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Showing papers on "Fractional Fourier transform published in 1997"


Proceedings ArticleDOI
17 Sep 1997
TL;DR: In this paper, the authors used the Fourier transform to calculate the impulse response of free space propagation, which is then Fourier transformed or the free space transfer function is defined immediately.
Abstract: Optically generated holograms can be recorded on CCD-arrays if the sampling theorem is obeyed The digitized and quantized holograms are processed digitally for the reconstruction of intensity and phase of the real or virtual image This digital reconstruction consists in a numerical realization of the diffraction integral One approach is the Fresnel approximation employing a single Fourier transform, the other is the interpretation of the diffraction formula as a convolution integral and calculation of this convolution by a double or triple Fourier transform In this convolution approach the impulse response of free space propagation has to be defined which is then Fourier transformed or the free space transfer function is defined immediately Impulse response as well as transfer function can be defined exactly or in an approximated version The main difference between the Fresnel and the convolution approach is the different size of the resulting images Furthermore in the Fresnel case this size depends on the wavelength and the distance of the object from the CCD, in the other case it does not In this paper consequences on the reconstructed wavefields and on the interference phase distributions of holographic interferometry are indicated and demonstrated by experimental results

244 citations


Journal ArticleDOI
TL;DR: An improved DFRFT is proposed that provides transforms similar to those of the continuous fractional Fourier transform and also retains the rotation properties.
Abstract: The fractional Fourier transform is a useful mathematical operation that generalizes the well-known continuous Fourier transform. Several discrete fractional Fourier transforms (DFRFT's) have been developed, but their results do not match those of the continuous case. We propose a new DFRFT. This improved DFRFT provides transforms similar to those of the continuous fractional Fourier transform and also retains the rotation properties.

185 citations


Journal ArticleDOI
TL;DR: In this article, the authors proposed a linear canonical transform with three free parameters, as opposed to the fractional Fourier transform which has only one free parameter, and the ordinary Fourier transformation which has none.

165 citations


Journal ArticleDOI
TL;DR: In this paper, the wavelet transform is used to decompose a function using basis functions that, unlike the Fourier transform, have finite extent in both frequency and time for ground-roll suppression.
Abstract: Low-frequency, high-amplitude ground roll is an old problem in land-based seismic field records. Current processing techniques aimed at ground-roll suppression, such as frequency filtering, f - k filtering, and f - k filtering with time-offset windowing, use the Fourier transform, a technique that assumes that the basic seismic signal is stationary. A new alternative to the Fourier transform is the wavelet transform, which decomposes a function using basis functions that, unlike the Fourier transform, have finite extent in both frequency and time. Application of a filter based on the wavelet transform to land seismic shot records suppresses ground roll in a time-frequency sense; unlike the Fourier filter, this filter does not assume that the signal is stationary. The wavelet transform technique also allows more effective time-frequency analysis and filtering than current processing techniques and can be implemented using an algorithm as computationally efficient as the fast Fourier transform. This new filtering technique leads to the improvement of shot records and considerably improves the final stack quality.

160 citations


Journal ArticleDOI
TL;DR: The purpose of this paper is to introduce extensions of the FT¿s convolution theorem, dealing with the FRFT of a product and of a convolution of two functions.
Abstract: The fractional Fourier transform (FRFT) is a generalization of the classical Fourier transform (FT). It has recently found applications in several areas, including signal processing and optics. Many properties of this transform are already known, but an extension of the FT?s convolution theorem is still missing. The purpose of this paper is to introduce extensions of this theorem, dealing with the FRFT of a product and of a convolution of two functions.

150 citations


Journal ArticleDOI
TL;DR: The wavelet transform, which has had a growing importance in signal and image processing, has been generalized by association with both the wavelettransform and the fractional Fourier transform.
Abstract: The wavelet transform, which has had a growing importance in signal and image processing, has been generalized by association with both the wavelet transform and the fractional Fourier transform. Possible implementations of the new transformation are in image compression, image transmission, transient signal processing, etc. Computer simulations demonstrate the abilities of the novel transform. Optical implementation of this transform is briefly discussed.

