Showing papers on "Fractional Fourier transform published in 2013"

Discrete Signal Processing on Graphs

Abstract: In social settings, individuals interact through webs of relationships. Each individual is a node in a complex network (or graph) of interdependencies and generates data, lots of data. We label the data by its source, or formally stated, we index the data by the nodes of the graph. The resulting signals (data indexed by the nodes) are far removed from time or image signals indexed by well ordered time samples or pixels. DSP, discrete signal processing, provides a comprehensive, elegant, and efficient methodology to describe, represent, transform, analyze, process, or synthesize these well ordered time or image signals. This paper extends to signals on graphs DSP and its basic tenets, including filters, convolution, z-transform, impulse response, spectral representation, Fourier transform, frequency response, and illustrates DSP on graphs by classifying blogs, linear predicting and compressing data from irregularly located weather stations, or predicting behavior of customers of a mobile service provider.

Core Transform Design in the High Efficiency Video Coding (HEVC) Standard

Abstract: This paper describes the core transforms specified for the high efficiency video coding (HEVC) standard. Core transform matrices of various sizes from 4 × 4 to 32 × 32 were designed as finite precision approximations to the discrete cosine transform (DCT). Also, special care was taken to allow implementation friendliness, including limited bit depth, preservation of symmetry properties, embedded structure and basis vectors having almost equal norm. The transform design has the following properties: 16 bit data representation before and after each transform stage (independent of the internal bit depth), 16 bit multipliers for all internal multiplications, no need for correction of different norms of basis vectors during quantization/de-quantization, all transform sizes above 4 × 4 can reuse arithmetic operations for smaller transform sizes, and implementations using either pure matrix multiplication or a combination of matrix multiplication and butterfly structures are possible. The transform design is friendly to parallel processing and can be efficiently implemented in software on SIMD processors and in hardware for high throughput processing.

Information Transmission using the Nonlinear Fourier Transform

Abstract: The central objective of this thesis is to suggest and develop one simple, unified method for communication over optical fiber networks, valid for all values of dispersion and nonlinearity parameters, and for a single-user channel or a multiple-user network. The method is based on the nonlinear Fourier transform (NFT), a powerful tool in soliton theory and exactly solvable models for solving integrable partial differential equations governing wave propagation in certain nonlinear media. The NFT decorrelates signal degrees of freedom in such models, in much the same way that the Fourier transform does for linear systems. In this thesis, this observation is exploited for data transmission over integrable channels such as optical fibers, where pulse propagation is governed by the nonlinear Schr\"odinger (NLS) equation. In this transmission scheme, which can be viewed as a nonlinear analogue of orthogonal frequency-division multiplexing commonly used in linear channels, information is encoded in the nonlinear spectrum of the signal. Just as the (ordinary) Fourier transform converts a linear convolutional channel into a number of parallel scalar channels, the nonlinear Fourier transform converts a nonlinear dispersive channel described by a \emph{Lax convolution} into a number of parallel scalar channels. Since, in the spectral coordinates the NLS equation is multiplicative, users of a network can operate in independent nonlinear frequency bands with no deterministic inter-channel interference. Unlike most other fiber-optic transmission schemes, this technique deals with both dispersion and nonlinearity directly and unconditionally without the need for dispersion or nonlinearity compensation methods. This thesis lays the foundations of such a nonlinear frequency-division multiplexing system.%%%%PhD

Novel image encryption/decryption based on quantum Fourier transform and double phase encoding

Abstract: A novel gray-level image encryption/decryption scheme is proposed, which is based on quantum Fourier transform and double random-phase encoding technique. The biggest contribution of our work lies in that it is the first time that the double random-phase encoding technique is generalized to quantum scenarios. As the encryption keys, two phase coding operations are applied in the quantum image spatial domain and the Fourier transform domain respectively. Only applying the correct keys, the original image can be retrieved successfully. Because all operations in quantum computation must be invertible, decryption is the inverse of the encryption process. A detailed theoretical analysis is given to clarify its robustness, computational complexity and advantages over its classical counterparts. It paves the way for introducing more optical information processing techniques into quantum scenarios.

