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548
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 121 (22 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.
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 116 (44 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 111 (36 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...
MINIMIZERS OF COSTFUNCTIONS INVOLVING NONSMOOTH DATAFIDELITY TERMS. APPLICATION TO THE PROCESSING OF OUTLIERS
, 2002
"... We present a theoretical study of the recovery of an unknown vector x ∈ Rp (such as a signal or an image) from noisy data y ∈ Rq by minimizing with respect to x a regularized costfunction F(x, y) = Ψ(x, y) + αΦ(x), where Ψ is a datafidelity term, Φ is a smooth regularization term, and α> 0 i ..."
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Cited by 103 (19 self)
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We present a theoretical study of the recovery of an unknown vector x ∈ Rp (such as a signal or an image) from noisy data y ∈ Rq by minimizing with respect to x a regularized costfunction F(x, y) = Ψ(x, y) + αΦ(x), where Ψ is a datafidelity term, Φ is a smooth regularization term, and α> 0 is a parameter. Typically, Ψ(x, y) = ‖Ax − y‖2, where A is a linear operator. The datafidelity terms Ψ involved in regularized costfunctions are generally smooth functions; only a few papers make an exception to this and they consider restricted situations. Nonsmooth datafidelity terms are avoided in image processing. In spite of this, we consider both smooth and nonsmooth datafidelity terms. Our goal is to capture essential features exhibited by the local minimizers of regularized costfunctions in relation to the smoothness of the datafidelity term. In order to fix the context of our study, we consider Ψ(x, y) = i ψ(aTi x − yi), where aTi are the rows of A and ψ is Cm on R \ {0}. We show that if ψ′(0−) < ψ′(0+), then typical data y give rise to local minimizers x ̂ of F(., y) which fit exactly a certain number of the data entries: there is a possibly large set h ̂ of indexes such that aTi x ̂ = yi for every i ∈ ĥ. In contrast, if ψ is
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 55 (12 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.
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 54 (6 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...
Electric Field Imaging
, 1999
"... The physical user interface is an increasingly significant factor limiting the effectiveness of our interactions with and through technology. This thesis introduces Electric Field Imaging, a new physical channel and inference framework for machine perception of human action. Though electric field se ..."
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Cited by 43 (6 self)
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The physical user interface is an increasingly significant factor limiting the effectiveness of our interactions with and through technology. This thesis introduces Electric Field Imaging, a new physical channel and inference framework for machine perception of human action. Though electric field sensing is an important sensory modality for several species of fish, it has not been seriously explored as a channel for machine perception. Technological applications of field sensing, from the Theremin to the capacitive elevator button, have been limited to simple proximity detection tasks. This thesis presents a solution to the inverse problem of inferring geometrical information about the configuration and motion of the human body from electric field measurements. It also presents simple, inexpensive hardware and signal processing techniques for making the field measurements, and several new applications of electric field sensing. The signal
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 35 (5 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
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 31 (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