subspace averaging and order determination for source enumeration in this paper we address the problem of subspace averaging with special emphasis placed on the question of estimating the dimension of the average the results suggest that the enumeration of sources in a multi sensor array which is a problem of estimating the dimension of the array manifold and as a consequence the number of radiating sources may be cast as a problem of averaging subspaces this point of view stands in contrast to conventional approaches which cast the problem as one of identifiying covariance models in a factor model we present a robust formulation of the proposed order fitting rule based on majorization minimization algorithms a key element of the proposed method is to construct a bootstrap procedure based on a newly proposed discrete distribution on the manifold of projection matrices for stochastically generating subspaces from a function of experimentally determined eigenvalues in this way the proposed subspace averaging sa technique determines the order based on the eigenvalues of an average projection matrix rather than on the likelihood of a covariance model penalized by functions of the model order by means of simulation examples we show that the proposed sa criterion is especially effective in high dimensional scenarios with low sample support

a bayesian approach to geometric subspace estimation this paper presents a geometric approach to estimating subspaces as elements of the complex grassmann manifold with each subspace represented by its unique complex projection matrix variation between the subspaces is modeled by rotating their projection matrices via the action of unitary matrices elements of the unitary group u n subspace estimation or tracking then corresponds to inferences on u n taking a bayesian approach a posterior density is derived on u n and certain expectations under this posterior are empirically generated for the choice of the hilbert schmidt norm on u n to define estimation errors an optimal mmse estimator is derived it is shown that this estimator achieves a lower bound on the expected squared errors associated with all possible estimators the estimator and the bound are computed using metropolis adjusted langevin s diffusion algorithm for sampling from the posterior for use in subspace tracking a prior model on subspace rotation that utilizes newtonian dynamics is suggested

intrinsic inference on the mean geodesic of planar shapes and tree discrimination by leaf growth for planar landmark based shapes taking into account the non euclidean geometry of the shape space a statistical test for a common mean first geodesic principal component gpc is devised which rests on one of two asymptotic scenarios for both scenarios strong consistency and central limit theorems are established along with an algorithm for the computation of a ziezold mean geodesic in application this allows to verify the geodesic hypothesis for leaf growth of canadian black poplars and to discriminate genetically different trees by observations of leaf shape growth over brief time intervals with a test based on procrustes tangent space coordinates not involving the shape space s curvature neither can be achieved

principal geodesic analysis for the study of nonlinear statistics of shape a primary goal of statistical shape analysis is to describe the variability of a population of geometric objects a standard technique for computing such descriptions is principal component analysis however principal component analysis is limited in that it only works for data lying in a euclidean vector space while this is certainly sufficient for geometric models that are parameterized by a set of landmarks or a dense collection of boundary points it does not handle more complex representations of shape we have been developing representations of geometry based on the medial axis description or m rep while the medial representation provides a rich language for variability in terms of bending twisting and widening the medial parameters are not elements of a euclidean vector space they are in fact elements of a nonlinear riemannian symmetric space in this paper we develop the method of principal geodesic analysis a generalization of principal component analysis to the manifold setting we demonstrate its use in describing the variability of medially defined anatomical objects results of applying this framework on a population of hippocampi in a schizophrenia study are presented

covariance subspace and intrinsic crame spl acute r rao bounds crame spl acute r rao bounds on estimation accuracy are established for estimation problems on arbitrary manifolds in which no set of intrinsic coordinates exists the frequently encountered examples of estimating either an unknown subspace or a covariance matrix are examined in detail the set of subspaces called the grassmann manifold and the set of covariance positive definite hermitian matrices have no fixed coordinate system associated with them and do not possess a vector space structure both of which are required for deriving classical crame spl acute r rao bounds intrinsic versions of the crame spl acute r rao bound on manifolds utilizing an arbitrary affine connection with arbitrary geodesics are derived for both biased and unbiased estimators in the example of covariance matrix estimation closed form expressions for both the intrinsic and flat bounds are derived and compared with the root mean square error rmse of the sample covariance matrix scm estimator for varying sample support k the accuracy bound on unbiased covariance matrix estimators is shown to be about 10 log 10 n k sup 1 2 db where n is the matrix order remarkably it is shown that from an intrinsic perspective the scm is a biased and inefficient estimator and that the bias term reveals the dependency of estimation accuracy on sample support observed in theory and practice the rmse of the standard method of estimating subspaces using the singular value decomposition svd is compared with the intrinsic subspace crame spl acute r rao bound derived in closed form by varying both the signal to noise ratio snr of the unknown p dimensional subspace and the sample support in the simplest case the crame spl acute r rao bound on subspace estimation accuracy is shown to be about p n p sup 1 2 1 2 snr sup 1 2 rad for p dimensional subspaces it is seen that the svd based method yields accuracies very close to the crame spl acute r rao bound establishing that the principal invariant subspace of a random sample provides an excellent estimator of an unknown subspace the analysis approach developed is directly applicable to many other estimation problems on manifolds encountered in signal processing and elsewhere such as estimating rotation matrices in computer vision and estimating subspace basis vectors in blind source separation

statistics of toppling wave boundaries in deterministic and stochastic sandpile models we study numerically the statistics of curves which form the boundaries of toppling wave clusters in the deterministic bak tang and wiesenfeld sandpile model and stochastic manna model on a square lattice we consider the abelian version of each model multiple tests show that the boundary of toppling wave clusters in both deterministic and stochastic models can be described by sle curves with diffusivity 2

the complex watson distribution and shape analysis the complex watson distribution is an important simple distribution on the complex sphere which is used in statistical shape analysis we describe the density obtain the integrating constant and provide large sample approximations maximum likelihood estimation and hypothesis testing procedures for one and two samples are described the particular connection with shape analysis is discussed and we consider an application examining shape differences between normal and schizophrenic brains we make some observations about bayesian shape inference and finally we describe a more general rotationally symmetric family of distributions

