Showing papers on "Non-uniform discrete Fourier transform published in 2000"

The discrete fractional Fourier transform

Abstract: We propose and consolidate a definition of the discrete fractional Fourier transform that generalizes the discrete Fourier transform (DFT) in the same sense that the continuous fractional Fourier transform generalizes the continuous ordinary Fourier transform. This definition is based on a particular set of eigenvectors of the DFT matrix, which constitutes the discrete counterpart of the set of Hermite-Gaussian functions. The definition is exactly unitary, index additive, and reduces to the DFT for unit order. The fact that this definition satisfies all the desirable properties expected of the discrete fractional Fourier transform supports our confidence that it will be accepted as the definitive definition of this transform.

Determination of Two-Dimensional Correlation Spectra Using the Hilbert Transform

Abstract: A computationally efficient numerical procedure to generate twodimensional (2D) correlation spectra from a set of spectral data collected at certain discrete intervals of an external physical variable, such as time, temperature, pressure, etc., is proposed. The method is based on the use of a discrete Hilbert transform algorithm which carries out the time-domain orthogonal transformation of dynamic spectra. The direct computation of a discrete Hilbert transform provides a definite computational advantage over the more traditional fast Fourier transform route, as long as the total number of discrete spectral data traces does not significantly exceed 40. Furthermore, the mathematical equivalence between the Hilbert transform approach and the original formal definition based on the Fourier transform offers an additional useful insight into the true nature of the asynchronous 2D spectrum, which may be regarded as a time-domain cross-correlation function between orthogonally transformed dynamic spectral intensity variations.

Closed-form discrete fractional and affine Fourier transforms

Abstract: The discrete fractional Fourier transform (DFRFT) is the generalization of discrete Fourier transform. Many types of DFRFT have been derived and are useful for signal processing applications. We introduce a new type of DFRFT, which are unitary, reversible, and flexible; in addition, the closed-form analytic expression can be obtained. It works in performance similar to the continuous fractional Fourier transform (FRFT) and can be efficiently calculated by the FFT. Since the continuous FRFT can be generalized into the continuous affine Fourier transform (AFT) (the so-called canonical transform), we also extend the DFRFT into the discrete affine Fourier transform (DAFT). We derive two types of the DFRFT and DAFT. Type 1 is similar to the continuous FRFT and AFT and can be used for computing the continuous FRFT and AFT. Type 2 is the improved form of type 1 and can be used for other applications of digital signal processing. Meanwhile, many important properties continuous FRFT and AFT are kept in the closed-form DFRFT and DAFT, and some applications, such as filter design and pattern recognition, are also discussed. The closed-form DFRFT we introduce has the lowest complexity among all current DFRFTs that is still similar to the continuous FRFT.

Double random fractional Fourier domain encoding for optical security

Abstract: We propose a new optical encryption technique using the fractional Fourier transform. In this method, the data are encrypted to a stationary white noise by two statistically independent random phase masks in fractional Fourier domains. To decrypt the data correctly, one needs to specify the fractional domains in which the input plane, encryp- tion plane, and output planes exist, in addition to the key used for en- cryption. The use of an anamorphic fractional Fourier transform for the encryption of two-dimensional data is also discussed. We suggest an optical implementation of the proposed idea. Results of a numerical simulation to analyze the performance of the proposed method are pre- sented. © 2000 Society of Photo-Optical Instrumentation Engineers. (S0091-3286(00)01811-0)

Optical image encryption based on multifractional Fourier transforms

Abstract: We propose a new image encryption algorithm based on a generalized fractional Fourier transform, to which we refer as a multifractional Fourier transform. We encrypt the input image simply by performing the multifractional Fourier transform with two keys. Numerical simulation results are given to verify the algorithm, and an optical implementation setup is also suggested.

The discrete harmonic oscillator, Harper's equation, and the discrete fractional Fourier transform

Abstract: Certain solutions to Harper's equation are discrete analogues of (and approximations to) the Hermite-Gaussian functions. They are the energy eigenfunctions of a discrete algebraic analogue of the harmonic oscillator, and they lead to a definition of a discrete fractional Fourier transform (FT). The discrete fractional FT is essentially the time-evolution operator of the discrete harmonic oscillator.

On fractional Fourier transform moments

Abstract: Based on the relation between the ambiguity function represented in a quasipolar coordinate system and the fractional power spectra, the fractional Fourier transform (FT) moments are introduced. Important equalities for the global second order fractional FT moments are derived, and their applications for signal analysis are discussed. The connection between the local moments and the angle derivative of the fractional power spectra is established. This permits us to solve the phase retrieval problem if only two close fractional power spectra are known.

