Showing papers on "Fourier transform published in 1993"

Efficient Similarity Search In Sequence Databases

Abstract: We propose an indexing method for time sequences for processing similarity queries. We use the Discrete Fourier Transform (DFT) to map time sequences to the frequency domain, the crucial observation being that, for most sequences of practical interest, only the first few frequencies are strong. Another important observation is Parseval's theorem, which specifies that the Fourier transform preserves the Euclidean distance in the time or frequency domain. Having thus mapped sequences to a lower-dimensionality space by using only the first few Fourier coefficients, we use R * -trees to index the sequences and efficiently answer similarity queries. We provide experimental results which show that our method is superior to search based on sequential scanning. Our experiments show that a few coefficients (1–3) are adequate to provide good performance. The performance gain of our method increases with the number and length of sequences.

Measurement of the Wigner distribution and the density matrix of a light mode by using optical homodyne tomography: application to squeezed states and vacuum

Abstract: We report measurements of the Wigner distribution and the density matrix of an electromagnetic field mode, for both vacuum and quadrature-squeezed states. As proposed by Vogel and Risken,1 we obtained the Wigner distribution by tomographic inversion of a set of probability distributions of field-quadrature amplitudes measured by using balanced homodyne detection. A Fourier transform of the Wiener function yields the density matrix, which, according to the standard interpretation of quantum mechanics, contains all knowable information about a given quantum system.

Image rotation, Wigner rotation, and the fractional Fourier transform

Abstract: In this study the degree p = 1 is assigned to the ordinary Fourier transform The fractional Fourier transform, for example with degree P = 1/2, performs an ordinary Fourier transform if applied twice in a row Ozaktas and Mendlovic [ “ Fourier transforms of fractional order and their optical implementation,” Opt Commun (to be published)] introduced the fractional Fourier transform into optics on the basis of the fact that a piece of graded-index (GRIN) fiber of proper length will perform a Fourier transform Cutting that piece of GRIN fiber into shorter pieces corresponds to splitting the ordinary Fourier transform into fractional transforms I approach the subject of fractional Fourier transforms in two other ways First, I point out the algorithmic isomorphism among image rotation, rotation of the Wigner distribution function, and fractional Fourier transforming Second, I propose two optical setups that are able to perform a fractional Fourier transform

Fractional Fourier transforms and their optical implementation. II

Abstract: The linear transform kernel for fractional Fourier transforms is derived. The spatial resolution and the space–bandwidth product for propagation in graded-index media are discussed in direct relation to fractional Fourier transforms, and numerical examples are presented. It is shown how fractional Fourier transforms can be made the basis of generalized spatial filtering systems: Several filters are interleaved between several fractional transform stages, thereby increasing the number of degrees of freedom available in filter synthesis.

Constant depth circuits, Fourier transform, and learnability

Abstract: In this paper, Boolean functions in ,4C0 are studied using harmonic analysis on the cube. The main result is that an ACO Boolean function has almost all of its "power spectrum" on the low-order coefficients. An important ingredient of the proof is Hastad's switching lemma (8). This result implies several new properties of functions in -4C(': Functions in AC() have low "average sensitivity;" they may be approximated well by a real polynomial of low degree and they cannot be pseudorandom function generators. Perhaps the most interesting application is an O(n POIYIOg(n ')-time algorithm for learning func- tions in ACO. The algorithm observes the behavior of an AC'" function on O(nPO'Y'Og(n)) randomly chosen inputs, and derives a good approximation for the Fourier transform of the function. This approximation allows the algorithm to predict, with high probability, the value of the function on other randomly chosen inputs.