height correlations in the abelian sandpile model the authors study the distribution of heights in the self organized critical state of the abelian sandpile model on a d dimensional hypercubic lattice they calculate analytically the concentration of sites having minimum allowed value in the critical state they also calculate in the critical state the probability that the heights at two sites separated by a distance r would both have minimum values and show that the lowest order r dependent term in it varies as r 2d for large r

a weighted procrustes criterion the procrustes criterion is a common measure for the distance between two matricesx andy and can be interpreted as the sum of squares of the euclidean distances between their respective column vectors often a weighted procrustes criterion using for example a weighted sum of the squared distances between the column vectors is called for this paper describes and analyzes the performance of an algorithm for rotating a matrixx such that the column weighted procrustes distance toy is minimized the problem of rotatingx intoy such that an aggregate measure of tucker s coefficient of congruence is maximized is also discussed

non equilibrium critical behaviour on fractal lattices a simple one component surface reaction model exhibiting a critical point has been studied on deterministic fractal lattices with fractal dimension 1 df 2 results from time dependent simulations show a change in the value of the critical exponents with df two of the exponents interpolate very nicely between the values in one and two dimensions

surface bounded growth modeling applied to human mandibles from a set of longitudinal three dimensional scans of the same anatomical structure the authors have accurately modeled the temporal shape and size changes using a linear shape model on a total of 31 computed tomography scans of the mandible from six patients 14 851 semilandmarks are found automatically using shape features and a new algorithm called geometry constrained diffusion the semilandmarks are mapped into procrustes space principal component analysis extracts a one dimensional subspace which is used to construct a linear growth model the worst case mean modeling error in a cross validation study is 3 7 mm

self organized criticality and the lattice topology abstract we examine exhaustively the behavior of avalanches in critical height sandpile models based in two and three dimensional lattices of various topologies we get that for two dimensional lattices the spatial and temporal distributions characterizing bulk avalanches do not depend on the lattice topology for the three dimensional case we detect a small dependence of the topology for the temporal distribution while the spatial ones are independent the two dimensional lattices studied are the plane r 2 the cylinder s 1 r and the mobius strip m and the three dimensional are r 3 s 1 r 2 s 1 s 1 r m r s 2 r k r and rp r where k and rp are respectively the klein bottle and the real projective plane

parameter orthogonality and approximate conditional inference on propose une statistique du rapport de vraisemblance construite a partir de la distribution conditionnelle des observations etant donne les estimateurs du maximum de vraisemblance pour les parametres de nuisance

kronecker products unitary matrices and signal processing applications discrete unitary transforms are extensively used in many signal processing applications and in the development of fast algorithms kronecker products have proved quite useful in this semitutorial paper we briefly review properties of kronecker products and direct sums of matrices which provide a compact notation in treating patterned matrices a generalized matrix product which inherits some useful algebraic properties from the standard kronecker product and allows a large class of discrete unitary transforms to be generated from a single recursion formula is then introduced the notation is intimately related to sparse matrix factorizations and examples are included illustrating the utility of the new notation in signal processing applications finally some novel characteristics of hadamard transforms and polyadic permutations are derived in the framework of kronecker products

geometry and multiple direction estimation abstract we consider the problem of the estimation of multiple target directions given the output of a phased array of identical elements the natural mathematical setting for this problem is the grassmannian manifold in order to understand the problems considered here a knowledge of exterior forms lie groups and differential geometry is essential we review some of the essential ideas relevant to the problem including the maurer cartan equations algebraic varieties in projective space and the invariant volume integral for symmetric spaces our purpose is to study probability measures on grassmannians in order to pose and solve the bayes problem on grassmannians

continuous generalized procrustes analysis two dimensional shape models have been successfully applied to solve many problems in computer vision such as object tracking recognition and segmentation typically 2d shape models are learned from a discrete set of image landmarks corresponding to projection of 3d points of an object after applying generalized procustes analysis gpa to remove 2d rigid transformations however the standard gpa process suffers from three main limitations firstly the 2d training samples do not necessarily cover a uniform sampling of all the 3d transformations of an object this can bias the estimate of the shape model secondly it can be computationally expensive to learn the shape model by sampling 3d transformations thirdly standard gpa methods use only one reference shape which can might be insufficient to capture large structural variability of some objects to address these drawbacks this paper proposes continuous generalized procrustes analysis cgpa cgpa uses a continuous formulation that avoids the need to generate 2d projections from all the rigid 3d transformations it builds an efficient in space and time non biased 2d shape model from a set of 3d model of objects a major challenge in cgpa is the need to integrate over the space of 3d rotations especially when the rotations are parameterized with euler angles to address this problem we introduce the use of the haar measure finally we extended cgpa to incorporate several reference shapes experimental results on synthetic and real experiments show the benefits of cgpa over gpa highlightscontinuous formulation of the generalized procrustes analysis cgpa avoids the need to generate 2d projections from all 3d rigid transformations cgpa builds an efficient non biased 2d shape model from 3d objects cgpa uses the haar measure to integrate over the space of 3d rotations experimental results building 2d shape models for different public datasets

introduction to the sandpile model abstract this article is based on a talk given by one of us evi at the conference statphys taipei 1997 it overviews the exact results in the theory of the sandpile model and discusses briefly the yet unsolved problem of calculation of avalanche distribution exponents the key ingredients include the analogy with the critical reaction diffusion system the spanning tree representation of height configurations and the decomposition of the avalanche process into waves of topplings

shape appearance correlated active appearance model among the challenges faced by current active shape or appearance models facial feature localization in the wild with occlusion in a novel face image i e in a generic environment is regarded as one of the most difficult computer vision tasks in this paper we propose an active appearance model aam to tackle the problem of generic environment firstly a fast face model initialization scheme is proposed based on the idea that the local appearance of feature points can be accurately approximated with locality constraints nearest neighbors which have similar poses and textures to a test face are retrieved from a training set for constructing the initial face model to further improve the fitting of the initial model to the test face an orthogonal cca occa is employed to increase the correlation between shape features and appearance features represented by principal component analysis pca with these two contributions we propose a novel aam namely the shape appearance correlated aam sac aam and the optimization is solved by using the recently proposed fast simultaneous inverse compositional fast sic algorithm experiment results demonstrate a 5 10 improvement on controlled and semi controlled datasets and with around 10 improvement on wild face datasets in terms of fitting accuracy compared to other state of the art aam models a fast face model initialization scheme is proposed for aams orthogonal cca is employed to increase the correlation between shape and appearance a shape appearance correlated aam is proposed to tackle the generic problem of localizing feature points on unseen faces

