Two-dimensional Fast Fourier Transform Research Articles

Two-Dimensional Fast Fourier Transform and Power Spectrum for Surface Roughness in three Dimensions

Two-dimensional power spectrums of engineering surfaces contain plenty of information that is important and valuable for surface characterization. However, the characteristics of the two-dimensional spectrums are largely unknown and the algorithm to implement them is not familiar to many engineers or researchers. This paper describes a detailed procedure to implement the two-dimensional fast Fourier transform and power spectrum for surface roughness in three dimensions. Methods used to extract information from the spectrums are introduced. In order to perform two-dimensional spectral analysis and to have a comprehensive understanding of the characteristics of engineering surfaces, an atlas of the two-dimensional spectrums of representative engineering surfaces are presented. The properties of the spectrums are discussed in conjunction with theoretical analysis and visual characterization of the presented spectrums.

Backprojection filtering for variable orbit fan-beam tomography

Backprojection filtering algorithms are presented for three variable orbit fan-beam geometries. Expressions for the fan-beam projection and backprojection operators are given for a flat detector fan-beam geometry with fixed focal length, with variable focal length, and with fixed focal length and off-center focusing. Backprojection operators are derived for each geometry using transformation of coordinates to transform from a parallel geometry backprojector to a fan-beam backprojector for the appropriate geometry. The backprojection operator includes a factor which is a function of the coordinates of the projection ray and the coordinates of the pixel in the backprojected image. The backprojection filtering algorithm first backprojects the variable orbit fan-beam projection data using the appropriately derived backprojector to obtain a 1/r blurring of the original image, then takes the two-dimensional (2D) Fast Fourier Transform (FFT) of the backprojected image, then multiples the transformed image by the 2D ramp filter function, and finally takes the inverse 2D FFT to obtain the reconstructed image. Computer simulations verify that backprojectors with appropriate weighting give artifact free reconstructions of simulated line integral projections. Also, it is shown that it is not necessary to assume a projection model of line integrals, but the projector and backprojector can be defined to model the physics of the imaging detection process. A backprojector for variable orbit fan-beam tomography with fixed focal length is derived which includes an additional factor which is a function of the flux density along the flat detector. It is shown that the impulse response for the composite of the projection and backprojection operations is equal to 1/r.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Measuring Particle Size Distribution by Digital Image Processing with Inverse Fourier-Bessel Transformation

An analytical approach based on digital image processing is proposed for measuring the size distribution of circular particles. From an analogy with the light diffraction pattern of a circular aperture, a reduced spectrum of the particles' image can be obtained using a two-dimensional fast Fourier transformation. The reduced spectrum, which is equivalent to the diffraction pattern, is analyzed using the theory of inverse light scattering to evaluate the size distribution of the particles. The usefulness of this method is confirmed not only in binary images but also in gray-scale images created by computer simulation.

Egg Shell Inspection Using Global Image Analysis

A global image analysis procedure was developed to analyze the overall appearance of objects in frequency domain and applied to egg shell inspection. The egg images were analyzed using a two-dimensional Fast Fourier Transform (FFT). The FFT was condensed into 33 feature signatures representing position and orientation invariant morphological features for global analysis. Multivariate discriminant analysis was used to develop classification criteria. The global analysis procedure was applied to egg shell inspection, and various inspection models demonstrated an 88% success ratio. While the global analysis is suitable for inspecting overall appearances with ambiguous criteria, conventional morphological analysis has to complement the global analysis when an application requires specific dimensional measurement or precise location of the object.

A Massively Parallel Digital Processor for Spotlight Synthetic Aperture Radar

Near-real-time digital formation of large synthetic aper ture radar (SAR) images requires the computational throughput that only a dedicated special processor or a massively parallel computer can offer. This article documents the implementation of digital spotlight SAR processing components on three commercially avail able massively parallel computers: the Connection Ma chine (CM-2), the nCUBE 2, and the Intel iPSC/860. The three basic spotlight SAR processing components, the polar reformatter, the two-dimensional fast Fourier transformation, and autofocus, are briefly discussed to provide a technical background for the implementation issues that apply to massively parallel computers. As pects of the SAR components that can exploit features of a SIMD or MIMD architecture are also presented. Finally, timing test results on various computers are provided for evaluation and comparison.

