computational analysis and improvement of sirt iterative x ray computed tomography ct algorithms have the potential for producing high quality images but are computationally very demanding especially when applied to high resolution problems focusing on simultaneous iterative reconstruction technique sirt we provide an eigenvalue based scheme for automatically determining a near optimal value of the relaxation parameter this accelerates the convergence rate of sirt to the point where only half the number of iterations normally required is needed we also modify the way sirt uses preconditioning to solve a weighted least squares problem the resulting algorithm which we call psirt is associated with a smaller memory footprint and calls for less data to be communicated in a distributed memory implementation experimental residual norm and timing results are provided based on cone beam micro ct mouse data including for an ordered subsets study

global rigidity of higher rank abelian anosov algebraic actions we show that all c infty anosov mathbb z r actions on tori and nilmanifolds without rank one factor actions are up to c infty conjugacy actions by automorphisms

the liapunov dimension of strange attractors abstract many papers have been published recently on studies of dynamical processes in which the attracting sets appear quite strange in this paper the question of estimating the dimension of the attractor is addressed while more general conjectures are made here particular attention is paid to the idea that if the jacobian determinant of a map is greater than one and a ball is mapped into itself then generically the attractor will have positive two dimensional measure and most of this paper is devoted to presenting cases with such jacobians for which the attractors are proved to have non empty interior

an improved algorithm for reprojecting rays through pixel images it is often desired to calculate line integrals through a field of reconstructed ct density pixels for the purpose of improving ct image quality two algorithms widely published and discussed in the past are known to either degrade spatial resolution or generate errors in the results due to the discontinuous square pixel modeling of the reconstructed image an algorithm is described based on linear interpolation between pixels which provides superior accuracy without unnecessary loss of resolution it was tested on simulated data for a head section and on a narrow gaussian density distribution the experimental results demonstrated improved performance the method is expected to prove useful for many types of post reconstruction processing including beam hardening missing data and noise supression algorithms

a dimension formula for bernoulli convolutions we present a dynamical approach to the study of the singularity of infinitely convolved bernoulli measuresv for the golden section we introducev as the transverse measure of the maximum entropy measure on the repelling set invariant for contracting maps of the square the fat baker s transformation our approach strongly relies on the markov structure of the underlying dynamical system indeed if golden mean the fat baker s transformation has a very simple markov coding the ambiguity of order two of this coding which appears when projecting on the line due to passages for the central overlapping zone can be expressed by means of products of matrices of order two this product has a markov distribution inherited by the markov structure of the map the dimension of the projected measure is therefore associated to the growth of this product our dimension formula appears in a natural way as a version of the furstenberg guivarch formula our technique provides an explicit dimension formula and most important provides a formalism well suited for the multifractal analysis of this measure as we will show in a forthcoming paper

the hierarchical basis multigrid method we derive and analyze the hierarchical basis multigrid method for solving discretizations of self adjoint elliptic boundary value problems using piecewise linear triangular finite elements the method is analyzed as a block symmetric gau seidel iteration with inner iterations but it is strongly related to 2 level methods to the standard multigridv cycle and to earlier jacobi like hierarchical basis methods the method is very robust and has a nearly optimal convergence rate and work estimate it is especially well suited to difficult problems with rough solutions discretized using highly nonuniform adaptively refined meshes

a wavelet based method for multiscale tomographic reconstruction the authors represent the standard ramp filter operator of the filtered back projection fbp reconstruction in different bases composed of haar and daubechies compactly supported wavelets the resulting multiscale representation of the ramp filter matrix operator is approximately diagonal the accuracy of this diagonal approximation becomes better as wavelets with larger numbers of vanishing moments are used this wavelet based representation enables the authors to formulate a multiscale tomographic reconstruction technique in which the object is reconstructed at multiple scales or resolutions a complete reconstruction is obtained by combining the reconstructions at different scales the authors multiscale reconstruction technique has the same computational complexity as the fbp reconstruction method it differs from other multiscale reconstruction techniques in that 1 the object is defined through a one dimensional multiscale transformation of the projection domain and 2 the authors explicitly account for noise in the projection data by calculating maximum a posteriori probability map multiscale reconstruction estimates based on a chosen fractal prior on the multiscale object coefficients the computational complexity of this maximum a posteriori probability map solution is also the same as that of the fbp reconstruction this result is in contrast to commonly used methods of statistical regularization which result in computationally intensive optimization algorithms