effects of bulk dissipation on the critical exponents of a sandpile bulk dissipation of a sandpile on a square lattice with the periodic boundary condition is investigated through a dissipating probability f during each toppling process we find that the power law behavior is broken for f 10 1 and not evident for 10 1 f 10 2 in the range 10 2 or f or 10 5 numerical simulations for the toppling size exponents of all dissipative and last waves have been studied two kinds of definitions for exponents are considered the exponents obtained from the direct fitting of data and the exponents defined by the simple scaling our result shows that the exponents from these two definitions may be different furthermore we propose analytic expressions of the exponents for the direct fitting and it is consistent with the numerical result finally we point out that small dissipation drives the behavior of this model toward the simple scaling

large sample theory of intrinsic and extrinsic sample means on manifolds ii this article develops nonparametric inference procedures for estimation and testing problems for means on manifolds a central limit theorem for frechet sample means is derived leading to an asymptotic distribution theory of intrinsic sample means on riemannian manifolds central limit theorems are also obtained for extrinsic sample means w r t an arbitrary embedding of a differentiable manifold in a euclidean space bootstrap methods particularly suitable for these problems are presented applications are given to distributions on the sphere s d directional spaces real projective space rp n 1 axial spaces complex projective space cp k 2 planar shape spaces w r t veronese whitney embeddings and a three dimensional shape space 4 3

procrustean normal distribution for non rigid structure from motion a well defined deformation model can be vital for non rigid structure from motion nrsfm most existing methods restrict the deformation space by assuming a fixed rank or smooth deformation which are not exactly true in the real world and they require the degree of deformation to be predetermined which is impractical meanwhile the errors in rotation estimation can have severe effects on the performance i e these errors can make a rigid motion be misinterpreted as a deformation in this paper we propose an alternative to resolve these issues motivated by an observation that non rigid deformations excluding rigid changes can be concisely represented in a linear subspace without imposing any strong constraints such as smoothness or low rank this observation is embedded in our new prior distribution the procrustean normal distribution pnd which is a shape distribution exclusively for non rigid deformations because of this unique characteristic of the pnd rigid and non rigid changes can be strictly separated which leads to better performance the proposed algorithm em pnd fits a pnd to given 2d observations to solve nrsfm without any user determined parameters the experimental results show that em pnd gives the state of the art performance for the benchmark data sets confirming the adequacy of the new deformation model

tangent bundle maps on the grassmann manifold application to empirical arithmetic averaging the present paper elaborates on tangent bundle maps on the grassmann manifold with application to subspace arithmetic averaging in particular the present contribution elaborates on the work about retraction lifting maps devised for the stiefel manifold in the recently published paper t kaneko s fiori and t tanaka empirical arithmetic averaging over the compact stiefel manifold ieee trans signal process vol 61 no 4 pp 883 894 february 2013 and discusses the extension of such maps to the grassmann manifold tangent bundle maps are devised on the basis of the thin qr matrix decomposition the polar matrix decomposition and the exponential map also tangent bundle pseudo maps based on the matrix cayley transform are devised theoretical and numerical comparisons about the devised tangent bundle maps are performed in order to get an insight into their relative merits and demerits with special emphasis to their computational burden the averaging algorithm based on the thin qr decomposition maps stands out as it exhibits the best trade off between numerical precision and computational burden such algorithm is further compared with two grassmann averaging algorithms drawn from the scientific literature on an handwritten digits recognition data set the thin qr tangent bundle maps based algorithm exhibits again numerical features that make it preferable over such algorithms

maximum a posteriori estimation of linear shape variation with application to vertebra and cartilage modeling the estimation of covariance matrices is a crucial step in several statistical tasks especially when using few samples of a high dimensional representation of shapes the standard maximum likelihood estimation ml of the covariance matrix can be far from the truth is often rank deficient and may lead to unreliable results in this paper we discuss regularization by prior knowledge using maximum a posteriori map estimates we compare ml to map using a number of priors and to tikhonov regularization we evaluate the covariance estimates on both synthetic and real data and we analyze the estimates influence on a missing data reconstruction task where high resolution vertebra and cartilage models are reconstructed from incomplete and lower dimensional representations our results demonstrate that our methods outperform the traditional ml method and tikhonov regularization

learning mixtures of point distribution models with the em algorithm this paper demonstrates how the em algorithm can be used for learning and matching mixtures of point distribution models we make two contributions first we show how shape classes can be learned in an unsupervised manner we present a fast procedure for training point distribution models using the em algorithm rather than estimating the class means and covariance matrices needed to construct the pdm the method iteratively refines the eigenvectors of the covariance matrix using a gradient ascent technique second we show how recognition by alignment can be realised by fitting a mixture of linear shape deformations we evaluate the method on the problem of learning the class structure and recognising arabic characters

a monotonically convergent algorithm for orthogonal congruence rotation brokken has proposed a method for orthogonal rotation of one matrix such that its columns have a maximal sum of congruences with the columns of a target matrix this method employs an algorithm for which convergence from every starting point is not guaranteed in the present paper an iterative majorization algorithm is proposed which is guaranteed to converge from every starting point specifically it is proven that the function value converges monotonically and that the difference between subsequent iterates converges to zero in addition to the better convergence properties another advantage of the present algorithm over brokken s one is that it is easier to program the algorithms are compared on 80 simulated data sets and it turned out that the new algorithm performed well in all cases whereas brokken s algorithm failed in almost half the cases the derivation of the algorithm is given in full detail because it involves a series of inequalities that can be of use to derive similar algorithms in different contexts

