Showing papers on "Harmonic wavelet transform published in 1987"
TL;DR: The main features of so-called wavelet transforms are illustrated through simple mathematical examples and the first applications of the method to the recognition and visualisation of characteristic features of speech and of musical sounds are presented.
Abstract: This paper starts with a brief discussion of so-called wavelet transforms, i.e., decompositions of arbitrary signals into localized contributions labelled by a scale parameter. The main features of the method are first illustrated through simple mathematical examples. Then we present the first applications of the method to the recognition and visualisation of characteristic features of speech and of musical sounds.
622 citations
TL;DR: In this article, it was shown that it is possible to reconstruct a complex-valued object from the modulus of its Fourier transform if one has a sufficiently strong support constraint.
Abstract: Previously it was shown that one can reconstruct an object from the modulus of its Fourier transform (solve the phase-retrieval problem) by using the iterative Fourier-transform algorithm if one has a nonnegativity constraint and a loose support constraint on the object. In this paper it is shown that it is possible to reconstruct a complex-valued object from the modulus of its Fourier transform if one has a sufficiently strong support constraint. Sufficiently strong support constraints include certain special shapes and separated supports. Reconstruction results are shown, including the effect of tapered edges on the object’s support.
529 citations
TL;DR: The present invention is a nonlinear joint transform image correlator which employs a spatial modulator operating in a binary mode at the Fourier plane which produces a correlation output formed by an inverse Fourier transform of this binarized Fouriers transform interference intensity.
Abstract: The present invention is a nonlinear joint transform image correlator which employs a spatial modulator operating in a binary mode at the Fourier plane. The reference and input images are illuminated by a coherent light at the object plane of a Fourier transform lens system. A image detection device, such as a charge coupled device, is disposed at the Fourier plane of this Fourier transform lens system. A thresholding network detects the median intensity level of the imaging cells of the charge coupled device at the Fourier plane and binarizes the Fourier transform interference intensity. The correlation output is formed by an inverse Fourier transform of this binarized Fourier transform interference intensity. In the preferred embodiment this is achieved via a second Fourier transform lens system. This binary data is then applied to spatial light modulator device operating in a binary mode located at the object plane of a second Fourier transform lens system. This binary mode spatial light modulator device is illuminated by coherent light producing the correlation output at the Fourier plane of the second Fourier transform lens system. The inverse Fourier transform may also be formed via a computer. In an alternative embodiment, the Fourier transform interference intensity is thresholded into one of three ranges. An inverse Fourier transform of this trinary Fourier transform interference intensity produces the correlation output.
333 citations
TL;DR: A single-mode star network, made from polarization-preserving components, can perform the spatial discrete Fourier transform of coherent light patterns presented at the inputs.
Abstract: A single-mode star network, made from polarization-preserving components, can perform the spatial discrete Fourier transform of coherent light patterns presented at the inputs. This can be accomplished with passive components, such as 2 x 2 couplers, and propagation delays. The Hadamard transform can be performed similarly.
83 citations
TL;DR: The theoretical basis of the selective Fourier transform technique is developed and experimental results are presented, including comparisons of spectral localization using either the selective fourier transform method or conventional multidimensional Fouriertransform chemical‐shift imaging.
Abstract: We have introduced the selective Fourier transform technique for spectral localization. This technique allows the acquisition of a high-resolution spectrum from a selectable location with control over the shape and size of the spatial response function. The shape and size of the spatial response are defined during data acquisition and the location is selectable through processing after the data acquisition is complete. The technique uses pulsed-field-gradient phase encoding to define the spatial coordinates. In this paper the theoretical basis of the selective Fourier transform technique is developed and experimental results are presented, including comparisons of spectral localization using either the selective Fourier transform method or conventional multidimensional Fourier transform chemical-shift imaging. © Academic Press, Inc.
79 citations
TL;DR: It is shown that the lower bound for the computation of the multidimensional transform is O(n2 log2 n) and an optimal architecture based on arrays of processors computing one-dimensional Fourier transforms and a rotation network or rotation array is proposed.
Abstract: It is often desirable in modern signal processing applications to perform two-dimensional or three-dimensional Fourier transforms. Until the advent of VLSI it was not possible to think about one chip implementation of such processes. In this paper several methods for implementing the multidimensional Fourier transform together with the VLSI computational model are reviewed and discussed. We show that the lower bound for the computation of the multidimensional transform is O(n2 log2 n). Existing nonoptimal architectures suitable for implementing the 2-D transform, the RAM array transposer, mesh connected systolic array, and the linear systolic matrix vector multiplier are discussed for area time tradeoff. For achieving a higher degree of concurrency we suggest the use of rotators for permutation of data. With ``hybrid designs'' comprised of a rotator and one-dimensional arrays which compute the one-dimensional Fourier transform we propose two methods for implementation of multidimensional Fourier transform. One design uses the perfect shuffle for rotations and achieves an AT2 p of O(n2 log2 n· log2 N). An optimal architecture for calculation of multidimensional Fourier transform is proposed in this paper. It is based on arrays of processors computing one-dimensional Fourier transforms and a rotation network or rotation array. This architecture realizes the AT2 p lower bound for the multidimensional FT processing.
