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479
Fast Discrete Curvelet Transforms
, 2005
"... This paper describes two digital implementations of a new mathematical transform, namely, the second generation curvelet transform [12, 10] in two and three dimensions. The first digital transformation is based on unequallyspaced fast Fourier transforms (USFFT) while the second is based on the wrap ..."
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Cited by 128 (9 self)
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This paper describes two digital implementations of a new mathematical transform, namely, the second generation curvelet transform [12, 10] in two and three dimensions. The first digital transformation is based on unequallyspaced fast Fourier transforms (USFFT) while the second is based on the wrapping of specially selected Fourier samples. The two implementations essentially differ by the choice of spatial grid used to translate curvelets at each scale and angle. Both digital transformations return a table of digital curvelet coefficients indexed by a scale parameter, an orientation parameter, and a spatial location parameter. And both implementations are fast in the sense that they run in O(n 2 log n) flops for n by n Cartesian arrays; in addition, they are also invertible, with rapid inversion algorithms of about the same complexity. Our digital transformations improve upon earlier implementations—based upon the first generation of curvelets—in the sense that they are conceptually simpler, faster and far less redundant. The software CurveLab, which implements both transforms presented in this paper, is available at
Internet Tomography
 IEEE Signal Processing Magazine
, 2002
"... Today's Internet is a massive, distributed network which continues to explode in size as ecommerce and related activities grow. The heterogeneous and largely unregulated structure of the Internet renders tasks such as dynamic routing, optimized service provision, service level verification, and ..."
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Cited by 120 (13 self)
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Today's Internet is a massive, distributed network which continues to explode in size as ecommerce and related activities grow. The heterogeneous and largely unregulated structure of the Internet renders tasks such as dynamic routing, optimized service provision, service level verification, and detection of anomalous/malicious behavior increasingly challenging tasks. The problem is compounded by the fact that one cannot rely on the cooperation of individual servers and routers to aid in the collection of network traffic measurements vital for these tasks. In many ways, network monitoring and inference problems bear a strong resemblance to other "inverse problems" in which key aspects of a system are not directly observable. Familiar signal processing problems such as tomographic image reconstruction, system identification, and array processing all have interesting interpretations in the networking context. This article introduces the new field of network tomography, a field which we believe will benefit greatly from the wealth of signal processing theory and algorithms.
High contrast impedance tomography
 INVERSE PROBLEMS
, 1996
"... We introduce an output leastsquares method for impedance tomography problems that have regions of high conductivity surrounded by regions of lower conductivity. The high conductivity is modeled on network approximation results from an asymptotic analysis and its recovery is based on this model. The ..."
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Cited by 59 (6 self)
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We introduce an output leastsquares method for impedance tomography problems that have regions of high conductivity surrounded by regions of lower conductivity. The high conductivity is modeled on network approximation results from an asymptotic analysis and its recovery is based on this model. The smoothly varying part of the conductivity is recovered by a linearization process as is usual. We present the results of several numerical experiments that illustrate
Sharp Adaptation for Inverse Problems With Random Noise
, 2000
"... We consider a heteroscedastic sequence space setup with polynomially increasing variances of observations that allows to treat a number of inverse problems, in particular multivariate ones. We propose an adaptive estimator that attains simultaneously exact asymptotic minimax constants on every ellip ..."
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Cited by 58 (7 self)
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We consider a heteroscedastic sequence space setup with polynomially increasing variances of observations that allows to treat a number of inverse problems, in particular multivariate ones. We propose an adaptive estimator that attains simultaneously exact asymptotic minimax constants on every ellipsoid of functions within a wide scale (that includes ellipoids with polynomially and exponentially decreasing axes) and, at the same time, satisfies asymptotically exact oracle inequalities within any class of linear estimates having monotone nondecreasing weights. As application, we construct sharp adaptive estimators in the problems of deconvolution and tomography.
Determining a function from its mean values over a family of spheres
 SIAM J. Math. Anal
"... Abstract. Suppose D is a bounded, connected, open set in Rn and f is a smooth function on Rn with support in D. We study the recovery of f from the mean values of f over spheres centered on a part or the whole boundary of D. For strictly convex D, we prove uniqueness when the centers are restricted ..."
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Cited by 47 (2 self)
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Abstract. Suppose D is a bounded, connected, open set in Rn and f is a smooth function on Rn with support in D. We study the recovery of f from the mean values of f over spheres centered on a part or the whole boundary of D. For strictly convex D, we prove uniqueness when the centers are restricted to an open subset of the boundary. We provide an inversion algorithm (with proof) when the mean values are known for all spheres centered on the boundary of D, with radii in the interval [0, diam(D)/2]. We also give an inversion formula when D is a ball in Rn, n ≥ 3 and odd, and the mean values are known for all spheres centered on the boundary.
Mathematics of thermoacoustic tomography
 European Journal Applied Mathematics
"... The paper presents a survey of mathematical problems, techniques, and challenges arising in the Thermoacoustic (also called Photoacoustic or Optoacoustic) Tomography. 1 ..."
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Cited by 34 (6 self)
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The paper presents a survey of mathematical problems, techniques, and challenges arising in the Thermoacoustic (also called Photoacoustic or Optoacoustic) Tomography. 1
Locally regularized spatiotemporal modeling and model comparison for functional MRI
 Neuroimage
, 2001
"... In this work we treat fMRI data analysis as a spatiotemporal system identification problem and address issues of model formulation, estimation, and model comparison. We present a new model that includes a physiologically based hemodynamic response and an empirically derived lowfrequency noise model ..."
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Cited by 29 (2 self)
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In this work we treat fMRI data analysis as a spatiotemporal system identification problem and address issues of model formulation, estimation, and model comparison. We present a new model that includes a physiologically based hemodynamic response and an empirically derived lowfrequency noise model. We introduce an estimation method employing spatial regularization that improves the precision of spatially varying noise estimates. We call the algorithm locally regularized spatiotemporal (LRST) modeling. We develop a new model selection criterion and compare our model to the SPMGLM method. Our findings suggest that our method offers a better approach to identifying appropriate statistical models for fMRI studies.
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 25 (4 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.
Stable Iterative Reconstruction Algorithm For Nonlinear Traveltime Tomography
 Inverse Problems
, 1990
"... Reconstruction of acoustic, seismic, or electromagnetic wave speed distribution from first arrival traveltime data is the goal of traveltime tomography. The reconstruction problem is nonlinear, because the ray paths that should be used for tomographic backprojection techniques can depend strongly on ..."
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Cited by 25 (14 self)
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Reconstruction of acoustic, seismic, or electromagnetic wave speed distribution from first arrival traveltime data is the goal of traveltime tomography. The reconstruction problem is nonlinear, because the ray paths that should be used for tomographic backprojection techniques can depend strongly on the unknown wave speeds. In our analysis, Fermat's principle is used to show that trial wave speed models which produce any ray paths with traveltime smaller than the measured traveltime are not feasible models. Furthermore, for a given set of trial ray paths, nonfeasible models can be classified by their total number of "feasibility violations", i.e., the number of ray paths with traveltime less than that measured. Fermat's principle is subsequently used to convexify the fully nonlinear traveltime tomography problem. In principle, traveltime tomography could be accomplished by solving a multidimensional nonlinear constrained optimization problem based on counting the number of ray paths th...