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242
Penalized Weighted LeastSquares Image Reconstruction for Positron Emission Tomography
 IEEE TR. MED. IM
, 1994
"... This paper presents an image reconstruction method for positronemission tomography (PET) based on a penalized, weighted leastsquares (PWLS) objective. For PET measurements that are precorrected for accidental coincidences, we argue statistically that a leastsquares objective function is as approp ..."
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Cited by 86 (38 self)
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This paper presents an image reconstruction method for positronemission tomography (PET) based on a penalized, weighted leastsquares (PWLS) objective. For PET measurements that are precorrected for accidental coincidences, we argue statistically that a leastsquares objective function is as appropriate, if not more so, than the popular Poisson likelihood objective. We propose a simple databased method for determining the weights that accounts for attenuation and detector efficiency. A nonnegative successive overrelaxation (+SOR) algorithm converges rapidly to the global minimum of the PWLS objective. Quantitative simulation results demonstrate that the bias/variance tradeoff of the PWLS+SOR method is comparable to the maximumlikelihood expectationmaximization (MLEM) method (but with fewer iterations), and is improved relative to the conventional filtered backprojection (FBP) method. Qualitative results suggest that the streak artifacts common to the FBP method are nearly eliminat...
Mean and Variance of Implicitly Defined Biased Estimators (such as Penalized Maximum Likelihood): Applications to Tomography
 IEEE Tr. Im. Proc
, 1996
"... Many estimators in signal processing problems are defined implicitly as the maximum of some objective function. Examples of implicitly defined estimators include maximum likelihood, penalized likelihood, maximum a posteriori, and nonlinear leastsquares estimation. For such estimators, exact analyti ..."
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Cited by 84 (30 self)
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Many estimators in signal processing problems are defined implicitly as the maximum of some objective function. Examples of implicitly defined estimators include maximum likelihood, penalized likelihood, maximum a posteriori, and nonlinear leastsquares estimation. For such estimators, exact analytical expressions for the mean and variance are usually unavailable. Therefore investigators usually resort to numerical simulations to examine properties of the mean and variance of such estimators. This paper describes approximate expressions for the mean and variance of implicitly defined estimators of unconstrained continuous parameters. We derive the approximations using the implicit function theorem, the Taylor expansion, and the chain rule. The expressions are defined solely in terms of the partial derivatives of whatever objective function one uses for estimation. As illustrations, we demonstrate that the approximations work well in two tomographic imaging applications with Poisson sta...
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.
Statistical image reconstruction for polyenergetic Xray computed tomography
 IEEE Transactions on Medical Imaging
, 2002
"... Abstract—This paper describes a statistical image reconstruction method for Xray computed tomography (CT) that is based on a physical model that accounts for the polyenergetic Xray source spectrum and the measurement nonlinearities caused by energydependent attenuation. We assume that the object c ..."
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Cited by 31 (10 self)
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Abstract—This paper describes a statistical image reconstruction method for Xray computed tomography (CT) that is based on a physical model that accounts for the polyenergetic Xray source spectrum and the measurement nonlinearities caused by energydependent attenuation. We assume that the object consists of a given number of nonoverlapping materials, such as soft tissue and bone. The attenuation coefficient of each voxel is the product of its unknown density and a known energydependent mass attenuation coefficient. We formulate a penalizedlikelihood function for this polyenergetic model and develop an orderedsubsets iterative algorithm for estimating the unknown densities in each voxel. The algorithm monotonically decreases the cost function at each iteration when one subset is used. Applying this method to simulated Xray CT measurements of objects containing both bone and soft tissue yields images with significantly reduced beam hardening artifacts. Index Terms—Beam hardening, penalized likelihood, statistical reconstruction, Xray CT. I.
W.: Tomographic reconstruction of transparent objects
 In: Eurographics Symposium on Rendering (2006
"... The scanning of 3D geometry has become a popular way of capturing the shape of realworld objects. Transparent objects, however, pose problems for traditional scanning methods. We present a tomographic method for recovering the shape of objects made of ..."
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Cited by 24 (4 self)
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The scanning of 3D geometry has become a popular way of capturing the shape of realworld objects. Transparent objects, however, pose problems for traditional scanning methods. We present a tomographic method for recovering the shape of objects made of
Estimation of Local Modeling Error and GoalOriented Adaptive Modeling of Heterogeneous Materials; Part I : Error Estimates and Adaptive Algorithms
 of Heterogeneous Materials; Part I : Error Estimates and Adaptive
"... . A theory of a posteriori estimation of modeling errors in local quantities of interest in the analysis of heterogeneous elastic solids is presented. These quantities may, for example, represent averaged stresses on the surface of inclusions or mollications of pointwise stresses or displacements, o ..."
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Cited by 23 (2 self)
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. A theory of a posteriori estimation of modeling errors in local quantities of interest in the analysis of heterogeneous elastic solids is presented. These quantities may, for example, represent averaged stresses on the surface of inclusions or mollications of pointwise stresses or displacements, or, in general, local features of the \nescale" solution characterized by continuous linear functionals. These estimators are used to construct goaloriented adaptive procedures in which models of the microstructure are adapted so as to deliver local features to a preset level of accuracy. Algorithms for implementing these procedures are discussed and preliminary numerical results are given. 1 Introduction The idea of automatically adapting characteristics of mathematical and computational models of heterogeneous media so as to obtain results of a specied level of accuracy was advanced in recent work on hierarchical modeling [11, 7]. In these papers, a posteriori bounds on the error in s...
