Showing papers on "Fractional Fourier transform published in 1995"
TL;DR: The proposed chirplets are generalizations of wavelets related to each other by 2-D affine coordinate transformations in the time-frequency plane, as opposed to wavelets, which are related to Each other by 1-D affirmations in thetime domain only.
Abstract: We consider a multidimensional parameter space formed by inner products of a parameterizable family of chirp functions with a signal under analysis. We propose the use of quadratic chirp functions (which we will call q-chirps for short), giving rise to a parameter space that includes both the time-frequency plane and the time-scale plane as 2-D subspaces. The parameter space contains a "time-frequency-scale volume" and thus encompasses both the short-time Fourier transform (as a slice along the time and frequency axes) and the wavelet transform (as a slice along the time and scale axes). In addition to time, frequency, and scale, there are two other coordinate axes within this transform space: shear in time (obtained through convolution with a q-chirp) and shear in frequency (obtained through multiplication by a q-chirp). Signals in this multidimensional space can be obtained by a new transform, which we call the "q-chirplet transform" or simply the "chirplet transform". The proposed chirplets are generalizations of wavelets related to each other by 2-D affine coordinate transformations (translations, dilations, rotations, and shears) in the time-frequency plane, as opposed to wavelets, which are related to each other by 1-D affine coordinate transformations (translations and dilations) in the time domain only.
460 citations
TL;DR: An explicit approximation of the Fourier Transform of generalized functions of functions with singularities based on projecting such functions on a subspace of Multiresolution Analysis is obtained and a fast algorithm based on its evaluation is developed.
359 citations
TL;DR: In this paper, the amplitude distributions of light on two spherical surfaces of given radii and separation are modeled as a process of continual fractional Fourier transform transformation, where the amplitude distribution evolves through fractional transforms of increasing order.
Abstract: There exists a fractional Fourier-transform relation between the amplitude distributions of light on two spherical surfaces of given radii and separation. The propagation of light can be viewed as a process of continual fractional Fourier transformation. As light propagates, its amplitude distribution evolves through fractional transforms of increasing order. This result allows us to pose the fractional Fourier transform as a tool for analyzing and describing optical systems composed of an arbitrary sequence of thin lenses and sections of free space and to arrive at a general class of fractional Fourier-transforming systems with variable input and output scale factors.
276 citations
TL;DR: A continuum of “fractional” domains making arbitrary angles with the time and frequency domains is considered, derived by the fractional Fourier transform, to derive transformation, commutation, and uncertainty relations among coordinate multiplication, differentiation, translation, and phase shift operators between domains making arbitrarily angles with each other.
133 citations
09 May 1995
TL;DR: The optimal fractional Fourier domain filter is derived that minimizes the MSE for given non-stationary signal and noise statistics, and time-varying distortion kernel.
Abstract: The ordinary Fourier transform is suited best for analysis and processing of time-invariant signals and systems. When we are dealing with time-varying signals and systems, filtering in fractional Fourier domains might allow us to estimate signals with smaller minimum mean square error (MSE). We derive the optimal fractional Fourier domain filter that minimizes the MSE for given non-stationary signal and noise statistics, and time-varying distortion kernel. We present an example for which the MSE is reduced by a factor of 50 as a result of filtering in the fractional Fourier domain, as compared to filtering in the conventional Fourier or time domains. We also discuss how the fractional Fourier transformation can be computed in O(N log N) time, so that the improvement in performance is achieved with little or no increase in computational complexity.
130 citations
TL;DR: In this paper, the same rules can be applied to create a new type of fractional-order Fourier transform which results in a smooth transition of a function when transformed between the real and Fourier spaces.
112 citations
TL;DR: In this paper, the authors considered the inversion of the 3D X-ray transform with a limited data set containing the line integrals which have two intersections with the lateral surface of a cylindrical detector.
