preconditioned conjugate gradient methods for solving electromagnetic problems it is shown that appropriately chosen preconditioners can significantly improve the convergence rate of the conjugate gradient cg algorithm as applied to electromagnetic problems preconditioners are constructed for the problems of scattering from frequency selective surfaces and scattering from infinite cylinders

estimation and statistical bounds for three dimensional polar shapes in diffuse optical tomography voxel based reconstructions in diffuse optical tomography dot using a quadratic regularization functional tend to produce very smooth images due to the attenuation of high spatial frequencies this then causes difficulty in estimating the spatial extent and contrast of anomalous regions such as tumors given an assumption that the target image is piecewise constant we can employ a parametric model to estimate the boundaries and contrast of an inhomogeneity directly in this paper we describe a method to directly reconstruct such a shape boundary from diffuse optical measurements we parameterized the object boundary using a spherical harmonic basis and derived a method to compute sensitivities of measurements with respect to shape parameters we introduced a centroid constraint to ensure uniqueness of the combined shape center parameter estimate and a projected newton method was utilized to optimize the object center position and shape parameters simultaneously using the shape jacobian we also computed the cramer rao lower bound on the theoretical estimator accuracy given a particular measurement configuration object shape and level of measurement noise knowledge of the shape sensitivity matrix and of the measurement noise variance allows us to visualize the shape uncertainty region in three dimensions giving a confidence region for our shape estimate we have implemented our shape reconstruction method using a finite difference based forward model to compute the forward and adjoint fields reconstruction results are shown for a number of simulated target shapes and we investigate the problem of model order selection using realistic levels of measurement noise assuming a signal to noise ratio in the amplitude measurements of 40 db and a standard deviation in the phase measurements of 0 1deg we are able to estimate an object represented with an eighth order spherical harmonic model having an absorption contrast of 0 15 cm 1 and a volume of 4 82 cm3 with errors of less than 10 in object volume and absorption contrast we also investigate the robustness of our shape based reconstruction approach to a violation of the assumption that the medium is purely piecewise constant

flexible skew symmetric shape model for shape representation classification and sampling skewness of shape data often arises in applications e g medical image analysis and is usually overlooked in statistical shape models in such cases a gaussian assumption is unrealistic and a formulation of a general shape model which accounts for skewness is in order in this paper we present a novel statistical method for shape modeling which we refer to as the flexible skew symmetric shape model fssm the model is sufficiently flexible to accommodate a departure from gaussianity of the data and is fairly general to learn a mean shape template with a potential for classification and random generation of new realizations of a given shape robustness to skewness results from deriving the fssm from an extended class of flexible skew symmetric distributions in addition we demonstrate that the model allows us to extract principal curves in a point cloud the idea is to view a shape as a realization of a spatial random process and to subsequently learn a shape distribution which captures the inherent variability of realizations provided they remain with high probability within a certain neighborhood range around a mean specifically given shape realizations fssm is formulated as a joint bimodal distribution of angle and distance from the centroid of an aggregate of random points mean shape is recovered from the modes of the distribution while the maximum likelihood criterion is employed for classification

covering of a finite lattice by a random walk abstract we study the covering process by a simple random walk of a d dimensional periodic hypercubic lattice of n sites in d 1 the probability l n x for site x to be the last site visited in this covering process does not depend on x as long as x is not the starting point of the walk we argue that in dimensions d 2 the probability l n x approaches a constant value according to a coulomb law l n x 1 2 1 const x d 2 valid for x small on the scale n 1 d whereas it behaves logarithmically in d 2 also there is a dimension dependent characteristic time scale on which the last site is visited the structure of the set of sites not yet visited on this characteristic time scale is fractal like in d 2 in d 3 on the other hand this set is essentially distributed randomly through the lattice

asymptotic global confidence regions for 3 d parametric shape estimation in inverse problems this paper derives fundamental performance bounds for statistical estimation of parametric surfaces embedded in ropf3 unlike conventional pixel based image reconstruction approaches our problem is reconstruction of the shape of binary or homogeneous objects the fundamental uncertainty of such estimation problems can be represented by global confidence regions which facilitate geometric inference and optimization of the imaging system compared to our previous work on global confidence region analysis for curves two dimensional 2 d shapes computation of the probability that the entire surface estimate lies within the confidence region is more challenging because a surface estimate is an inhomogeneous random field continuously indexed by a 2 d variable we derive an asymptotic lower bound to this probability by relating it to the exceedence probability of a higher dimensional gaussian random field which can in turn be evaluated using the tube formula due to sun simulation results demonstrate the tightness of the resulting bound and the usefulness of the three dimensional global confidence region approach