128 citations


Journal ArticleDOI
TL;DR: In this article, a unified and concise solution for inverse lattice problems is presented, which uses Ramanujan's sum rule, and a uniformly sampled arithmetic Fourier transform is presented in this work.
Abstract: The present work develops a unified and concise solution for inverse lattice problems. Also, a uniformly sampled arithmetic Fourier transform is presented in this work which uses Ramanujan's sum rule.

115 citations


Journal ArticleDOI
TL;DR: The so-called Linear-time Legendre Transform (LLT) improves all previously known Fast Legendre transform algorithms by reducing their log-linear worst-case time complexity to linear.
Abstract: A new algorithm to compute the Legendre–Fenchel transform is proposed and investigated. The so-called Linear-time Legendre Transform (LLT) improves all previously known Fast Legendre Transform algorithms by reducing their log-linear worst-case time complexity to linear. Since the algorithm amounts to computing several convex hulls and sorting, any convex hull algorithm well-suited for a particular problem gives a corresponding LLT algorithm. After justifying the convergence of the Discrete Legendre Transform to the Legendre–Fenchel transform, an extended computation time complexity analysis is given and confirmed by numerical tests. Finally, the LLT is illustrated with several examples and a LLT MATLAB package is described.

99 citations


Journal ArticleDOI
TL;DR: A fast pattern matching algorithm with a large set of templates based on the typical template matching speeded up by the dual decomposition; the Fourier transform and the Karhunen-Loeve transform that is appropriate for the search of an object with unknown distortion within a short period.
Abstract: We present a fast pattern matching algorithm with a large set of templates. The algorithm is based on the typical template matching speeded up by the dual decomposition; the Fourier transform and the Karhunen-Loeve transform. The proposed algorithm is appropriate for the search of an object with unknown distortion within a short period. Patterns with different distortion differ slightly from each other and are highly correlated. The image vector subspace required for effective representation can be defined by a small number of eigenvectors derived by the Karhunen-Loeve transform. A vector subspace spanned by the eigenvectors is generated, and any image vector in the subspace is considered as a pattern to be recognized. The pattern matching of objects with unknown distortion is formulated as the process to extract the portion of the input image, find the pattern most similar to the extracted portion in the subspace, compute normalized correlation between them at each location in the input image, and find the location with the best score. Searching for objects with unknown distortion requires vast computation. The formulation above makes it possible to decompose highly correlated reference images into eigenvectors, as well as to decompose images in frequency domain, and to speed up the process significantly.

98 citations


Journal ArticleDOI
TL;DR: The capabilities and flexibility of a discrete-dipole code implementing the two-dimensional fast Fourier transform technique are demonstrated with scattering results from circuit features on surfaces.
Abstract: A two-dimensional fast Fourier transform technique is proposed for accelerating the computation of scattering characteristics of features on surfaces by using the discrete-dipole approximation. The two-dimensional fast Fourier transform reduces the CPU execution time dependence on the number of dipoles N from O(N2) to O(N log N). The capabilities and flexibility of a discrete-dipole code implementing the technique are demonstrated with scattering results from circuit features on surfaces.

96 citations


Journal Article
TL;DR: In this paper, the authors considered the initial value problem for the spatially homogeneous Boltzmann equation with Maxwell molecules and developed a difference scheme for this problem using the Fast Fourier Transform and a study of different numerical effects.
Abstract: The initial value problem for the spatially homogeneous Boltzmann equation with Maxwell molecules is considered. A difference scheme for this problem is developed using the Fast Fourier Transform and a study of different numerical effects is presented.

Journal ArticleDOI
TL;DR: In this article, the authors characterize the radial functions for which the a priori inequality holds with constant independent of k. The condition is for V to have the X-rays transform everywhere bounded.

