derivations with values in ideals of semifinite von neumann algebras let m be a semifinite von neumann algebra with a faithful semifinite normal trace and let a be an arbitrary c subalgebra of m assume that e is a fully symmetric function space on 0 having fatou property and order continuous norm and e m is the corresponding symmetric operator space we prove that every derivation a e m e m m is inner strengthening earlier results by kaftal and weiss 28 in the case when m is a semifinite non finite factor we show that our assumptions on e 0 are sharp

embedded trees estimation of gaussian processes on graphs with cycles graphical models provide a powerful general framework for encoding the structure of large scale estimation problems however the graphs describing typical real world phenomena contain many cycles making direct estimation procedures prohibitively costly in this paper we develop an iterative inference algorithm for general gaussian graphical models it operates by exactly solving a series of modified estimation problems on spanning trees embedded within the original cyclic graph when these subproblems are suitably chosen the algorithm converges to the correct conditional means moreover and in contrast to many other iterative methods the tree based procedures we propose can also be used to calculate exact error variances although the conditional mean iteration is effective for quite densely connected graphical models the error variance computation is most efficient for sparser graphs in this context we present a modeling example suggesting that very sparsely connected graphs with cycles may provide significant advantages relative to their tree structured counterparts thanks both to the expressive power of these models and to the efficient inference algorithms developed herein the convergence properties of the proposed tree based iterations are characterized both analytically and experimentally in addition by using the basic tree based iteration to precondition the conjugate gradient method we develop an alternative accelerated iteration that is finitely convergent simulation results are presented that demonstrate this algorithm s effectiveness on several inference problems including a prototype distributed sensing application

weak compactness and fixed point property for affine mappings abstract it is shown that a closed convex bounded subset of a banach space is weakly compact if and only if it has the generic fixed point property for continuous affine mappings the class of continuous affine mappings can be replaced by the class of affine mappings which are uniformly lipschitzian with some constant m 1 in the case of c0 the class of affine mappings which are uniformly lipschitzian with some constant m 6 in the case of quasi reflexive james space j and the class of nonexpansive affine mappings in the case of l embedded spaces

scene segmentation from visual motion using global optimization this paper presents results from computer experiments with an algorithm to perform scene disposition and motion segmentation from visual motion or optic flow the maximum a posteriori map criterion is used to formulate what the best segmentation or interpretation of the scene should be where the scene is assumed to be made up of some fixed number of moving planar surface patches the bayesian approach requires first specification of prior expectations for the optic flow field which here is modeled as spatial and temporal markov random fields and secondly a way of measuring how well the segmentation predicts the measured flow field the markov random fields incorporate the physical constraints that objects and their images are probably spatially continuous and that their images are likely to move quite smoothly across the image plane to compute the flow predicted by the segmentation a recent method for reconstructing the motion and orientation of planar surface facets is used the search for the globally optimal segmentation is performed using simulated annealing

property t and rigidity for actions on banach spaces we study property t and the fixed point property for actions on lp and other banach spaces we show that property t holds when l2 is replaced by lp and even a subspace quotient of lp and that in fact it is independent of 1 p we show that the fixed point property for lp follows from property t when 1 p 2 e for simple lie groups and their lattices we prove that the fixed point property for lp holds for any 1 p if and only if the rank is at least two finally we obtain a superrigidity result for actions of irreducible lattices in products of general groups on superreflexive banach spaces

a karhunen loeve like expansion for 1 f processes via wavelets while so called 1 f or scaling processes emerge regularly in modeling a wide range of natural phenomena as yet no entirely satisfactory framework has been described for the analysis of such processes orthonormal wavelet bases are used to provide a new construction for nearly 1 f processes from a set of uncorrelated random variables

a generalized levinson algorithm for covariance extension with application to multiscale autoregressive modeling efficient computation of extensions of banded partially known covariance matrices is provided by the classical levinson algorithm one contribution of this paper is the introduction of a generalization of this algorithm that is applicable to a substantially broader class of extension problems this generalized algorithm can compute unknown covariance elements in any order that satisfies certain graph theoretic properties which we describe this flexibility which is not provided by the classical levinson algorithm is then harnessed in a second contribution of this paper the identification of a multiscale autoregressive mar model for the maximum entropy me extension of a banded partially known covariance matrix the computational complexity of mar model identification is an order of magnitude below that of explicitly computing a full covariance extension and is comparable to that required to build a standard autoregressive ar model using the classical levinson algorithm

