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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 84 (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.
Reconstruction of nonuniformly sampled bandlimited signals by means of digital fractional delay filters
 IEEE Trans. Signal Processing
, 2002
"... Abstract – This paper deals with reconstruction of nonuniformly sampled bandlimited continuoustime signals using timevarying discretetime FIR filters. The points of departures are that the signal is slightly oversampled as to the average sampling frequency and that the sampling instances are know ..."
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Cited by 23 (1 self)
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Abstract – This paper deals with reconstruction of nonuniformly sampled bandlimited continuoustime signals using timevarying discretetime FIR filters. The points of departures are that the signal is slightly oversampled as to the average sampling frequency and that the sampling instances are known. Under these assumptions, a representation of the reconstructed sequence is derived that utilizes a timefrequency function. This representation enables a proper utilization of the oversampling and reduces the reconstruction problem to a design problem that resembles an ordinary filter design problem. Furthermore, for an important special case, corresponding to a certain type of periodic nonuniform sampling, it is shown that the reconstruction problem can be posed as a filterbank design problem, thus with requirements on a distortion transfer function and a number of aliasing transfer functions. 1.
DirectFourier Reconstruction In Tomography And Synthetic Aperture Radar
 Intl. J. Imaging Sys. and Tech
, 1998
"... We investigate the use of directFourier (DF) image reconstruction in computerized tomography and synthetic aperture radar (SAR). One of our aims is to determine why the convolutionbackprojection (CBP) method is favored over DF methods in tomography, while DF methods are virtually always used in SAR ..."
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Cited by 9 (0 self)
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We investigate the use of directFourier (DF) image reconstruction in computerized tomography and synthetic aperture radar (SAR). One of our aims is to determine why the convolutionbackprojection (CBP) method is favored over DF methods in tomography, while DF methods are virtually always used in SAR. We show that the CBP algorithm is equivalent to DF reconstruction using a Jacobianweighted 2D periodic sinckernel interpolator. This interpolation is not optimal in any sense, which suggests that DF algorithms utilizing optimal interpolators may surpass CBP in image quality. We consider use of two types of DF interpolation: a windowed sinc kernel, and the leastsquares optimal Yen interpolator. Simulations show that reconstructions using the Yen interpolator do not possess the expected visual quality, because of regularization needed to preserve numerical stability. Next, we show that with a concentricsquares sampling scheme, DF interpolation can be performed accurately and efficiently...
Interpolation and denoising of nonuniformly sampled data using wavelet domain processing
 in Proc. IEEE Int. Conf. on Acoust., Speech, Signal Proc.  ICASSP '99
, 1999
"... In this paper, we link concepts from nonuniform sampling, smoothness function spaces, interpolation, and denoising to derive a suite of multiscale, maximumsmoothness interpolation algorithms. We formulate the interpolation problem as the optimization of finding the signal that matches the given sam ..."
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Cited by 6 (3 self)
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In this paper, we link concepts from nonuniform sampling, smoothness function spaces, interpolation, and denoising to derive a suite of multiscale, maximumsmoothness interpolation algorithms. We formulate the interpolation problem as the optimization of finding the signal that matches the given samples with smallest norm in a function smoothness space. For signals in the Besov space B, " (Lp), the optimization corresponds to convex programming in the wavelet domain; for signals in the Sobolev space We(&), the optimization reduces to a simple weighted leastsquares problem. An optional wavelet shrinkage regularization step makes the algorithm suitable for even noisy sample data, unlike classical approaches such as bandlimited and spline interpolation. 1.
Reconstruction of nonuniformly sampled bandlimited signals using a differentiatormultiplier cascade
 IEEE Trans. Circuits Syst. I, Reg. Papers
, 2008
"... Abstract—This paper considers the problem of reconstructing a bandlimited signal from its nonuniform samples. Based on a discretetime equivalent model for nonuniform sampling, we propose the differentiator–multiplier cascade, a multistage reconstruction system that recovers the uniform samples from ..."
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Cited by 5 (4 self)
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Abstract—This paper considers the problem of reconstructing a bandlimited signal from its nonuniform samples. Based on a discretetime equivalent model for nonuniform sampling, we propose the differentiator–multiplier cascade, a multistage reconstruction system that recovers the uniform samples from the nonuniform samples. Rather than using optimally designed reconstruction filters, the system improves the reconstruction performance by cascading stages of linearphase finite impulse response (FIR) filters and timevarying multipliers. Because the FIR filters are designed as differentiators, the system works for the general nonuniform sampling case and is not limited to periodic nonuniform sampling. To evaluate the reconstruction performance for a sinusoidal input signal, we derive the signaltonoiseratio at the output of each stage for the twoperiodic and the general nonuniform sampling case. The main advantage of the system is that once the differentiators have been designed, they are implemented with fixed multipliers, and only some general multipliers have to be adapted when the sampling pattern changes; this reduces implementation costs substantially, especially in an application like timeinterleaved analogtodigital converters (TIADCs) where the timing mismatches among the ADCs may change during operation. Index Terms—Discretetime differentiator, Farrow structure, nonuniform sampling, Taylor series expansion, timeinterleaved analogtodigital converter (TIADC), timevarying multiplier. I.