Integration and differentiation to a variable fractional order

Abstract: Interpretation and differentiation of functions to a variable order (d/dx)nf(x) is studied in two ways: 1) using the Riemann-Liouville definition, 2) using Fourier transforms Some properties and the inversion formula are obtained

Local error estimates for radial basis function interpolation of scattered data

Abstract: Introducing a suitable variational formulation for the local error of scattered data intepolation by radial basis functions φ(r), the error can be bounded by a term depending on the Fourier transform of the interpolated function f and a certain «Kriging function», which allows a formulation as an integral involving the Fourier transform of φ. The explicit construction of locally well-behaving admissible coefficient vectors makes the Kriging function bounded by some power of the local density h of data points

Learning decision trees using the Fourier spectrum

Abstract: This work gives a polynomial time algorithm for learning decision trees with respect to the uniform distribution. (This algorithm uses membership queries.) The decision tree model that is considered is an extension of the traditional boolean decision tree model that allows linear operations in each node (i.e., summation of a subset of the input variables over $GF(2)$).This paper shows how to learn in polynomial time any function that can be approximated (in norm $L_2 $) by a polynomially sparse function (i.e., a function with only polynomially many nonzero Fourier coefficients). The authors demonstrate that any function f whose $L_1 $-norm (i.e., the sum of absolute value of the Fourier coefficients) is polynomial can be approximated by a polynomially sparse function, and prove that boolean decision trees with linear operations are a subset of this class of functions. Moreover, it is shown that the functions with polynomial $L_1 $-norm can be learned deterministically.The algorithm can also exactly identi...

Frequency tracking in power networks in the presence of harmonics

Abstract: Three new techniques for frequency measurement are proposed. The first is a modified zero-crossing method using curve fitting of voltage samples. The second method is based on polynomial fitting of the discrete Fourier transform (DFT) quasi-stationary phasor data for calculation of the rate of change of the positive sequence phase angle. The third method operates on a complex signal obtained by the standard technique of quadrature demodulation. All three methods are characterized by immunity to reasonable amounts of noise and harmonics in power systems. The performance of the proposed techniques is illustrated for several scenarios by computer simulation. >

Discrete Gabor transform

Abstract: A feasible algorithm for implementing the Gabor expansion, the coefficients of which are computed by the discrete Gabor transform (DGT), is presented. For a given synthesis window and sampling pattern, computing the auxiliary biorthogonal function of the DGT is nothing more than solving a linear system. The DGT presented applies for both finite as well as infinite sequences. By exploiting the nonuniqueness of the auxiliary biorthogonal function at oversampling an orthogonal like DGT is obtained. As the discrete Fourier transform (DFT) is a discrete realization of the continuous-time Fourier transform, similarly, the DGT introduced provides a feasible vehicle to implement the useful Gabor expansion. >

Dithered quantizers

Abstract: A theory of overall quantization noise for nonsubtractive dither was originally developed in unpublished work by J.N. Wright and by T.J. Stockham and subsequently expanded by L.K. Brinton, S.P. Lipshitz, J. Vanderkooy, and R.A. Wannamaker. It is suggested that since these latter results are not as well known as the original results, misunderstanding persists in the literature. New proofs of the properties of quantizer dither, both subtractive and nonsubtractive, are provided. The new proofs are based on elementary Fourier series and Rice's characteristic function method and do not require the use of generalized functions (impulse trains of Dirac delta functions) and sampling theorem arguments. The goal is to provide a unified derivation and presentation of the two forms of dithered quantizer noise based on elementary Fourier techniques. >

Wavelet transforms versus Fourier transforms

Abstract: This note is a very basic introduction to wavelets. It starts with an orthogonal basis of piecewise constant functions, constructed by dilation and translation. The ``wavelet transform'' maps each $f(x)$ to its coefficients with respect to this basis. The mathematics is simple and the transform is fast (faster than the Fast Fourier Transform, which we briefly explain), but approximation by piecewise constants is poor. To improve this first wavelet, we are led to dilation equations and their unusual solutions. Higher-order wavelets are constructed, and it is surprisingly quick to compute with them --- always indirectly and recursively. We comment informally on the contest between these transforms in signal processing, especially for video and image compression (including high-definition television). So far the Fourier Transform --- or its 8 by 8 windowed version, the Discrete Cosine Transform --- is often chosen. But wavelets are already competitive, and they are ahead for fingerprints. We present a sample of this developing theory.