application of the fourier pseudospectral time domain method in orthogonal curvilinear coordinates for near rigid moderately curved surfaces the fourier pseudospectral time domain method is an efficient wave based method to model sound propagation in inhomogeneous media one of the limitations of the method for atmospheric sound propagation purposes is its restriction to a cartesian grid confining it to staircase like geometries a transform from the physical coordinate system to the curvilinear coordinate system has been applied to solve more arbitrary geometries for applicability of this method near the boundaries the acoustic velocity variables are solved for their curvilinear components the performance of the curvilinear fourier pseudospectral method is investigated in free field and for outdoor sound propagation over an impedance strip for various types of shapes accuracy is shown to be related to the maximum grid stretching ratio and deformation of the boundary shape and computational efficiency is reduced relative to the smallest grid cell in the physical domain the applicability of the curvilinear fourier pseudospectral time domain method is demonstrated by investigating the effect of sound propagation over a hill in a nocturnal boundary layer with the proposed method accurate and efficient results for sound propagation over smoothly varying ground surfaces with high impedances can be obtained

a flag representation for finite collections of subspaces of mixed dimensions abstract given a finite set of subspaces of r n perhaps of differing dimensions we describe a flag of vector spaces i e a nested sequence of vector spaces that best represents the collection based on a natural optimization criterion and we present an algorithm for its computation the utility of this flag representation lies in its ability to represent a collection of subspaces of differing dimensions when the set of subspaces all have the same dimension d the flag mean is related to several commonly used subspace representations for instance the d dimensional subspace in the flag corresponds to the extrinsic manifold mean when the set of subspaces is both well clustered and equidimensional of dimension d then the d dimensional component of the flag provides an approximation to the karcher mean an intermediate matrix used to construct the flag can also be used to recover the canonical components at the heart of multiset canonical correlation analysis two examples utilizing the carnegie mellon university pose illumination and expression database cmu pie serve as visual illustrations of the algorithm

extrinsic mean of brownian distributions on compact lie groups this paper studies brownian distributions on compact lie groups these are defined as the marginal distributions of brownian processes and are intended as a natural extension of the well known normal distributions to compact lie groups it is shown that this definition preserves key properties of normal distributions in particular brownian distributions transform in a nice way under group operations and satisfy an extension of the central limit theorem brownian distributions on a compact lie group g belong to one of two parametric families nl g c and nr g c g g and c a positive definite symmetric matrix in particular the parameter g appears as a location parameter an approach based on the extrinsic mean for estimation of the parameters g and c is studied in detail it is shown that g is the unique extrinsic mean for a brownian distribution nl g c or nr g c resulting estimates are proved to be consistent and asymptotically normal while they may also be used to simultaneously estimate g and c it is seen this requires that g be embedded into a higher dimensional matrix lie group going beyond brownian distributions it is shown the extrinsic mean can be used to recover the location parameter for a wider class of distributions arising more generally from levy processes the compact lie group structure places limitations on the analogy between normal distributions and brownian distributions this is illustrated by the study of multivariate brownian distributions these are introduced as brownian distributions on some product group e g g g this paper describes their covariance structure and considers its transformation under group operations

representation and detection of deformable shapes we describe some techniques that can be used to represent and detect deformable shapes in images the main difficulty with deformable template models is the very large or infinite number of possible nonrigid transformations of the templates this makes the problem of finding an optimal match of a deformable template to an image incredibly hard using a new representation for deformable shapes we show how to efficiently find a global optimal solution to the nonrigid matching problem the representation is based on the description of objects using triangulated polygons our matching algorithm can minimize a large class of energy functions making it applicable to a wide range of problems we present experimental results of detecting shapes in medical images and images of natural scenes we also consider the problem of learning a nonrigid shape model for a class of objects from examples we show how to learn good models while constraining them to be in the form required by the matching algorithm

the statistical analysis of shape data summary the paper deals with the statistical shape analysis of landmark data bookstein 1986 introduced a plausible model and obtained a useful normal approximation for three landmarks we use exact distribution theory and a normal approximation to consider maximum likelihood estimation the fisher information matrix for the normal approximation is given and we consider the hypothesis of shape change using these results we analyse a practical example based on digitized shapes this example motivated the work and the general direction for future work is indicated

analysis of planar shapes using geodesic paths on shape spaces for analyzing shapes of planar closed curves we propose differential geometric representations of curves using their direction functions and curvature functions shapes are represented as elements of infinite dimensional spaces and their pairwise differences are quantified using the lengths of geodesics connecting them on these spaces we use a fourier basis to represent tangents to the shape spaces and then use a gradient based shooting method to solve for the tangent that connects any two shapes via a geodesic using the surrey fish database we demonstrate some applications of this approach 1 interpolation and extrapolations of shape changes 2 clustering of objects according to their shapes 3 statistics on shape spaces and 4 bayesian extraction of shapes in low quality images

perturbation theory for generalized procrustes analysis summary perturbation theory for procrustes statistics is developed when both configurations are subject to independent normal errors the total sum of squares is shown to partition into sums of squares for translation rotation reflection and scaling plus a residual these results are extended to generalized procrustes analysis this gives a theoretical justification to gower s analysis of variance summary of the results from a generalized procrustes analysis

fourier continuation methods for high fidelity simulation of nonlinear acoustic beams on the basis of recently developed fourier continuation fc methods and associated efficient parallelization techniques this text introduces numerical algorithms that due to very low dispersive errors can accurately and efficiently solve the types of nonlinear partial differential equation pde models of nonlinear acoustics in hundred wavelength domains as arise in the simulation of focused medical ultrasound as demonstrated in the examples presented in this text the fc approach can be used to produce solutions to nonlinear acoustics pdes models with significantly reduced discretization requirements over those associated with finite difference finite element and finite volume methods especially in cases involving waves that travel distances that are orders of magnitude longer than their respective wavelengths in these examples the fc methodology is shown to lead to improvements in computing times by factors of hundreds and even thousands over those required by the standard approaches a variety of one and two dimensional examples presented in this text demonstrate the power and capabilities of the proposed methodology including an example containing a number of scattering centers and nonlinear multiple scattering events

