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Nonuniform Fast Fourier Transforms Using MinMax Interpolation
 IEEE Trans. Signal Process
, 2003
"... The FFT is used widely in signal processing for efficient computation of the Fourier transform (FT) of finitelength signals over a set of uniformlyspaced frequency locations. However, in many applications, one requires nonuniform sampling in the frequency domain, i.e.,a nonuniform FT . Several pap ..."
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Cited by 83 (13 self)
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The FFT is used widely in signal processing for efficient computation of the Fourier transform (FT) of finitelength signals over a set of uniformlyspaced frequency locations. However, in many applications, one requires nonuniform sampling in the frequency domain, i.e.,a nonuniform FT . Several papers have described fast approximations for the nonuniform FT based on interpolating an oversampled FFT. This paper presents an interpolation method for the nonuniform FT that is optimal in the minmax sense of minimizing the worstcase approximation error over all signals of unit norm. The proposed method easily generalizes to multidimensional signals. Numerical results show that the minmax approach provides substantially lower approximation errors than conventional interpolation methods. The minmax criterion is also useful for optimizing the parameters of interpolation kernels such as the KaiserBessel function.
Fast slant stack: A notion of Radon transform for data in a Cartesian grid which is rapidly computible, algebraically exact, geometrically faithful and invertible
 SIAM J. Sci. Comput
, 2001
"... Abstract. We define a notion of Radon Transform for data in an n by n grid. It is based on summation along lines of absolute slope less than 1 (as a function either of x or of y), with values at nonCartesian locations defined using trigonometric interpolation on a zeropadded grid. The definition i ..."
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Cited by 48 (11 self)
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Abstract. We define a notion of Radon Transform for data in an n by n grid. It is based on summation along lines of absolute slope less than 1 (as a function either of x or of y), with values at nonCartesian locations defined using trigonometric interpolation on a zeropadded grid. The definition is geometrically faithful: the lines exhibit no ‘wraparound effects’. For a special set of lines equispaced in slope (rather than angle), we describe an exact algorithm which uses O(N log N) flops, where N = n2 is the number of pixels. This relies on a discrete projectionslice theorem relating this Radon transform and what we call the Pseudopolar Fourier transform. The Pseudopolar FT evaluates the 2D Fourier transform on a nonCartesian pointset, which we call the pseudopolar grid. Fast Pseudopolar FT – the process of rapid exact evaluation of the 2D Fourier transform at these nonCartesian grid points – is possible using chirpZ transforms. This Radon transform is onetoone and hence invertible on its range; it is rapidly invertible to any degree of desired accuracy using a preconditioned conjugate gradient solver. Empirically, the numerical conditioning is superb; the singular value spread of the preconditioned Radon transform turns out numerically to be less than 10%, and three iterations of the conjugate gradient solver typically suffice for 6 digit accuracy. We also describe a 3D version of the transform.
Iterative tomographic image reconstruction using Fourierbased forward and back projectors
 IEEE Trans. Med. Imag
, 2004
"... Fourierbased reprojection methods have the potential to reduce the computation time in iterative tomographic image reconstruction. Interpolation errors are a limitation of Fourierbased reprojection methods. We apply a minmax interpolation method for the nonuniform fast Fourier transform (NUFFT) t ..."
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Cited by 23 (4 self)
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Fourierbased reprojection methods have the potential to reduce the computation time in iterative tomographic image reconstruction. Interpolation errors are a limitation of Fourierbased reprojection methods. We apply a minmax interpolation method for the nonuniform fast Fourier transform (NUFFT) to minimize the interpolation errors. Numerical results show that the minmax NUFFT approach provides substantially lower approximation errors in tomographic reprojection and backprojection than conventional interpolation methods.
Fast and accurate Polar Fourier transform
 Appl. Comput. Harmon. Anal.
, 2006
"... In a wide range of applied problems of 2D and 3D imaging a continuous formulation of the problem places great emphasis on obtaining and manipulating the Fourier transform in Polar coordinates. However, the translation of continuum ideas into practical work with data sampled on a Cartesian grid is pr ..."
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Cited by 17 (1 self)
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In a wide range of applied problems of 2D and 3D imaging a continuous formulation of the problem places great emphasis on obtaining and manipulating the Fourier transform in Polar coordinates. However, the translation of continuum ideas into practical work with data sampled on a Cartesian grid is problematic. In this article we develop a fast high accuracy Polar FFT. For a given twodimensional signal of size N × N, the proposed algorithm’s complexity is O(N^2 log N), just like in a Cartesian 2DFFT. A special feature of our approach is that it involves only 1D equispaced FFT’s and 1D interpolations. A central tool in our method is the pseudoPolar FFT, an FFT where the evaluation frequencies lie in an oversampled set of nonangularly equispaced points. We describe the concept of pseudoPolar domain, including fast forward and inverse transforms. For those interested primarily in Polar FFT’s, the pseudoPolar FFT plays the role of a halfway point—a nearlyPolar system from which conversion to Polar coordinates uses processes relying purely on 1D FFT’s and interpolation operations. We describe the conversion process, and give an error analysis of it. We compare accuracy results obtained by a Cartesianbased unequallysampled FFT method to ours, both algorithms using a smallsupport interpolation and no precompensating, and show marked advantage to the use of the pseudoPolar initial grid.