Tilings of the time-frequency plane: construction of arbitrary orthogonal bases and fast tiling algorithms

Abstract: The authors consider expansions which give arbitrary orthonormal tilings of the time-frequency plane. These differ from the short-time Fourier transform, wavelet transform, and wavelet packets tilings in that they change over time. They show how this can be achieved using time-varying orthogonal tree structures, which preserve orthogonality, even across transitions. The method is based on the construction of boundary and transition filters; these allow us to construct essentially arbitrary tilings. Time-varying modulated lapped transforms are a special case, where both boundary and overlapping solutions are possible with filters obtained by modulation. They present a double-tree algorithm which for a given signal decides on the best binary segmentation in both time and frequency. That is, it is a joint optimization of time and frequency splitting. The algorithm is optimal for additive cost functions (e.g., rate-distortion), and results in time-varying best bases, the main application of which is for compression of nonstationary signals. Experiments on test signals are presented. >

Roughness spectrum and surface width of self-affine fractal surfaces via the K-correlation model.

Abstract: Theoretical expressions for the height-height correlation function of self-affine fractal surfaces are discussed in comparison with scanning tunneling microscopy, correlation and surface-width data obtained from rough silver and gold films. Fourier transformations are used to compare with equilibrium phenomena, and lead to a correlation model with an associated roughness spectrum of analytic form.

Pattern recognition system

Abstract: A method and apparatus under software control for pattern recognition utilizes a neural network implementation to recognize two dimensional input images which are sufficiently similar to a database of previously stored two dimensional images. Images are first image processed and subjected to a Fourier transform which yields a power spectrum. An in-class to out-of-class study is performed on a typical collection of images in order to determine the most discriminatory regions of the Fourier transform. A feature vector consisting of the highest order (most discriminatory) magnitude information from the power spectrum of the Fourier transform of the image is formed. Feature vectors are input to a neural network having preferably two hidden layers, input dimensionality of the number of elements in the feature vector and output dimensionality of the number of data elements stored in the database. Unique identifier numbers are preferably stored along with the feature vector. Application of a query feature vector to the neural network will result in an output vector. The output vector is subjected to statistical analysis to determine if a sufficiently high confidence level exists to indicate that a successful identification has been made. Where a successful identification has occurred, the unique identifier number may be displayed.

Spatio-temporal image processing

Abstract: and overview.- Image sequence acquisition.- Kinematics and dynamics of motion.- Motion in space-time images.- Fourier transform methods.- Differential methods.- Quadrature filter set methods.- Tensor methods.- Correlation methods.- Phase methods.- Implementation.- Experimental results.

Real time optical correlation system

Abstract: An optical correlation unit (10) for correlating the images of an inspection object and a reference object. The unit uses two phase modulating reflective spatial light modulators (12a, 12b). A first spatial light modulator (12a) receives electronic input in the form of image data representing the inspection object. It modulates incoming light (15) with this input and reflects the modulated output to a first Fourier transform lens (13). This lens provides the optical input to second spatial light modulator (12b), whose electronic input is transform data presenting the complex conjugate of the Fourier transform of the reference image. The electronic input modulates the optical input, resulting in the Fourier product of the two images, which is then transformed to provide correlation data.