Journal ArticleDOI
TL;DR: It is shown that the Hermite transform can take many alternative forms, all of which have their specific advantages, and a systematic theory is described, based on well-known principles from vector spaces, for deriving such alternative forms.
Abstract: We introduce new theoretical results on how the local differential structure in images can be described, processed, and coded efficiently by means of the Hermite transform. It is shown that the Hermite transform can take many alternative forms, all of which have their specific advantages. The various forms of the Hermite transform correspond to different ways of coding the local orientation in the image, and we describe a systematic theory, based on well-known principles from vector spaces, for deriving such alternative forms. One specific case, called the sampled Hermite transform, is shown to be especially interesting for adaptive image processing and coding. An application in the field of adaptive noise reduction is included by way of illustration.

01 Jun 1997
TL;DR: In this article, a method for evaluating the finite Fourier transform using cubic interpolation of sampled time domain data for high accuracy, and the chirp z-transform for arbitrary frequency resolution is presented.
Abstract: Many system identification and signal processing procedures can be done advantageously in the frequency domain. A required preliminary step for this approach is the transformation of sampled time domain data into the frequency domain. The analytical tool used for this transformation is the finite Fourier transform. Inaccuracy in the transformation can degrade system identification and signal processing results. This work presents a method for evaluating the finite Fourier transform using cubic interpolation of sampled time domain data for high accuracy, and the chirp z-transform for arbitrary frequency resolution. The accuracy of the technique is demonstrated in example cases where the transformation can be evaluated analytically. Arbitrary frequency resolution is shown to be important for capturing details of the data in the frequency domain. The technique is demonstrated using flight test data from a longitudinal maneuver of the F-18 High Alpha Research Vehicle.

Jont B. Allen1
01 Jan 1997
TL;DR: In this article, a theory of short term spectral analysis, synthesis, and modification is presented with an attempt at pointing out certain practical and theoretical questions, which are useful in designing filter banks when the filter bank outputs are to be used for synthesis after multiplicative modifications are made to the spectrum.
Abstract: A theory of short term spectral analysis, synthesis, and modification is presented with an attempt at pointing out certain practical and theoretical questions. The methods discussed here are useful in designing filter banks when the filter bank outputs are to be used for synthesis after multiplicative modifications are made to the spectrum.

Journal ArticleDOI
TL;DR: In this article, a new technique based on double affine Hecke algebras was applied to the Harish-Chandra theory of spherical zonal functions, and the formulas for the Fourier transforms of the multiplications by the coordinates were obtained as well as a simple proof of the inversion theorem using the Opdam transform.
Abstract: We apply a new technique based on double affine Hecke algebras to the Harish-Chandra theory of spherical zonal functions. The formulas for the Fourier transforms of the multiplications by the coordinates are obtained as well as a simple proof of the Harish-Chandra inversion theorem using the Opdam transform.

Journal ArticleDOI
TL;DR: An extension of the Fresnel transform to first-order optical systems that can be represented by an ABCD matrix is analyzed in this article, which is recognized to belong to the class of linear canonical transforms.
Abstract: An extension of the Fresnel transform to first-order optical systems that can be represented by an ABCD matrix is analyzed. We present and discuss a definition of the generalized transform, which is recognized to belong to the class of linear canonical transforms. A general mathematical characterization is performed by listing a number of meaningful theorems that hold for this operation and can be exploited for simplyfying the analysis of optical systems. The relevance to physics of this transform and of the theorems is stressed. Finally, a comprehensive number of possible decompositions of the generalized transform in terms of elementary optical transforms is discussed to obtain further insight into this operation.

Proceedings ArticleDOI
21 Apr 1997
TL;DR: A new four-parameter atomic decomposition of chirplets is developed for compact representation of signals with chirp components and provides a more compact and precise representation of chiral components as compared to the three- parameter ones.
Abstract: A new four-parameter atomic decomposition of chirplets is developed for compact representation of signals with chirp components. The four-parameter atom is obtained by scaling the Gaussian function, and then applying the fractional Fourier transform (FRFT), time-shift and frequency-shift operators to the scaled Gaussian. The decomposition is realized by extending the matching pursuit algorithm to four parameters. For this purpose, the four-parameter space is discretized to obtain a dense subset in the Hilbert space. Also, a related time-frequency distribution is developed for clear visualization of the signal components. The decomposition provides a more compact and precise representation of chirp components as compared to the three-parameter ones.