structure of two dimensional sandpile i height probabilities the height probabilities of the two dimensional abelian sandpile model are the fractionial numbers of lattice sites having heights 1 2 3 4 a combinatorial method for evaluation of these quantities is proposed the method is based on mapping the set of allowed sandpile configurations onto the set of spanning trees covering a given lattice exact analytical expressions for all probabilities are obtained

directions and projective shapes this paper deals with projective shape analysis which is a study of finite configurations of points modulo projective transformations the topic has various applications in machine vision we introduce a convenient projective shape space as well as an appropriate coordinate system for this shape space for generic configurations of k points in m dimensions the resulting projective shape space is identified as a product of k m 1 copies of axial spaces rp m this identification leads to the need for developing multivariate directional and multivariate axial analysis and we propose parametric models as well as nonparametric methods for these areas in particular we investigate the frechet extrinsic mean for the multivariate axial case asymptotic distributions of the appropriate parametric and nonparametric tests are derived we illustrate our methodology with examples from machine vision

the geometry of shape spaces kendall introduced shape spaces which classified n tuples of points in r k and explained their role for statistics the aim of this paper is to study such shape spaces and in particular to explore the corresponding spaces for points on spheres the ultimate purpose of the work is to be able to apply statistical techniques to observations where only the shape of the observations is significant the geometry of the shape space is critical in determining the type of statistical technique which one should use therefore the major part of this paper is devoted to an examination of the geometry and the statistics is only introduced in a simple case when considering random processes on the shape space

hilbert schmidt lower bounds for estimators on matrix lie groups for atr deformable template representations of observed imagery model the variability of target pose via the actions of the matrix lie groups on rigid templates in this paper we study the construction of minimum mean squared error estimators on the special orthogonal group so n for pose estimation due to the nonflat geometry of so n the standard bayesian formulation of optimal estimators and their characteristics requires modifications by utilizing hilbert schmidt metric defined on gl n a larger group containing so n a mean squared criterion is defined on so n the hilbert schmidt estimate hse is defined to be a minimum mean squared error estimator restricted to so n the expected error associated with the hse is shown to be a lower bound called the hilbert schmidt bound hsb on the error incurred by any other estimator analysis and algorithms are presented for evaluating the hse and the hsb in cases of both ground based and airborne targets

a robust comparison of biological shapes summary localized differences in the form of two related animal skeletons are more effectively determined when resistant fitting techniques are used rather than least squares the repeated median resistant fitting algorithm is introduced the methods are tested by comparing hominid skulls with those of other primates and the male ostracode with the female in each case the resistant fit is superior to the least squares fit in that the difierences are more readily identified and agl ee more closely with the structural differences perceived on biological grounds le llltroductioll studies of allometric growth and of morphological change often run into difficulties in identifying and measuring the location and extent of relative shape deformations a reference system is needed for measuring these deformations and in the absence of an inherent fixed reference system one must be estimated the problem of estimating shape change in animal morphology provides an elegant setting for the comparison of least squares with recently developed robust and resistant statistical techniques being robust these new methods are protected against departures from the assumptions of analysis such as independent identically and normally distributed errors being resistant they are also protected against the potentially strong influences of atypical or incorrect data values two shapes can rarely be superimposed perfectly different fitting criteria will generally yield different results by allowing regions with large deformations to have a large impact on the fit least squares methods can minimize true shape differences and thereby obscure them a resistant technique however limits the influence of large deformations and the resulting fit is close in similar regions and not close in relatively deformed regions in this way resistant techniques can help to identify similarities and diflerences in form more effectively than least squares methods when sneath 1967 used least squares as a basis for some of the first quantitative estimates of relative deformation and so extended the fundamental descriptive work of thompson 1917 useful robust methods were not generally available sneath did in fact consider using the available robust methods but concluded that alternative methods that are less affected by wild values are not attractive in practice since then however the development of new theory and methods as reported by huber 1972 1981 hampel

robust multidimensional matched subspace classifiers based on weighted least squares we propose and design two classes of robust subspace classifiers for classification of multidimensional signals our classifiers are based on robust m estimators and the least median of squares principle and we show that they may be unified as iterated reweighted oblique subspace classifiers the performance of the proposed classifiers are demonstrated by two examples noncoherent detection of space time frequency shift keying signals and shape classification of partially occluded two dimensional 2 d objects in both cases the proposed robust subspace classifiers outperform the conventional subspace classifiers

shape procrustes tangent projections and bilateral symmetry summary in shape analysis with concentrated data the procrustes tangent projection of shapes plays a fundamental role since this projection can be used to convert shape analysis to standard multivariate analysis in view of its importance a systematic and rigorous treatment of the procrustes tangent projection is developed here to facilitate its use in applications one important application involves an analytical description of bilateral symmetry of objects together with some related distributional results an explicit expression for the tangent projection is also derived

n site approximations and coherent anomaly method analysis for a stochastic sandpile n site cluster approximations for a stochastic sandpile in one dimension are developed a height restriction is imposed to limit the number of states each site can harbor at most two particles height z i l 2 this yields a considerable simplification over the unrestricted case in which the number of states per site is unbounded on the basis of results for nl 11 sites the critical particle density as ensuremath zeta c 0 930 1 is estimated in good agreement with simulations a coherent anomaly analysis yields estimates for the order parameter exponent ensuremath beta 0 41 1 and the relaxation time exponent ensuremath nu ensuremath simeq 2 5

self organized critical scaling at surfaces at dissipative boundaries models of self organized criticality show peculiar scalings different from the bulk ones in the distributions characterizing avalanches for abelian models with dirichlet boundary conditions evidence of this is obtained by a mean field approach to semi infinite sandpiles and by numerical simulations in two and three dimensions on the other hand within the mean field description closed neumann conditions restore bulk scaling exponents also at the border numerical results are consistent with this property also at finite d

