numerical simulations are exhibited in order to validate our system and support the fact that our EMD and CSPM based cryptosystem is extremely suitable for securing images.

Non-linear Cryptosystem for Image Encryption Using Radial Hilbert Mask in Fractional Fourier Transform Domain

Abstract: An asymmetric image encryption scheme has been proposed in the fractional Fourier transform (FRT) domain, using a radial Hilbert mask in the input plane and a random phase mask based in the frequency plane. The use of a radial Hilbert mask provides an addition of extra encryption parameter along with the asymmetric scheme which is non-linear where the encryption and decryption keys are different. The encrypted image resulting from the application of FRT is attenuated by a factor and combined with the asymmetric scheme to provide an encrypted image. The decryption process is the reverse of the encryption. The designed scheme has been implemented digitally using MATLAB R2014a (8.3.0.532). By analysing the decryption results using input images the strength and efficacy of the proposed scheme has been established. The performance assessment of the method has been evaluated in terms of peak signal-to-noise ratio, mean-squared-error (MSE). The proposed scheme provides increased security.

An efficient lossless robust watermarking scheme by integrating redistributed invariant wavelet and fractional Fourier transforms

Abstract: An ensemble lossless watermarking scheme is proposed in the present study by integrating different concepts like redistributed invariant wavelet transform, discrete fractional Fourier transform, singular value decomposition (SVD) and visual cryptography within the framework of a single algorithm The invariant wavelet transform helps to obtain the transform domain, which is invariant to flipping and rotation of image, this is followed by discrete fractional Fourier transform to obtain the translation invariant domain Finally, embedding positions are selected based on a key and reliable features are extracted by performing SVD on a window centered at these positions Based on these reliable features a binary map is generated through which a master share is created The corresponding ownership share is produced from the master share and the watermark In verification process the same operations of the embedding process are applied to the test image to obtain the master share and the watermark is recovered by stacking it over the ownership share There are two main features of the proposed scheme (1) The quality of the image to be watermarked do not degrade during the process and (2) the extracted watermark can still be identified even from a seriously distorted image These findings are also demonstrated with the help of a comparative study with several related schemes

3-D Interferometric Inverse Synthetic Aperture Radar Imaging of Ship Target With Complex Motion

Abstract: A novel algorithm for 3-D interferometric inverse synthetic aperture radar (InISAR) imaging of ship target with complex motion via orthogonal double baseline is presented. For the ship target with a certain translational velocity and 3-D rotation, the distance between any scatterers on the target and the radar is analyzed in detail, and the keystone transform is used to reduce the impact of migration through resolution cell of ship target with big size. Then, the fractional Fourier transform is adopted to achieve the 2-D ISAR image of the target, and the mismatch of the ISAR images achieved by the three radars is solved by the image coregistration method according to the 1-D range profile. Finally, the 3-D InISAR image of the ship target is achieved with the interferometric operation with the three ISAR images. The effectiveness of the proposed method is proved by some simulation results, and the influence of different motion parameters on the 3-D imaging of ship target is analyzed simultaneously in this paper.

Optimized SVD-based robust watermarking in the fractional Fourier domain

Abstract: Digital watermarking is one of the most effective methods for protecting multimedia from different kind of threats. It has been used for many purposes, like copyright protection, ownership identification, tamper detection, etc. Many watermarking applications require embedding techniques that provide robustness against common watermarking attacks, like compression, noise, filtering, etc. In this paper, an optimized robust watermarking method is proposed using Fractional Fourier Transform and Singular Value Decomposition. The approach provides a secure way for watermarking through the embedding parameters that are required for the watermark extraction. It is a block-based method, where each watermark bit is embedded in its corresponding image block. First, the transform is applied to each block, and then the singular values are evaluated through which the embedding modification is performed. The optimum fractional powers, of the transform, and the embedding strength factor are evaluated through a Meta-heuristic optimization to optimize the watermark imperceptibility and robustness. The Artificial Bee Colony is used as the Meta-heuristic optimization method. A fitness function is employed, at the optimization process, through which the maximum achievable robustness can be provided without degrading the watermarking quality below a predetermined quality threshold Qth. The effectiveness of the proposed method is demonstrated through a comparison with recent watermarking techniques in terms of the watermarking performance. The watermarking quality and robustness are evaluated for different quality threshold values. Experimental results show that the proposed approach achieves a better quality compared to that of other existing watermarking methods. On the other hand, the robustness is examined against the most common applied attacks. It is noticed that the proposed method can achieve a higher robustness degree when decreasing the quality threshold value.