Journal ArticleDOI
TL;DR: In this paper, the concept of an extended fractional Fourier transform (FRT) is proposed and the physical meaning of any optical Fresnel diffraction through a lens is explained.
Abstract: The concept of an extended fractional Fourier transform (FRT) is suggested. Previous FRT’s and complex FRT’s are only its subclasses. Then, through this concept and its method, we explain the physical meaning of any optical Fresnel diffraction through a lens: It is just an extended FRT; a lens-cascaded system can equivalently be simplified to a simple analyzer of the FRT; the two-independent-parameter FRT of an object illuminated with a plane wave can be readily implemented by a lens of arbitrary focal length; when cascading, the function of each lens unit and the relationship between the adjacent ones are clear and simple; and more parameters and fewer restrictions on cascading make the optical design easy.

Proceedings ArticleDOI
19 Oct 1997
TL;DR: In this article, a parametric modeling of the short-time Fourier transform is proposed to improve the estimation of frequency, amplitude and phase of the partials of a sound.
Abstract: A new method which improves the estimation of frequency, amplitude and phase of the partials of a sound is presented. It allows the reduction of the analysis-window size from four periods to two periods. It therefore gives better accuracy in parameter determination, and has proved to remain efficient at low signal-to-noise ratios. The basic idea consists of using a parametric modeling of the short-time Fourier transform. The method alternately estimates the complex amplitudes and the frequencies starting from the result of the classical analysis method. It uses the least-square procedure and a first-order limited expansion of the model around previous estimations. This method leads us to design new windows which do not have any sidelobes in order to help the convergence. Finally an analysis algorithm which has been built according to the observed behavior of the method for various kinds of sound is presented.

Journal ArticleDOI
TL;DR: A novel fast computational procedure of the quadratic phase transform (QPT) for joint phase parameter estimation of multicomponent chirp signals and explicit expressions for the arithmetic operation count are derived.

Journal ArticleDOI
TL;DR: A modified Fourier transform method for interferogram fringe pattern analysis is proposed, which eliminates the assumptions of slowly varying phase variation in the test section and the constant spatial carrier frequency and extends the frequency bandwidth.
Abstract: A modified Fourier transform method for interferogram fringe pattern analysis is proposed. While it retains most of the advantages of the Fourier transform method, the new method overcomes some drawbacks of the previous method. It eliminates the assumptions of slowly varying phase variation in the test section and the constant spatial carrier frequency. It also extends the frequency bandwidth and avoids phase distortion caused by discreteness of the sampling frequency. Both numerical simulation and experimental examination are performed to evaluate the performance of the method.

Patent
20 Oct 1997
TL;DR: In this paper, a pre-emphasis step is performed to perform gross decorrelation, followed by an adaptive linear prediction to perform further decorrelation and a transform is performed on the residual of the linear prediction, to obtain transform coefficients representing the residual in the frequency domain.
Abstract: Audio source data is subjected to a pre-emphasis step (302) to perform gross decorrelation, followed by an adaptive linear prediction (306) to perform further decorrelation. A transform is performed on the residual of the linear prediction, to obtain transform coefficients representing the residual in the frequency domain. A number of tonal components are identified (310), subtracted from the transform coefficients and encoded by vector quantization. The transform coefficients are then grouped into sub-bands, and each sub-band encoded in the frequency domain by vector quantization. The sub-bands are of uniform width on an auditory scale, so that each vector may comprise a different number of transform coefficients.

