graphical models for statistical inference and data assimilation in data assimilation for a system which evolves in time one combines past and current observations with a model of the dynamics of the system in order to improve the simulation of the system as well as any future predictions about it from a statistical point of view this process can be regarded as estimating many random variables which are related both spatially and temporally given observations of some of these variables typically corresponding to times past we require estimates of several others typically corresponding to future times graphical models have emerged as an efiective formalism for assisting in these types of inference tasks particularly for large numbers of random variables graphical models provide a means of representing dependency structure among the variables and can provide both intuition and e ciency in estimation and other inference computations we provide an overview and introduction to graphical models and describe how they can be used to represent statistical dependency and how the resulting structure can be used to organize computation the relation between statistical inference using graphical models and optimal sequential estimation algorithms such as kalman flltering is discussed we then give several additional examples of how graphical models can be applied to climate dynamics speciflcally estimation using multi resolution models of large scale data sets such as satellite imagery and learning hidden markov models to capture rainfall patterns in space and time

multigrid tomographic inversion with variable resolution data and image spaces a multigrid inversion approach that uses variable resolutions of both the data space and the image space is proposed since the computational complexity of inverse problems typically increases with a larger number of unknown image pixels and a larger number of measurements the proposed algorithm further reduces the computation relative to conventional multigrid approaches which change only the image space resolution at coarse scales the advantage is particularly important for data rich applications where data resolutions may differ for different scales applications of the approach to bayesian reconstruction algorithms in transmission and emission tomography with a generalized gaussian markov random field image prior are presented both with a poisson noise model and with a quadratic data term simulation results indicate that the proposed multigrid approach results in significant improvement in convergence speed compared to the fixed grid iterative coordinate descent method and a multigrid method with fixed data resolution

regularization in tomographic reconstruction using thresholding estimators in tomographic medical devices such as single photon emission computed tomography or positron emission tomography cameras image reconstruction is an unstable inverse problem due to the presence of additive noise a new family of regularization methods for reconstruction based on a thresholding procedure in wavelet and wavelet packet wp decompositions is studied this approach is based on the fact that the decompositions provide a near diagonalization of the inverse radon transform and of prior information in medical images a wp decomposition is adaptively chosen for the specific image to be restored corresponding algorithms have been developed for both two dimensional and full three dimensional reconstruction these procedures are fast noniterative and flexible numerical results suggest that they outperform filtered back projection and iterative procedures such as ordered subset expectation maximization

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

image reconstruction from projections under wavelet constraints first the authors discuss how the wavelet transform can be used to perform spatially varying filtering of an image suppressing noise locally in smooth regions of the image and discuss detection of such regions in a noise corrupted image second they show how to compute the minimum mean square estimate of an image given 1 noisy projections of the image 2 statistics of additive noise in the projections and 3 constraints on wavelet coefficients of the image examples illustrate the resulting procedure

continuous spatial semigroups of endomorphisms of ℌ to each continuous semigroup of endomorphisms a of z s with an intertwining semigroup of isometries there is associated a representation 7t of the domain 2d 6 of the generator of ac it is shown that the arveson index d a is the number of times the representation 7t contains the identity representation of 2 6 this result is obtained from an analysis of the relation between two semigroups of isometries u and s satisfying the condition s t u t e ti for t 0 and 0

the car parking problem at high densities abstract the radial distribution functions of random 1 d systems of sequential hard rods have been studied in the range of very high densities it is found that as the number of samples rejected before completion increases anomalies in the pairwise distribution functions arise these are discussed using analytical solutions for systems of three rods and numerical simulations with twelve rods the probabilities of different spatial orderings with respect to the sequential order are examined

estimation of fractal signals from noisy measurements using wavelets the role of the wavelet transformation as a whitening filter for 1 f processes is exploited to address problems of parameter and signal estimations for 1 f processes embedded in white background noise robust computationally efficient and consistent iterative parameter estimation algorithms are derived based on the method of maximum likelihood and cramer rao bounds are obtained included among these algorithms are optimal fractal dimension estimators for noisy data algorithms for obtaining bayesian minimum mean square signal estimates are also derived together with an explicit formula for the resulting error these smoothing algorithms find application in signal enhancement and restoration the parameter estimation algorithms find application in signal enhancement and restoration the parameter estimation algorithms in addition to solving the spectrum estimation problem and to providing parameters for the smoothing process are useful in problems of signal detection and classification results from simulations are presented to demonstrated the viability of the algorithms

on the car parking problem the radial distribution functions of randomly distributed one dimensional spheres are studied the differences between equilibrium fluids and sequentially constructed systems are discussed in terms of the corresponding configuration spaces and numerical simulations it is shown that for the sequentially constructed systems which correspond to the car parking problem a memory effect occurs such that the sequential spheres are temporally distinguishable exact solutions for some simple cases and numerical simulations for large systems are shown discussed and compared with the equilibrium fluid results

