Abstract: We present an assortment of both standard and advanced Fourier techniques that are useful in the analysis of astrophysical time series of very long duration—where the observation time is much greater than the time resolution of the individual data points. We begin by reviewing the operational characteristics of Fourier transforms of time-series data, including power-spectral statistics, discussing some of the differences between analyses of binned data, sampled data, and event data, and we briefly discuss algorithms for calculating discrete Fourier transforms (DFTs) of very long time series. We then discuss the response of DFTs to periodic signals and present techniques to recover Fourier amplitude "lost" during simple traditional analyses if the periodicities change frequency during the observation. These techniques include Fourier interpolation, which allows us to correct the response for signals that occur between Fourier frequency bins. We then present techniques for estimating additional signal properties such as the signal's centroid and duration in time, the first and second derivatives of the frequency, the pulsed fraction, and an overall estimate of the significance of a detection. Finally, we present a recipe for a basic but thorough Fourier analysis of a time series for well-behaved pulsations.

Abstract: The Fourier spectrum of a periodic signal may be obtained by fast Fourier transform algorithms, but, as is well known, special care must be taken to avoid severe distortions introduced by the sampling process. The main problem is the leakage generated by the truncation required to obtain a finite length sampled data. The usual procedure to reduce leakage is to multiply the sampled signal by a weighting window. Several kinds of windows have been proposed in the literature, and today they are also included in many commercial instruments. A simple alternative procedure is proposed in this paper. It is implemented with a PC compatible data acquisition board (DAQ) and consists of an algorithm that uses decimation and interpolation techniques. This algorithm is equivalent to the use of an adjustable sampling frequency and correspondingly an adjustable window size. Results obtained by this method on both harmonic and polyharmonic signals are empirically analyzed and compared with those given by an instrument with built-in FFT capabilities.

Abstract: We present a new unified approach to FFT based image registration. Prior works divided the registration process into two stages: the first was based on phase correlation (PC) which provides pixel accurate registration [5], while the second step provides subpixel registration accuracy [1, 3]. By extending the PC method we derive a FFT based image registration algorithm which is able to estimate large translations with subpixel accuracy. The algorithm's properties resemble those of the Gradient Methods [4] while outperforming it by exhibiting superior convergence range.

Abstract: Plane waves are transmitted at different incident angles, and the radio frequency echo waveforms received by the elements in an array are processed with a Fourier transform. This method is capable of generating one frame of “raw” image for each transmitted plane wave. The formation of each raw image includes: temporal Fourier transform of radio frequency echo signal from each element; phase rotation; spatial Fourier transform; complex interpolation; and an inverse spatial-temporal Fourier transform. This method does not require the synthesis of limited diffraction beams and is computationally more efficient compared to conventional delay-and-sum approach. These raw images are combined coherently to enlarge spatial frequency coverage and enhance lateral resolution. The resolution-enhanced images are further combined incoherently to achieve speckle reduction.

Abstract: Polarimetric imagery that is collected from time-sequential and multiple image format sensors all have potential for image misregistration. Since polarization is usually measured as small differences between radiometric measurements, it is highly sensitive to misregistration, especially at regions of high contrast. The general consensus in the polarization community is that image misregistration on the order of 1/10th of a pixel can introduce artifacts in polarization images. If the registration is not achieved and maintained to this resolution, the data must be registered in software. Typically, rotation and translation (horizontal and vertical) are the main transformations that need to be corrected. It is desirable to have a registration algorithm that determines rotations and translations to 1/10th of a pixel, does not require user intervention, takes minimal computation time, and is based on analytical (non-iterative), automated calculations. This paper details an analytical, automated registration algorithm that corrects for rotation and translations by using a Fourier transform technique. Examples of images registered with this algorithm, and estimates of residual misregistrations are presented. Typical processing times are also given.

Abstract: A system to deal with gray level images applying complex-valued networks has already been proposed The proposed system combines complex-valued networks with a 2-dimensional discrete Fourier Transform, and is based on the idea of phase matrix image representation This paper is intended to build pre-processing and post-processing based on network architecture in the system and propose a novel complex-valued neuron to transform gray level images to the phase matrices in the pre-processing The phase and amplitude of an input for the complex-valued neuron determine its output phase by shifting the input phase by the quantity, which is proportional to the input amplitude Introducing such neurons enables us easily to deal with gray level images using complex-valued networks Simulation results on the image representation ability through the pre-processing are also presented

Abstract: A non-destructive method for discriminating between different types of paper has been proposed, using image analysis, Fourier transformation, and cross-correlation matching. A fast Fourier transform (FFT) is used to extract the periodicity in the structure of paper that results from the manufacturing processes. The light-transmission images of the paper to be Fourier transformed are obtained from a flatbed image scanner. The similarity between the power spectrum of the FFT of the sample and that of a reference is quantified using a cross-correlation matching method. An advantage of using frequency analysis is that periodicity can be detected even if the sample is damaged or is printed on. The technique works on samples as small as 2 cm 2 .