Abstract: We consider the inversion of the three-dimensional (3D) X-ray transform with a limited data set containing the line integrals which have two intersections with the lateral surface of a cylindrical detector The usual solution to this problem is based on 3D filtered-backprojection, but this method is slow This paper presents a new algorithm which factors the 3D reconstruction problem into a set of independent 2D radon transforms for a stack of parallel slices Each slice is then reconstructed using standard 2D filtered-backprojection The algorithm is based on the application of the stationary-phase approximation to the 2D Fourier transform of the data, and is an extension to three dimensions of the frequency-distance relation derived by Edholm et al(1986) for the 2D radon transform Error estimates are also obtained
95 citations
TL;DR: A derivation of the Mellin transform given the Fourier transform that permits closed-form derivations of the temporal moments for various simple geometries is presented and it is demonstrated that the computational cost to produce the nth moment is the same as producing the first n temporal samples of the original function.
Abstract: Modeling of the full temporal behavior of photons propagating in diffusive materials is computationally costly. Rather than deriving intensity as a function of time to fine sampling, we may consider methods that derive a transform of this function. To derive the Fourier transform involves calculation in the (complex) frequency domain and relates to intensity-modulated experiments. We consider instead the Mellin transform and show that this relates to the moments of the original temporal distribution. A derivation of the Mellin transform given the Fourier transform that permits closed-form derivations of the temporal moments for various simple geometries is presented. For general geometries a finite-element method is presented, and it is demonstrated that the computational cost to produce the nth moment is the same as producing the first n temporal samples of the original function.
93 citations
TL;DR: In this article, a collocation method based on an expansion in rational eigenfunctions of the Hilbert transform operator is proposed, which is implemented through the Fast Fourier Transform.
Abstract: We introduce a new method for computing the Hilbert transform on the real line. It is a collocation method, based on an expansion in rational eigenfunctions of the Hilbert transform operator, and implemented through the Fast Fourier Transform. An error analysis is given, and convergence rates for some simple classes of functions are established. Numerical tests indicate that the method compares favorably with existing methods
82 citations
Book•
31 Mar 1995TL;DR: The formal complex Fourier transform (CFT) as mentioned in this paper is an extension of the Fourier Transform (FT) and is used in signal analysis and communication theory, as well as in spectroscopy.
Abstract: Preface to the first edition Preface to the second edition 1. Physics and Fourier transforms 2. Useful properties and theorems 3. Applications I: Fraunhofer diffraction 4. Applications II: signal analysis and communication theory 5. Applications III: spectroscopy and spectral line shapes 6. Two-dimensional Fourier transforms 7. Multi-dimensional Fourier transforms 8. The formal complex Fourier transform 9. Discrete and digital Fourier transform Appendix Bibliography.
TL;DR: The fake zoom lens as mentioned in this paper consists of two singlet lenses with focal powers of + 1 f 1 and − 1 f 2, respectively, which can be used for optical implementation of the fractional Fourier transform.
TL;DR: In this article, the authors present general design formulae for optically implementing the two-dimensional fractional Fourier transform in two orthogonal dimensions and specify the two orders and the input, output scale parameters simultaneously.
TL;DR: The fractional Fourier transform can also be helpful for lens design, especially for specifying a lens cascade, according to its role in wave propagation and signal processing.
Abstract: The fractional Fourier transform has been used in optics so far for wave propagation and for signal processing. Now we show that this new transform can also be helpful for lens design, especially for specifying a lens cascade.
TL;DR: This work shows that the original bulk-optics configuration for performing the fractional-Fourier-transform operation provides a scaled output using a fixed lens and suggests an asymmetrical setup for obtaining a non-scaled output.
Abstract: Recently two optical interpretations of the fractional Fourier transform operator were introduced. We address implementation issues of the fractional-Fourier-transform operation. We show that the original bulk-optics configuration for performing the fractional-Fourier-transform operation [J. Opt. Soc. Am. A 10, 2181 (1993)] provides a scaled output using a fixed lens. For obtaining a non-scaled output, an asymmetrical setup is suggested and tested. For comparison, computer simulations were performed. A good agreement between computer simulations and experimental results was obtained.