isomorphisms between spaces of lipschitz functions abstract we develop tools for proving isomorphisms of normed spaces of lipschitz functions over various doubling metric spaces and banach spaces in particular we show that lip 0 z d lip 0 r d for all d n more generally we e g show that lip 0 lip 0 g where is from a large class of finitely generated nilpotent groups and g is its mal cev closure or that lip 0 l p lip 0 l p for all 1 p we leave a large area for further possible research

the derivation problem for group algebras if g is a locally compact group then for each derivation d from l 1 g into l 1 g there is a bounded measure m g with d a a a for a l 1 g derivation problem of b e johnson

multiscale recursive estimation data fusion and regularization we describe a framework for modeling stochastic phenomena at multiple scales and for their efficient estimation or reconstruction given partial and or noisy measurements which may also be at several scales in particular multiscale signal representations lead naturally to pyramidal or tree like data structures in which each level in the tree corresponds to a particular scale of representation a class of multiscale dynamic models evolving on dyadic trees is introduced the main focus of this paper is on the description analysis and application of an extremely efficient optimal estimation algorithm for this class of models this algorithm consists of a fine to coarse filtering sweep followed by a coarse to fine smoothing step corresponding to the dyadic tree generalization of kalman filtering and rauch tung striebel smoothing the kalman filtering sweep consists of the recursive application of three steps a measurement update step a fine to coarse prediction step and a fusion step we illustrate the use of our methodology for the fusion of multiresolution data and for the efficient solution of fractal regularizations of ill posed signal and image processing problems encountered

likelihood calculation for a class of multiscale stochastic models with application to texture discrimination a class of multiscale stochastic models based on scale recursive dynamics on trees has previously been introduced theoretical and experimental results have shown that these models provide an extremely rich framework for representing both processes which are intrinsically multiscale e g 1 f processes as well as 1d markov processes and 2d markov random fields moreover efficient optimal estimation algorithms have been developed for these models by exploiting their scale recursive structure the authors exploit this structure in order to develop a computationally efficient and parallelizable algorithm for likelihood calculation they illustrate one possible application to texture discrimination and demonstrate that likelihood based methods using the algorithm achieve performance comparable to that of gaussian markov random field based techniques which in general are prohibitively complex computationally

image analysis using multigrid relaxation methods image analysis problems posed mathematically as variational principles or as partial differential equations are amenable to numerical solution by relaxation algorithms that are local iterative and often parallel although they are well suited structurally for implementation on massively parallel locally interconnected computational architectures such distributed algorithms are seriously handi capped by an inherent inefficiency at propagating constraints between widely separated processing elements hence they converge extremely slowly when confronted by the large representations of early vision application of multigrid methods can overcome this drawback as we showed in previous work on 3 d surface reconstruction in this paper we develop multiresolution iterative algorithms for computing lightness shape from shading and optical flow and we examine the efficiency of these algorithms using synthetic image inputs the multigrid methodology that we describe is broadly applicable in early vision notably it is an appealing strategy to use in conjunction with regularization analysis for the efficient solution of a wide range of ill posed image analysis problems

multiscale methods for the segmentation and reconstruction of signals and images this paper addresses the problem of both segmenting and reconstructing a noisy signal or image the work is motivated by large problems arising in certain scientific applications such as medical imaging two objectives for a segmentation and denoising algorithm are laid out it should be computationally efficient and capable of generating statistics for the errors in the reconstruction and estimates of the boundary locations the starting point for the development of a suitable algorithm is a variational approach to segmentation shah 1992 this paper then develops a precise statistical interpretation of a one dimensional 1 d version of this variational approach to segmentation the 1 d algorithm that arises as a result of this analysis is computationally efficient and capable of generating error statistics a straightforward extension of this algorithm to two dimensions would incorporate recursive procedures for computing estimates of inhomogeneous gaussian markov random fields such procedures require an unacceptably large number of operations to meet the objective of developing a computationally efficient algorithm the use of previously developed multiscale statistical methods is investigated this results in the development of an algorithm for segmenting and denoising which is not only computationally efficient but also capable of generating error statistics as desired