Multiple-image encryption based on phase mask multiplexing in fractional Fourier transform domain

Abstract: A multiple-image encryption scheme is proposed based on the phase retrieval process and phase mask multiplexing in the fractional Fourier transform domain. First, each original gray-scale image is encoded into a phase only function by using the proposed phase retrieval process. Second, all the obtained phase functions are modulated into an interim, which is encrypted into the final ciphertext by using the fractional Fourier transform. From a plaintext image, a group of phase masks is generated in the encryption process. The corresponding decrypted image can be recovered from the ciphertext only with the correct phase mask group in the decryption process. Simulation results show that the proposed phase retrieval process has high convergence speed, and the encryption algorithm can avoid cross-talk; in addition, its encrypted capacity is considerably enhanced.

An Analytic Wavelet Transform With a Flexible Time-Frequency Covering

Abstract: We develop a rational-dilation wavelet transform for which the dilation factor, the Q-factor and the redundancy can be easily specified. The introduced transform contains Hilbert transform pairs of atoms, therefore it is also suitable for oscillatory signal processing. The transform may be modified to obtain a tight chirplet frame for discrete-time signals. A fast implementation, that makes use of an equivalent filter bank, makes the transform suitable for long signals. Examples on natural signals are provided to demonstrate the utility of the transform.

Sparse Time–Frequency Decomposition and Some Applications

Abstract: In this paper, time-frequency (TF) decomposition (TFD) is studied in the framework of sparse regularization theory. The short-time Fourier transform is first formulated as a convex constrained optimization where a mixed l1-l2 norm of the coefficients is minimized subject to a data fidelity constraint. Such formulation leads to a novel invertible decomposition with adjustable TF resolution. Then, a fast and efficient algorithm based on the alternating split Bregman technique is proposed to carry out the optimization with computational complexity [N2 log(N)]. Window length is a key parameter in windowed Fourier transform which affects the TF resolution; a novel method is also presented to determine the optimum window length for a given signal resulting to maximum compactness of energy in the TF domain. Numerical experiments show that the proposed sparsity-based TFD generates high-resolution TF maps for a wide range of signals having simple to complicated patterns in the TF domain. The performance of the proposed algorithm is also shown on real oil industry examples, such as ground roll noise attenuation and direct hydrocarbon detection from seismic data.

Sample-optimal average-case sparse Fourier Transform in two dimensions

Abstract: We present the first sample-optimal sublinear time algorithms for the sparse Discrete Fourier Transform over a two-dimensional √n × √n grid. Our algorithms are analyzed for the average case signals. For signals whose spectrum is exactly sparse, we present algorithms that use O(k) samples and run in O(k log k) time, where k is the expected sparsity of the signal. For signals whose spectrum is approximately sparse, we have an algorithm that uses O(k log n) samples and runs in O(k log2 n) time, for k = Θ(√n). All presented algorithms match the lower bounds on sample complexity for their respective signal models.

Flexible multiple-image encryption algorithm based on log-polar transform and double random phase encoding technique

Abstract: We present a novel multiple-image encryption algorithm by combining log-polar transform with double random phase encoding in the fractional Fourier domain. In this algorithm, the original images are transformed to annular domains by inverse log-polar transform and then the annular domains are merged into one image. The composite image is encrypted by the classical double random phase encoding method. The proposed multiple-image encryption algorithm takes advantage of the data compression characteristic of log-polar transform to obtain high encryption efficiency and avoids cross-talk in the meantime. Optical implementation of the proposed algorithm is demonstrated and numerical simulation results verify the feasibility and the validity of the proposed algorithm.

Single-channel color image encryption based on iterative fractional Fourier transform and chaos

Abstract: A single-channel color image encryption is proposed based on iterative fractional Fourier transform and two-coupled logistic map. Firstly, a gray scale image is constituted with three channels of the color image, and permuted by a sequence of chaotic pairs which is generated by two-coupled logistic map. Firstly, the permutation image is decomposed into three components again. Secondly, the first two components are encrypted into a single one based on iterative fractional Fourier transform. Similarly, the interim image and third component are encrypted into the final gray scale ciphertext with stationary white noise distribution, which has camouflage property to some extent. In the process of encryption and description, chaotic permutation makes the resulting image nonlinear and disorder both in spatial domain and frequency domain, and the proposed iterative fractional Fourier transform algorithm has faster convergent speed. Additionally, the encryption scheme enlarges the key space of the cryptosystem. Simulation results and security analysis verify the feasibility and effectiveness of this method.

