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13
A WaveletBased Method For Multiscale Tomographic Reconstruction
, 1995
"... We represent the standard ramp filter operator of the filtered backprojection (FBP) reconstruction in different bases composed of Haar and Daubechies compactly supported wavelets. The resulting multiscale representation of the ramp filter matrix operator is approximately diagonal. The accuracy of t ..."
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Cited by 32 (4 self)
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We represent the standard ramp filter operator of the filtered backprojection (FBP) reconstruction in different bases composed of Haar and Daubechies compactly supported wavelets. The resulting multiscale representation of the ramp filter matrix operator is approximately diagonal. The accuracy of this diagonal approximation becomes better as wavelets with larger number of vanishing moments are used. This waveletbased representation enables us to formulate a multiscale tomographic reconstruction technique wherein the object is reconstructed at multiple scales or resolutions. A complete reconstruction is obtained by combining the reconstructions at different scales. Our multiscale reconstruction technique has the same computational complexity as the FBP reconstruction method. It differs from other multiscale reconstruction techniques in that 1) the object is defined through a multiscale transformation of the projection domain, and 2) we explicitly account for noise in the projection da...
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...
Direct Algebraic Reconstruction and Optimal Sampling in Vector Field Tomography
, 1997
"... Vector field tomography has been proven to be a very powerful technique for the noninvasive determination of vector field distribution such as in the case of a fluid velocity field. We show that classical tomographic sampling conditions can essentially be applied to vector field tomography. Thus, e ..."
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Cited by 6 (2 self)
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Vector field tomography has been proven to be a very powerful technique for the noninvasive determination of vector field distribution such as in the case of a fluid velocity field. We show that classical tomographic sampling conditions can essentially be applied to vector field tomography. Thus, essentially the same sampling schemes are obtained, and the interlaced scheme is also shown to be the most efficient scheme in vector field tomography. We then propose a Direct Algebraic approach for vector field tomography, with an efficient and robust algorithm for interlaced schemes. Numerical experiments showing the superiority of interlaced schemes are provided. Keywords Vector tomography, efficient sampling, algebraic reconstruction algorithm EDICS number SP 2.5.1 We grant permission for publication of the abstract separate from the text. DRAFT August 14, 1997 5 I. Introduction By developing vector field tomography in recent years, a very powerful device has become available for det...
Efficient Parallel Sampling in Vector Field Tomography
 Inverse Problems
, 1995
"... Ultrasound techniques allow us to measure the integral of the inner product of a vector field with the propagation direction. Efficient sampling schemes obtained in scalar tomography are extended to vector field tomography. Reconstructions from interlaced sampling are compared with reconstructio ..."
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Cited by 3 (1 self)
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Ultrasound techniques allow us to measure the integral of the inner product of a vector field with the propagation direction. Efficient sampling schemes obtained in scalar tomography are extended to vector field tomography. Reconstructions from interlaced sampling are compared with reconstructions from the corresponding standard sampling. 2 1 Introduction Vector field tomography is essentially considered for measuring the velocity of a fluid in a domain, such as the fluid velocity in a tube, see [1, 14, 15, 19, 20] and references therein for practical applications. For that purpose differential ultrasonic timeofflight measurements are made. Let us denote ~v the vector field considered (the velocity of a fluid in our application).\Omega is a 2dimensional bounded domain (a crosssection of a tube). In this paper\Omega is the unit disk. If the celerity of the medium is much larger than the velocity of the fluid then the sum and the difference of the propagation time between ...
Sampling In ParallelBeam Tomography
 in: Inverse Problems and Imaging, A.G. Ramm (editor), Plenum
, 1998
"... We pesent Shannon sampling theory for functions defined on T \Theta IR, where T denotes the circle group, prove a new estimate for the aliasing error, and apply the result to parallelbeam diffraction tomography. The class of admissible sampling lattices is characterized and general sampling conditi ..."
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Cited by 3 (1 self)
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We pesent Shannon sampling theory for functions defined on T \Theta IR, where T denotes the circle group, prove a new estimate for the aliasing error, and apply the result to parallelbeam diffraction tomography. The class of admissible sampling lattices is characterized and general sampling conditions are derived which lead to the identification of new efficient sampling schemes. Corresponding results for xray tomography are obtained in the highfrequency limit. 1. INTRODUCTION Sampling theorems provide interpolation formulas for functions whose Fourier transform is compactly supported. If the Fourier transform does not have compact support, a socalled aliasing error occurs. In this paper we pesent a new estimate for the aliasing error for functions defined on T \Theta IR, where T denotes the circle group, and work out its application to computed tomography. In computed tomography (CT) an object is exposed to radiation which is measured after passing through the object. From the...
Efficient Sampling on Coarse Grids in Tomography
, 1993
"... In tomography we have to give an estimate of a function from a finite number of its intergrals along straight lines or on strips. Under very reasonable conditions, the interlaced sampling is well known to be the most efficient scheme for this problem. In this paper we examine some pertubations on th ..."
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Cited by 2 (0 self)
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In tomography we have to give an estimate of a function from a finite number of its intergrals along straight lines or on strips. Under very reasonable conditions, the interlaced sampling is well known to be the most efficient scheme for this problem. In this paper we examine some pertubations on the interlaced scheme. Using a theorem due to A. Faridani, we show that sampling on coarse grids leads to efficient schemes, allowing to consider a lot of different sampling geometries. Some of them could be in practice much more easily generated than the interlaced one. New efficient sampling schemes of the Radon transform are proposed. Numerical experiments in the case of integrals on strips show the efficiency of these new schemes.
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 ...
RESOLUTION IN DYNAMIC EMISSION TOMOGRAPHY
 SIAM J. MATH. ANAL
, 2000
"... Based on a twodimensional (2D) Fourier analysis of the attenuated Radon transform and a 2D version of the Shannon sampling theorem, we investigate the problem of resolution in dynamic emission tomography. As a result we provide guidelines on how to acquire and on how to filter the projection data ..."
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Cited by 1 (0 self)
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Based on a twodimensional (2D) Fourier analysis of the attenuated Radon transform and a 2D version of the Shannon sampling theorem, we investigate the problem of resolution in dynamic emission tomography. As a result we provide guidelines on how to acquire and on how to filter the projection data.