Frequency Domain Volume Rendering by the Wavelet Xray Transform
 IEEE TRANSACTIONS ON IMAGE PROCESSING
, 2000
"... We describe a waveletbased Xray rendering method in the frequency domain with a smaller time complexity than wavelet splatting. Standard Fourier volume rendering is summarized and interpolation and accuracy issues are briefly discussed. We review the implementation of the fast wavelet transform in ..."
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Cited by 20 (9 self)
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We describe a waveletbased Xray rendering method in the frequency domain with a smaller time complexity than wavelet splatting. Standard Fourier volume rendering is summarized and interpolation and accuracy issues are briefly discussed. We review the implementation of the fast wavelet transform in the frequency domain. The wavelet Xray transform is derived, and the corresponding Fourierwavelet volume rendering algorithm (FWVR) is introduced. FWVR uses Haar or Bspline wavelets and linear or cubic spline interpolation. Various combinations are tested and compared with wavelet splatting (WS). We use medical MR and CT scan data, as well as a 3D analytical phantom to assess the accuracy, time complexity, and memory cost of both FWVR and WS. The di#erences between both methods are enumerated.
Inversion Of LargeSupport IllPosed Linear . . .
 IEEE TRANSACTIONS ON IMAGE PROCESSING
, 1998
"... We propose a method for the reconstruction of signals and images observed partially through a linear operator with a large support (e.g., a Fourier transform on a sparse set). This inverse problem is illposed and we resolve it by incorporating the prior information that the reconstructed object ..."
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Cited by 20 (12 self)
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We propose a method for the reconstruction of signals and images observed partially through a linear operator with a large support (e.g., a Fourier transform on a sparse set). This inverse problem is illposed and we resolve it by incorporating the prior information that the reconstructed objects are composed of smooth regions separated by sharp transitions. This feature is modeled by a piecewise Gaussian (PG) Markov random field (MRF), known also as the weakstring in one dimension and the weakmembrane in two dimensions. The reconstruction is defined as the maximum a posteriori estimate. The prerequisite
Gradientbased 2D/3D rigid registration of fluoroscopic Xray to CT
 IEEE Trans. Med. Imag
, 2003
"... Abstract—We present a gradientbased method for rigid registration of a patient preoperative computed tomography (CT) to its intraoperative situation with a few fluoroscopic Xray images obtained with a tracked Carm. The method is noninvasive, anatomybased, requires simple user interaction, and inc ..."
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Cited by 19 (0 self)
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Abstract—We present a gradientbased method for rigid registration of a patient preoperative computed tomography (CT) to its intraoperative situation with a few fluoroscopic Xray images obtained with a tracked Carm. The method is noninvasive, anatomybased, requires simple user interaction, and includes validation. It is generic and easily customizable for a variety of routine clinical uses in orthopaedic surgery. Gradientbased registration consists of three steps: 1) initial pose estimation; 2) coarse geometrybased registration on bone contours, and; 3) fine gradient projection registration (GPR) on edge pixels. It optimizes speed, accuracy, and robustness. Its novelty resides in using volume gradients to eliminate outliers and foreign objects in the fluoroscopic Xray images, in speeding up computation, and in achieving higher accuracy. It overcomes the drawbacks of intensitybased methods, which are slow and have a limited convergence range, and of geometrybased methods, which depend on the image segmentation quality. Our simulated, in vitro, and cadaver experiments on a human pelvis CT, dry vertebra, dry femur, fresh lamb hip, and human pelvis under realistic conditions show a mean 0.5–1.7 mm (0.5–2.6 mm maximum) target registration accuracy. Index Terms—Fluoroscopic Xray to CT registration, gradient based, image registration, 2D/3D rigid registration. I.
Resolution properties of regularized image reconstruction methods
 of EECS, Univ. of Michigan, Ann Arbor, MI
, 1995
"... This paper examines the spatial resolution properties of penalizedlikelihood image reconstruction methods by analyzing the local impulse response. The analysis shows that standard regularization penalties induce spacevariant local impulse response functions, even for spaceinvariant tomographic s ..."
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Cited by 18 (12 self)
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This paper examines the spatial resolution properties of penalizedlikelihood image reconstruction methods by analyzing the local impulse response. The analysis shows that standard regularization penalties induce spacevariant local impulse response functions, even for spaceinvariant tomographic systems. Paradoxically, for emission image reconstruction the local resolution is generally poorest in highcount regions. We show that the linearized local impulse response induced by quadratic roughness penalties depends on the object only through its projections. This analysis leads naturally to a modified regularization penalty that yields reconstructed images with nearly uniform resolution. The modified penalty also provides a very practical method for choosing the regularization parameter to obtain a specified resolution in images reconstructed by penalizedlikelihood methods.