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163
Wavelet Methods For The Inversion Of Certain Homogeneous Linear Operators In The Presence Of Noisy Data
, 1994
"... In this dissertation we explore the use of wavelets in certain linear inverse problems with discrete, noisy data. We observe discrete samples of a process y(u) = (Kf)(u)+ z(u), where K is a linear operator, z is a noise process, and f is a function we wish to recover from the data. In the problems ..."
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Cited by 27 (1 self)
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In this dissertation we explore the use of wavelets in certain linear inverse problems with discrete, noisy data. We observe discrete samples of a process y(u) = (Kf)(u)+ z(u), where K is a linear operator, z is a noise process, and f is a function we wish to recover from the data. In the problems that we consider, the inverse of K, K \Gamma1 , either does not exist or is poorly behaved. Such problems are termed illposed i.e., ones in which small changes in the data may lead to large changes in the recovered version of f . Our methods are most effective for problems where the operator K is homogeneous with respect to dilations, such as integration, fractional integration, convolution, and the Radon transform. The theoretical framework in which we work is that of Donoho's (1992) WaveletVaguelette Decomposition (WVD). The WVD uses wavelets and vaguelettes (almost wavelets) to decompose the operator K. Although this formally resembles the Singular Value Decomposition (SVD), the use o...
Overview of methods for image reconstruction from projections in emission computed tomography
 PROC. IEEE
, 2003
"... Emission computed tomography (ECT) is a technology for medical imaging whose importance is increasing rapidly. There is a growing appreciation for the value of the functional (as opposed to anatomical) information that is provided by ECT and there are significant advancements taking place, both in t ..."
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Cited by 27 (2 self)
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Emission computed tomography (ECT) is a technology for medical imaging whose importance is increasing rapidly. There is a growing appreciation for the value of the functional (as opposed to anatomical) information that is provided by ECT and there are significant advancements taking place, both in the instrumentation for data collection, and in the computer methods for generating images from the measured data. These computer methods are designed to solve the inverse problem known as “image reconstruction from projections.” This paper uses the various models of the data collection process as the framework for presenting an overview of the wide variety of methods that have been developed for image reconstruction in the major subfields of ECT, which are positron emission tomography (PET) and singlephoton emission computed tomography (SPECT). The overall sequence of the major sections in the paper, and the presentation within each major section, both proceed from the more realistic and general models to those that are idealized and application specific. For most of the topics, the description proceeds from the threedimensional case to the twodimensional case. The paper presents a broad overview of algorithms for PET and SPECT, giving references to the literature where these algorithms and their applications are described in more detail.
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 26 (5 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.
Discretization of the Radon Transform and of its Inverse by Spline Convolutions
, 2002
"... We present an explicit formula for Bspline convolution kernels; these are defined as the convolution of several Bsplines of variable widths hi and degrees rzl. We apply our results to derive splineconvolutionbased algorithms for two closely related problems: the computation of the Radon transfor ..."
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Cited by 18 (7 self)
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We present an explicit formula for Bspline convolution kernels; these are defined as the convolution of several Bsplines of variable widths hi and degrees rzl. We apply our results to derive splineconvolutionbased algorithms for two closely related problems: the computation of the Radon transform and of its inverse. First, we present an efficient discrete implementation of the Radon transform that is optimal in the leastsquares sense. We then consider the reverse problem and introduce a new splineconvolution version of the filtered backprojection algorithm for tomographic reconstruction. In both cases, our explicit kernel formula allows for the use of highdegree splines; these offer better approximation performance than the conventional lowerdegree formulations (e.g., piecewise constant or piecewise linear models). We present multiple experiments to validate our approach and to find the parameters that give the best tradeoff between image quality and computational complexity. In particular, we find that it can be computationally more efficient to increase the approximation degree than to increase the sampling rate.
A fast and accurate multilevel inversion of the radon transform
 SIAM J. Appl. Math
, 1999
"... Abstract. A number of imaging technologies reconstruct an image function from its Radon projection using the convolution backprojection method. The convolution is an O(N 2 log N) algorithm, where the image consists of N ×N pixels, while the backprojection is an O(N 3) algorithm, thus constituting th ..."
