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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.
Accelerating the nonuniform Fast Fourier Transform
 SIAM REVIEW
, 2004
"... The nonequispaced Fourier transform arises in a variety of application areas, from medical imaging to radio astronomy to the numerical solution of partial differential equations. In a typical problem, one is given an irregular sampling of N data in the frequency domain and one is interested in recon ..."
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Cited by 36 (2 self)
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The nonequispaced Fourier transform arises in a variety of application areas, from medical imaging to radio astronomy to the numerical solution of partial differential equations. In a typical problem, one is given an irregular sampling of N data in the frequency domain and one is interested in reconstructing the corresponding function in the physical domain. When the sampling is uniform, the fast Fourier transform (FFT) allows this calculation to be computed in O(N log N) operations rather than O(N 2) operations. Unfortunately, when the sampling is nonuniform, the FFT does not apply. Over the last few years, a number of algorithms have been developed to overcome this limitation and are often referred to as nonuniform FFTs (NUFFTs). These rely on a mixture of interpolation and the judicious use of the FFT on an oversampled grid [A. Dutt and V. Rokhlin, SIAM J. Sci. Comput., 14 (1993), pp. 1368–1383]. In this paper, we observe that one of the standard interpolation or “gridding ” schemes, based on Gaussians, can be accelerated by a significant factor without precomputation and storage of the interpolation weights. This is of particular value in two and threedimensional settings, saving either 10dN in storage in d dimensions or a factor of about 5–10 in CPUtime (independent of dimension).
FOURIER VOLUME RENDERING OF IRREGULAR DATA SETS
, 2002
"... Examining irregularly sampled data sets usually requires gridding that data set. However, examination of a data set at one particular resolution may not be adequate since either fine details will be lost, or coarse details will be obscured. In either case, the original data set has been lost. We p ..."
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Cited by 6 (0 self)
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Examining irregularly sampled data sets usually requires gridding that data set. However, examination of a data set at one particular resolution may not be adequate since either fine details will be lost, or coarse details will be obscured. In either case, the original data set has been lost. We present an algorithm to create a regularly sampled data set from an irregular one. This new data set is not only an approximation to the original, but allows the original points to be accurately recovered, while still remaining relatively small. This result is accompanied by an efficient ‘zooming ’ operation that allows the user to increase the resolution while gaining new details, all without regridding the data. The technique is presented in Ndimensions, but is particularly well suited to Fourier Volume Rendering, which is the fastest known method of direct volume rendering. Together, these techniques allow accurate and efficient, multiresolution exploration of volume data.
Efficient DualTone MultiFrequency Detection Using the NonUniform Discrete Fourier Transform
 IEEE Signal Processing Letters
, 1998
"... The International Telecommunication Union (ITU) recommendations for dualtone multifrequency (DTMF) signaling are not met by conventional DTMF detectors. We present an efficient DTMF detection algorithm based on the nonuniform discrete Fourier transform that meets all of the ITU recommendations. T ..."
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Cited by 6 (1 self)
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The International Telecommunication Union (ITU) recommendations for dualtone multifrequency (DTMF) signaling are not met by conventional DTMF detectors. We present an efficient DTMF detection algorithm based on the nonuniform discrete Fourier transform that meets all of the ITU recommendations. The key innovations are the use of two sliding windows and development of sophisticated timing tests. Our algorithm requires no buffering of input samples. To perform DTMF detection on n telephone channels, our algorithm requires approximately n MIPS on a digital signal processor (DSP), 120 + 30 n words of data memory, and 1000 words of program memory. Using the new algorithm, a single fixedpoint DSP can perform ITUcompliant DTMF detection on the 24 telephone channels of a T1 timedivision multiplexed telecommunications line. The authors are with the Department of Electrical and Computer Engineering, The University of Texas at Austin, Austin, TX 787121084 (email: felder@ece.utexas.edu, ma...
Design Of PolarSeparable Fir Filters By Radial Slice Approximations
 in IEEE International Conference on Acoustics, Speech, and Signal Processing
, 1997
"... We introduce the design of polarseparable 2D FIR filters by radial slice approximations (RSA). It is a two step procedure. First, 1D filters for the radial and the angular components are designed. Then the desired filter response is approximated on many radial slices in a weighted mean square sen ..."