the extended fourier pseudospectral time domain method for atmospheric sound propagation an extended fourier pseudospectral time domain pstd method is presented to model atmospheric sound propagation by solving the linearized euler equations in this method evaluation of spatial derivatives is based on an eigenfunction expansion evaluation on a spatial grid requires only two spatial points per wavelength time iteration is done using a low storage optimized six stage runge kutta method this method is applied to two dimensional non moving media models one with screens and one for an urban canyon with generally high accuracy in both amplitude and phase for a moving atmosphere accurate results have been obtained in models with both a uniform and a logarithmic wind velocity profile over a rigid ground surface and in the presence of a screen the method has also been validated for three dimensional sound propagation over a screen for that application the developed method is in the order of 100 times faster than the second order accurate fdtd solution to the linearized euler equations the method is found to be well suited for atmospheric sound propagation simulations where effects of complex meteorology and straight rigid boundary surfaces are to be investigated

eigenshape analysis of a cut grow mapping for triangles and its application to phyllotaxis in plants phyllotaxis the regular arrangement of leaves in plants has been extensively studied geometrically and through theories of mechanism recent observations of the biophysics of cell growth and division at the plant apex during leaf formation lead to a cut and grow algorithm in which a newly formed leaf is first cut off from the meristem and then grows rapidly deforming the adjacent tissue to a new configuration the properties of the sequence of shapes constructed by a cut grow map of triangles are discussed the cut grow map excises a triangular segment of a triangle and dilates the newly formed edge in shape space the cut grow map is equivalent to a reflection followed by inversion each map has a conjugate pair of eigenshapes in the limiting case when the eigenshapes are equal the shapes converge with successive iterations at rate 1 n however an arbitrarily small perturbation from the eigenshape can initialize a divergent sequence of arbitrary length before the shapes begin to converge

landmark based shape encoding and sparse dictionary learning in the continuous domain we provide a generic framework to learn shape dictionaries of landmark based curves that are defined in the continuous domain we first present an unbiased alignment method that involves the construction of a mean shape as well as training sets whose elements are subspaces that contain all affine transformations of the training samples the alignment relies on orthogonal projection operators that have a closed form we then present algorithms to learn shape dictionaries according to the structure of the data that needs to be encoded 1 projection based functional principal component analysis for homogeneous data and 2 continuous domain sparse shape encoding to learn dictionaries that contain imbalanced data outliers or different types of shape structures through parametric spline curves we provide a detailed and exact implementation of our method we demonstrate that it requires fewer parameters than purely discrete methods and that it is computationally more efficient and accurate we illustrate the use of our framework for dictionary learning of structures in biomedical images as well as for shape analysis in bioimaging

strings variational deformable models of multivariate continuous boundary features we propose a new image segmentation technique called strings a string is a variational deformable model that is learned from a collection of example objects rather than built from a priori analytical or geometrical knowledge as opposed to existing approaches an object boundary is represented by a one dimensional multivariate curve in functional space a feature function rather than by a point in vector space in the learning phase feature functions are defined by extraction of multiple shape and image features along continuous object boundaries in a given learning set the feature functions are aligned then subjected to functional principal components analysis and functional principal regression to summarize the feature space and to model its content respectively also a mahalanobis distance model is constructed for evaluation of boundaries in terms of their feature functions taking into account the natural variations seen in the learning set in the segmentation phase an object boundary in a new image is searched for with help of a curve the curve gives rise to a feature function a string that is weighted by the regression model and evaluated by the mahalanobis model the curve is deformed in an iterative procedure to produce feature functions with minimal mahalanobis distance strings have been compared with active shape models on 145 vertebra images showing that strings produce better results when initialized close to the target boundary and comparable results otherwise

patterned and disordered continuous abelian sandpile model we study critical properties of the continuous abelian sandpile model with anisotropies in toppling rules that produce ordered patterns on it also we consider the continuous directed sandpile model perturbed by a weak quenched randomness study critical behavior of the model using perturbative conformal field theory and show that the model has a random fixed point

equivalence between the abelian sandpile model and the q 0 limit of the potts model abstract we establish an equivalence between the undirected abelian sandpile model and the q 0 limit of the q state potts model the equivalence is valid for arbitrary finite graphs two dimensional abelian sandpile models thus correspond to a conformal field theory with central charge c 2 the equivalence also gives a monte carlo algorithm to generate random spanning trees we study the growth process of the spread of fire under the burning algorithm in the background of a random recurrent configuration of the abelian sandpile model the average number of sites burnt upto time t varies at ta in two dimensions our numerically determined value of a agrees with the theoretical prediction a 8 5 we relate this exponent to the conventional exponents characterizing the distributions of avalanche sizes

properties of euclidean and non euclidean distance matrices abstract a distance matrix d of order n is symmetric with elements 1 2 d ij 2 where dii 0 d is euclidean when the 1 2 n n 1 quantities dij can be generated as the distances between a set of n points x n p in a euclidean space of dimension p the dimensionality of d is defined as the least value of p rank x of any generating x in general p 1 and p 2 are also acceptable but may include imaginary coordinates even when d is euclidean basic properties of euclidean distance matrices are established in particular when rank d it is shown that depending on whether etd e is not or is zero the generating points lie in either p 1 dimensions in which case they lie on a hypersphere or in p 2 dimensions in which case they do not the notation e is used for a vector all of whose values are one when d is non euclidean its dimensionality p r s will comprise r real and s imaginary columns of x and r s are invariant for all generating x of minimal rank higher ranking representations can arise only from p 1 r 1 s or p 1 r s 1 or p 2 r 1 s 1 so that not only are r s invariant but they are both minimal for all admissible representations x

boundary height fields in the abelian sandpile model we study the abelian sandpile model on the upper half plane and reconsider the correlations of the four height variables lying on the boundary for more convenience we carry out the analysis in the dissipative massive extension of the model and identify the boundary scaling fields corresponding to the four heights we find that they all can be accounted for by the massive pertubation of a c 2 logarithmic conformal field theory comment 19 pages 3 figures