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Cited by 15 (2 self)
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Abstract. A number of imaging technologies reconstruct an image function from its Radon projection using the convolution backprojection method. The convolution is an O(N 2 log N) algorithm, where the image consists of N ×N pixels, while the backprojection is an O(N 3) algorithm, thus constituting the major computational burden of the convolution backprojection method. An O(N 2 log N) multilevel backprojection method is presented here. When implemented with a Fourierdomain postprocessing technique, also presented here, the resulting image quality is similar or superior to the image quality of the classical backprojection technique. Key words. Radon transform, inversion of the Radon transform, computed tomography, convolution backprojection, multilevel, Fourierdomain postprocessing AMS subject classifications. 92C55, 44A12, 65R10, 68U10 PII. S003613999732425X 1. Background. Reconstruction of a function of two or three variables from its Radon transform has proven vital in computed tomography (CT), nuclear magnetic resonance imaging, astronomy, geophysics, and a number of other fields [13]. One of the best known reconstruction algorithms is the convolution backprojection method (CB), which is widely used in commercial CT devices [13] (with rebinning for divergentbeam projections [18]). Recently, it has been applied to spotlightmode synthetic aperture radar image reconstruction [14, 23] in which the conventional method is the direct Fourier method (DF), i.e., Fourierdomain interpolation followed by twodimensional (2D) FFT [21]. Originally, CB was preferred to DF since the former provided better images [18, 20]. However, since the backprojection part of CB raises the computational complexity of the method to O(N 3), while DF’s complexity is O(N 2 log N), there has been
AntiAliased ThreeDimensional ConeBeam Reconstruction of LowContrast Objects with Algebraic Methods
 IEEE Trans. Med. Imag
, 1999
"... This paper examines the use of the algebraic reconstruction technique (ART) and related techniques to reconstruct 3D objects from a relatively sparse set of conebeam projections. Although ART has been widely used for conebeam reconstruction of highcontrast objects, e.g., in computed angiography, ..."
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Cited by 15 (2 self)
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This paper examines the use of the algebraic reconstruction technique (ART) and related techniques to reconstruct 3D objects from a relatively sparse set of conebeam projections. Although ART has been widely used for conebeam reconstruction of highcontrast objects, e.g., in computed angiography, the work presented here explores the more challenging lowcontrast case which represents a littleinvestigated scenario for ART. Preliminary experiments indicate that for cone angles greater than 20 ffiffiffi , traditional ART produces reconstructions with strong aliasing artifacts. These artifacts are in addition to the usual offmidplane inaccuracies of conebeam tomography with planar orbits. We find that the source of these artifacts is the nonuniform reconstruction grid sampling and correction by the conebeam rays during the ART projectionbackprojection procedure. A new method to compute the weights of the reconstruction matrix is devised, which replaces the usual constantsize in...
Realistic analytical phantoms for parallel Magnetic Resonance Imaging
 IEEE Trans. Med. Imaging
"... Abstract—The quantitative validation of reconstruction algorithms requires reliable data. Rasterized simulations are popular but they are tainted by an aliasing component that impacts the assessment of the performance of reconstruction. We introduce analytical simulation tools that are suited to par ..."
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Abstract—The quantitative validation of reconstruction algorithms requires reliable data. Rasterized simulations are popular but they are tainted by an aliasing component that impacts the assessment of the performance of reconstruction. We introduce analytical simulation tools that are suited to parallel magnetic resonance imaging and allow one to build realistic phantoms. The proposed phantoms are composed of ellipses and regions with piecewisepolynomial boundaries, including spline contours, Bézier contours, and polygons. In addition, they take the channel sensitivity into account, for which we investigate two possible models. Our analytical formulations provide welldefined data in both the spatial and kspace domains. Our main contribution is the closedform determination of the Fourier transforms that are involved. Experiments validate the proposed implementation. In a typical parallel magnetic resonance imaging reconstruction experiment, we quantify the bias in the overly optimistic results obtained with rasterized simulations—the inversecrime situation. We provide a package that implements the different simulations and provide tools to guide the design of realistic phantoms. Index Terms—Fourier analytical simulation, inverse crime, magnetic resonance imaging (MRI), Shepp–Logan. I.