Quaternion-Fourier transforms for analysis of two-dimensional linear time-invariant partial differential systems

Abstract: Hamilton's hypercomplex, or quaternion, extension to the complex numbers provides a means to algebraically analyze systems whose dynamics can be described by a system of partial differential equations. The Quaternion-Fourier transformation, defined in this work, associates two dimensional linear time-invariant (2D-LTI) systems of partial differential equations with the geometry of a sphere. This transform provides a generalized gain-phase frequency response analysis technique. It shows full utility in the algebraic reduction of 2D-LTI systems described by the double convolution of their Green's functions. The standard two dimensional complex Fourier transfer function has a phase associated with each frequency axis and does not describe clearly how each axis interacts with the other. The Quaternion-Fourier transfer function gives an exact measure of this interaction by a single phase angle that may be used as a measure of the relative stability of the system. This extended Fourier transformation provides an exquisite tool for the analysis of 2D-LTI systems. >

Fourier volume rendering

Abstract: In computer graphics we have traditionally rendered images of data sets specified spatially, Here, we present a volume rendering technique that operates on a frequency domain representation of the data set and that efficiently generates line integral projections of the spatial data it represents, The motivation for this approach is that the Fourier Projection-Slice Theorem allows us to compute 2-D projections of 3-D data seta using only a 2-D slice of the data in the frequency domain. In general, these “X-ray-like” images can be rendered at a significantly lower computational cost than images generated by current volume rendering techniques, Additionally, assurances of image accuracy can he made.

Convergence of the coupled-wave method for metallic lamellar diffraction gratings

Abstract: Numerical evidence is presented that shows that, for metallic lamellar gratings in TM polarization, the coupled-wave method formulated by Moharam and Gaylord [ J. Opt. Soc. Am. A3, 1780 ( 1986)] converges slowly. (In some cases, for achieving a relative error of less than 1% in diffraction efficiencies, the number of spatial harmonics retained in the computation must be much greater than 100.) By classification of the modal methods for analyzing diffraction gratings into two distinct categories, the cause for the slow convergence is analyzed and attributed to the use of Fourier expansions to represent the permittivity and the electromagnetic fields in the grating region. The eigenvalues and the eigenfunctions of the modal fields in the grating region, whose accurate determination is crucial to the success of the coupled-wave method, are shown to converge slowly as a result of the use of these Fourier expansions. Despite its versatility and simplicity, the coupled-wave method should be used with caution for metallic surface-relief gratings in TM polarization.

Cross‐correlated random field generation with the direct Fourier Transform Method

Abstract: This paper presents a computer algorithm that is capable of cogenerating pairs of three-dimensional, cross-correlated random fields. The algorithm produces random fields of real variables by the inverse Fourier transform of a randomized, discrete three-dimensional spectral representations of the variables. The randomization is done in the spectral domain in a way that preserves the direct power and cross-spectral density structure. Two types of cross spectra were examined. One type specifies a linear relationship between the two fields, which produces the same correlation scales for both variables but different variances. The second cross spectrum is obtained from a specified transfer function and the two power spectra, and it produces fields with different correlation scales. For both models the degree of correlation is specified by the coherency. A delay vector can also be specified to produce an out-of-phase correlation between the two fields. The algorithm is very efficient computationally, is relatively easy to use, and does not produce the lineation problems that can be encountered with the turning bands method. Perhaps most important, this random field generator is capable of co-generating cross-correlated random fields.

The FROG PC series: programs for electron-density and model investigations for proteins

Abstract: A set of computer programs, developed for IBM-compatible personal computers and aimed at crystallographic use, is described. The programs have user-friendly interfaces and allow the calculation of various Fourier syntheses, which can be vizualized and compared. The possibility of obtaining a synthesis and an atomic model together and performing the model image rotations and translations with respect to the synthesis also exists.

Image representation via a finite Radon transform

Abstract: A model of finite Radon transforms composed of Radon projections is presented. The model generalizes to finite group projections in the classical Radon transform theory. The Radon projector averages a function on a group over cosets of a subgroup. Reconstruction formulae that were formally similar to the convolved backprojection ones are derived, and an iterative reconstruction technique is found to converge after a finite number of steps. Applying these results to the group Z/sub 2//sup P/, new computationally favorable image representations have been obtained. A numerical study of the transform coding aspects is attached. >