Detection of low observable moving target in sea clutter via fractal characteristics in fractional fourier transform domain

Abstract: Effective detection of low observable moving target at sea is important for remote sensing and radar signal processing. The non-Gaussian property of sea clutter and lack of accurate model make the detection difficult for statistics based detectors. Also the fractal techniques in time domain cannot achieve high detection probability in heavy sea clutter. To help solve the problems, fractal characteristics of IPIX datasets in fractional Fourier transform (FRFT) domain are analysed making use of the fluctuation of FRFT amplitudes and moving target detection algorithms are proposed based on the fractal characteristics in FRFT domain. Firstly, fractal model in FRFT domain is established with fractional Brownian motion model and two judgment and extraction methods are employed for calculating the fractal characteristics in FRFT domain. It is found that sea clutter of different polarisations exhibit fractal behaviours in FRFT domain, that is, self-similarity property, within its corresponding scale-invariant interval. Then, we find that four specific fractal statistics in the best FRFT domain can provide valuable information for developing simple and effective detectors. Finally, traditional amplitude detector and Hurst exponent detector in time domain are compared and the results prove the superior detection ability of low observable moving target without complex computations.

A Tighter Uncertainty Principle for Linear Canonical Transform in Terms of Phase Derivative

Abstract: This study devotes to uncertainty principles under the linear canonical transform (LCT) of a complex signal. A lower-bound for the uncertainty product of a signal in the two LCT domains is proposed that is sharper than those in the existing literature. We also deduce the conditions that give rise to the equal relation of the new uncertainty principle. The uncertainty principle for the fractional Fourier transform is a particular case of the general result for LCT. Examples, including simulations, are provided to show that the new uncertainty principle is truly sharper than the latest one in the literature, and illustrate when the new and old lower bounds are the same and when different.

Exact inversion of the conical Radon transform with a fixed opening angle

Abstract: We study a new class of Radon transforms defined on circular cones called the conical Radon transform. In $\mathbb{R}^3$ it maps a function to its surface integrals over circular cones, and in $\mathbb{R}^2$ it maps a function to its integrals along two rays with a common vertex. Such transforms appear in various mathematical models arising in medical imaging, nuclear industry and homeland security. This paper contains new results about inversion of conical Radon transform with fixed opening angle and vertical central axis in $\mathbb{R}^2$ and $\mathbb{R}^3$. New simple explicit inversion formulae are presented in these cases. Numerical simulations were performed to demonstrate the efficiency of the suggested algorithm in 2D.

SAR Image Categorization With Log Cumulants of the Fractional Fourier Transform Coefficients

Abstract: With the advent of high-resolution (HR) synthetic aperture radar (SAR) images from satellites like TerraSAR-X and TanDEM-X, interest is now on patch-oriented image categorization in contrast to the pixel-based classification in low-resolution SAR images. SAR image categorization requires the generation of a compact feature descriptor that accurately defines the content of the image patch under consideration. As phase information plays a critical role in SAR images, this paper proposes the use of a chirplet-derived transform, i.e., the fractional Fourier transform (FrFT), for generating a compact feature descriptor for single-look complex (SLC) SAR images. Representing a SAR signal in rotated joint time-frequency planes via the FrFT allows discovering the underlying backscattering phenomenon of the objects on the ground. SAR image projections on different planes of the joint time-frequency space using the FrFT provide a simple statistical response that is easier to analyze. The proposed method has been compared with a multiscale approach, i.e., Gabor filter banks, a second-order-statistics-based method (as gray-level co-occurrence matrices), and a spectral descriptor method. We demonstrate the suitability of the FrFT-based method for image categorization on the basis of backscattering behavior, whereas the Gabor-filter-bank-based method is found mainly suitable for images with a strong texture. This paper demonstrates enhancement in the separability for most of the considered categories when using logarithmic cumulants instead of linear moments for both the FrFT-based and Gabor-filter-bank-based methods. The experimental database consists of 2000 image patches (of size 200 × 200 pixels) extracted from SLC HR TerraSAR-X scenes.