multiresolution tomographic reconstruction using wavelets shows how the separable two dimensional wavelet representation leads naturally to an efficient multiresolution tomographic reconstruction algorithm this algorithm is similar to the conventional filtered backprojection algorithm except that the filters are now angle dependent and the backprojection gives the wavelet coefficients of the reconstruction which are then used to synthesize the reconstruction at various resolution levels by reconstructing only a small localized region at high resolution the authors show how radiation exposure and computation can be significantly reduced compared to a standard reconstruction

a general framework for nonlinear multigrid inversion a variety of new imaging modalities such as optical diffusion tomography require the inversion of a forward problem that is modeled by the solution to a three dimensional 3d partial differential equation for these applications image reconstruction is particularly difficult because the forward problem is both nonlinear and computationally expensive to evaluate we propose a general framework for nonlinear multigrid inversion that is applicable to a wide variety of inverse problems the multigrid inversion algorithm results from the application of recursive multigrid techniques to the solution of optimization problems arising from inverse problems the method works by dynamically adjusting the cost functionals at different scales so that they are consistent with and ultimately reduce the finest scale cost functional in this way the multigrid inversion algorithm efficiently computes the solution to the desired fine scale inversion problem importantly the new algorithm can greatly reduce computation because both the forward and inverse problems are more coarsely discretized at lower resolutions an application of our method to bayesian optical diffusion tomography with a generalized gaussian markov random field image prior model shows the potential for very large computational savings numerical data also indicates robust convergence with a range of initialization conditions for this nonconvex optimization problem

krylov subspace iterative techniques on the detection of brain activity with electrical impedance tomography in this paper we review some numerical techniques based on the linear krylov subspace iteration that can be used for the efficient calculation of the forward and the inverse electrical impedance tomography problems exploring their computational advantages in solving large scale systems of equations we specifically address their implementation in reconstructing localized impedance changes occurring within the human brain if the conductivity of the head tissues is assumed to be real the preconditioned conjugate gradients pcgs algorithm can be used to calculate efficiently the approximate forward solution to a given error tolerance the performance and the regularizing properties of the pcg iteration for solving ill conditioned systems of equations pcgns is then explored and a suitable preconditioning matrix is suggested in order to enhance its convergence rate for image reconstruction the nonlinear inverse problem is considered based on the gauss newton method for solving nonlinear problems we have developed two algorithms that implement the pcgn iteration to calculate the linear step solution using an anatomically detailed model of the human head and a specific scalp electrode arrangement images of a simulated impedance change inside brain s white matter have been reconstructed

universal behaviour of extreme value statistics for selected observables of dynamical systems the main results of the extreme value theory developed for the investigation of the observables of dynamical systems rely up to now on the block maxima approach in this framework extremes are identified with the block maxima of the time series of the chosen observable in the limit of infinitely long blocks it has been proved that assuming suitable mixing conditions for the underlying dynamical systems the extremes of a specific class of observables are distributed according to the so called generalised extreme value gev distribution direct calculations show that in the case of quasi periodic dynamics the block maxima are not distributed according to the gev distribution in this paper we show that considering the exceedances over a given threshold instead of the block maxima approach it is possible to obtain a generalised pareto distribution also for extremes computed in systems which do not satisfy mixing conditions requiring that the invariant measure locally scales with a well defined exponent the local dimension we show that the limiting distribution for the exceedances of the observables previously studied with the block maxima approach is a generalised pareto distribution where the parameters depend only on the local dimensions and the values of the threshold but not on the number of observations considered we also provide connections with the results obtained with the block maxima approach in order to provide further support to our findings we present the results of numerical experiments carried out considering the well known chirikov standard map

representations of linear periodically time varying and multirate systems the linear switched time varying lstv polyphase blocked and alias component representations of linear periodically time varying lptv systems are studied in particular alias components are related to the time shifted versions of the system it is shown that in general a filterbank is equivalent to the cascade connection of two lptv systems by generalizing some of the results of the lstv representation of lptv systems it is shown that for relatively coprime integers p and m a p channel filterbank that has lptv filters with period m is equivalent to an mp channel filterbank with lti filters the representation of multirate systems that have lptv kernels is discussed next due to the presence of the upsampler and downsampler there are some degrees of freedom in the choice of the kernel this redundancy is dealt with by choosing from various subclasses so that there is a one to one relationship between a multirate system and its kernel then we find the lptv kernel that has the least period

some relations between dimension and lyapounov exponents we consider differentiable maps and compact invariant sets we introduce dimensional quantities related to the ergodic invariant measures and prove some simple relations

