Abstract: Fourier transforms are the basis for conventional radar range-Doppler imaging. Target rotation during the coherent integration time results in a time-varying Doppler frequency shift that produces, after Fourier transform, a smeared Doppler spectrum and a blurred image. Sophisticated motion compensation algorithms must be applied to obtain focused images using Fourier techniques. However, image blurring can be mitigated without resorting to sophisticated focusing algorithms by using time-varying spectral analysis in place of the Fourier analysis for Doppler processing. Various methods of time-varying spectral analysis are described and compared. Range-Doppler images of manoeuvring aircraft produced by conventional Fourier analysis are then compared with those produced by time-varying spectral analysis. A quantitative simulation illustrates the benefit of using the latter.

Abstract: A method and an apparatus for registering frames of imaging data using Fourier phase matching. Successive image frames of pixel data are processed using an image registration algorithm. Either image frames or the frames after edge detection are transformed into polar-logarithmic coordinates in the frequency domain and the phases of the Fourier transform of the resulting image representations, after inverse Fast Fourier transformation, are matched. The coordinates at the maximum phase difference are used to determine the scaling and rotation parameters needed for image frame registration. The frequency domain representation of one of the image frames is then scaled and rotated using an anti-aliasing algorithm in accordance with those parameters. The phase of the result is matched with the phase of the frequency domain representation of the other image frame to determine the translation parameters needed for image frame registration.

Abstract: We first discuss the discrete fractional Fourier transform and present some essential properties. We then propose a recursive algorithm to implement phase retrieval from two intensities in the fractional Fourier transform domain. This approach can significantly simplify computational manipulations and does not need an initial phase estimate compared with conventional iterative algorithms. Simulation results show that this approach can successfully recover the phase from two intensities.

Abstract: We propose a method for maximising the affine correlation between images. The method is more global in its search than for example steepest descent based methods. In a first approximation, there is no need to compute any derivatives and it is shown that the results are very good. The method is based on certain changes of coordinates in the images and extensive use of the fast Fourier transformation (FFT). This makes the method very fast, when implemented on a computer.

Abstract: A nonuniform inverse fast Fourier transform (NU-IFFT) for nonuniformly sampled data is realised by combining the conjugate-gradient fast Fourier transform (CG-FFT) method with the newly developed nonuniform fast Fourier transform (NUFFT) algorithms. An example application of the algorithm in computational electromagnetics is presented.

Abstract: An optically implemented wide bandwidth correlation system ( 10 ) that employs a multi-mode imaging device ( 42 ) and a particular modulation format to provide both in-phase and quadrature phase correlation components in a single correlation process. The correlation system ( 10 ) includes an optical source ( 12 ) that generates a laser beam ( 14 ) that is split into a first beam path ( 18 ) and a second beam path ( 20 ). The first split beam and a first electrical signal are applied to a first modulator ( 22 ) in the first path ( 18 ) and the second split beam and the second electrical signal are applied to a second modulator ( 24 ) in the second path ( 20 ). The modulated beams are then applied to the optical imaging device ( 42 ) that causes the beams to interfere with each other within an optical cavity ( 48 ). Four optical outputs are connected to the optical cavity ( 48 ) at strategic locations to provide a zero phase output and a π phase output that represent the in-phase correlation component, and a π/2 quadrature phase output and a 3π/2 quadrature phase output that represent the quadrature phase correlation component. A photodetector (60-66) detects each of the output signals from the imaging device ( 42 ) to provide electrical signals indicative of the phase outputs. A first differential amplifier ( 68 ) receives the electrical signals of the in-phase component and a second differential amplifier ( 70 ) receives the electrical signals of the quadrature phase component. The differential amplifier outputs are applied to separate integrators ( 72,74 ) to sum the signals for the correlation process. The modulators can be Mach-Zehnder interferometer modulators to provide a single sideband suppressed carrier modulation so that the in-phase and quadrature phase correlation components can be simultaneously generated.

Abstract: We report the measurement of transient bending waves with double-pulsed-subtraction TV holography The correlation fringe patterns are automatically quantitatively analyzed by the application of Fourier methods A novel optical setup with two different object-beam optical paths is demonstrated for the generation of carrier fringes The proposed system is highly immune to environmental disturbances because the optical setup imposes no lower limit on the time separation between laser pulses One removes the linear phase distribution due to the spatial carrier in the spatial domain by subtracting the phase of the undeformed carrier fringes from the phase of the modulated fringes Experimental results obtained with an aluminum plate excited by the impact of a piezoelectric translator are presented

Abstract: We present the formulation and implementation of an image stabilization system capable of stabilizing video with very large displacements between frames. A coarse-to-fine technique is applied in resolution and in model spaces. The registration algorithm uses phase correlation to obtain an initial estimate for translation between images; then Levenberg-Marquardt method for nonlinear optimization is applied to refine the solution. Registration is performed in color space, using a subset of the pixels selected by a gradient-based sub-sampling criterion. This software implementation runs at 5 Hz on non-dedicated hardware (Silicon Graphics R10000 workstation).