Efficient implementation of the Fourier dual reciprocity boundary element method using two-dimensional fast Fourier transforms

The Poisson equation is one of the field equations which arises often in aerospace applications, and its efficient solution is of great importance to aerodynamic studies. The boundary element method (BEM) is a powerful method for the solution of such field problems as it reduces the dimension of the problem by one and leads to boundary only discretization for linear problems without distributed source/sink terms. The Fourier Dual Reciprocity Boundary Element Method (FDRBEM) was developed to retain the boundary only discretization character of the BEM. The basis of the method is the expansion of the generation term in a two-dimensional Fourier series which is in turn used to transform the BEM area integrals into contour integrals. In this paper, a two-dimensional Fast Fourier Transform (FFT) algorithm is developed to efficiently and accurately extract the Fourier coefficients of the two-dimensional series expansion of the source term. Practical concerns in the implementation of the FDRBEM are discussed. Four numerical examples are presented to validate the approach.

Application of digital image processing to a β-gauge for determining mass concentration of suspending particulate matter in atmosphere

A two-dimensional image of the mass concentration of suspending particulate matter (SPM) collected on Millipore filter paper was photographed with Ultrofilm- 3H. The printed paper image was transformed into a digital image (256 × 256 pixels) with 256 gray levels. Two rresults were obtained. The averaged values of gray level over all pixels of the digital image was found to correlate with the mass value measured by a β-gauge. The characteristical range of the digital image which was transformed to frequency by two-dimensional fast fourier transformation was found in the low frequency. It was presumed to relate to SPM from anthropogenic sources because the SPMs usually show higher density and smaller particle size than SPMs from natural sources.

A two-dimensional FFT algorithm for three-phase inverter-fed systems

A spectrum analyzer for three-phase inverter-fed balanced systems which is capable of calculating up to 24 harmonic components of the line currents every 130 mu s is presented. The method is based on a synchronized sampling technique and on a highly efficient fast Fourier transform (FFT) for three-phase systems. The latter consists of a two-dimensional six-point discrete Fourier transform (DFT) followed by a two-dimensional four-point DFT. The total FFT algorithm has been successfully implemented on a TMS32010 digital signal processor.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

An instruction systolic array implementation of the two-dimensional Fast Fourier Transform

An Instruction Systolic Array (ISA) implementation of the two-dimensional Fast Fourier Transform (FFT) algorithm is presented in this paper. The ISA is characterised by a systolic flow of instructions instead of data as in ordinary systolic array. The ISA implementation of the two-dimensional FFT algorithm is derived following a systematic design methodology. The derived ISA for the two-dimensional FFT is modular and uses identical processing elements, with local and regular interconnections among the processing elements. For a √ N × √ N point two-dimensional FFT, the time-complexity of the proposed ISA design is O(√ fN) with N processing elements.

Surface quality assessment using computer vision methods

An assessment of surface quality in turned and milled specimens is made utilizing a PC-based digital imaging system. Two different methods of surface finish measurement are examined and compared. The first approach, as pioneered by Luk and Huynh, utilizes the intensity histogram of the surface image to characterize surface roughness and quality. Changes in the statistical descriptors of the histogram serve to quantify the surface finish. A second method utilizes the two-dimensional Fast Fourier Transform of the digitized surface image. The magnitude and frequency information obtained from the FFT are used as measurement parameters of the surface finish. Advantages and disadvantages of these two methods are discussed and compared.

Surface quality assessment using computer vision methods

An assessment of surface quality in turned and milled specimens is made utilizing a PC-based digital imaging system. Two different methods of surface finish measurement are examined and compared. The first approach, as pioneered by Luk and Huynh, utilizes the intensity histogram of the surface image to characterize surface roughness and quality. Changes in the statistical descriptors of the histogram serve to quantify the surface finish. A second method utilizes the two-dimensional Fast Fourier Transform of the digitized surface image. The magnitude and frequency information obtained from the FFT are used as measurement parameters of the surface finish. Advantages and disadvantages of these two methods are discussed and compared.

A SAR processor based on two-dimensional FFT codes

A synthetic aperture radar (SAR) processor approach based on two-dimensional fast Fourier transform (FFT) codes coupled with an asymptotic evaluation of the unit response function is presented. For the latter, no approximation is made to the distance function, so that the full range of geometric aberrations is analytically considered, enabling an effective reference filter to be designed. The two-dimensional FFTs were designed as to run on computers of very limited memory: the required FFT is computed by means of FFTs of lower order. Two FFT codes were considered: one is faster and allows full or reduced (quick look or multilook) resolution performance to be obtained easily; the second is slower but allows the use of a space-varying filter and/or investigations on limited portions (zoom) of the image. Both codes are suited to parallel processing, e.g. by a transputer net. A full discussion on computer memory and time requirements is presented as well as first examples of image processing results.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

A connection between bit reversal and matrix transposition: hardware and software consequences