riemann and weierstrass walks revisited abstract the weierstrass and riemann walks are non trivial discrete random processes to model and characterize the underlying noise in the dynamics of fluctuations for out of equilibrium systems and in more general contexts to simulate complex dynamics like order disorder phase transitions and anomalous diffusion properties in physical biological and financial systems in this work simple algorithms implemented in gnu r for both riemann and weierstrass discrete processes are presented explicit formulas for the probability distributions of n steps are obtained finally a way to connect both random processes is commented

nirs spm statistical parametric mapping for near infrared spectroscopy near infrared spectroscopy nirs is a non invasive method to measure brain activity via changes in the degree of hemoglobin oxygenation through the intact skull as optically measured hemoglobin signals strongly correlate with bold signals simultaneous measurement using nirs and fmri promises a significant mutual enhancement of temporal and spatial resolutions although there exists a powerful statistical parametric mapping tool in fmri current public domain statistical tools for nirs have several limitations related to the quantitative analysis of simultaneous recording studies with fmri in this paper a new public domain statistical toolbox known as nirs spm is described it enables the quantitative analysis of nirs signal more specifically nirs data are statistically analyzed based on the general linear model glm and sun s tube formula the p values are calculated as the excursion probability of an inhomogeneous random field on a representation manifold that is dependent on the structure of the error covariance matrix and the interpolating kernels nirs spm not only enables the calculation of activation maps of oxy deoxy hemoglobin and total hemoglobin but also allows for the super resolution localization which is not possible using conventional analysis tools extensive experimental results using finger tapping and memory tasks confirm the viability of the proposed method

on the survival probability of a random walk in a finite lattice with a single trap we consider the survival of a random walker in a finite lattice with periodic boundary conditions the initial position of the random walker is uniformly distributed on the lattice with respect to the trap we show that the survival of a random walker un can be exactly related to the expected number of distinct sites visted on a trap free lattice by un 1 sn nd wherend is the number of lattice points ind dimensions we then analyze the behavior of sn in any number of dimensions by using tauberian methods we find that at sufficiently long times sn decays exponentially withn in all numbers of dimensions ind 1 and 2 dimensions there is an intermediate behavior which can be calculated and is valid forn2 n 1 whend 1 andn lnn n 1 whend 2 no such crossover exists when z 3 the form of suggests that the single trap approximation is indeed a valid low concentration limit for survival on an infinite lattice with a finite concentration of traps

on hardly visited points of the brownian motion letl x t be the local time process of a standard wiener process w t t 0 denote begin gathered t r inf left t 0 l left 0 t right geqq r right hfill mu left r q right mu left y 0 l left y t r right q right hfill end gathered the almost sure behavior of the random measure r q is investigated

cramer rao bounds for parametric shape estimation in inverse problems we address the problem of computing fundamental performance bounds for estimation of object boundaries from noisy measurements in inverse problems when the boundaries are parameterized by a finite number of unknown variables our model applies to multiple unknown objects each with its own unknown gray level or color and boundary parameterization on an arbitrary known background while such fundamental bounds on the performance of shape estimation algorithms can in principle be derived from the cramer rao lower bounds very few results have been reported due to the difficulty of computing the derivatives of a functional with respect to shape deformation we provide a general formula for computing cramer rao lower bounds in inverse problems where the observations are related to the object by a general linear transform followed by a possibly nonlinear and noisy measurement system as an illustration we derive explicit formulas for computed tomography fourier imaging and deconvolution problems the bounds reveal that highly accurate parametric reconstructions are possible in these examples using severely limited and noisy data

electromagnetic wave scattering by an infinite plane metallic grating in case of oblique incidence and arbitrary polarization a novel spectral domain method combined with a sampling theorem is applied to the rigorous analysis of this quasi two dimensional problem the accuracy and convergence of the method are examined numerically taking into account errors in comparison with extrapolated values errors in the power relation and the duality relationship it is found that the numerical results exhibit good convergence another formulation related to babinet s principle is discussed and a relationship between the fields of an original and its complementary gratings is demonstrated numerical calculations are carried out showing that the method provides precise numerical results not only for the far fields such as transmitted and reflected powers but also for the near fields such as surface current distribution on a strip and electric field intensity in a slit some numerical results related to the dual problem are provided