Abstract: The effectiveness and limitations of medical image processing using analog and digital methods are studied. Several types of errors introduced during the image processing are analyzed. For the analog optical Fourier transform, errors are introduced by the vignetting effect and lens aberration. For the digital Fourier transform, errors are introduced by the aliasing effect and the band limit. To compare the results obtained by the two techniques, a set of x-ray images was processed both optically and digitally. The former was achieved by an optical system containing a large Fourier telephoto lens and the latter by a personal computer using a Fourier transform algorithm. The veracity of both the optical and digital Fourier spectra is analyzed. Our results indicate that the optical method has high speed due to parallel processing. High veracity can be achieved in high frequency regions by using an optimal optical system. In comparison, the digital method has the advantages of high processing precision and programmability, but has low processing speed. The comparison of the two different techniques presented in this article can provide a basis for selection of the processing method in different clinical settings. Even with today's fast computers, the optical method is still suitable for many clinical applications. The best choice lies in an analog-digital combination.

Abstract: This paper describes a rose variety recognition project and proposes the modified Fourier descriptor. Based on the modified descriptor, a new measurement, angle measurement, is proposed for image shape analysis. We use it for rose flower shape analysis and the experimental results show that this measurement is quite efficient for image shape analysis.

Abstract: Iterative image reconstruction algorithms play an increasingly important role in modern tomographic systems, especially in emission tomography. With the fast increase of the sizes of the tomographic data, reduction of the computation demands of the reconstruction algorithms is of great importance. Fourier-based forward and back-projection methods have the potential to considerably reduce the computation time in iterative reconstruction. Additional substantial speed-tip of those approaches can be obtained utilizing powerful and cheap off-the-shelf FFT processing hardware. The Fourier reconstruction approaches are based on the relationship between the Fourier transform or the image and Fourier transformation of the parallel-ray projections. The critical two steps are the estimations of the samples of the projection transform, on the central section through the origin of Fourier space, from the samples of the transform of the image, and vice versa for back-projection. Interpolation errors are a limitation of Fourier-based reconstruction methods. We have applied min-max optimized Kaiser-Bessel interpolation within the nonuniform Fast Fourier transform (NUFFT) framework. This approach is particularly well suited to the geometries of PET scanners. Numerical and computer simulation results show that the min-max NUFFT approach provides substantially lower approximation errors in tomographic forward and back-projection than conventional interpolation methods, and that it is a viable candidate for fast iterative image reconstruction.

Abstract: Phase correlation based only on the phase information is a highly accurate alignment technique which is insensitive to various types of noise and geometric distortion and immune to nonuniform illumination. But only pixel-level matching resolution can be obtained. This paper presents an efficient and practical image matching algorithm with sub-pixel accuracy. The algorithm is based upon phase correlation with rotational compensation and paraboloid surface fit and can determine the translational and rotational parameters needed to match the two images. The validity and accuracy of the algorithm are also discussed.

Abstract: This work presents a fast exhaustive robust matching technique for aligning translated images. The key idea is to express a robust matching surface function in terms of correlation operations. Speed is obtained from computing correlations in the frequency domain. Different sized images and arbitrary shapes may be matched. The method outputs a matching surface representing the quality of match at each translatory position. Experimental results of a comparison with the standard method of phase correlation are shown.

Abstract: In this paper, the theory of the unified discrete Fourier-Hartley transform (UDFHT) is developed. The UDFHT includes the discrete Fourier and Hartley transforms of types I-IV as special cases, and with little modification, it also includes the discrete cosine and sine transforms of types II-IV. A unified efficient structure using fast Fourier transform is proposed. The UDFHT is then generalized (GDFHT) to the case of non-constant coefficients. Sufficient conditions for orthogonality of the transform are presented. The GDFHT can be used to compute the modified discrete cosine transform for both non-window and window modes as illustrated in the paper.