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Cited by 2 (2 self)
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We introduce the design of polarseparable 2D FIR filters by radial slice approximations (RSA). It is a two step procedure. First, 1D filters for the radial and the angular components are designed. Then the desired filter response is approximated on many radial slices in a weighted mean square sense. In the case of circular filters, RSA outperforms other design procedures in terms of ripple size and circularity of the passband. Examples of filters with nonconstant angular functions prove the flexibility of the new method. 1. INTRODUCTION The design of 2D FIR filters for signal and image processing is an important and difficult problem. Several image processing techniques [1, 2] require FIR filters that are polarseparable in the ideal case. We present here a general two step procedure called Radial Slice Approximations (RSA) to design FIR filters with arbitrary angular and radial specifications in the frequency domain. First 1D filters for the radial and angular components are de...
Mixed FourierRadon reconstruction of irregularly and sparsely sampled seismic data
, 1997
"... Seismic data irregularly sampled in two dimensions is transformed to the Fourier/Radon domain using a least squares formulation where the inverse transform, from the Fourier/Radon to the spatial domain is used as a forward model. By a proper choice of the region of support, the total number of para ..."
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Cited by 2 (0 self)
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Seismic data irregularly sampled in two dimensions is transformed to the Fourier/Radon domain using a least squares formulation where the inverse transform, from the Fourier/Radon to the spatial domain is used as a forward model. By a proper choice of the region of support, the total number of parameters is limited (yet such that the actual data is optimally contained), leading to a stable inversion. Subsequently the data can be transformed to any desired grid in the spatial domain. Also, using suitable transforms, signal and noise map to different parts of the transform domain and can be separated. The method is applied to synthetic and marine data. I. Introduction In exploration seismology structural information of the subsurface is obtained by recording the wavefield generated by a source (e.g. dynamite), using many receivers. In 2D seismics, these receivers are positioned along a line at the surface with sampling interval \Deltax r and starting at a certain 'offset' x 0 from the...
FREQUENCY WARPING IN LOW DELAY AUDIO CODING
"... The goal of the schemes we present in this paper is to obtain an ultra low delay audio coder with a good performance even at low bit rates (around 64 kb/s). The problem to be solved is to gain sufficient frequency resolution at low frequencies for precise low frequency psychoacoustics and quantizat ..."
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The goal of the schemes we present in this paper is to obtain an ultra low delay audio coder with a good performance even at low bit rates (around 64 kb/s). The problem to be solved is to gain sufficient frequency resolution at low frequencies for precise low frequency psychoacoustics and quantization noise shaping, because the ear has a higher frequency resolution at lower frequencies. Our approach is to use a warped linear noise shaping pre and postfilter, and a short DFT for the psychoacoustic model (length 256), but with frequency warping. We compare four different psychoacoustic versions: DFT with no warping, DFT without warping using warped pre and postfilters, warping with the socalled NDFT (WDFT), and a DFT with an allpass delay chain preprocessing. Listening tests show that the best performance is obtained using the WDFT. 1.
unknown title
, 2003
"... Performance of dual tone multifrequency signal decoding algorithm using the subband nonuniform discrete Fourier transform on the ADSP2192 processor ..."
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Performance of dual tone multifrequency signal decoding algorithm using the subband nonuniform discrete Fourier transform on the ADSP2192 processor
Warped Polynomials and their Applications in Signal and Image Processing
"... This document summarizes the author's research on adaptive image and signal representations using warped polynomial approximations and their applications in (mainly) coding and filtering. Some of the research results described in this document are explained in much greater detail in the author's Ph. ..."
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This document summarizes the author's research on adaptive image and signal representations using warped polynomial approximations and their applications in (mainly) coding and filtering. Some of the research results described in this document are explained in much greater detail in the author's Ph.D. thesis. However, this document additionally presents many recent new results concerning, e.g., adaptive filtering, and image and contour coding. It also discusses a few possible extensions and new applications of the theory. Acknowledgments
Parabolic Radon transform and xsquared fktransform  aliasing and efficiency
, 1997
"... this paper the aliasing is studied by comparing the PRT with the Nonuniform Discrete Fourier Transform (NDFT). Although we find the same main principals, there are some differences. Moreover, the new points of view can provide a better understanding of the results found. The similarity between the P ..."
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this paper the aliasing is studied by comparing the PRT with the Nonuniform Discrete Fourier Transform (NDFT). Although we find the same main principals, there are some differences. Moreover, the new points of view can provide a better understanding of the results found. The similarity between the PRT and the NDFT can also be used for the utilization of fast algorithms.