noniterative manipulation of discrete energy based models for image analysis abstract with emphasis on the graph structure of energy based models devoted to image analysis we investigate efficient procedures for sampling and inferring we show that triangulated graphs whom trees are simple instances of always support causal models for which noniterative procedures can be devised to minimize the energy to extract probabilistic descriptions to sample from corresponding prior and posterior distributions or to infer from local marginals the relevance and efficiency of these procedures are illustrated for classification problems

fixed point properties for semigroups of nonlinear mappings and amenability abstract in this paper we study fixed point properties for semitopological semigroup of nonexpansive mappings on a bounded closed convex subset of a banach space we also study a schauder fixed point property for a semitopological semigroup of continuous mappings on a compact convex subset of a separated locally convex space such semigroups properly include the class of extremely left amenable semitopological semigroups the free commutative semigroup on one generator and the bicyclic semigroup s 1 a b a b 1

random groups random graphs and eigenvalues of p laplacians abstract we prove that a random group in the triangular density model has for density larger than 1 3 fixed point properties for actions on l p spaces affine isometric and more generally 2 2 1 2 p uniformly lipschitz with p varying in an interval increasing with the set of generators in the same model we establish a double inequality between the maximal p for which l p fixed point properties hold and the conformal dimension of the boundary in the gromov density model we prove that for every p 0 2 for a sufficiently large number of generators and for any density larger than 1 3 a random group satisfies the fixed point property for affine actions on l p spaces that are 2 2 1 2 p uniformly lipschitz and this for every p 2 p 0 to accomplish these goals we find new bounds on the first eigenvalue of the p laplacian on random graphs using methods adapted from kahn and szemeredi s approach to the 2 laplacian these in turn lead to fixed point properties using arguments of bourdon and gromov which extend to l p spaces previous results for kazhdan s property t established by uk and ballmann wiatkowski

a fixed point theorem for l1 spaces we prove a fixed point theorem for a family of banach spaces notably for l 1 applications include the optimal answer to the derivation problem for group algebras which originated in the 1960s

superrigidity of actions on finite rank median spaces abstract finite rank median spaces are a simultaneous generalisation of finite dimensional cat 0 cube complexes and real trees if is an irreducible lattice in a product of rank one simple lie groups we show that every action of on a complete finite rank median space has a global fixed point this is in sharp contrast with the behaviour of actions on infinite rank median spaces where even proper cocompact actions can arise the fixed point property is obtained as corollary to a superrigidity result the latter holds for irreducible lattices in arbitrary products of compactly generated topological groups we exploit roller compactifications of median spaces these were introduced in 48 and generalise a well known construction in the case of cube complexes we show that the haagerup cocycle provides a reduced 1 cohomology class that detects group actions with a finite orbit in the roller compactification this is new even for cat 0 cube complexes and has interesting consequences involving shalom s property h f d for instance in gromov s density model random groups at low density do not have h f d

image registration using wavelet based motion model an image registration algorithm is developed to estimate dense motion vectors between two images using the coarse to fine wavelet based motion model this motion model is described by a linear combination of hierarchical basis functions proposed by cai and wang siam numer anal 33 3 937 970 1996 the coarser scale basis function has larger support while the finer scale basis function has smaller support with these variable supports in full resolution the basis functions serve as large to small windows so that the global and local information can be incorporated concurrently for image matching especially for recovering motion vectors containing large displacements to evaluate the accuracy of the wavelet based method two sets of test images were experimented using both the wavelet based method and a leading pyramid spline based method by szeliski et al international journal of computer vision 22 3 199 218 1996 one set of test images taken from barron et al international journal of computer vision 12 43 77 1994 contains small displacements the other set exhibits low texture or spatial aliasing after image blurring and contains large displacements the experimental results showed that our wavelet based method produced better motion estimates with error distributions having a smaller mean and smaller standard deviation