the metric entropy of diffeomorphisms part ii relations between entropy exponents and dimension on considere f m m m m un c 2 diffeomorphisme f d une variete de riemann compacte m preservant une mesure de probabilite de borel m soit hm f l entropie metrique de f et 1 x r x x les exposants de lyapunov distincts en x on etablit une formule qui relie h m f et les r x x

fourier analysis of gmres m preconditioned by multigrid this paper deals with convergence estimates of gmres m saad and schultz siam j sci statist comput 7 1986 pp 856 869 preconditioned by multigrid brandt math comp 31 1977 pp 333 390 hackbusch multi grid methods and applications springer berlin 1985 fourier analysis is a well known and useful tool in the multigrid community for the prediction of two grid convergence rates brandt math comp 31 1977 pp 333 390 stuben and trottenberg in multigrid methods lecture notes in math 960 k stuben and u trottenberg eds springer berlin pp 1 176 this analysis is generalized here to the situation in which multigrid is a preconditioner since it is possible to obtain the whole spectrum of the two grid iteration matrix a preconditioned krylov subspace acceleration method like gmres m implicitly builds up a minimal residual polynomial the determination of the polynomial coefficients is easily possible and can be done explicitly since from fourier analysis a simple block diagonal two grid iteration matrix results based on the gmres m polynomial sharp theoretical convergence estimates can be obtained which are compared with estimates based on the spectrum of the iteration matrix several numerical scalar test problems are computed in order to validate the theoretical predictions

dynamics of horizontal like maps in higher dimension we study the regularity of the green currents and of the equilibrium measure associated to a horizontal like map in under a natural assumption on the dynamical degrees we estimate the speed of convergence towards the green currents the decay of correlations for the equilibrium measure and the lyapounov exponents we show in particular that the equilibrium measure is hyperbolic we also show that the green currents are the unique invariant vertical and horizontal positive closed currents the results apply in particular to henon like maps to regular polynomial automorphisms of and to their small perturbations

inverting periodic filters the authors consider linear periodic filters they give simple necessary and sufficient conditions for the filter to be invertible and a simple formula to compute its inverse if the filter is not invertible they propose a method to compute its optimal approximate inverse an illustrative example is given

multilevel image reconstruction with natural pixels the sampled radon transform of a two dimensional 2d function can be represented as a continuous linear map a l 2 r n where au j u j and j is the characteristic function of a strip through approximating the set of line integrals in the sample the image reconstruction problem is given a vector b r n find an image or density function u x y such that au b in general there are infinitely many solutions we seek the solution with minimal 2 norm which leads to a matrix equation bw b where b is a square dense matrix with several convenient properties we analyze the use of gauss seidel iteration applied to the problem observing that while the iteration formally converges there exists a near null space into which the error vectors migrate after which the iteration stalls the null space and near null space of b are characterized in order to develop a multilevel scheme based on the principles of the multilevel projection method pml this scheme leads to somewhat improved performance its primary utility however is that it facilitates the development of a pml based method for spotlight tomography that is local grid refinement over a portion of the image in which features of interest can be resolved at finer scale than is possible globally

a multi level preconditioned krylov method for the efficient solution of algebraic tomographic reconstruction problems classical iterative methods for tomographic reconstruction include the class of algebraic reconstruction techniques art convergence of these stationary linear iterative methods is however notably slow in this paper we propose the use of krylov solvers for tomographic linear inversion problems these advanced iterative methods feature fast convergence at the expense of a higher computational cost per iteration causing them to be generally uncompetitive without the inclusion of a suitable preconditioner combining elements from standard multigrid mg solvers and the theory of wavelets a novel wavelet based multi level wmg preconditioner is introduced which is shown to significantly speed up krylov convergence the performance of the wmg preconditioned krylov method is analyzed through a spectral analysis and the approach is compared to existing methods like the classical simultaneous iterative reconstruction technique sirt and unpreconditioned krylov methods on a 2d tomographic benchmark problem numerical experiments are promising showing the method to be competitive with the classical algebraic reconstruction techniques in terms of convergence speed and overall performance cpu time as well as precision of the reconstruction

algebraic multilevel krylov methods in erlangga and nabben siam j sci comput 30 2008 pp 1572 1595 we developed a new type of multilevel method called the multilevel krylov mk method to solve linear systems of equations the basic idea of this type of method is to shift small eigenvalues that are responsible for slow convergence of krylov methods to an a priori fixed constant this shifting of the eigenvalues is similar to projection type methods and is achieved via the solution of subspace or coarse level systems numerical results show that mk works very well for the two dimensional 2d poisson and convection diffusion equation i e the convergence can be made almost independent of the grid size h and the physical parameter involved in a follow up paper we have used mk in the context of the preconditioned helmholtz equation for this problem we show that the convergence can be made only mildly dependent on the wavenumber even though the coarse level system of mk is algebraically related to that of multigrid the const