Multiphoton discrete fractional Fourier dynamics in waveguide beam splitters

Abstract: We demonstrate that when a waveguide beam splitter (BS) is excited by N indistinguishable photons, the arising multiphoton states evolve in a way as if they were coupled to each other with coupling strengths that are identical to the ones exhibited by a discrete fractional Fourier system. Based on the properties of the fractional Fourier transform, we then derive a multiphoton suppression law for 50/50 BSs, thereby generalizing the Hong–Ou–Mandel effect. Furthermore, we examine the possibility of performing simultaneous multiphoton quantum random walks by using a single waveguide BS in combination with photon-number-resolving detectors. We anticipate that the multiphoton lattice-like structures unveiled in this work will be useful to identify new effects and applications of high-dimensional multiphoton states.

Comparative performance assessment between FFT-based and FRFT-based MIMO-OFDM systems in underwater acoustic communications

Abstract: In this study, the performance of multiple-input multiple-output (MIMO) systems based on the fractional Fourier transform (FRFT) and the fast Fourier transform (FFT) in underwater acoustic (UWA) communication channels is evaluated and compared for various conditions including the number of subcarriers, modulation schemes, the number of paths between the transmitter and the receiver, Doppler frequencies, and MIMO modes including diversity and multiplexing. The study demonstrates while the computational complexity of the FRFT is in the order of the FFT, the performance of the FRFT-based UWA system is superior to that of the FFT-based system for all multipath scenarios, and for the flat fading channel, they achieve the same performance.

Asymmetric image encryption using phase-truncated discrete multiple-parameter fractional Fourier transform

Abstract: An asymmetric image encryption scheme is proposed using a phase-truncated discrete multiple-parameter fractional Fourier transform (DMPFRFT). After applying a pixel-scrambling operation and random-phase mask, an asymmetric ciphertext with stationary white noise can be obtained using phase truncation in the DMPFRFT domain. Using the phase key, an inverse pixel-scrambling operation, and the parameters of the DMPFRFT, the original image can be successfully retrieved. Numerical simulations were conducted to demonstrate the validity and the security of the proposed method, and electro-optical hybrid setups are suggested for encryption and decryption.

The Fractional Fourier Transform on Graphs: Sampling and Recovery

Abstract: Signal processing on graphs expands discrete signal processing theory and techniques to signals supported on graphs. In this paper, we study the sampling and recovery of graph signals under the graph fractional Fourier transform. We show that a-bandlimited signals in the graph fractional Fourier domain can be perfectly recovered. Experimentally designed sampling strategy is used to generate optimal fractional sampling operators on graphs. We give numerical examples, and test the semi-supervised classification of online blogs and handwritten digits using fractional sampling on graphs, and compare it with GFT sampling. We find that fractional sampling on graphs can lead to better classification accuracy at an optimal fractional order.

Pansharpening scheme using filtering in two-dimensional discrete fractional Fourier transform

Abstract: The aim of the pansharpening scheme is to improve the spatial information of multispectral images using the panchromatic (PAN) image. In this study, a novel pansharpening scheme based on two-dimensional discrete fractional Fourier transform (2D-DFRFT) is proposed. In the proposed scheme, PAN and intensity images are transformed using 2D-DFRFT and filtered by highpass filters, respectively. The filtered images are inverse transformed and further used to generate the pansharpened image using appropriate fusion rule. The additional degree of freedom in terms of its angle parameters associated with the 2D-DFRFT is exploited for obtaining better results in the proposed pansharpening scheme. Simulation results of the proposed technique carried out in MATLAB are presented for IKONOS and GeoEye-1 satellite images and compared with existing fusion methods in terms of both visual observation and quality metrics. It is seen that the proposed pansharpening scheme has improved spectral and spatial resolution as compared to the existing schemes.