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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.
Nonuniform fast Fourier transform
 Geophysics
, 1999
"... The nonuniform discrete Fourier transform (NDFT) can be computed with a fast algorithm, referred to as the nonuniform fast Fourier transform (NFFT). In L dimensions, the NFFT requires O(N(ln #) L + ( Q L #=1 M # ) P L #=1 log M # ) operations, where M # is the number of Fourier components ..."
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Cited by 44 (1 self)
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The nonuniform discrete Fourier transform (NDFT) can be computed with a fast algorithm, referred to as the nonuniform fast Fourier transform (NFFT). In L dimensions, the NFFT requires O(N(ln #) L + ( Q L #=1 M # ) P L #=1 log M # ) operations, where M # is the number of Fourier components along dimension #, N is the number of irregularly spaced samples, and # is the required accuracy. This is a dramatic improvement over the O(N Q L #=1 M # ) operations required for the direct evaluation (NDFT). The performance of the NFFT depends on the lowpass filter used in the algorithm. A truncated Gauss pulse, proposed in the literature, is optimized. A newly proposed filter, a Gauss pulse tapered with a Hanning window, performs better than the truncated Gauss pulse and the Bspline, also proposed in the literature. For small filter length, a numerically optimized filter shows the best results. Numerical experiments for 1D and 2D implementations confirm the theoretically predicted ...
Seismic interferometry by crosscorrelation and by multidimensional deconvolution: a systematic comparison
 Geophysical Journal International
, 2011
"... Seismic interferometry, also known as Green’s function retrieval by crosscorrelation, has a wide range of applications, ranging from surface wave tomography using ambient noise, to creating virtual sources for improved reflection seismology. Despite its successful applications, the crosscorrelation ..."
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Cited by 5 (2 self)
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Seismic interferometry, also known as Green’s function retrieval by crosscorrelation, has a wide range of applications, ranging from surface wave tomography using ambient noise, to creating virtual sources for improved reflection seismology. Despite its successful applications, the crosscorrelation approach also has its limitations. The main underlying assumptions are that the medium is lossless and that the wave field is equipartitioned. These assumptions are in practice often violated: the medium of interest is often illuminated from one side only, the sources may be irregularly distributed, and, particularly for EM applications, losses may be significant. These limitations may partly be overcome by reformulating seismic interferometry as a multidimensional deconvolution (MDD) process. We present a systematic analysis of seismic interferometry by crosscorrelation and by MDD. We show that for the nonideal situations mentioned above, the correlation function is proportional to a Green’s function with a blurred source. The source blurring is quantified by a socalled pointspread function which, like the correlation function, can be derived from the observed data (i.e., without the need to know the sources and the medium). The source of the Green’s function obtained by the correlation method can be deblurred by deconvolving the correlation function for the pointspread function. This is the essence of seismic interferometry by MDD. We illustrate the crosscorrelation and MDD methods for controlledsource and passive data applications with numerical examples and discuss the advantages and limitations of both methods. Key words: seismic interferometry, crosscorrelation, deconvolution, ambient noise, virtual source 1
2006a, Application of stable signal recovery to seismic interpolation
 Presented at the SEG International Exposition and 76th Annual Meeting
"... We propose a method for seismic data interpolation based on 1) the reformulation of the problem as a stable signal recovery problem and 2) the fact that seismic data is sparsely represented by curvelets. This method does not require information on the seismic velocities. Most importantly, this formu ..."
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Cited by 4 (2 self)
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We propose a method for seismic data interpolation based on 1) the reformulation of the problem as a stable signal recovery problem and 2) the fact that seismic data is sparsely represented by curvelets. This method does not require information on the seismic velocities. Most importantly, this formulation potentially leads to an explicit recovery condition. We also propose a largescale problem solver for the ℓ1regularization minimization involved in the recovery and successfully illustrate the performance of our algorithm on 2D synthetic and real examples.
Improved wavefield reconstruction from randomized sampling via weighted onenorm
"... minimization ..."
1 Z99 Seismic Data Regularization with AntiLeakage Fourier Transform
"... In the theory of Fourier reconstruction from discrete seismic data, it aims to estimate the spatial frequency content on an irregularly sampled grid. After obtaining the Fourier coefficients, the data can be reconstructed on any desired grid. For this type of transform, difficulties arise from the n ..."
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In the theory of Fourier reconstruction from discrete seismic data, it aims to estimate the spatial frequency content on an irregularly sampled grid. After obtaining the Fourier coefficients, the data can be reconstructed on any desired grid. For this type of transform, difficulties arise from the nonorthogonality of the global basis functions on an irregular grid. As a consequence, energy from one Fourier coefficient leaks onto other coefficients. This wellknown phenomenon is called “spectral leakage”. In this paper, we present an algorithm, called antileakage Fourier transform, for seismic data reconstruction from an irregularly sampled grid to a regular grid that overcomes these difficulties. The key to resolving the spectral leakage is to reduce the leakages among Fourier coefficients in the original data before the calculation of subsequent components. We demonstrate the robustness and effectiveness of this technique with both synthetic and real data examples.
CWP689 Seismic
"... interferometry by crosscorrelation and by multidimensional deconvolution: a systematic comparison ..."
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interferometry by crosscorrelation and by multidimensional deconvolution: a systematic comparison