restriction of a markov random field on a graph and multiresolution statistical image modeling the association of statistical models and multiresolution data analysis in a consistent and tractable mathematical framework remains an intricate theoretical and practical issue several consistent approaches have been proposed previously to combine markov random field mrf models and multiresolution algorithms in image analysis renormalization group subsampling of stochastic processes mrfs defined on trees or pyramids etc for the simulation or a practical use of these models in statistical estimation an important issue is the preservation of the local markovian property of the representation at the different resolution levels it is shown that this key problem may be studied by considering the restriction of a markov random field defined on some simple finite nondirected graph to a part of its original site set several general properties of the restricted field are derived the general form of the distribution of the restriction is given locality of the field is studied by exhibiting a neighborhood structure with respect to which the restricted field is an mrf sufficient conditions for the new neighborhood structure to be minimal are derived several consequences of these general results related to various multiresolution mrf based modeling approaches in image analysis are presented

multiscale representations of markov random fields recently a framework for multiscale stochastic modeling was introduced based on coarse to fine scale recursive dynamics defined on trees this model class has some attractive characteristics which lead to extremely efficient statistically optimal signal and image processing algorithms the authors show that this model class is also quite rich in particular they describe how 1 d markov processes and 2 d markov random fields mrfs can be represented within this framework the recursive structure of 1 d markov processes makes them simple to analyze and generally leads to computationally efficient algorithms for statistical inference on the other hand 2 d mrfs are well known to be very difficult to analyze due to their noncausal structure and thus their use typically leads to computationally intensive algorithms for smoothing and parameter identification in contrast their multiscale representations are based on scale recursive models and thus lead naturally to scale recursive algorithms which can be substantially more efficient computationally than those associated with mrf models in 1 d the multiscale representation is a generalization of the midpoint deflection construction of brownian motion the representation of 2 d mrfs is based on a further generalization to a midline deflection construction the exact representations of 2 d mrfs are used to motivate a class of multiscale approximate mrf models based on one dimensional wavelet transforms they demonstrate the use of these latter models in the context of texture representation and in particular they show how they can be used as approximations for or alternatives to well known mrf texture models

an investigation of smoothness constraints for the estimation of displacement vector fields from image sequences a mapping between one frame from an image sequence and the preceding or following frame can be represented as a displacement vector field in most situations the mere gray value variations do not provide sufficient information in order to estimate such a displacement vector field supplementary constraints are necessary for example the postulate that a displacement vector field varies smoothly as a function of the image position taken as a general requirement this creates difficulties at gray value transitions which correspond to occluding contours nagel therefore introduced the oriented smoothness requirement which restricts variations of the displacement vector field only in directions with small or no variation of gray values this contribution reports results of an investigation about how such an oriented smoothness constraint may be formulated and evaluated

optimal approximations by piecewise smooth functions and associated variational problems abstract this reprint will introduce and study the most basic properties of three new variational problems which are suggested by applications to computer vision in computer vision a fundamental problem is to appropriately decompose the domain r of a function g x y of two variables this problem starts by describing the physical situation which produces images assume that a three dimensional world is observed by an eye or camera from some point p and that g1 rho represents the intensity of the light in this world approaching the point sub 1 from a direction rho if one has a lens at p focusing this light on a retina or a film in both cases a plane domain r in which we may introduce coordinates x y then let g x y be the strength of the light signal striking r at a point with coordinates x y g x y is essentially the same as sub 1 rho possibly after a simple transformation given by the geometry of the imaging syste the function g x y defined on the plane domain r will be called an image what sort of function is g the light reflected off the surfaces si of various solid objects o sub i visible from p will strike the domain r in various open subsets r sub i when one object o1 is partially in front of another object o2 as seen from p but some of object o2 appears as the background to the sides of o1 then the open sets r1 and r2 will have a common boundary the edge of object o1 in the image defined on r and one usually expects the image g x y to be discontinuous along this boundary jhd

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

the remainder term for analytic functions of gauss radau and gauss lobatto quadrature rules with multiple end points a study is undertaken of the kernels in the contour integral representations of the remainder terms for gauss radau and gauss lobatto quadrature rules over the interval 1 1 it is assumed that

averaged projections angles between groups and strengthening of banach property t recently lafforgue introduced a new strengthening of banach property t which he called strong banach property t and showed that this property has implications regarding fixed point properties and banach expanders in this paper we introduce a new strengthening of banach property t called robust banach property t which is weaker than strong banach property t but is still strong enough to ensure similar applications using the method of averaged projections in banach spaces and introducing a new notion of angles between projections we establish a criterion for robust banach property t and show several examples of groups in which this criterion is fulfilled we also derive several applications regarding fixed point properties and banach expanders and give examples of these applications

