Abstract not found

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 unequally-spaced 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

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.

We introduce an output least-squares 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

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 non-decreasing weights. As application, we construct sharp adaptive estimators in the problems of deconvolution and tomography.

We develop an algorithm to reconstruct the wavelet coefficients of an image from the Radon transform data. The proposed method uses the properties of wavelets to localize the Radon transform and can be used to reconstruct a local region of the cross section of a body, using almost completely local data which significantly reduces the amount of exposure and computations in X-ray tomography. This property which distinguishes our algorithm from the previous algorithms is based on the observation that for some wavelet basis with sufficiently many vanishing moments, the ramp-filtered version of the scaling function as well as the wavelet function has extremely rapid decay. We show that the variance of the elements of the null-space is negligible in the locally reconstructed image. Also we find an upper bound for the reconstruction error in terms of the amount of data used in the algorithm. To reconstruct a local region 20 pixels in radius in a 256 \Theta 256 image we require 12:5% of full e...

Tikhonov-Phillips regularization is one of the bestknown regularization methods for inverse problems. A posteriori criteria for determining the regularization parameter ff require solving (A A+ ffI)x = A y ffi () for different values of ff. We investigate two methods for accelerating the standard CG-algorithm for solving the family of systems (). The first one utilizes a stopping criterion for the CG-iterations which depends on ff and ffi. The second method exploits the shifted structure of the linear systems (), which allows to solve () simultaneously for different values of ff. We present numerical experiments for three test problems which illustrate the practical efficiency of the new methods. The experiments as well as theoretical considerations show that run times are accelerated by a factor of at least 3.

We describe a wavelet-based X-ray 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 X-ray transform is derived, and the corresponding Fourier-wavelet volume rendering algorithm (FWVR) is introduced. FWVR uses Haar or B-spline 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 3-D analytical phantom to assess the accuracy, time complexity, and memory cost of both FWVR and WS. The di#erences between both methods are enumerated.

. We use the theory of the continuous wavelet transform to derive inversion formulas for the Radon transform on L 1 " L 2 (R d ). These inversion formulas turn out to be local in even dimensions in the following sense. In order to recover a function f from its Radon transform in a ball of radius R ? 0 about a point x to within error ffl, we can find ff(ffl) ? 0 such that this can be accomplished by knowing the projections of f only on lines passing through a ball of radius R+ ff(ffl) about x. We give explicit a priori estimates on the error in the L 2 and L 1 norms. in Proceedings of the Conference: 75 Years of the Radon Transform (Vienna, 1992), International Press Co., Ltd., 38--58 (1994). 0. Introduction. Given a function f defined on R d , its Radon transform, Rf , is defined by R ` f(s) = Z ` ? f(s` + y) dy; where ` 2 S d\Gamma1 and s 2 R. Rf(`; s) is the integral of f on the hyperplane in R d defined by fx: hx; `i = sg. The backprojection operator is given by...

Fourier-based reprojection methods have the potential to reduce the computation time in iterative tomographic image reconstruction. Interpolation errors are a limitation of Fourier-based reprojection methods. We apply a min-max interpolation method for the nonuniform fast Fourier transform (NUFFT) to minimize the interpolation errors. Numerical results show that the min-max NUFFT approach provides substantially lower approximation errors in tomographic reprojection and backprojection than conventional interpolation methods.