01 Sep 1995
TL;DR: By implementing the FFT algorithm on a custom computing machine (CCM) called Splash-2, a computation speed of 180 Mflops and a speed-up of 23 times over a Sparc-10 workstation is achieved.
Abstract: The two dimensional fast Fourier transform (2-D FFT) is an indispensable operation in many digital signal processing applications but yet is deemed computationally expensive when performed on a conventional general purpose processors This paper presents the implementation and performance figures for the Fourier transform on a FPGA-based custom computer The computation of a 2-D FFT requires O(N2log2N) floating point arithmetic operations for an NxN image By implementing the FFT algorithm on a custom computing machine (CCM) called Splash-2, a computation speed of 180 Mflops and a speed-up of 23 times over a Sparc-10 workstation is achieved
TL;DR: Based on the fractional Fourier-transform operation, a new space-frequency chart definition is introduced, and by the application of various geometric operations on this new chart, optical systems may be designed or analyzed.
Abstract: The fractional Fourier transform is a mathematical operation that generalizes the well-known Fourier transform. This operation has been shown to have physical and optical fundamental meanings, and it has been experimentally implemented by relatively simple optical setups. Based on the fractional Fourier-transform operation, a new space-frequency chart definition is introduced. By the application of various geometric operations on this new chart, such as radial and angular shearing and rotation, optical systems may be designed or analyzed. The field distribution, as well as full information about the spectrum and the space–bandwidth product, can be easily obtained in all the stages of the optical system.
Book•
01 May 1995
TL;DR: In this paper, Fourier Spectra for Non-Periodic Functions One-Dimensional Fourier series and Spectra in One-Dimension for Functions of Finite Period One Dimensional.
Abstract: Partial table of contents: Some of the How and Why of Fourier Analysis. Fourier Series and Spectra in One--Dimension for Functions of Finite Period. Fourier Series and Spectra for Functions of Infinite Period One Dimensional. Fourier Spectra for Non--Periodic Functions One--Dimensional. The Diffraction of Light and Fourier Transforms in Two Dimensions. A Brief Summary of Linear Systems Theory Applied to Optical Imaging. Fourier Optical Transformations by Computer. Apodization and Super--Resolution, Phase from Shift, and Multiple Apertures. Complex Apertures. Operations in the Fourier Transform Plane. Other Interesting and Related Topics. References. A Selected Bibliography. Index.
TL;DR: It is demonstrated that the definition of a fractional-order Fourier transform can be extended into the complexorder regime and the beam width of a Gaussian beam subjected to the complex- order Fouriertransform is obtained analytically with the ABCD matrix approach.
Abstract: It is demonstrated that the definition of a fractional-order Fourier transform can be extended into the complexorder regime. A complex-order Fourier transform deals with the imaginary part as well as the real part of the exponential function in the integral. As a result, while the optical implementation of fractional-order Fourier transform involves gradient-index media or spherical lenses, the optical interpretation of complex-order Fourier transform is practically based on the convolution operation and Gaussian apertures. The beam width of a Gaussian beam subjected to the complex-order Fourier transform is obtained analytically with the ABCD matrix approach.
TL;DR: By analysing a windowing signal with Fourier transform, the leakage-induced phase error is investigated, and the phase error distribution is indicated, and a practical approach to correct leakage in a discrete frequency signal to obtain accurate phase information is presented.
TL;DR: In this article, a new mathematical tool, the fractional Fourier transform (FRT) was introduced into the optical field, which makes shift-variant object recognition possible.
Abstract: Recently a new mathematical tool, the fractional Fourier transform (FRT) was introduced into the optical field [ J. Opt. Soc. Am. A10, 1875 and 2521 ( 1993)]. So far, theoretical investigations have been done in order to prove the potential of the FRT operations. On the basis of the FRT operation, the conventional correlator has been generalized through definition of the fractional correlation (FC) operation. We demonstrate the experimental evaluation of the FC operation. Because of the shift-variance property of the FRT, FC makes shift-variant object recognition possible. The laboratory experiments are supplemented by computer simulations, and a reasonable agreement between the results is demonstrated.