Method for subpixel registration of images

Abstract: Methods for registering first and second images which are offset by an x and/or y displacement in sub-pixel locations are provided. A preferred implementation of the methods includes the steps of: multiplying the first image by a window function to create a first windowed image; transforming the first windowed image with a Fourier transform to create a first image Fourier transform; multiplying the second image by the window function to create a second windowed image; transforming the second windowed image with a Fourier transform to create a second image Fourier transform; computing a collection of coordinate pairs from the first and second image Fourier transforms such that at each coordinate pair the values of the first and second image Fourier transforms are likely to have very little aliasing noise; computing an estimate of a linear Fourier phase relation between the-first and second image Fourier transforms using the Fourier phases of the first and second image Fourier transforms at the coordinate pairs in a minimum-least squares sense; and computing the displacements in the x and/or y directions from the linear Fourier phase relationship. Also provided are a computer program having computer readable program code and program storage device having a program of instructions for executing and performing the methods of the present invention, respectively.

Discrete fractional Hilbert transform

Abstract: The Hilbert transform plays an important role in the theory and practice of signal processing. A generalization of the Hilbert transform, the fractional Hilbert transform, was recently proposed, and it presents physical interpretation in the definition. In this paper, we develop the discrete fractional Hilbert transform, and apply the proposed discrete fractional Hilbert transform to the edge detection of digital images.

Fractional Fourier transforms in two dimensions.

Abstract: We analyze the fractionalization of the Fourier transform (FT), starting from the minimal premise that repeated application of the fractional Fourier transform (FrFT) a sufficient number of times should give back the FT. There is a qualitative increase in the richness of the solution manifold, from U(1) (the circle S1) in the one-dimensional case to U(2) (the four-parameter group of 2 × 2 unitary matrices) in the two-dimensional case [rather than simply U(1)×U(1)]. Our treatment clarifies the situation in the N-dimensional case. The parameterization of this manifold (a fiber bundle) is accomplished through two powers running over the torus T2=S1×S1 and two parameters running over the Fourier sphere S2. We detail the spectral representation of the FrFT: The eigenvalues are shown to depend only on the T2 coordinates; the eigenfunctions, only on the S2 coordinates. FrFT’s corresponding to special points on the Fourier sphere have for eigenfunctions the Hermite–Gaussian beams and the Laguerre–Gaussian beams, while those corresponding to generic points are SU(2)-coherent states of these beams. Thus the integral transform produced by every Sp(4, R) first-order system is essentially a FrFT.

Applications of nonuniform fast transform algorithms in numerical solutions of differential and integral equations

Abstract: We review our efforts to apply the nonuniform fast Fourier transform (NUFFT) and related fast transform algorithms to numerical solutions of Maxwell's equations in the time and frequency domains. The NUFFT is a fast algorithm to perform the discrete Fourier transform of data sampled nonuniformly (NUDFT). Through oversampling and fast interpolation, the forward and inverse NUFFTs can be achieved with O(N log/sub 2/ N) arithmetic operations, asymptotically the same as the regular fast Fourier transform (FFT) algorithms. Using the NUFFT scheme, we develop nonuniform fast cosine transform (NUFCT) and fast Hankel transform (NUFHT) algorithms. These algorithms provide an efficient tool for numerical differentiation and integration, the key in the solutions to differential equations and volume integral equations. We present sample applications of these nonuniform fast transform algorithms in the numerical solution to Maxwell's equations.

An industrially useful means for decomposition and differentiation of harmonic components of periodic waveforms

Abstract: This paper presents efficient methods to estimate the spectral content of (noisy) periodic waveforms that are common in industrial processes The techniques presented, which are based on the recursive discrete Fourier transform, are especially useful in computing high-order derivatives of such waveforms Unlike conventional differentiating techniques, the methods presented differentiate in the frequency domain and thus are quite immune to uncorrelated measurement noise This paper also shows the theoretical relationship between the proposed methods and those of well-known resonant filters

Simplified fractional Fourier transforms

Abstract: The fractional Fourier transform (FRFT) has been used for many years, and it is useful in many applications. Most applications of the FRFT are based on the design of fractional filters (such as removal of chirp noise and the fractional Hilbert transform) or on fractional correlation (such as scaled space-variant pattern recognition). In this study we introduce several types of simplified fractional Fourier transform (SFRFT). Such transforms are all special cases of a linear canonical transform (an affine Fourier transform or an ABCD transform). They have the same capabilities as the original FRFT for design of fractional filters or for fractional correlation. But they are simpler than the original FRFT in terms of digital computation, optical implementation, implementation of gradient-index media, and implementation of radar systems. Our goal is to search for the simplest transform that has the same capabilities as the original FRFT. Thus we discuss not only the formulas and properties of the SFRFT’s but also their implementation. Although these SFRFT’s usually have no additivity properties, they are useful for the practical applications. They have great potential for replacing the original FRFT’s in many applications.