pair correlation function in random sequential adsorption processes an analytical expression of the pair distribution function is derived for the car parking problem and for the random sequential adsorption of k mers onto a one dimensional lattice both on a lattice and in the continuum limit super exponential decay of the distribution function is observed a comparison of the spatial correlations with those at equilibrium demonstrates the influence of the irreversibility of the rsa process

is a local description of stable 2d random packing possible local models of stable 2d random packing by williams 1985 and by uhler and schilling 1985 are discussed in terms of car parking and equilibrium fluid 1d distributions a numerical simulation of a simple 2d model is also presented whose results are different from the previous two it is concluded that those simple minded approaches to the problem of random 2d packing are questionable and accurate answers to the problem will have to come from more realistic models

kinetics and fractal properties of the random sequential adsorption of line segments the random sequential adsorption rsa of infinitely thin line segments is studied by computer simulation it is shown that the exponent for the time dependence of the surface coverage increases beyond the value 0 33 reported by sherwood 1990 to a value of 0 38 at later times it is also argued that the fractal dimension of the adsorbed lines is about 1 8 making use of a previous result on the rsa of rectangular objects

tomographic traveltime inversion using natural pixels traditionally in the problem of tomographic traveltime inversion the model is divided into a number of rectangular cells of constant slowness inversion consists of finding these constant values using the measured traveltimes the inversion process can demand a large computational effort if a high resolution result is desired we show how to use a different kind of parameterization of the model based on beam propagation paths this parameterization is obtained within the framework of reconstruction in hilbert spaces by minimizing the error between the true model and the estimated model the traveltimes are interpreted as the projections of the slowness along the beampaths although the actual beampaths are described by complicated spatial functions we simplify the computations by approximating these functions with functions of constant width and height i e fat rays which collectively form a basis set of natural pixels with a simple numerical example we demonstrate that the main advantage of this

fast wavelet transforms and numerical algorithms i a class of algorithms is introduced for the rapid numerical application of a class of linear operators to arbitrary vectors previously published schemes of this type utilize detailed analytical information about the operators being applied and are specific to extremely narrow classes of matrices in contrast the methods presented here are based on the recently developed theory of wavelets and are applicable to all calderon zygmund and pseudo differential operators the algorithms of this paper require order o n or o n log n operations to apply an n n matrix to a vector depending on the particular operator and the version of the algorithm being used and our numerical experiments indicate that many previously intractable problems become manageable with the techniques presented here

fractal feature analysis and classification in medical imaging following b b mandelbrot s fractal theory 1982 it was found that the fractal dimension could be obtained in medical images by the concept of fractional brownian motion an estimation concept for determination of the fractal dimension based upon the concept of fractional brownian motion is discussed two applications are found 1 classification 2 edge enhancement and detection for the purpose of classification a normalized fractional brownian motion feature vector is defined from this estimation concept it represented the normalized average absolute intensity difference of pixel pairs on a surface of different scales the feature vector uses relatively few data items to represent the statistical characteristics of the medial image surface and is invariant to linear intensity transformation for edge enhancement and detection application a transformed image is obtained by calculating the fractal dimension of each pixel over the whole medical image the fractal dimension value of each pixel is obtained by calculating the fractal dimension of 7 7 pixel block centered on this pixel

efficient multiscale regularization with applications to the computation of optical flow a new approach to regularization methods for image processing is introduced and developed using as a vehicle the problem of computing dense optical flow fields in an image sequence the solution of the new problem formulation is computed with an efficient multiscale algorithm experiments on several image sequences demonstrate the substantial computational savings that can be achieved due to the fact that the algorithm is noniterative and in fact has a per pixel computational complexity that is independent of image size the new approach also has a number of other important advantages specifically multiresolution flow field estimates are available allowing great flexibility in dealing with the tradeoff between resolution and accuracy multiscale error covariance information is also available which is of considerable use in assessing the accuracy of the estimates in particular these error statistics can be used as the basis for a rational procedure for determining the spatially varying optimal reconstruction resolution furthermore if there are compelling reasons to insist upon a standard smoothness constraint the new algorithm provides an excellent initialization for the iterative algorithms associated with the smoothness constraint problem formulation finally the usefulness of the approach should extend to a wide variety of ill posed inverse problems in which variational techniques seeking a smooth solution are generally used

on the product system of a completely positive semigroup abstract given a w continuous semigroup of unital normal completely positive maps of b h we introduce its continuous tensor product system e if is a minimal dilation e 0 semigroup of with arveson product system f then e is canonically isomorphic to f we apply this construction to a class of semigroups of b l 2 r arising from a modified weyl moyal quantization of convolution semigroups of borel probability measures on r 2 this class includes the heat flow on the ccr algebra studied recently by arveson we prove that the minimal dilations of all such semigroups are completely spatial and additionally we prove that the minimal dilation of the heat flow is cocyle conjugate to the car ccr flow of index two