TL;DR: An additional degree of freedom is introduced to fractional-Fourier-transform systems by use of anamorphic optics and preliminary results show preliminary results that demonstrate the proposed theory.
Abstract: An additional degree of freedom is introduced to fractional-Fourier-transform systems by use of anamorphic optics. A different fractional Fourier order along the orthogonal principal directions is performed. A laboratory experimental system shows preliminary results that demonstrate the proposed theory. Applications such as anamorphic fractional correlation and multiplexing in fractional domains are briefly suggested.
TL;DR: In this paper, fast Fourier transform algorithms (FFTs) are constructed for wreath products of the form G[Sn] and their iterates, often identified as automorphism groups of spherically homogeneous rooted trees.
TL;DR: In this paper, the authors introduce an extended distance transform which may be used to capture more of the symmetries of a shape and describe the relationship of this extended distance transformation to the skeletal shape descriptors themselves, and other geometric phenomena related to the boundary of the curve.
TL;DR: These architectures are demonstrated to be flexible in practical spatially variant filtering systems that employ cascaded multiple stages of fractional Fourier transforms, because of their capabilities of changing the standard focal length.
Abstract: General optical setups that implement the fractional Fourier transforms are proposed by use of the impulse response theory. These architectures are demonstrated to be flexible in practical spatially variant filtering systems that employ cascaded multiple stages of fractional Fourier transforms, because of their capabilities of changing the standard focal length.
01 May 1995
TL;DR: In this article, the authors present a design algorithm for the design of multilevel signals realising an arbitrarily defined set of Fourier specifications and show that the properties of these signals make them a strong candidate for linear system identification, as well as for detection of nonlinear effects.
Abstract: The paper reviews the existing multifrequency signals available for linear system identification in the frequency domain. The motivation for an excitation with a specified Fourier amplitude spectrum, yet having a small number of signal levels, is then outlined. A design algorithm is described for the design of multilevel signals realising an arbitrarily defined set of Fourier specifications. It is shown that the properties of the multilevel signals make them a strong candidate for linear system identification, as well as for the detection of nonlinear effects. >
TL;DR: In this article, it is shown that any optical system can be decomposed into its roots of any order, by a merger of the ray matrix representation and the canonical operator representation of first-order optical systems.
Abstract: If the performance of an optical system A can be executed by a cascade of n identical optical systems B, we term the system B the nth root of A. At the same time A is the nth power of B. It is shown that, in principle, any optical system can be decomposed into its roots of any order. The procedure is facilitated by a merger of the ray matrix representation and the canonical operator representation of first-order optical systems. The results are demonstrated by several examples, including the fractional Fourier transform, which is just one special case in a complete group structure. Moreover, it is shown that the root and power transformations themselves represent special cases of a much more general family of transformations. Application in optical design, optical signal processing, and resonator theory can be envisaged.
TL;DR: In this article, a phase-space coordinate rotation of the Wigner distribution function associated with the input signal is defined by performing three successive shearing processes, which are reduced to a free-space Fresnel diffraction originated by a scaled version of the input object illuminated with a spherical wave.
TL;DR: A lens system that is able to perform equidistant fractional Fourier transforms that cover the whole range of orders and that consist of a minimum number of modules is introduced.
Abstract: The fractional Fourier transform is a new topic in optics. To make use of the fractional Fourier transform as an experimental tool, I design a fractional Fourier transformer of variable order: I introduce a lens system that is able to perform equidistant fractional Fourier transforms that cover the whole range of orders and that consist of a minimum number of modules. By module, I mean an elementary fractional Fourier transform of certain order that consists of a lens between two free-space lengths. Because of the commutative additivity of the transform, various fractional orders can be achieved by means of different constellations of the modules. It is possible to perform a large variety of fractional Fourier transforms with a small number of modules.