Signal Processing Issues In Synthetic Aperture Radar And Computer Tomography
, 1998
"... This paper also proposed another reconstruction method based on a direct approximation of the Fourier inversion formula using a twodimensional (2D) trapezoidal rule. In addition, the possibility of reconstruction from a concentricsquares raster was discussed. Numerous simple interpolators have bee ..."
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Cited by 1 (0 self)
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This paper also proposed another reconstruction method based on a direct approximation of the Fourier inversion formula using a twodimensional (2D) trapezoidal rule. In addition, the possibility of reconstruction from a concentricsquares raster was discussed. Numerous simple interpolators have been tried in DF reconstruction with the results compared with CBP [33]. In [34] and [35], the concept of angular bandlimiting was used to interpolate the polar data onto a Cartesian grid. In [36], a DF reconstruction using bilinear interpolation for diffraction tomography provided image quality that was comparable to that produced by the CBP algorithm. Very good reconstruction quality was obtained in [37] and [38] using a spline interpolator, or a hybrid type of spline interpolator. The notion of "gridding" was introduced in [39] as a method of obtaining optimal inversion of Fourier data. An optimal gridding function was proposed, and successful results were obtained when applied to the tomographic reconstruction problem. In [40], several different gridding functions were tried for DF reconstruction, and the performances were compared. In [41, 42], the linogram reconstruction method was proposed as a form of DF reconstruction. The data collection grid in the linogram method is the same as in the concentricsquares sampling scheme. The inversion of the Fourier data in [41, 42] was accomplished by first applying the chirpz transform in one direction and then computing FFTs in the other direction. In CT, many of these attempts at DF reconstruction have given a poorer result than the CBP algorithm, due to the error incurred in the process of the polartoCartesian interpolation. The attraction of DF reconstruction, however, is that it is thought to require less computation than ...
Fast and Motion Robust Dynamic R ∗ 2 Reconstruction for Functional MRI
, 2009
"... Blood oxygen level dependent (BOLD) functional MRI (fMRI) imaging is the most common way of imaging neuronal activity in humans using MRI. The BOLD contrast is directly related to changes in vascular physiology associated with neuronal activity and can be directly linked to changes in cerebral blood ..."
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Blood oxygen level dependent (BOLD) functional MRI (fMRI) imaging is the most common way of imaging neuronal activity in humans using MRI. The BOLD contrast is directly related to changes in vascular physiology associated with neuronal activity and can be directly linked to changes in cerebral blood volume, blood flow and metabolic rate of oxygen. Conventional BOLD imaging is done by reconstructing T ∗ 2weighted images. T ∗ 2weighted images are unitless and even though they measure the magnitude of the BOLD contrast they are still nonquantifiable in terms of the vascular physiology. An alternative approach is to reconstruct R ∗ 2 maps which are quantifiable and can be directly linked to the vascular changes during activation. However, conventional R ∗ 2 mapping involves long readouts and generally ignores relaxation and offresonance during readout. Since fMRI data is usually acquired over a course of several minutes, where the same image volume is collected multiple times, it is important for the time series of each pixel to only reflect changes due to neuronal activity. However, BOLD imaging suffers from temporal drift/fluctuations and subject motion which can confound the findings. Conventionally,
THE FAST SINC TRANSFORM AND IMAGE RECONSTRUCTION FROM NONUNIFORM SAMPLES IN kSPACE
"... A number of problems in image reconstruction and image processing can be addressed, in principle, using the sinc kernel. Since the sinc kernel decays slowly, however, it is generally avoided in favor of some more local but less precise choice. In this paper, we describe the fast sinc transform, an a ..."
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A number of problems in image reconstruction and image processing can be addressed, in principle, using the sinc kernel. Since the sinc kernel decays slowly, however, it is generally avoided in favor of some more local but less precise choice. In this paper, we describe the fast sinc transform, an algorithm which computes the convolution of arbitrarily spaced data with the sinc kernel in O.N log N / operations, where N denotes the number of data points. We briefly discuss its application to the construction of optimal density compensation weights for Fourier reconstruction and to the iterative approximation of the pseudoinverse of the signal equation in MRI. 1.
Author manuscript, published in "SAMPTA'09, Marseille: France (2009)" Reconstruction of signals in a shiftinvariant space from nonuniform samples
, 2010
"... Abstract In this paper, we consider the method for reconstructing a signal from a finite number of samples. In shiftinvariant space framework, we derive an aproximately minmax optimal interpolator to to reconstruct a signal on an interval. An effective noniterative algorithm for signal reconstru ..."
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Abstract In this paper, we consider the method for reconstructing a signal from a finite number of samples. In shiftinvariant space framework, we derive an aproximately minmax optimal interpolator to to reconstruct a signal on an interval. An effective noniterative algorithm for signal reconstruction is given also. Numerical examples show the effectiveness of the proposed method. Index Terms–sampling, signal reconstruction, scaling function, shiftinvariant space Ⅰ.