A fast Hough transform for the parametrisation of straight lines using fourier methods

Abstract: The Hough transform is a useful technique in the detection of straight lines and curves in an image. Due to the mathematical similarity of the Hough transform and the forward Radon transform, the Hough transform can be computed using the Radon transform which, in turn, can be evaluated using the central slice theorem. This involves a two-dimensional Fourier transform, an x-y to r-? mapping and a ID Fourier transform. This can be implemented in specialized hardware to take advantage of the computational savings of the fast Fourier transform. In this paper, we outline a fast and efficient method for the computation of the Hough transform using Fourier methods. The maxima points generated in the Radon space, corresponding to the parametrisation of straight lines, can be enhanced with a post transform convolutional filter. This can be applied as a ID filtering operation on the resampled data whilst in the Fourier space, so further speeding the computation. Additionally, any edge enhancement or smoothing operations on the input function can be combined into the filter and applied as a net filter function.

Continuous and discrete fractional Fourier domain decomposition

Abstract: We introduce the fractional Fourier domain decomposition for continuous and discrete signals and systems. A procedure called pruning, analogous to truncation of the singular-value decomposition, underlies a number of potential applications, among which we discuss the fast implementation of space-variant linear systems.

Measurement scale for non-uniform data sampling in n dimensions

Abstract: A method to perform the sampling step of application specific processing for signal processing, primarily digital signal processing of signals derived directly from measurement of a one dimensional or multi-dimensional physical quantity or after processing the physical signal into an N dimensional derived signal. The method extracts information from the signal at discrete points and the sampled values are fed into rigorous mathematically provable correct algorithms, to perform nonuniform sampling. The sampled values may be used to calculate the coefficients of a Fourier or other transform of the signal.

Quasidistributions, tomography, and fractional Fourier transform in signal analysis

Abstract: The quasidistribution functions like the Ville-Wigner function, the Husimi-Kano function, and the coherent-state approach of Glauber and Sudarshan are reviewed in their application to signal analysis and information processing. Noncommutative tomography of an analytic signal and its relation to the fractional Fourier transform is discussed. The analogy of the analytic signal and the quantum wave function is used to show the identity of the fractional Fourier transform and the Green function of the harmonic oscillator. The approach is discussed for both time-dependent signals and spatial signals.

The discrete fractional Fourier transform and Harper's equation

Abstract: It is shown that the discrete fractional Fourier transform recovers the continuum fractional Fourier transform via a limiting process whereby inner products are preserved.

Perspective projections in the space-frequency plane and fractional Fourier transforms

Abstract: Perspective projections in the space-frequency plane are analyzed, and it is shown that under certain conditions they can be approximately modeled in terms of the fractional Fourier transform. The region of validity of the approximation is examined. Numerical examples are presented.

Decimated Signal Diagonalization for Fourier Transform Spectroscopy

Abstract: A nonlinear parameter estimator with frequency-windowing for signal processing, called Decimated Signal Diagonalization (DSD), is presented. This method is used to analyze exponentially damped time signals of arbitrary length corresponding to spectra that are sums of pure Lorentzians. Such time signals typically arise in many experimental measurements, e.g., ion cyclotron resonance (ICR), nuclear magnetic resonance or Fourier transform infrared spectroscopy, etc. The results are compared with the standard spectral estimator, the Fast Fourier Transform (FFT). It is shown that the needed absorption spectra can be constructed simply, without any supplementary experimental work or concern about the phase problems that are known to plague FFT. Using a synthesized signal with known parameters, as well as experimentally measured ICR time signals, excellent results are obtained by DSD with significantly shorter acquisition time than that which is needed with FFT. Moreover, for the same signal length, DSD is demonstrated to exhibit a better resolving power than FFT.