Inversion of noisy Radon transform by SVD based needlets
, 2009
"... A linear method for inverting noisy observations of the Radon transform is developed based on decomposition systems (needlets) with rapidly decaying elements induced by the Radon transform SVD basis. Upper bounds of the risk of the estimator are established in L p (1 ≤ p ≤ ∞) norms for functions wit ..."
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Cited by 12 (6 self)
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A linear method for inverting noisy observations of the Radon transform is developed based on decomposition systems (needlets) with rapidly decaying elements induced by the Radon transform SVD basis. Upper bounds of the risk of the estimator are established in L p (1 ≤ p ≤ ∞) norms for functions with Besov space smoothness. A practical implementation of the method is given and several examples are discussed.
1 A Nonlocal TransformDomain Filter for Volumetric Data Denoising
"... Abstract—We present an extension of the BM3D filter to volumetric data. The proposed algorithm, denominated BM4D, implements the grouping and collaborative filtering paradigm, where mutually similar ddimensional patches are stacked together in a (d + 1)dimensional array and jointly filtered in tra ..."
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Cited by 11 (3 self)
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Abstract—We present an extension of the BM3D filter to volumetric data. The proposed algorithm, denominated BM4D, implements the grouping and collaborative filtering paradigm, where mutually similar ddimensional patches are stacked together in a (d + 1)dimensional array and jointly filtered in transform domain. While in BM3D the basic data patches are blocks of pixels, in BM4D we utilize cubes of voxels, which are stacked into a fourdimensional “group”. The fourdimensional transform applied on the group simultaneously exploits the local correlation present among voxels in each cube and the nonlocal correlation between the corresponding voxels of different cubes. Thus, the spectrum of the group is highly sparse, leading to very effective separation of signal and noise through coefficients shrinkage. After inverse transformation, we obtain estimates of each grouped cube, which are then adaptively aggregated at their original locations. We evaluate the algorithm on denoising of volumetric data corrupted by Gaussian and Rician noise, as well as on reconstruction of phantom data from sparse Fourier measurements. Experimental results demonstrate the stateoftheart denoising performance of BM4D, and its effectiveness when exploited as a regularizer in volumetric data reconstruction. Index Terms—Volumetric data denoising, volumetric data reconstruction, compressed sensing, magnetic resonance imaging, computed tomography, nonlocal methods, adaptive transforms I.
Fast And Accurate ThreeDimensional Reconstruction From ConeBeam Projection Data Using Algebraic Methods
, 1998
"... Conebeam computed tomography (CT) is an emerging imaging technology, as it provides all projections needed for threedimensional (3D) reconstruction in a single spin of the Xray sourcedetector pair. This facilitates fast, lowdose data acquisition as required for imaging fast moving objects, such ..."
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Cited by 10 (1 self)
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Conebeam computed tomography (CT) is an emerging imaging technology, as it provides all projections needed for threedimensional (3D) reconstruction in a single spin of the Xray sourcedetector pair. This facilitates fast, lowdose data acquisition as required for imaging fast moving objects, such as the heart, and intraoperative CT applications. Current conebeam reconstruction algorithms mainly employ the FilteredBackprojection (FBP) approach. In this dissertation, a different class of reconstruction algorithms is studied: the algebraic reconstruction methods. Algebraic reconstruction starts from an initial guess for the reconstructed object and then performs a sequence of iterative grid projections and correction backprojections until the reconstruction has converged. Algebraic methods have many advantages over FBP, such as better noise tolerance and better handling of sparse and nonuniformly distributed projection datasets. So far, the main repellant for using algebraic methods...