Journal ArticleDOI
TL;DR: In this paper, the limit mixed Hodge structure of the cohomology of the fibers in terms of the Fourier transform of the Gauss-Manin system associated to a smooth quasi-projective variety is computed.
Abstract: Let/:C/-*C be a regular function on a smooth quasi-projective variety U. We compute the limit mixed Hodge structure (when t —>• oo) of the cohomology of the fibers /= t in terms of the Fourier transform of the Gauss-Manin system associated to /

Journal ArticleDOI
TL;DR: Optical implementation of a three-dimensional (3-D) Fourier transform is proposed and demonstrated and a 3-D joint transform correlator is described that is capable of recognizing targets in the 3- D space.
Abstract: Optical implementation of a three-dimensional (3-D) Fourier transform is proposed and demonstrated. A spatial 3-D object, as seen from the paraxial zone, is transformed to the 3-D spatial frequency space. Based on the new procedure, a 3-D joint transform correlator is described that is capable of recognizing targets in the 3-D space.

Journal ArticleDOI
TL;DR: The resulting adaptive STFT shares many desirable properties with the adaptive CKD, such as the ability to adapt to transient as well as long-term signal components, making it competitive in complexity with nonadaptive time-frequency algorithms.
Abstract: This article presents a method of adaptively adjusting the window length used in short-time Fourier analysis, related to our earlier work in which we developed a means of adaptively optimizing the performance of the cone kernel distribution (CKD). The optimal CKD cone length is, by definition, a measure of the interval over which the signal has constant or slowly changing frequency structure. The article shows that this length can also be used to compute a time-varying short-time Fourier transform (STFT). The resulting adaptive STFT shares many desirable properties with the adaptive CKD, such as the ability to adapt to transient as well as long-term signal components. The optimization requires O(N) operations per step, less than the fast Fourier transform (FFT) used in computing each time slice, making it competitive in complexity with nonadaptive time-frequency algorithms.

Journal ArticleDOI
TL;DR: A new linear integral transform is defined, which is called the exponential chirp transform, which provides frequency domain image processing for space-variant image formats, while preserving the major aspects of the shift-invariant properties of the usual Fourier transform.
Abstract: Space-variant (or foveating) vision architectures are of importance in both machine and biological vision. In this paper, we focus on a particular space-variant map, the log-polar map, which approximates the primate visual map, and which has been applied in machine vision by a number of investigators during the past two decades. Associated with the log-polar map, we define a new linear integral transform, which we call the exponential chirp transform. This transform provides frequency domain image processing for space-variant image formats, while preserving the major aspects of the shift-invariant properties of the usual Fourier transform. We then show that a log-polar coordinate transform in frequency provides a fast exponential chirp transform. This provides size and rotation, in addition to shift, invariant properties in the transformed space. Finally, we demonstrate the use of the fast exponential chirp algorithm on a database of images in a template matching task, and also demonstrate its uses for spatial filtering.

Proceedings ArticleDOI
14 Jul 1997
TL;DR: The discrete quaternion Fourier transform (DQFT) as discussed by the authors is a generalization of the two-dimensional complex DFT to the case of images with more than two (but not more than four) image components.
Abstract: This paper presents a two-dimensional discrete Fourier transform (DFT), applicable to colour images (and multispectral images generally). It is based on quaternion numbers, which may be thought of as a generalisation of the complex numbers with one real part and three imaginary parts. The transform is referred to the discrete quaternion Fourier transform (DQFT). The DQFT is a useful generalisation of the two-dimensional complex DFT to the case of images with more than two (but not more than four) image components. Natural colour images are a significant example of such images and their three components, for example, red, green and blue, may now be transformed as a whole. There are some important further theoretical steps needed, however, to make generalisation of monochrome frequency domain techniques in image processing widely applicable to colour and other multispectral images.


Journal ArticleDOI
TL;DR: This work shows how to implement the fractional Hilbert transform for two-dimensional inputs, which is now suitable for image processing.
Abstract: The classical Hilbert transform can be implemented optically as a spatial-filtering process, whereby half the Fourier spectrum is π-phase shifted. Recently the Hilbert transform was generalized. The generalized version, called the fractional Hilbert transform, is quite easy to implement optically if the input is one dimensional. Here we show how to implement the fractional Hilbert transform for two-dimensional inputs. Hence the new transform is now suitable for image processing.