orthonormal bases of compactly supported wavelets we construct orthonormal bases of compactly supported wavelets with arbitrarily high regularity the order of regularity increases linearly with the support width we start by reviewing the concept of multiresolution analysis as well as several algorithms in vision decomposition and reconstruction the construction then follows from a synthesis of these different approaches

minimal periodic realizations of transfer matrices periodic controllers designed based on the so called lifting technique are usually represented by transfer matrices real operations require that the controllers be implemented as periodic systems the problem of realizing an nn sub o nn sub i proper rational transfer matrix as an n sub i input n sub o output n periodic discrete time system is studied an algorithm for obtaining periodic realizations which have a minimal number of states is proposed the result can also be used to remove any redundant states that exist in a periodic system

the probability distribution and hausdorff dimension of self affine functions a simple natural measure is found with respect to which the probability distribution of a continuous self affine functionf in the sense of kono is absolutely continuous as an immediate corollary we obtain the result of kono that provides a necessary and sufficient condition for this distribution to be absolutely continuous with respect to lebesgue measure for the class of continuous self affine functions one proves the conjecture of t bedford which says in this context that the hausdorff dimension of the graph off is equal to its box dimension if and only if the probability distribution off is absolutely continuous with respect to lebesgue measure

optimal approximate inverse of linear periodic filters we propose a method for constructing optimal causal approximate inverse for discrete time single input single output siso causal periodic filters in the presence of measurement noise the analysis is based on block signals and multi input multi output mimo time invariant models for periodic filters the objective function to be minimized is the asymptotic block mean square error the optimization problem is formulated in terms of transfer matrices as an optimal model matching problem with nonsquare model and plant based on an inner outer factorization on the transpose of the plant rational matrix it is shown that the problem can be further reduced to one with a lower dimensional square model and plant which is then solved in the time domain and a closed form solution is obtained a lower bound on the objective function is given it is shown that the lower bound can be asymptotically achieved as the order of the optimal transfer matrix increases the proposed method is extended to mimo periodic systems numerical examples are used to illustrate the performance of the proposed approximate inverse

a multigrid method enhanced by krylov subspace iteration for discrete helmholtz equations standard multigrid algorithms have proven ineffective for the solution of discretizations of helmholtz equations in this work we modify the standard algorithm by adding gmres iterations at coarse levels and as an outer iteration we demonstrate the algorithm s effectiveness through theoretical analysis of a model problem and experimental results in particular we show that the combined use of gmres as a smoother and outer iteration produces an algorithm whose performance depends relatively mildly on wave number and is robust for normalized wave numbers as large as 200 for fixed wave numbers it displays grid independent convergence rates and has costs proportional to the number of unknowns

on the dimension of the graph of the classical weierstrass function abstract this paper examines dimension of the graph of the famous weierstrass non differentiable function w b x n 0 n cos 2 b n x for an integer b 2 and 1 b 1 we prove that for every b there exists explicitly given b 1 b 1 such that the hausdorff dimension of the graph of w b is equal to d 2 log log b for every b 1 we also show that the dimension is equal to d for almost every on some larger interval this partially solves a well known thirty year old conjecture furthermore we prove that the hausdorff dimension of the graph of the function f x n 0 n b n x for an integer b 2 and 1 b 1 is equal to d for a typical z periodic c 3 function

dimension theory for invariant measures of endomorphisms we establish the exact dimensional property of an ergodic hyperbolic measure for a c 2 non invertible but non degenerate endomorphism on a compact riemannian manifold without boundary based on this we give a new formula of lyapunov dimension of ergodic measures and show it coincides with the dimension of hyperbolic ergodic measures in a setting of random endomorphisms our results extend several well known theorems of barreira et al ann math 149 755 783 1999 and ledrappier and young commun math phys 117 4 529 548 1988 for diffeomorphisms to the case of endomorphisms

a unified approach to scrambling filter design most speech scrambling systems are either linear periodic filters or can be modeled as such it is well known that from an input output point of view a periodic filter is equivalent to a multi input multi output linear time invariant system and thus a rational transfer matrix in this paper we propose a framework based on transfer matrices and frequency domain descriptions for analysis and design of speech scrambling filters we derive necessary and sufficient conditions for a scrambling system to be insensitive to frame synchronization error we propose a procedure for the design of scrambling filters that are insensitive to synchronization error and have zero bandwidth expansion an illustrative design example is given