Abstract: A connection is established between the multidimensional Fourier transform of a radial function f from a given class and the one-dimensional Fourier transform of a related function. This is applied to give an asymptotic formula for the Fourier transform of 1 . The function class in question is compared with related classes already considered in the literature.

Abstract: A system that optically performs complex transforms, such as Fourier transforms. The system includes a pixel addressable spatial modulator that, in parallel, adjusts the phase of the light of each pixel. The modulator can be a reflective or transmissive type device. A transform lens, such as a Fourier lens, performs a two dimensional transform of the pixels outputs. This operation is repeated for the characteristic function (real and imaginary) of the function. The transformed outputs of the characteristic functions are sampled by a light detector and processed by a computer using simple fast operations, such as addition, into the final transform.

Abstract: The feasibility of using a three-dimensional fast Fourier transform as the basis for determining the crystal cell and orientation from a single monochromatic oscillation image of a protein crystal has been investigated. The results of tests, carried out for both simulated and real data-sets, are presented and confirm that a practical method of auto-indexing can be developed.

Abstract: An image processing method is presented that enables an efficient and robust separation of straight line-like structures from an isotropic background texture. The algorithm is based on a transform se- quence mapping both image components onto two nearly disjoint and parameter invariant areas in the resulting transform domain. Applying a linear notch filter to the transformed image enables us to ascertain for each component of the discrete Fourier spectrum of the image, whether it contributes more to the line-like structures or more to the background. Based on this decision, the spectrum is split into two complementary and disjoint spectra, the inverse Fourier transforms of which are taken as estimates for both image components. In practical applications, the al- gorithm behaves robustly with respect to its parameters as well as to the assumptions made about the images. © 1998 Society of Photo-Optical Instru- mentation Engineers. (S0091-3286(98)01410-X)

Abstract: Free-form 3-D surfaces registration is a fundamental problem in 3-D imaging, typically approached by extensions or variations of the ICP algorithm. This work presents a new frequency domain technique for 3-D view registration, totally different form any other techniques for 3-D motion estimation also, based on the Fourier transform. The proposed method can give a non-feature-based method for unsupervised registration of 3-D views. The obtained results are useful "per se" in applications targeted to visual quality or can serve as good starting point for the ICP algorithm when a higher precision is needed.

Abstract: The retrieval of phase information from only the magnitude of the Fourier transform of a signal remains an important problem for many applications. We present an algorithm for phase retrieval when there exist two related sets of Fourier-transform magnitude data. The data are assumed to come from a single object observed in two different polarizations through a distorting medium, so the phase component of the Fourier transform of the object is corrupted. Phase retrieval is accomplished by minimization of a suitable criterion function, which can take three different forms.

Abstract: Based on the modified fractional order Fourier transform (MFROFT), we introduce a modified fractional correlation operation (MFCO). We propose two systems to implement the MFCO experimentally. These are analyzed theoretically to conclude that the MFCO of two objects equals the conventional correlation between the two objects modified by multiplying each by a quadratic phase factor that contains the parameter that determines the fractional order. The MFCO is applied to a rectangular slit analytically, numerically, and experimentally. We show that the MFCO results in superior correlation peaks compared to those of the conventional correlation operation.

Abstract: Wright State UniversityComputer Science and EngineeringDepartmentDayton, Ohio 45435E-mail: aawwal@cs.wright.eduAbstract. A new implementation of the joint Fourier transform correla-tion (JFTC) utilizing the phase information in the Fourier plane is pro-posed. When a phase-based thresholding scheme is used on the jointFourier spectrum, the computation is significantly accelerated. We obtainalmost double normalized correlation peaks with comparable discrimina-tion peak ratio as that of binarized JFTC.

Abstract: In ultrasound elastography, the resolution and accuracy of strain estimation are bounded by the sampling period when the echo signal is digitized. Here, the authors describe three new algorithms which based on multiresolution, time-shift property of Fourier transform and sinc function interpolation, respectively, for improving the precision of peak position estimation of cross-correlation function between echo signals obtained pre- and post- target tissue compression. Experimental results demonstrate that these methods bring about good effects.