variational optical flow computation in real time this paper investigates the usefulness of bidirectional multigrid methods for variational optical flow computations although these numerical schemes are among the fastest methods for solving equation systems they are rarely applied in the field of computer vision we demonstrate how to employ those numerical methods for the treatment of variational optical flow formulations and show that the efficiency of this approach even allows for real time performance on standard pcs as a representative for variational optic flow methods we consider the recently introduced combined local global method it can be considered as a noise robust generalization of the horn and schunck technique we present a decoupled as well as a coupled version of the classical gau spl szlig seidel solver and we develop several multigrid implementations based on a discretization coarse grid approximation in contrast with standard bidirectional multigrid algorithms we take advantage of intergrid transfer operators that allow for nondyadic grid hierarchies as a consequence no restrictions concerning the image size or the number of traversed levels have to be imposed in the experimental section we juxtapose the developed multigrid schemes and demonstrate their superior performance when compared to unidirectional multigrid methods and nonhierachical solvers for the well known 316 spl times 252 yosemite sequence we succeeded in computing the complete set of dense flow fields in three quarters of a second on a 3 06 ghz pentium4 pc this corresponds to a frame rate of 18 flow fields per second which outperforms the widely used gau spl szlig seidel method by almost three orders of magnitude

the laplacian pyramid as a compact image code we describe a technique for image encoding in which local operators of many scales but identical shape serve as the basis functions the representation differs from established techniques in that the code elements are localized in spatial frequency as well as in space pixel to pixel correlations are first removed by subtracting a lowpass filtered copy of the image from the image itself the result is a net data compression since the difference or error image has low variance and entropy and the low pass filtered image may represented at reduced sample density further data compression is achieved by quantizing the difference image these steps are then repeated to compress the low pass image iteration of the process at appropriately expanded scales generates a pyramid data structure the encoding process is equivalent to sampling the image with laplacian operators of many scales thus the code tends to enhance salient image features a further advantage of the present code is that it is well suited for many image analysis tasks as well as for image compression fast algorithms are described for coding and decoding

multiply rooted multiscale models for large scale estimation divide and conquer or multiscale techniques have become popular for solving large statistical estimation problems the methods rely on defining a state which conditionally decorrelates the large problem into multiple subproblems each more straightforward than the original however this step cannot be carried out for asymptotically large problems since the dimension of the state grows without bound leading to problems of computational complexity and numerical stability in this paper we propose a new approach to hierarchical estimation in which the conditional decorrelation of arbitrarily large regions is avoided and the problem is instead addressed piece by piece the approach possesses promising attributes it is not a local method the estimate at every point is based on all measurements it is numerically stable for problems of arbitrary size and the approach retains the benefits of the multiscale framework on which it is based a broad class of statistical models a stochastic realization theory an algorithm to calculate statistical likelihoods and the ability to fuse local and nonlocal measurements

beyond independent components trees and clusters we present a generalization of independent component analysis ica where instead of looking for a linear transform that makes the data components independent we look for a transform that makes the data components well fit by a tree structured graphical model this tree dependent component analysis tca provides a tractable and flexible approach to weakening the assumption of independence in ica in particular tca allows the underlying graph to have multiple connected components and thus the method is able to find clusters of components such that components are dependent within a cluster and independent between clusters finally we make use of a notion of graphical models for time series due to brillinger 1996 to extend these ideas to the temporal setting in particular we are able to fit models that incorporate tree structured dependencies among multiple time series

a canonical correlations approach to multiscale stochastic realization we develop a realization theory for a class of multiscale stochastic processes having white noise driven scale recursive dynamics that are indexed by the nodes of a tree given the correlation structure of a 1 d or 2 d random process our methods provide a systematic way to realize the given correlation as the finest scale of a multiscale process motivated by akaike s use of canonical correlation analysis to develop both exact and reduced order models for time series we too harness this tool to develop multiscale models we apply our realization scheme to build reduced order multiscale models for two applications namely linear least squares estimation and generation of random field sample paths for the numerical examples considered least squares estimates are obtained having nearly optimal mean square errors even with multiscale models of low order although both field estimates and field sample paths exhibit a visually distracting blockiness this blockiness is not an important issue in many applications for such applications our approach to multiscale stochastic realization holds promise as a valuable general tool