numerical evidence against a conjecture on the cover time of planar graphs we investigate a conjecture on the cover times of planar graphs by means of large monte carlo simulations the conjecture states that the cover time g n of a planar graph g n of n vertices and maximal degree d is lower bounded by g n c d n lnn 2 with c d d 4 tan d with equality holding for some geometries we tested this conjecture on the regular honeycomb d 3 regular square d 4 regular elongated triangular d 5 and regular triangular d 6 lattices as well as on the nonregular union jack lattice dmin 4 dmax 8 indeed the monte carlo data suggest that the rigorous lower bound may hold as an equality for most of these lattices with an interesting issue in the case of the union jack lattice the data for the honeycomb lattice however violate the bound with the conjectured constant the empirical probability distribution function of the cover time for the square lattice is also briefly presented since very little is known about cover time probability distribution functions in general

the range of stable random walks limit theorems are proved for the range of d dimensional random walks in the domain of attraction of a stable process of index p3 the range rn is the number of distinct sites of zd visited by the random walk before time n our results depend on the value of the ratio 3 d the most interesting results are obtained for 2 3 1 which can only occur if d 1 we prove the convergence in distribution of rn e rn toward some constant times the lebesgue measure of the range of the limiting stable process some of our results require regularity assumptions on the characteristic function of x

a generating function for the second moment of the distinct number of sites visited by an n step lattice random walk we derive a generating function for the second moment of the distinct number of sites visited by an n step lattice random walk the formalism allows us to find asymptotic forms for the second moment as previously given in the mathematical literature using much more complicated techniques the general technique can in principle be utilized in a derivation of higher moments

spectral domain analysis of circular patch frequency selective surfaces a computationally efficient method is presented for analyzing the scattering from frequency selective surfaces fss comprised of circular metal patches the formulation is carried out in the spectral domain where the convolution form of the integral equation for the induced current reduces to an algebraic one and the spectral galerkin technique is used to solve the resulting equation entire domain basis functions that satisfy the edge condition are introduced to expand the unknown induced current on the metal patches calculated results using this procedure show good agreement with data reported by other authors

fluctuation results for the wiener sausage on demontre des resultats de fluctuation pour le volume de la saucisse de wiener associe au mouvement brownien a d dimensions et a un ensemble compact de capacite positive

the aspherity of random walks the asphericity a of a random walk is defined and calculated for non self avoiding walks the definition of a generalises to self avoiding walks percolating clusters and other fractal objects and thus provides a generalised quantitative measure of the departure from spherical symmetry of the gross shape of these objects

statistics of the one dimensional riemann walk the riemann walk is the lattice version of the levy flight for the one dimensional riemann walk of levy exponent 0 2 we study the statistics of the support i e set of visited sites after t steps we consider a wide class of support related observables m t including the number s t of visited sites and the number i t of sequences of adjacent visited sites for t we obtain the asymptotic power laws for the averages variances and correlations of these observables logarithmic correction factors appear for 2 3 and 1 bulk and surface observables have different power laws for 1 2 fluctuations are shown to be universal for 2 3 2 this means that in the limit t the deviations from average m t m t m 0304 t 0304 are fully described either by a single m independent stochastic process when 2 3 1 or by two such processes one for the bulk and one for the surface observables when 1 2

two dimensional polymer networks near percolation we report an extensive finite size study of polymer networks near the percolation threshold using numerical techniques the polymers are modeled by random walks occupying the bonds of a two dimensional square lattice we measure the percolation threshold and critical exponents of the networks for various polymer lengths we find that the critical occupation probability is a decreasing function of the polymer length and the percolation of polymers with a fixed polymer length belongs to the same universality class as ordinary bond percolation by adding particles to the lattice cells we can study the diffusion process in polymer networks measuring the current near percolation we observe that its critical exponent is also independent of polymer length

the detection of local shape changes via the geometry of hotelling s t 2 fields this paper is motivated by the problem of detecting local changes or differences in shape between two samples of objects via the nonlinear deformations required to map each object to an atlas standard local shape changes are then detected by high values of the random field of hotelling s t 2 statistics for detecting a change in mean of the vector deformations at each point in the object to control the null probability of detecting a local shape change we use the recent result of adler that the probability that a random field crosses a high threshold is very accurately approximated by the expected euler characteristic ec of the excursion set of the random field above the threshold we give an exact expression for the expected ec of a hotelling s t 2 field and we study the behavior of the field near local extrema this extends previous results for gaussian random fields by adler and x 2 t and f fields by worsley and cao for illustration these results are applied to the detection of differences in brain shape between a sample of 29 males and 23 females