Abstract: X-ray reflectometry allows the determination of the thickness, density and roughness of thin layers on a substrate from several Angstroms to some hundred nanometres. The thickness is determined by simulation with trial-and-error methods after extracting initial values of the layer thicknesses from the result of a classical Fast Fourier Transform (FFT) of the reflectivity data. However, the order information of the layers is lost during classical FFT. The order of the layers has then to be known a priori. In this paper, it will be shown that the order of the layers can be obtained by a sliding-window Fourier transform, the so-called Gabor representation. This joint time-frequency analysis allows the direct determination of the order of the layers and, therefore, the use of a more appropriate starting model for refining simulations. A simulated and a measured example show the interest of this method.

Abstract: The present invention provides a method and system (1) for pattern recognition and processing. Information representative of physical characteristics or representations of physical characteristics is transformed into a Fourier series in Fourier space (12) within an input context of the physical characteristics that is encoded in time as delays corresponding to modulation of the Fourier series at corresponding frequencies. Associations (14) are formed between Fourier series by filtering the Fourier series and by using a spectral similarity between the filtered Fourier series to determine the association based on Poissonian probability. The associated Fourier series are added to form strings (16) of Fourier series. Each string is ordered by filtering it with multiple selected filters to form multiple time order formatted subset Fourier series, and by establishing the order through associations with one or more initially ordered strings to form an ordered string. Associations are formed between the ordered strings to form complex ordered strings (18) that relate similar items of interest.

Abstract: Free-form 3-D surfaces registration is a fundamental problem in 3-D imaging, typically approached by extensions or variations of the ICP algorithm. This work presents an alternative method for 3-D motion estimation based on the Fourier transform of the range data. The frequency domain techniques for estimating motion parameters are nonfeature-based methods suitable for unsupervised registration of 3-D views. The proposed method can give unsupervised registration of range data within 1/spl deg/ of accuracy of the angular parameters, useful "per se" in applications where this kind of precision is adequate or which can serve as good starting point for the ICP algorithm when a higher precision is needed.

Abstract: In this study, Fourier descriptor method and energy continuity concept are combined to establish an image deformable model. The deformable model developed in this paper can reconstruct the data of organization structure of a body that is given in three-dimensional sectional images. The reconstructed data can be provided for more medical applications based on a few slices obtained from a set of sectional images. For a more exact and efficient reconstruction of the medical images, the authors' model combines Fourier descriptor, Fourier transform theory, and the energy continuity concept of the image in the frequency domain to analyse and synthesise the object's image information. The Fourier descriptor and its parameterized deformation capture the characteristics of the objects which one wants to present in visualization. The energy continuity concept was introduced into the model to estimate the deformable images that lie between any two known images. The method of this study integrates the spatial coordinates, energy continuity concept and Fourier descriptor to describe the objects for time sequence or spatially related images. So that, the model of this work can estimates the interpolational images that lie between the given images and reconstructs any three-dimensional object efficiently and accurately.

Abstract: In Airborne Laser (ABL) like systems, the phase information is typically reconstructed from wavefront tilt sensor data using a least-squares approach. The solution to the phase reconstruction problem provides a smooth unwrapped phase surface that is not necessarily equal to the original wrapped phase surface in an optical propagation sense. Still, the computed phase surface furnishes information that can be used in determining the required phase conjugation for partial correction of an atmospherically distorted laser source. In this paper, we examine the effect of two phase reconstruction algorithms; unweighted least-squares and intensity weighted least-squares on the performance of an idealized ABL-like adaptive optics systems under ABL-like atmospheric conditions. The performance of each phase reconstruction algorithm will be investigated using MZA's Airborne Laser Simulation under atmospheres with Rytov numbers ranging from 0.233 to 3.725. Since the unwrapping process creates a phase surface without the 2(pi) discontinuities of the original phase, the r0 and phase variance can be calculated from the unwrapped phase. These predictors of the phase correlation performance will be related to the performance measures; Strehl and power-in- the-bucket at the target.© (1998) COPYRIGHT SPIE--The International Society for Optical Engineering. Downloading of the abstract is permitted for personal use only.

Abstract: We give a unified treatment of Fast Fourier Transforms for UDMD systems which contains, as special cases, Fast Fourier algorithms for character groups of many subgroups associated with binary fields.

Abstract: Pattern comparison and pattern alignment are related tasks. The former requires the use of a similarity measure for discriminating between patterns of different classes, while the latter requires a measure of the degree of alignment of two similar patterns. This paper presents an experimental evaluation of two promising measures, Fourier phase correlation and mutual information, which can be used for quantifying the similarity as well as the degree of alignment of two patterns. It is shown that these measures are quite stable to a range of pattern intensity variations, while exhibiting gradual monotonic decrease in the presence of geometric transformations. Phase correlation is more resistant to pattern noise and interference, whereas mutual information is more resistant to intensity variations and geometric distortions.