computation of component image velocity from local phase information we present a technique for the computation of 2d component velocity from image sequences initially the image sequence is represented by a family of spatiotemporal velocity tuned linear filters component velocity computed from spatiotemporal responses of identically tuned filters is expressed in terms of the local first order behavior of surfaces of constant phase justification for this definition is discussed from the perspectives of both 2d image translation and deviations from translation that are typical in perspective projections of 3d scenes the resulting technique is predominantly linear efficient and suitable for parallel processing moreover it is local in space time robust with respect to noise and permits multiple estimates within a single neighborhood promising quantiative results are reported from experiments with realistic image sequences including cases with sizeable perspective deformation

multiscale systems kalman filters and riccati equations an algorithm analogous to the rauch tung striebel algorithm spl minus consisting of a fine to coarse kalman filter like sweep followed by a coarse to fine smoothing step spl minus was developed previously by the authors ibid vol 39 no 3 p 464 78 1994 in this paper they present a detailed system theoretic analysis of this filter and of the new scale recursive riccati equation associated with it while this analysis is similar in spirit to that for standard kalman filters the structure of the dyadic tree leads to several significant differences in particular the structure of the kalman filter error dynamics leads to the formulation of an ml version of the filtering equation and to a corresponding smoothing algorithm based on triangularizing the hamiltonian for the smoothing problem in addition the notion of stability for dynamics requires some care as do the concepts of reachability and observability using these system theoretic constructs the stability and steady state behavior of the fine to coarse kalman filter and its riccati equation are analysed

multimodal estimation of discontinuous optical flow using markov random fields the estimation of dense velocity fields from image sequences is basically an ill posed problem primarily because the data only partially constrain the solution it is rendered especially difficult by the presence of motion boundaries and occlusion regions which are not taken into account by standard regularization approaches in this paper the authors present a multimodal approach to the problem of motion estimation in which the computation of visual motion is based on several complementary constraints it is shown that multiple constraints can provide more accurate flow estimation in a wide range of circumstances the theoretical framework relies on bayesian estimation associated with global statistical models namely markov random fields the constraints introduced here aim to address the following issues optical flow estimation while preserving motion boundaries processing of occlusion regions fusion between gradient and feature based motion constraint equations deterministic relaxation algorithms are used to merge information and to provide a solution to the maximum a posteriori estimation of the unknown dense motion field the algorithm is well suited to a multiresolution implementation which brings an appreciable speed up as well as a significant improvement of estimation when large displacements are present in the scene experiments on synthetic and real world image sequences are reported

acylindrically hyperbolic groups with exotic properties abstract we prove that every countable family of countable acylindrically hyperbolic groups has a common finitely generated acylindrically hyperbolic quotient as an application we obtain an acylindrically hyperbolic group q with strong fixed point properties q has property f l p for all p 1 and every action of q on a finite dimensional contractible topological space has a fixed point in addition q has other properties which are rather unusual for groups exhibiting hyperbolic like behavior e g q is not uniformly non amenable and has finite generating sets with arbitrary large balls consisting of torsion elements

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

2m weak amenability of group algebras a common fixed point property for semigroups is applied to show that the group algebra l1 g of a locally compact group g is 2m weakly amenable for each integer m 1

wavelet representations of stochastic processes and multiresolution stochastic models deterministic signal analysis in a multiresolution framework through the use of wavelets has been extensively studied very successfully in recent years in the context of stochastic processes the use of wavelet bases has not yet been fully investigated we use compactly supported wavelets to obtain multiresolution representations of stochastic processes with paths in l sup 2 defined in the time domain we derive the correlation structure of the discrete wavelet coefficients of a stochastic process and give new results on how and when to obtain strong decay in correlation along time as well as across scales we study the relation between the wavelet representation of a stochastic process and multiresolution stochastic models on trees proposed by basseville et al see ieee trans inform theory vol 38 p 766 784 mar 1992 we propose multiresolution stochastic models of the discrete wavelet coefficients as approximations to the original time process these models are simple due to the strong decorrelation of the wavelet transform experiments show that these models significantly improve the approximation in comparison with the often used assumption that the wavelet coefficients are completely uncorrelated