statistical bias of conic fitting and renormalization introducing a statistical model of noise in terms of the covariance matrix of the n vector we point out that the least squares conic fitting is statistically biased we present a new fitting scheme called renormalization for computing an unbiased estimate by automatically adjusting to noise relationships to existing methods are discussed and our method is tested using real and synthetic data

the stability of nonlinear least squares problems and the cramer rao bound a number of problems of interest in signal processing can be reduced to nonlinear parameter estimation problems the traditional approach to studying the stability of these estimation problems is to demonstrate finiteness of the cramer rao bound crb for a given noise distribution we review an alternate deterministic notion of stability for the associated nonlinear least squares nls problem from the realm of nonlinear programming i e that the global minimizer of the least squares problem exists and varies smoothly with the noise furthermore we show that under mild conditions identifiability of the parameters along with a finite crb for the case of gaussian noise is equivalent to the deterministic stability of the nls problem finally we demonstrate the application of our result which is general to the problems of multichannel blind deconvolution and sinusoid retrieval to generate new stability results for these problems with little additional effort

study of the shape of random walks ii inertia moment ratios and the two dimensional asphericity a study of some features of the quantities used to describe the shape of random walks focusing on the probability distribution of the asphericity of discrete random walks in two dimensions is presented a connection is established between the asphericity distribution and the probability distribution corresponding to the ratio of the principal inertia moments the probability distributions of arbitrary inertia moment ratios are analysed for varying spatial dimension and an analytic expression for them is found which presents excellent agreement with monte carlo data in all the cases considered this function is then used to obtain the corresponding two dimensional asphericity distribution which also proved to be a very good approximation to the true distribution obtained from an independent monte carlo simulation

in a random walk the number of unique experiences is two on the average for any simple random walk of length n 0 we consider the number u of places which are reached once and only once the remarkable fact which we alluded to in our title is that the expected value of u call it e n is always equal to 2 independent ofn

cramer rao bounds for 2 d target shape estimation in nonlinear inverse scattering problems with application to passive radar we present new methods for computing fundamental performance limits for two dimensional 2 d parametric shape estimation in nonlinear inverse scattering problems with an application to passive radar imaging we evaluate cramer rao lower bounds crb on shape estimation accuracy using the domain derivative technique from nonlinear inverse scattering theory the crb provides an unbeatable performance limit for any unbiased estimator and under fairly mild regularity conditions is asymptotically achieved by the maximum likelihood estimator mle the resultant crbs are used to define an asymptotic global confidence region centered around the true boundary in which the boundary estimate lies with a prescribed probability these global confidence regions conveniently display the uncertainty in various geometric parameters such as shape size orientation and position of the estimated target and facilitate geometric inferences numerical simulations are performed using the layer approach and the nystrom method for computation of domain derivatives and using fourier descriptors for target shape parameterization this analysis demonstrates the accuracy and generality of the proposed methods

wiener sausage volume moments the statistical characteristics of a spatial region visited by a spherical brownian particle during timet wiener sausage are investigated the expectation value and dispersion of this quantity are obtained for a space of arbitrary dimension in the one dimensional case the distribution of probability density and the moments of any order are determined for this quantity

lower bounds for parametric estimation with constraints a chapman robbins form of the barankin bound is used to derive a multiparameter cramer rao cr type lower bound on estimator error covariance when the parameter theta in r sup n is constrained to lie in a subset of the parameter space a simple form for the constrained cr bound is obtained when the constraint set theta sub c can be expressed as a smooth functional inequality constraint it is shown that the constrained cr bound is identical to the unconstrained cr bound at the regular points of theta sub c i e where no equality constraints are active on the other hand at those points theta in theta sub c where pure equality constraints are active the full rank fisher information matrix in the unconstrained cr bound must be replaced by a rank reduced fisher information matrix obtained as a projection of the full rank fisher matrix onto the tangent hyperplane of the full rank fisher matrix onto the tangent hyperplane of the constraint set at theta a necessary and sufficient condition involving the forms of the constraint and the likelihood function is given for the bound to be achievable and examples for which the bound is achieved are presented in addition to providing a useful generalization of the cr bound the results permit analysis of the gain in achievable mse performance due to the imposition of particular constraints on the parameter space without the need for a global reparameterization