Abstract: The 2D phase correlation and the 2D cepstrum technique are known as two solutions to the translation-based image registration problem However, both methods have a large computational load Satisfactory results were obtained in a smaller translation range by applying phase correlations to the 1D projections of images to reduce the computational load (Alliney, S and Morandi, C, 1986) The paper presents a new algorithm based on applying the cepstrum technique to image projections in a similar way An enhanced cepstrum technique is developed by subtracting the cepstrum of the projection differences from the cepstrum of the projection additions A superior performance compared to that of the 1D phase correlation can be obtained by whitening the power spectrum by the square-root instead of the logarithm and using a high-pass filter afterwards The algorithm yields larger translation ranges in registering noisy images and remarkably outperforms the phase correlation method in the case of different degrees of blurring between images

Abstract: It is well known that a SAR image is composed of two types of information: amplitude and phase. Nevertheless, the information contained in the phase is hardly exploited on its own. Indeed, the number of processes at work and the scale difference between the image resolution and the wavelength induce, with regard to the phase, a quasi-random spatial behavior. However, our recent work shows that the phase of one image can be spatially correlated. First, we define an estimator for the spatial correlation of the phase, and study its behavior with real data. We assess the phase correlation according to the resolution and the type of surface. Then, we set down the theoretical bases of a statistical model of this behavior. We highlight the conditions required with regard to the resolution, the sampling rate, and the impulse response. We therefore identify the best kinds of surfaces, so that the phenomenon occurs. Hence, we simulate the phase correlation for different cases according to the phase model defined. We choose suitable parameters to the conditions of the real data and compare measurements and simulations. Finally, we propose possible applications related to the use of this new source of information.

Abstract: We present a detailed study of the intensity variations of the joint Fourier spectrum of two input objects when one of them is rotated while the other remains fixed by determining the 2-D average contrast of the spectrum as a function of the angle of rotation. The contrast is found to be object dependent, decreases rapidly for small rotations, and has an almost flat response for larger rotations. We explain this behavior in both theoretical and numerical ways for two simple rectangular objects and only numerically using the fast Fourier transform, for binary and grayscale test objects whose analog rotations are acquired with an optomechanical system that uses a high-precision rotatory stage. As an application of the 2-D average contrast analysis a pair of similar cross section images of a biological organ are used, to obtain the lowest rotation angle between them, and as a consequence a quantitative aligning method for fine angular movements is presented that uses a 2 f optical-digital coherent processor to obtain the Fourier transform of the input objects. Experimental results are presented.

Abstract: Coherent reception depends upon matching of phase between the transmitted and received signal. Fast convolution techniques based on fast Fourier transform (FFT) are widely used for extracting time delay information from such matching. The latency in processing a large data window of the received signal is a serious overhead for mission critical real time applications. The implementation of a parallel algorithm for correlation of long data sequences in multiprocessor environment is demonstrated here. The algorithm does processing while acquiring the received signal and reduces the computation overhead considerably because of inherent parallelism.

Abstract: PROBLEM TO BE SOLVED: To provide a means for reducing or removing the effect caused by the phase noise of an interferometer system. SOLUTION: The phase noise can be removed at least partially by using a delay/phase correlation approach against two interferometers 12, 14 in an interferometer system 42. The correlation approach can be used for measuring the group delay 44 of a device to be measured. This approach includes determining differences 46, 48 between an output phase of each interferometer at a time t and the phase of the same output at a time subtracting the delay of the other interferometer from the time t. In an embodiment, a first phase difference is the difference between the phase of the output 52 of a test interferometer at the time t and the phase of the output of a test interferometer at a time offsetting the known delay of a reference interferometer from the time t. A second phase difference is calculated by using the same technique but a time offset is the delay expressing a relative delay of two optical transmissions in the test interferometer. COPYRIGHT: (C)2003,JPO

Abstract: A GPS receiving apparatus for receiving spectrum-spreading signal from a plurality of GPS satellites and calculating the position on the basis of the received signals comprising: fast Fourier transform means for performing a fast Fourier transform on the received signal having a carrier wave modulated with a signal obtained by the spectrum-spreading data with a spread code; a first memory in which a result of the fast Fourier transform of the received signal obtained by the fast Fourier transform means is stored; a second memory in which a result of fast Fourier transform of the spread code used in modulating the received signal is stored; multiplying means for multiplying the fast Fourier transform result of the received signal read from the first memory by the result of the fast Fourier transform of the spread code read from the second memory; inverse fast Fourier transform means for performing an inverse fast Fourier transform on a result obtained by the multiplying means to obtain a correlation detection output between the received signal and the spread code; and means for searching for a peak of a correlation between the received signal and the spread code based on the correlation detection output obtained by the inverse fast Fourier transform means to detect correlation point between the received signal and the spread code; and means for calculating the position of the GPS receiving apparatus based on the plurality of correlation points.