Pattern matching as a correlation on the discrete motion group, Computer Vision and Image Understanding 74
 25, 2005 15:58 WSPC/157IJCIA 00141 16
, 1999
"... In this paper we develop a correlation method for the template matching problem in pattern recognition which includes translations, rotations, and dilations in a natural way. The correlation method is implemented using Fourier analysis on the “discrete motion group ” and fast Fourier transform metho ..."
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Cited by 3 (2 self)
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In this paper we develop a correlation method for the template matching problem in pattern recognition which includes translations, rotations, and dilations in a natural way. The correlation method is implemented using Fourier analysis on the “discrete motion group ” and fast Fourier transform methods. A brief introduction to Fourier methods on the discrete motion group is given and the efficiency of these methods is discussed. Results of the numerical implementation are given for particular examples. c ○ 1999 Academic Press Key Words: pattern analysis; object recognition and indexing.
Highresolution ab initio threedimensional xray diffraction microscopy
 Journal of the Optical Society of America A
, 2006
"... Coherent Xray diffraction microscopy is a method of imaging nonperiodic isolated objects at resolutions only limited, in principle, by the largest scattering angles recorded. We demonstrate Xray diffraction imaging with high resolution in all three dimensions, as determined by a quantitative anal ..."
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Cited by 3 (1 self)
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Coherent Xray diffraction microscopy is a method of imaging nonperiodic isolated objects at resolutions only limited, in principle, by the largest scattering angles recorded. We demonstrate Xray diffraction imaging with high resolution in all three dimensions, as determined by a quantitative analysis of the reconstructed volume images. These images are retrieved from the 3D diffraction data using no a priori knowledge about the shape or composition of the object, which has never before been demonstrated on a nonperiodic object. We also construct 2D images of thick objects with infinite depth of focus (without loss of transverse spatial resolution). These methods can be used to image biological and materials science samples at high resolution using Xray undulator radiation, and establishes the techniques to be used in atomicresolution ultrafast imaging at Xray freeelectron laser sources. OCIS codes: 340.7460, 110.1650, 110.6880, 100.5070, 100.6890, 070.2590, 180.6900 1.
Accurate and Fast Discrete Polar Fourier Transform
 in Proc. 37th Asilomar Conf. Signals, Systems & Computers
, 2003
"... In this article we develop a fast high accuracy Polar FFT. For a given twodimensional signal of size N N , the proposed algorithm's complexity is O(N log N), just like in a Cartesian 2DFFT. A special feature of our approach is that it involves only 1D equispaced FFT's and 1D interpolations. ..."
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Cited by 2 (0 self)
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In this article we develop a fast high accuracy Polar FFT. For a given twodimensional signal of size N N , the proposed algorithm's complexity is O(N log N), just like in a Cartesian 2DFFT. A special feature of our approach is that it involves only 1D equispaced FFT's and 1D interpolations. A central tool in our approach is the pseudopolar FFT, an FFT where the evaluation frequencies lie in an oversampled set of nonangularly equispaced points. The pseudopolar FFT plays the role of a halfway point  a nearlypolar system from which conversion to Polar Coordinates uses processes relying purely on interpolation operations. We describe the conversion process, and compare accuracy results obtained by unequallysampled FFT methods to ours and show marked advantage to our approach.
An N²log N BackProjection Algorithm for SAR Image Formation
 IN THIRTYFORTH ASILOMAR CONFERENCE ON SIGNALS, SYSTEMS, AND COMPUTERS, OCTOBER 2000
, 2000
"... We propose a fast algorithm for farfield SAR imaging based on a new fast backprojection algorithm developed for tomography. We also modify the algorithm for the nearfield scenario. The fast backprojection algorithm for SAR has computational complexity O(N²log N). Compared to traditional FFTbase ..."
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We propose a fast algorithm for farfield SAR imaging based on a new fast backprojection algorithm developed for tomography. We also modify the algorithm for the nearfield scenario. The fast backprojection algorithm for SAR has computational complexity O(N²log N). Compared to traditional FFTbased methods, our new algorithm has potential advantages: the new algorithm does not need frequencydomain interpolation, which becomes complex for the wideangle case; the new approach is applicable to the nearfield scenario, taking into account wavefront curvature; and the backprojection algorithm can be easily adapted to parallel computing architectures. For some scenarios of interest, the computational cost of the new backprojection approach is similar to or less than that for FFTbased algorithms.
THREEDIMENSIONAL RECONSTRUCTION METHODS FOR MICROROTATION FLUORESCENCE MICROSCOPY
"... Microrotation fluorescence microscopy is a novel, optical imaging technique developed with a cell rotation system. The imaging system enables individual living cells to be rotated in suspension under microscopic dimensions, and allows us to acquire a series of images of the cells, simultaneously du ..."
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Microrotation fluorescence microscopy is a novel, optical imaging technique developed with a cell rotation system. The imaging system enables individual living cells to be rotated in suspension under microscopic dimensions, and allows us to acquire a series of images of the cells, simultaneously during the rotation. A challenging task in microrotation imaging is how to determine threedimensional (3D) cell structure from the image series. This thesis thus presents four alternative methods for reconstructing 3D objects from a series of microrotation images. The three former methods, the expectation maximisation (EM), the generalised SkillingBryan and the marginalisation methods, are iterative algorithms built on the Bayesian inversion theory, which is used to quantify uncertainties in data and model parameters, and also to