wavelet methods for inverting the radon transform with noisy data because the radon transform is a smoothing transform any noise in the radon data becomes magnified when the inverse radon transform is applied among the methods used to deal with this problem is the wavelet vaguelette decomposition wvd coupled with wavelet shrinkage as introduced by donoho 1995 we extend several results of donoho and others here first we introduce a new sufficient condition on wavelets to generate a wvd for a general homogeneous operator whose class includes the radon transform we show that a variant of donoho s method for solving inverse problems can be derived as the exact minimizer of a variational problem that uses a besov norm as the smoothing functional we give a new proof of the rate of convergence of wavelet shrinkage that allows us to estimate rather sharply the best shrinkage parameter needed to recover an image from noise corrupted data we conduct tomographic reconstruction computations that support the hypothesis that near optimal shrinkage parameters can be derived if one can estimate only two besov space parameters about an image f both theoretical and experimental results indicate that our choice of shrinkage parameters yields uniformly better results than kolaczyk s 1996 variant of donoho s method and the classical filtered backprojection method

wavelet localization of the radon transform the authors develop an algorithm which significantly reduces radiation exposure in x ray tomography when a local region of the body is to be imaged the algorithm uses the properties of wavelets to essentially localize the radon transform this algorithm differs from previous algorithms for doing local tomography because it recovers an approximation to the original image not the image module the nullspace of the local tomography operator or the lambda transform of the image this is possible because the authors do not truly invert the interior radon transform but rather sample the radon transform sparsely away from the local region of interest much attention in the field has been directed towards localized tomography the authors believe that this technique represents a significant contribution towards this effort

local unitary cocycles of e0 semigroups abstract this paper concerns the structure of the group of local unitary cocycles also called the gauge group of an e 0 semigroup the gauge group of a spatial e 0 semigroup has a natural action on the set of units by operator multiplication arveson has characterized completely the gauge group of e 0 semigroups of type i and as a consequence it is known that in this case the gauge group action is transitive in fact if the semigroup has index k then the gauge group action is transitive on the set of k 1 tuples of appropriately normalized independent units an action of the gauge group having this property is called k 1 fold transitive we construct examples of e 0 semigroups of type ii and index 1 which are not 2 fold transitive these new examples also illustrate that an e 0 semigroup of type ii k need not be a tensor product of an e 0 semigroup of type ii 0 and another of type i k

tomographic reconstruction and estimation based on multiscale natural pixel bases we use a natural pixel type representation of an object originally developed for incomplete data tomography problems to construct nearly orthonormal multiscale basis functions the nearly orthonormal behavior of the multiscale basis functions results in a system matrix relating the input the object coefficients and the output the projection data which is extremely sparse in addition the coarsest scale elements of this matrix capture any ill conditioning in the system matrix arising from the geometry of the imaging system we exploit this feature to partition the system matrix by scales and obtain a reconstruction procedure that requires inversion of only a well conditioned and sparse matrix this enables us to formulate a tomographic reconstruction technique from incomplete data wherein the object is reconstructed at multiple scales or resolutions in case of noisy projection data we extend our multiscale reconstruction technique to explicitly account for noise by calculating maximum a posteriori probability map multiscale reconstruction estimates based on a certain self similar prior on the multiscale object coefficients the framework for multiscale reconstruction presented can find application in regularization of imaging problems where the projection data are incomplete irregular and noisy and in object feature recognition directly from projection data

cellular automata approach to non equilibrium diffusion and gradient percolation the authors propose a deterministic approach to lattice diffusion in two dimensions this method is implemented on a cellular automaton special purpose computer in order to study the properties of the interface of particles diffusing from a source to a sink fractal properties of the diffusion front of this non equilibrium process are compared with results from percolation theory the observed agreement indicates that diffusion fronts and gradient percolation coincide asymptotically and that the cellular automata method is a viable alternative to standard simulations for this class of problems

a theory for multiresolution signal decomposition the wavelet representation multiresolution representations are effective for analyzing the information content of images the properties of the operator which approximates a signal at a given resolution were studied it is shown that the difference of information between the approximation of a signal at the resolutions 2 sup j 1 and 2 sup j where j is an integer can be extracted by decomposing this signal on a wavelet orthonormal basis of l sup 2 r sup n the vector space of measurable square integrable n dimensional functions in l sup 2 r a wavelet orthonormal basis is a family of functions which is built by dilating and translating a unique function psi x this decomposition defines an orthogonal multiresolution representation called a wavelet representation it is computed with a pyramidal algorithm based on convolutions with quadrature mirror filters wavelet representation lies between the spatial and fourier domains for images the wavelet representation differentiates several spatial orientations the application of this representation to data compression in image coding texture discrimination and fractal analysis is discussed