59 citations
01 Nov 1987
TL;DR: In the recursive Fourier transform, the data window can be chosen such that the number of computations required to update the transform at each frequency upon reception of a new data sample is independent of the transform block length.
Abstract: In the recursive Fourier transform, the data window can be chosen such that the number of computations required to update the transform at each frequency upon reception of a new data sample is independent of the transform block length.
40 citations
TL;DR: In this paper, it was shown that the two-dimensional Hartley transform is mathematically equivalent to the Fourier transform, but is real valued; amplitude alone fully represents everything.
Abstract: When the two-dimensional Fourier transformation is performed with a lens the optical amplitude and phase in the output plane represent the complex transform. It can be shown that the two-dimensional Hartley transform is mathematically equivalent to the Fourier transform, but is real valued; amplitude alone fully represents everything. This is significant because ordinary optical detectors do not respond to phase. Here we describe the construction of an optical system in the form of a modified Michelson interferometer which physically demonstrates that it is possible to produce the Hartley transform of a plane luminous object. It is thus possible to encode in the form of amplitude the half of the information in a diffraction pattern that normally is carried in the form of phase.
31 citations
01 Apr 1987
TL;DR: The computational method uses the sprit-radix algorithm which requires the least number of operations compared with other Hartley algorithms and is compared with those using the fast Fourier transform.
Abstract: The use of fast Hartley transform for fast discrete interpolation is considered. The computational method uses the sprit-radix algorithm which requires the least number of operations compared with other Hartley algorithms. Results from this method are compared with those using the fast Fourier transform.
27 citations
TL;DR: In this paper, the authors introduced the discrete Fourier transform (DFT) and showed that the DFT can produce a sequence of spectral components equally spaced in frequency, with a length equal to that of the original waveform.
Abstract: In part one of this tutorial (Jaffe 1987), we introduced the discrete Fourier transform (DFT). To review, the DFT takes a waveform as input and produces as output the spectrum of that waveform. One way to understand this process is to consider the samples of the waveform as a vector and to see the DFT as the projection of this vector onto a set of complex sinusoidal basis vectors. In this manner, the DFT produces a sequence of spectral components equally spaced in frequency, with a length equal to that of the original waveform. Each element of the spectrum is a coefficient of the projection given by the inner product of the waveform with one of the basis sinusoids. This coefficient can be represented in polar coordinates to give the amplitude and phase of the corresponding sinusoid. The equation for the DFT is:
TL;DR: In this article, the generalised matrix inverse problem is addressed for non-ideal conditions, such as non-uniform sampling, imaging in the presence of motion, deconvolution of T2 effects, resolution enhancement, and one-sided data reconstruction.
Abstract: Magnetic resonance imaging uses real and complex forms of the Fourier transform and to a lesser degree the Radon transform and their appropriate inverses. Under ideal conditions, these solutions are fast and optimal in the sense of signal-to-noise ratio (S/N). In practice, though, the phase of the signal may not be ideal so that the effective forward transform is no longer a Fourier transform. Using the inverse Fourier transform would then result in an image with artifacts. In this paper the generalised matrix inverse problem is addressed for such non-ideal conditions. Solutions are obtained for the following non-ideal circumstances: non-uniform sampling; imaging in the presence of motion; deconvolution of T2 effects; resolution enhancement; one-sided data reconstruction. The method is applicable to other deviant models as well. The goal is to maintain some specified property of the image, such as resolution, with minimal production of artifacts. A concomitant loss in S/N is inevitable for such trade-offs, but is often not a serious problem compared with the artifacts themselves.
TL;DR: This correspondence presents details of a new implementation of the prime factor FFT algorithm (PFA) for computing the discrete Fourier transform (DFT) that saves about 40 percent of the execution time of the conventional one.
Abstract: This correspondence presents details of a new implementation of the prime factor FFT algorithm (PFA) for computing the discrete Fourier transform (DFT). This implementation applies a program generation technique to the PFA algorithm and saves about 40 percent of the execution time of the conventional one.
06 Apr 1987
TL;DR: This paper addresses the problem of signal reconstruction from Fourier transform phase and presents the results of studies on reconstruction from partial phase and discusses the application of these results in speech analysis and coding.
Abstract: This paper addresses the problem of signal reconstruction from Fourier transform phase. In particular, we examine two aspects of this problem. First, we discuss signal reconstruction from the phase spectrum of the short-time Fourier transform(STFT). Next, we examine the problem of signal recovery from partial phase information. We present the results of our studies on reconstruction from partial phase and discuss the application of these results in speech analysis and coding.