Quasi‐Born Fourier migration

Abstract: Summary The Born approximation of the Lippman–Schwinger equation has recently been used to implement a recursive method for seismic migration of pressure wavefields. This Born-based method is stable only when the scattering from heterogeneities within an extrapolation depth interval is weak. To handle strong scattering accurately and efficiently, we propose a quasi-Born approximation of the Lippman–Schwinger equation to extrapolate pressure wavefields downwards recursively. We assume that the scattered wavefield is linearly related to the incident wavefield by a scalar function that varies slowly with lateral position within an extrapolation depth interval. The extrapolation is implemented as a dual-doma in procedure in the frequency–space and frequency–wavenumber domains. Fast Fourier transforms are used to transform data between these two domains. The quasi-Born-based depth-migration algorithm is termed the quasi-Born Fourier method. It can efficiently produce good-quality images of complex structures with strong lateral perturbations of slowness. It is stable for strong scattering and can accurately handle scattering and wave propagation along directions at large angles from the main propagation direction. Image quality obtained using the new method is similar to that of a dual-domain migration method that uses the Rytov approximation within each extrapolation depth interval, but the computational speed of the new method is approximately 27 per cent faster than the latter method for pre-stack migration of an industry standard data set—the Marmousi data set. Compared to the Born-based migration method, the quasi-Born Fourier method is slightly less efficient because it requires an additional multiplication and an additional division for each lateral gridpoint in each step of wavefield extrapolation. For weak scattering, the quasi-Born Fourier method converges to the Born-based method. To improve the efficiency of the quasi-Born Fourier method further without losing its accuracy, we propose a hybrid Born/quasi-Born Fourier method in which the Born-based method is used when the scattering within an extrapolation depth interval is weak, and the quasi-Born Fourier method is used for other cases. This hybrid method is approximately 32 per cent faster than the Rytov-based method for the pre-stack depth migration of the Marmousi data set, while the images obtained using both methods have almost the same quality.

Fast Fourier Transform for Hexagonal Aggregates

Abstract: Hexagonal aggregates are hierarchical arrangements of hexagonal cells These hexagonal cells may be efficiently addressed using a scheme known as generalized balanced ternary for dimension 2, or GBT_2 The objects of interest in this paper are digital images whose domains are hexagonal aggregates We define a discrete Fourier transform (DFT) for such images The main result of this paper is a radix-7, decimation-in-space fast Fourier transform (FFT) for images defined on hexagonal aggregates The algorithm has complexity N log_7 N It is expressed in terms of the p-product, a generalization of matrix multiplication Data reordering (also known as shuffle permutations) is generally associated with FFT algorithms However, use of the p-product makes data reordering unnecessary

Fast and efficient computation of cubic-spline interpolation for data compression

Abstract: A fast and efficient method and system for computation of cubic-spline interpolation for data compression is described. In one aspect, the present invention is a method and system for defining a cubic-spline filter; correlating the filter with the signal to obtain a correlated signal; autocorrelating the filter to obtain autocorrelated filter coefficients; computing a transform of the correlated signal and the autocorrelated filter coefficients; dividing the transform of the correlated signal by the transform of the autocorrelated filter coefficients to obtain a transform of a compressed signal; and computing an inverse transform of the transform of the compressed signal to obtain the compressed signal. The signal, the filter, and the transforms may be one dimensional or two dimensional. Further, the transforms may be a fast Fourier transform (FFT) or a Winograd discrete Fourier transform (WDFT) with an overlap-save scheme. Also, a zonal filter may be defined to simplify the steps of correlating and autocorrelating.

Discussion on spatial resolution and sensitivity of Fourier transform fringe detection

Abstract: The spatial resolution and sensitivity of the Fourier transform method for fringe detection is analyzed. The spatial resolution degradation due to Fourier transform is discussed through a signal processing technique. It is found that the upper limit of spatial resolution for displacement measurement is half the carrier fringe pitch, or half the grid pitch for the grid method. The formulation of sensitivity using signal processing and communication theory is also performed and analyzed. Measures to improve the spatial resolution sensitivity are discussed.

Adaptive Harmonic Fractional Fourier Transform

Abstract: A novel adaptive harmonic fractional Fourier transform is proposed for analysis of voiced speech signals It provides a higher concentration than STFT and avoids the cross interference components produced by the Wigner-Ville distribution and other bilinear representation The proposed method rotates the base tone and harmonics in time-frequency domain After the rotation, base tone and harmonics become parallel to the time axis in time-frequency domain so that a high concentration can be achieved

A modified discrete chirp-Fourier transform scheme

Abstract: The discrete chirp-Fourier transform (DCFT) proposed in Xia, can be used to detect both the linear frequencies and the frequency varying rate (chirp rate) at the same time. In this paper, we present a modified DCFT (MDCFT) scheme; some properties similar to DCFT in Xia are also studied. Compared to the DCFT in Xia, theoretical and simulation results have shown that the MDCFT can further improve the resolution of chirp rate of the detected signals. The study is of particular use in radar signal processing.