variance of the range of a random walk tn the expectation of the square of the number of distinct sites occupied by a random walk in steps 1 throughn is obtained from its relation to the dual first occupancy probabilityfij x x and the latter quantity is obtained from a recursion with the first occupancy probabilityf k x the variancevn of the number of distinct sites occupied is calculated directly from tn the procedure is illustrated by the calculation ofvn 4096 n and the derivation of asymptotic expansions forvn for a particular random walk in dimensions 1 through 3

recovering convex edges of an image from noisy tomographic data we consider the problem of recovering edges of an image from noisy tomographic data the original image is assumed to have a discontinuity jump edge along the boundary of a compact convex set the radon transform of the image is observed with noise and the problem is to estimate the edge we develop an estimation procedure which is based on recovering the support function of the edge it is shown that the proposed estimator is nearly optimal in order in a minimax sense numerical examples illustrate reasonable practical behavior of the estimation procedure

universal fluctuations in the support of the random walk a random walk starts from the origin of ad dimensional lattice the occupation numbern x t equals unity if aftert steps site x has been visited by the walk and zero otherwise we study translationally invariant sumsm t of observables defined locally on the field of occupation numbers examples are the numbers t of visited sites the areae t of the appropriately defined surface of the set of visited sites and in dimension d 3 the euler index of this surface ind 3 theaverages m t all increase linearly witht ast we show that in d 3 to leading order in an asymptotic expansion int thedeviations from average m t m t m t are up to a normalization allidentical to a single universal random variable this result resembles an earlier one in dimensiond 2 we show that this universality breaks down ford 3

statistical properties of the set of sites visited by the two dimensional random walk we study the support i e the set of visited sites of a t step random walk on a two dimensional square lattice in the large t limit a broad class of global properties m t of the support is considered including for example the number s t of its sites the length of its boundary the number of islands of unvisited sites that it encloses the number of such islands of given shape size and orientation and the number of occurrences in space of specific local patterns of visited and unvisited sites on a finite lattice we determine the scaling functions that describe the averages on appropriate lattice size dependent time scales on an infinite lattice we first observe that the all increase with t as where k is an m dependent positive integer we then consider the class of random processes constituted by the fluctuations around average we show that to leading order as t gets large these fluctuations are all proportional to a single universal random process normalized to for the probability law of tends to that of varadhan s renormalized local time of self intersections an implication is that in the long time limit all are proportional to

the excited random walk in one dimension we study the excited random walk in which a walk that is at a site that contains cookies eats one cookie and then hops to the right with probability p and to the left with probability q 1 p if the walk hops onto an empty site there is no bias for the 1 excited walk on the half line one cookie initially at each site the probability of first returning to the starting point at time t scales as t 2 p although the average return time to the origin is infinite for all p the walk eats on average only a finite number of cookies until this first return when p 3 4 where the walk is transient including a mean displacement that grows as t with 1 dependent on p and a breakdown of scaling for the probability distribution of the walk

law of the iterated logarithm for the range of random walks in two and three dimensions let s n be a random walk in z d and let r n be the range of s n we prove an almost sure invariance principle for r n when d 3 and a law of the iterated logarithm for r n when d 2

how a random walk covers a finite lattice abstract a random walker is confined to a finite periodic d dimensional lattice of n initially white sites when visited by the walk a site is colored black after t steps of the walk for t scaled appropriately with n we determined the structure of the set of white sites the variance of their number has a line of critical points in the td plane which separates a mean field region from a region with enhanced fluctuations at d 2 the critical point becomes a critical interval moreover for d 2 the set of white sites is fractal with a fractal dimensionality whose t dependence we determine

the fluctuation result for the multiple point range of two dimensional recurrent random walks we study the fluctuation problem for the multiple point range of random walks in the two dimensional integer lattice with mean 0 and finite variance the p multiple point range means the number of distinct sites with multiplicity p of random walk paths before time n the suitably normalized multiple point range is proved to converge to a constant which is independent of the multiplicity multiple of the renormalized self intersection local time of a planar brownian motion