simple conformal methods for finite difference time domain modeling of pressure release surfaces the finite difference time domain fdtd method provides a simple and accurate means of simulating a wide range of acoustic wave propagation problems unfortunately the method has a voracious appetite for computational resources for example to accurately model scattering from a continuously varying pressure release boundary an fdtd grid is typically required that has a much finer discretization than is necessary to model propagation in a homogeneous space such a fine discretization can become prohibitive when considering large scale problems two simple conformal techniques are presented for acoustic fdtd simulations of problems involving pressure release boundaries these techniques which rely upon splitting velocity cells adjacent to the pressure release boundary significantly improve the accuracy of the results over those of the standard staircase representation of the boundary these methods permit the use of a coarser fdtd grid than would otherwise be practical and yet add negligible computa

efficient calculation of the green function for acoustic propagation above a homogeneous impedance plane this paper is concerned with the problem of propagation from a monofrequency coherent line source above a plane of homogeneous surface impedance the solution of this problem occurs in the kernel of certain boundary integral equation formulations of acoustic propagation above an impedance boundary and the discussion of the paper is motivated by this application the paper starts by deriving representations as laplace type integrals of the solution and its first partial derivatives the evaluation of these integral representations by gauss laguerre quadrature is discussed and theoretical bounds on the truncation error are obtained specific approximations are proposed which are shown to be accurate except in the very near field for all angles of incidence and a wide range of values of surface impedance the paper finishes with derivations of partial results and analogous laplace type integral representations for the case of a point source

hybrid fourier pseudospectral discontinuous galerkin time domain method for wave propagation the fourier pseudospectral time domain fourier pstd method was shown to be an efficient way of modelling acoustic propagation problems as described by the linearized euler equations lee but is limited to real valued frequency independent boundary conditions and predominantly staircase like boundary shapes this paper presents a hybrid approach to solve the lee coupling fourier pstd with a nodal discontinuous galerkin dg method dg exhibits almost no restrictions with respect to geometrical complexity or boundary conditions the aim of this novel method is to allow the computation of complex geometries and to be a step towards the implementation of frequency dependent boundary conditions by using the benefits of dg at the boundaries while keeping the efficient fourier pstd in the bulk of the domain the hybridization approach is based on conformal meshes to avoid spatial interpolation of the dg solutions when transferring values from dg to fourier pstd while the data transfer from fourier pstd to dg is done utilizing spectral interpolation of the fourier pstd solutions the accuracy of the hybrid approach is presented for one and two dimensional acoustic problems and the main sources of error are investigated it is concluded that the hybrid methodology does not introduce significant errors compared to the fourier pstd stand alone solver an example of a cylinder scattering problem is presented and accurate results have been obtained when using the proposed approach finally no instabilities were found during long time calculation using the current hybrid methodology on a two dimensional domain a method for wave propagation in fluid domains with complex geometries is presented the method combines fourier pstd in the bulk of the domain and dg at boundaries the main application of interest is atmospheric sound propagation in cities the method shows no significant additional error compare with a fourier pstd solver no indication of instability in long time calculations has been detected

quantization on the grassmann manifold the problem of quantization in an euclidean space with unitary constraints can be formulated as an unconstrained problem on a grassmann manifold such constraints arise in areas such as wireless communication with multiple antennas at the transmitter and at the receiver due to the constraints the distortion rate analysis developed for euclidean spaces cannot be applied directly this paper extends gersho s asymptotic large rate small distortion distortion bounds to the case when the source is distributed on the complex grassmann manifold the special structure of the grassmann manifold and the distortion measures defined on it differentiate this problem from the traditional vector quantization in euclidean spaces

intrinsic analysis of statistical estimation the parametric statistical models with suitable regularity conditions have a natural riemannian manifold structure given by the information metric since the parameters are merely labels for the probability measures an inferential statement should be formulated through intrinsic objects invariant under reparametrizations in this context the estimators will be random objects valued on the manifold corresponding to the statistical model in spite of these considerations classical measures of an estimator s performance like the bias and the mean square error are clearly dependent on the statistical model parametrizations in this paper the authors work with extended notions of mean value and moments of random objects which take values on a hausdorff and connected manifold equipped with an affine connection in particular the riemannian manifold case is considered this extension is applied to the bias and the mean square error study in statistical point estimation theory under this approach an intrinsic version of the cramer rao lower bound is obtained a lower bound which depends on the intrinsic bias and the curvature of the statistical model for the mean square of the rao distance the invariant measure analogous to the mean square error further the behavior of the mean square of the rao distance of an estimator when conditioning with respect to a sufficient statistic is considered obtaining intrinsic versions of the rao blackwell and lehmann scheffe theorems asymptotic properties complete the study

on certain linear mappings between inner product and squared distance matrices abstract we obtain the spectral decomposition of four linear mappings the first is a mapping of the linear hull of all centered inner product matrices onto the linear hull of all the induced squared distance matrices it is based on the natural generalization of the cosine law of elementary euclidean geometry the other three mappings studied are 1 the adjoint and 1 extensions and applications particularly to multidimensional scaling are discussed in some detail

subspace procrustes analysis procrustes analysis pa has been a popular technique to align and build 2 d statistical models of shapes given a set of 2 d shapes pa is applied to remove rigid transformations later a non rigid 2 d model is computed by modeling the residual e g pca although pa has been widely used it has several limitations for modeling 2 d shapes occluded landmarks and missing data can result in local minima solutions and there is no guarantee that the 2 d shapes provide a uniform sampling of the 3 d space of rotations for the object to address previous issues this paper proposes subspace pa spa given several instances of a 3 d object spa computes the mean and a 2 d subspace that can model rigid and non rigid deformations of the 3 d object we propose a discrete dspa and continuous cspa formulation for spa assuming that 3 d samples of an object are provided dspa extends the traditional pa and produces unbiased 2 d models by uniformly sampling different views of the 3 d object cspa provides a continuous approach to uniformly sample the space of 3 d rotations being more efficient in space and time we illustrate the benefits of spa in two different applications first spa is used to learn 2 d face and body models from 3 d datasets experiments on the facewarehouse and cmu motion capture mocap datasets show the benefits of our 2 d models against the state of the art pa approaches and conventional 3 d models second spa learns an unbiased 2 d model from cmu mocap dataset and it is used to estimate the human pose on the leeds sports dataset