Proceedings Article•
01 Jan 1987
Journal Article•
TL;DR: A package of FORTRAN 77 subprograms to compute the discrete Fourier transform of one-dimensional data sets using a large collection of specialized modules, assembled in a top-down scheme, guarantees high flexibility and efficiency.
TL;DR: The least desirable characteristic of the 2-D Fourier transform is the lack of spatial variance, which limits its utility in phase shift migration and other tasks needing lateral parameter changes.
Abstract: The Fourier theorem has proven to be one of the most powerful algorithms in signal processing. In the two-dimensional format, the transform Is used to perform fast migration, dip filtering, DMO and noise filtering. The least desirable characteristic of the 2-D Fourier transform is the lack of spatial variance. This limits its utility in phase shift migration and other tasks needing lateral parameter changes.
01 Jan 1987
TL;DR: In this article, the authors provide an overview of transforms and transform properties for the analysis of data sequences, and discuss the reason for the periodicity of the discrete-time spectrum and presents DTFT properties.
Abstract: Publisher Summary This chapter provides an overview of transforms and transform properties. It reviews the nature of the sampled data and transforms and transform properties for the analysis of data sequences. The integrals defining the series coefficients correspond to the inverse discrete-time Fourier (IDTFT) and considers one- and two-dimensional series. The aliasing phenomenon leads to a periodic spectrum for data sequences so that the spectrum has a Fourier representation in terms of the data. This representation can be found from the data by using the discrete-time Fourier transform (DTFT). The DTFT is generalized to the z-transform, which is a powerful tool for data sequence analysis. The chapter discusses the reason for the periodicity of the discrete-time spectrum and presents DTFT properties. It also reviews the discrete Fourier transform (DFT) and highlights the Laplace transform. The chapter reviews discrete-time random sequences and discusses correlation and covariance sequences and their power spectral densities.
TL;DR: This correspondence investigates the symmetry and sparseness of the Walsh gain matrices and an efficient sparse matrix algorithm is used to calculate the Walsh Gain matrix.
Abstract: Zarowski and Yunik [1] demonstrated that an FIR filter can be realized with fewer multiplications in the fast Walsh transform (FWT) domain than in the fast Fourier transform (FFT) domain for some transform lengths. This correspondence investigates the symmetry and sparseness of the Walsh gain matrices. An efficient sparse matrix algorithm is used to calculate the Walsh gain matrix.
Proceedings Article•
01 Jan 1987
01 Jan 1987
TL;DR: Both the coherent parallel and converging beam Fourier transform methods are presented and the input objects chosen for this study are those of snow crystals.
Abstract: In this paper we describe two methods of constructing optical Fourier transforms for use in image processing. Both the coherent parallel and converging beam Fourier transform methods are presented. The converging beam set-up was used to obtain results presented in another paper (1) at this NATO school but its full discussion at that time was postponed until now. The input objects chosen for this study are those of snow crystals. They are most appropriate since their Fourier transforms contain both some high and low spatial frequencies, not to mention they are beautiful patterns to observe.
08 Apr 1987
TL;DR: The general motivation for the work comes from the need to validate the FFT algorithm when it newly implemented on a computer or when new techniques or devices are added to a computer facility to evaluate discrete Fourier transforms.
Abstract: : A method is described for validating fast Fourier transforms (FFTs) based on the use of simple input functions whose discrete Fourier transforms can be evaluated in closed form Explicit analytical results are developed for one dimensional and two dimensional discrete Fourier transforms The analytical results are easily generalized to higher dimensions The results offer a means for validating the FFT algorithm in one, two, or higher dimensional settings The general motivation for the work comes from the need to validate the FFT algorithm when it newly implemented on a computer or when new techniques or devices are added to a computer facility to evaluate discrete Fourier transforms Keywords: Computer Program Verification
01 Jan 1987
TL;DR: In this paper, a Joint Fourier Transform correlation method was proposed to recognize or classify automatically without visualization phase objects, which is applied to the recognition of different strain states in thin plastic samples (polymethyl methacrylate).
Abstract: Several methods can be used for visualizing phase objects, i.e. transparent planar objects whose index of refraction is a function of two spatial coordinates x, y and refractive properties are negligible1. Examples of phase objects are thin biological sections, flames and hot gases fluxes, pellicles, ultrasonic fields ... In this paper, we present a Joint Fourier Transform correlation method2 to recognize or classify automatically without visualization phase objects. The method is applied to the recognition of different strain states in thin plastic samples (polymethyl methacrylate). We also mention the results obtained on rapidly self modifying liquid pellicles which are other phase objects prepared by smearing microscopy cover glasses with a fast evaporating solvent.