Abstract: A technique is proposed that generates a depth-independent and narrow beamwidth 3D ultrasonic image. A high-frequency, wide-bandwidth spherical transducer is scanned in 2D along Cartesian coordinates. The received wideband ultrasonic pulses are dynamically focused by means of correcting the spatial spectrum of signals for various temporal frequencies. The main procedures of the algorithm consist in the direct and inverse fast Fourier transforms by time and by two spatial Cartesian coordinates.

Abstract: A Fourier-based solution to the problem of figure-ground segmentation in short baseline binocular image pairs is presented. Each image is modeled as an additive composite of two component images that exhibit a spatial shift due to the binocular parallax. The segmentation is accomplished by decoupling each Fourier component in one of the resultant additive images into its two constituent phasors, allocating each to its appropriate object-specific spectrum, and then reconstructing the foreground and background using the inverse Fourier transform. It is shown that the foreground and background shifts can be computed from the differences of the magnitudes and phases of the Fourier transform of the binocular image pair. While the model is based on translucent objects, it also works with occluding objects.

Abstract: A joint-transform phase correlator is presented for processing multiple-pattern recognition that uses the six-step phase-shifting interferometer with a wavelength-shifted laser diode. The correlation peaks are obtained by the numerical Fourier transformation of a measured phase in an on-axis joint power spectrum. Multiple-pattern phase correlation can be performed to demonstrate high discrimination without the intermodulation owing to zero optical path differences between multiple targets in the interferometer. The experimental results are shown.© (2002) COPYRIGHT SPIE--The International Society for Optical Engineering. Downloading of the abstract is permitted for personal use only.

Abstract: High accuracy correlation detection technology of object signal is to search and detect objects by optical correlation, and it can detect in real-time, recognize automatically and orientate precisely. Optical correlation processes images at light speed and its device is simple, and Fourier transform can be realized. In this article, we mainly study how to design optical Fourier transform system which can realize optical correlation.

Abstract: A method and system for registration system of a first and a corresponding second image are described The method comprises the steps of converting the spatial image data into frequency domain data using, for example, a fast Fourier transform (FFT); determining a correlation map between the two images by calculating the inverse transform of the complex conjugate product of the frequency domain data of the first and the second image; and identifying a translation between the two images on the basis of the correlation map

Abstract: This first chapter gives the definition and elementary properties of the Fourier transform of integrable functions, which constitute the specific calculus mentioned in the introduction. Besides linearity, the toolbox of this calculus contains the differentiation rule and the convolution—multiplication rule. The general problem of recovering a function from its Fourier transform then receives a partial answer that will be completed by the results on pointwise convergence of Chapter A3.

Abstract: A face recognition method based on discrete Fourier invariant features is presented. Starting from continuous Fourier transform, the properties of continuous Fourier transform are discussed, and the paper gives the properties of discrete Fourier transform. Then according to the properties of discrete Fourier transform, it deduces the discrete Fourier invariant features that are employed on facial image recognition. The experiment results show that the face recognition methods based on discrete Fourier invariantfeatures proposed have a good ability of recognition for the translation, scale and rotation-on-the-plane variant face images.

Abstract: We consider the spatial statistical characteristics of an HF signal and the interdependence of the correlation functions of the field amplitude and phase in the case of oblique reflection from the ionosphere. The amplitude and phase parameters are expressed either in terms of the parameter β of ionospheric turbidity, which can be determined most easily, or only in terms of the correlation coefficient of amplitudes at different points. We obtain the expressions for the amplitude and phase correlation functions, which describe well the experimental data.

Abstract: Many significant features of images are represented in their Fourier transform. The spectral phase of an image can often be measured more precisely than magnitude for frequencies of up to a few GHz. However, spectral magnitude is the only measurable data in many imaging applications. In this paper, the reconstruction of complex-valued images from either the phases or magnitudes of their Fourier transform is addressed. Conditions for unique representation of a complex-valued image by its spectral magnitude combined with additional spatial information is investigated and presented. Reconstruction algorithms of complex-valued images are developed and introduced. Three types of reconstruction algorithms are presented. (1) Algorithms that reconstruct a complex-valued image from the magnitude of its discrete Fourier transform and part of its spatial samples based on the autocorrelation function. (2) Iterative algorithms based on the Gerchberg and Saxton approach. (3) Algorithms that reconstruct a complex-valued image from its localized Fourier transform magnitude. The advantages of the proposed algorithms over the presently available approaches are presented and discussed.