A series formula for inversion of the V-line Radon transform in a disc

Abstract: The paper presents an exact formula for a Fourier series reconstruction of a function from its V-line Radon transform in a disc. This transform (often also called broken-ray Radon transform) appears in mathematical models of several imaging modalities, e.g. single-scattering optical tomography and @c-ray emission tomography. Our inversion formula relaxes the support restriction on the image function required in the previously discovered inversion technique (Ambartsoumian, 2012) [8], and uses data from only half of the set of broken rays required before. The general strategy of the current approach was outlined in (Ambartsoumian, 2012) [8].

Synchrosqueezed wave packet transform for 2D mode decomposition

Abstract: This paper introduces the synchrosqueezed wave packet transform as a method for analyzing two-dimensional images. This transform is a combination of wave packet transforms of a certain geometric scaling, a reallocation technique for sharpening phase space representations, and clustering algorithms for modal decomposition. For a function that is a superposition of several wave-like components with a highly oscillatory pattern satisfying certain separation conditions, we prove that the synchrosqueezed wave packet transform identifies these components and estimates their local wavevectors. A discrete version of this transform is discussed in detail, and numerical results are given to demonstrate the properties of the proposed transform.

Four-image encryption method based on spectrum truncation, chaos and the MODFrFT

Abstract: With the help of spectrum truncation, an encryption and decryption method for four images is proposed based on chaos and the multiple-order discrete fractional Fourier transform (MODFrFT). The spectra of four images gotten by the discrete cosine transform (DCT) are truncated and combined into a single array sequentially encrypted by the MODFrFT. The resulting performance is better than other similar algorithms in literature where the spectrum truncation is done in discrete Fourier transform(DFT) domain. Furthermore, the complex mode is introduced to reduce the data loss. The combined spectrum array is encoded with the MODFrFT twice and chaos is introduced to scramble the phases of complex matrix before each MODFrFT. The technology of rate-distortion control is introduced to balance the qualities of the decrypted images. Numerical simulations demonstrate the effectiveness and the security of the proposed four-image encryption algorithm.

Exact reconstruction formulas for a Radon transform over cones

Abstract: Inversion of Radon transforms is the mathematical foundation of many modern tomographic imaging modalities. In this paper we study a conical Radon transform, which is important for computed tomography taking Compton scattering into account. The conical Radon transform we study integrates a function in $\R^d$ over all conical surfaces having vertices on a hyperplane and symmetry axis orthogonal to this plane. As the main result we derive exact reconstruction formulas of the filtered back-projection type for inverting this transform.

Caputo-Based Fractional Derivative in Fractional Fourier Transform Domain

Abstract: This paper proposes a novel closed-form analytical expression of the fractional derivative of a signal in the Fourier transform (FT) and the fractional Fourier transform (FrFT) domain by utilizing the fundamental principles of the fractional order calculus. The generalization of the differentiation property in the FT and the FrFT domain to the fractional orders has been presented based on the Caputo's definition of the fractional differintegral, thereby achieving the flexibility of different rotation angles in the time-frequency plane with varying fractional order parameter. The closed-form analytical expression is derived in terms of the well-known higher transcendental function known as confluent hypergeometric function. The design examples are demonstrated to show the comparative analysis between the established and the proposed method for causal signals corrupted with high-frequency chirp noise and it is shown that the fractional order differentiating filter based on Caputo's definition is presenting good performance than the established results. An application example of a low-pass finite impulse response fractional order differentiating filter in the FrFT domain based on the definition of Caputo fractional differintegral method has also been investigated taking into account amplitude-modulated signal corrupted with high-frequency chirp noise.

Multiple-image encryption scheme based on cascaded fractional Fourier transform.

Abstract: A multiple-image encryption scheme based on cascaded fractional Fourier transform is proposed. In the scheme, images are successively coded into the amplitude and phase of the input by cascading stages, which ends up with an encrypted image and a series of keys. The scheme takes full advantage of multikeys and the cascaded relationships of all stages, and it not only realizes image encryption but also achieves higher safety and more diverse applications. So multiuser authentication and hierarchical encryption are achieved. Numerical simulation verifies the feasibility of the method and demonstrates the security of the scheme and decryption characteristics. Finally, flexibility and variability of the scheme in application are discussed, and the simple photoelectric mixed devices to realize the scheme are proposed.