A simple connection between bit reversal and matrix transposition is explained in the context of two-dimensional fast Fourier transform (FFT). This connection is then shown to be useful for hardware and software implementations of two-dimensional FFTs based on the row-column algorithm: the bit-reversal operations involved in the row and column one-dimensional transforms of length M can merge with the matrix transposition in between, to form a single bit reversal of length N=M/sup 2/, which is more efficient than considering these operations separately. Furthermore, this connection is also the base of new bit-reversal algorithms that are explained. The first one can be implemented in two different versions, depending on the availability of an auxiliary memory of size square root N. One of these versions turns out to be as fast as a recently proposed algorithm by D.M. Evans (1987), while the second is twice as fast as the usual one. The second algorithm is a divide-and-conquer approach to the problem that allows easy parallelization.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

A combined finite element-boundary integral formulation for solution of two-dimensional scattering problems via CGFFT

A novel technique is presented for computing the scattering by two-dimensional structures of arbitrary inhomogeneity. The proposed approach combines the usual finite-element (FE) method with the boundary-integral equation to formulate a discrete system. This is subsequently solved via the conjugate gradient (CG) algorithm. A particular characteristic of the method is the use of rectangular boundaries to enclose the scatterer. Several of the resulting boundary integrals are then convolutions and can be evaluated via the fast Fourier transform (FFT) in the implementation of the CG algorithm. The solution approach presented here offers the principal advantage of having O(N) memory demand and employs a one-dimensional FFT, as against the two-dimensional FFT required in a traditional implementation of the proposed CG-FFT algorithm. The speed of the proposed solution method is compared with that of the traditional CG-FFT algorithm. Results are presented for several rectangular composite cylinders and one perfectly conducting cylinder. These are shown to be in excellent agreement with the moment method.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Analysis of the power spectra of galactic images and spiral structures

view Abstract Citations (18) References (13) Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS Analysis of the power spectra of galactic images and spiral structures Krakow, W. ; Huntley, J. M. ; Seiden, P. E. Abstract Several digitized images of galaxies M83, MS 1, and NGC 1365 have been analyzed using two-dimensional Fast Fourier Transform techniques. Markedly different power spectra (i.e., the equivalent of optical diffractograms) are obtained depending on the galaxy chosen and the photometric band of the image. For M83, the Hα power spectrum reveals only long-range correlations characteristic of a distorted or segmented spiral shape. The power spectra of the ultraviolet and red spectral images, although different, show both long- and short-range order which would be present in well defined spiral arms. Using Fourier image-filtering techniques, it has also been possible to separate the size components in a galactic image. Low-pass filtering reveals the larger spatial frequencies of the fundamental spiral shapes, while high-pass filtering reveals small uncorrelated objects. A number of filtering experiments of this type have been performed. The second part of this study involves the Fourier analysis of computer-modeled ideal spirals with linear, logarithmic, and hyperbolic shapes as a function of radial distance. In order to approach the images of real galaxies, various modifications to these shapes have been evaluated. These modifications include: center smoothing, variable arm width, additive noise radial arm segmenting, and center feature reduction. The power spectrum of a hyperbolic model with the appropriate combination of these modifications most closely resembles that of M83 in the ultraviolet. Publication: The Astronomical Journal Pub Date: January 1982 DOI: 10.1086/113097 Bibcode: 1982AJ.....87..203K Keywords: Astronomical Spectroscopy; Digital Data; Fast Fourier Transformations; Image Processing; Power Spectra; Spiral Galaxies; Astronomical Photography; Data Smoothing; Fourier Analysis; Galactic Structure; H Alpha Line; Image Filters; Mathematical Models; Spectral Bands; Spectrum Analysis; Ultraviolet Spectra; Astrophysics full text sources ADS | data products NED (3)

Roundoff error in multidimensional generalized discrete transforms

The analysis of rounding error in the one-dimensional fast Fourier transform (FFT) is extended to a class of generalized orthogonal transforms [1] with a common fast algorithm similar to the FFT algorithm. This class includes the BInary FOurier REpresentation (BIFORE) transform (BT) [2], the complex BT (CBT) [3], and the discrete Fourier transform (DFT). Expressions for the mean square error (MSE) in the two-dimensional BT, CBT, and FFT are derived. In the case of white input data, the mean square error-to-signal ratio is derived for the multidimensional generalized transforms. The error-to-signal ratio for the one-dimensional FFT derived by Kaneko and Liu is modified with improvement. Some comparisons among BIFORE, DFT, and Haar transforms are also included. The theoretical results for the two-dimensional FFT and BIFORE have been verified experimentally. The experimental results are in good agreement with the theoretical results for lower order sequences, but deviate as the order increases due to the actual manner of rounding in the digital computer.