full wave analysis of layered aperture arrays the rigorous numerical analytical approach has been developed and used for study of the periodic aperture arrays containing multiple dissimilar apertures of complex shapes in stratified medium the analysis method is based on the solution of the system of integro differential equations by galerkin method with the entire domain basis functions explicitly accounting for the field singularities at the edges of the whole aperture except corners an analytical treatment of the green s function convolutions with the basis and test functions provides an efficient regularization and rapid uniform convergence of the solutions thickness of conducting screen has been taken into account with the aid of the approximation for effective aperture dimensions the characteristics of the quasioptical slot arrays simulated by the developed method are in excellent agreement with the results of measurements and alternative simulations an extensive parametric study of several aperture array configurations has been performed to explore properties of the single layer array arrangements with multiple apertures of complex shapes in a unit cell and stacked layered arrays effect of the aperture shape on the array performance and the physical mechanisms underlying the transverse magnetic wave resonances and luebbers anomaly in the two layer arrays are discussed in detail

repeated bond traversal probabilities for the simple random walk we consider the average number bm t of bonds traversed exactly m times by a t step simple random walk we determine bm t explicitly in the scaling limit t with m t fixed in dimension d 1 and m log t fixed in dimension d 2 the scaling function is an erfc in d 1 and an exponential in d 2

estimating the number of peaks in a random field using the hadwiger characteristic of excursion sets with applications to medical images excursion sets has been studied by adler who defined the ig integral geometry characteristic of excursion sets as a multidimensional analogue of the number of upcrossings of the threshold by a unidimensional process the ig characteristic equals the euler characteristic of an excursion set provided that the set does not touch the boundary of the volume and adler found the expected ig characteristic for a stationary random field inside a fixed volume worsley evans marrett and neelin used the ig characteristic as an estimator of the number of regions of activation of positron emission tomography pet images of blood flow in the brain unfortunately the ig characteristic is only defined on intervals it is not invariant under rotations and it can fail to count connected regions that touch the boundary this is important since activation often occurs in the cortical regions near the boundary of the brain in this paper we study the hadwiger characteristic which is defined on arbitrary sets is invariant under rotations and does count connected regions whether they touch the boundary or not our main result is a simple expression for the expected hadwiger characteristic for an isotropic stationary random field in two and three dimensions and on a smooth surface embedded in three dimensions results are applied to pet studies of pain perception and word recognition

asymptotic global confidence regions in parametric shape estimation problems we introduce confidence region techniques for analyzing and visualizing the performance of two dimensional parametric shape estimators assuming an asymptotically normal and efficient estimator for a finite parameterization of the object boundary cramer rao bounds are used to define an asymptotic confidence region centered around the true boundary computation of the probability that an entire boundary estimate lies within the confidence region is a challenging problem because the estimate is a two dimensional nonstationary random process we derive lower bounds on this probability using level crossing statistics the same bounds also apply to asymptotic confidence regions formed around the estimated boundaries lower bounding the probability that the entire true boundary lies within the confidence region the results make it possible to generate asymptotic confidence regions for arbitrary prescribed probabilities these asymptotic global confidence regions conveniently display the uncertainty in various geometric parameters such as shape size orientation and position of the estimated object and facilitate geometric inferences numerical simulations suggest that the new bounds are quite tight

random walks with spontaneous emission on lattices with periodically distributed imperfect traps abstract we study random walks on d dimensional lattices with periodically distributed traps in which the walker has a finite probability per step of disappearing from the lattice and a finite probability of escaping from a trap general expressions are derived for the total probability that the walk ends in a trap and for the moments of the number of steps made before this happens if it does happen the analysis is extended to lattices with more types of traps and to a model where the trapping occurs during special steps finally the green s function at the origin g 0 z for a finite lattice with periodic boundary conditions which enters into the main expressions is studied more closely a generalization of an expression for g 0 1 for the square lattice given by montroll to values of z different from but close to 1 is derived

a field theoretic approach to the wiener sausage the wiener sausage the volume traced out by a sphere attached to a brownian particle is a classical problem in statistics and mathematical physics initially motivated by a range of field theoretic technical questions we present a single loop renormalised perturbation theory of a stochastic process closely related to the wiener sausage which however proves to be exact for the exponents and some amplitudes the field theoretic approach is particularly elegant and very enjoyable to see at work on such a classic problem while we recover a number of known classical results the field theoretic techniques deployed provide a particularly versatile framework which allows easy calculation with different boundary conditions even of higher momenta and more complicated correlation functions at the same time we provide a highly instructive non trivial example for some of the technical particularities of the field theoretic description of stochastic processes such as excluded volume lack of translational invariance and immobile particles the aim of the present work is not to improve upon the well established results for the wiener sausage but to provide a field theoretic approach to it in order to gain a better understanding of the field theoretic obstacles to overcome