boundary height correlations in a two dimensional abelian sandpile boundary height distributions of the two dimensional abelian sandpile model are studied in the self organized critical state all height probabilities are calculated explicitly both at open and closed boundaries the leading asymptotic term of the corresponding correlation functions is observed to behave as r 4 when r to infinity on the basis of conformal field theory predictions the bulk height correlators are shown to have the same critical exponents as boundary ones all heights seem to be identified with appropriate counterparts of the local energy operator in the zero component potts model

a fast approach for dimensionality reduction with image data an important objective in image analysis is dimensionality reduction the most often used data exploratory technique with this objective is principal component analysis which performs a singular value decomposition on a data matrix of vectorized images when considering an array data or tensor instead of a matrix the high order generalization of pca for computing principal components offers multiple ways to decompose tensors orthogonally as an alternative we propose a new method based on the projection of the images as matrices and show that it leads to a better reconstruction of images than previous approaches

self organized critical state of sandpile automaton models we study a general bak tang wiesenfeld type automaton model of self organized criticality in which the toppling conditions depend on local height but not on its gradient we characterize the critical state and determine its entropy for an arbitrary finite lattice in any dimension the two point correlation function is shown to satisfy a linear equation the spectrum of relaxation times describing the approach to the critical state is also determined exactly

regression based cardiac motion prediction from single phase cta state of the art cardiac computed tomography ct enables the acquisition of imaging data of the heart over the entire cardiac cycle at concurrent high spatial and temporal resolution however in clinical practice acquisition is increasingly limited to 3 d images estimating the shape of the cardiac structures throughout the entire cardiac cycle from a 3 d image is therefore useful in applications such as the alignment of preoperative computed tomography angiography cta to intra operative x ray images for improved guidance in coronary interventions we hypothesize that the motion of the heart is partially explained by its shape and therefore investigate the use of three regression methods for motion estimation from single phase shape information quantitative evaluation on 150 4 d cta images showed a small but statistically significant increase in the accuracy of the predicted shape sequences when using any of the regression methods compared to shape independent motion prediction by application of the mean motion the best results were achieved using principal component regression resulting in point to point errors of 2 3 0 5 mm compared to values of 2 7 0 6 mm for shape independent motion estimation finally we showed that this significant difference withstands small variations in important parameter settings of the landmarking procedure

spherical principal curves this paper presents a new approach for dimension reduction of data observed on spherical surfaces several dimension reduction techniques have been developed in recent years for non euclidean data analysis as a pioneer work hauberg 2016 attempted to implement principal curves on riemannian manifolds however this approach uses approximations to process data on riemannian manifolds resulting in distorted results this study proposes a new approach to project data onto a continuous curve to construct principal curves on spherical surfaces our approach lies in the same line of hastie and stuetzle 1989 that proposed principal curves for data on euclidean space we further investigate the stationarity of the proposed principal curves that satisfy the self consistency on spherical surfaces the results on the real data analysis and simulation examples show promising empirical characteristics of the proposed approach

minimal sandpiles on hexagonal lattice we study the minimal recurrent configurations of the abelian sandpile model on the hexagonal lattice referred to the dynamics of a nonconservative sandpile model the one to one correspondence between these configurations and the set of maximally oriented spanning trees on the triangular sublattice is constructed we derive the correlation functions in minimal recurrent configurations on a quasi one dimensional 2 n lattice compare them with correlations for ordinary recurrent configurations and argue for asymptotic equivalence between them

an alternative approach to normalization and evaluation for gait patterns procrustes analysis applied to the cyclograms of sprinters and middle distance runners in order to compare gait patterns a common procedure is to normalize strides both in time and magnitude the stride duration is usually normalized to a time percentage before averaging curves as the timing of event occurrences may shift across strides the shape of the averaged curves is distorted and therefore the standard deviation is overvalued stride magnitude normalization is performed by means of dimensionless numbers however there is little agreement on which body size correction methods should be used the procrustes method describes curve shape and shape change in a mathematical and statistical framework independently of time and size factors the present study aims to explore how this technique may be used for time and magnitude stride normalization to reflect individual and group mean responses the procrustes method which combines quantitative and visual features is applied to the shape of the ankle and knee cyclograms superimposition of 25 cyclograms 10 for sprinters sp and 15 for middle distance runners mdr was supplemented by statistical procedures principal component analysis discriminant function to extract the main key events which vary according to the athletic specialities in comparison with the mdr poulaine shaped cyclogram the ovoid cyclogram of sp reveals the following gait indicators a short braking phase a rapid initial lower limb swing in the forward direction a fast upward movement of the knee and ankle and an active foot contact the procrustes approach could be used to describe other quasi periodic movements through relative motion plots e g cyclograms angle angle diagrams phase plane portraits

a shape distance based on the fisher rao metric and its application for shapes clustering abstract in the clustering of shapes which is a longstanding challenge in the framework of geometric morphometrics and shape analysis is crucial the selection and application of a suitable and appropriate measurement of distance among observations i e individuals the aim of this study is to model shapes from complex systems using information geometry tools it is well known that the fisher information endows the statistical manifold defined by a family of probability distributions with a riemannian metric called the fisher rao metric with respect to this geodesic paths are determined minimizing information in fisher sense the geodesic distance induced by the fisher rao metric can be used to define a shape metric which enables us to quantify differences between shapes the discriminative power of the proposed fisher rao distance is tested in the context of shapes clustering on both simulated and real data sets results show a better ability in recovering the true cluster structure with respect to the standard kendall s shape metric

monte carlo extrinsic estimators of manifold valued parameters monte carlo mc methods have become an important tool for inferences in non gaussian and non euclidean settings we study their use in those signal image processing scenarios where the parameter spaces are certain riemannian manifolds finite dimensional lie groups and their quotient sets we investigate the estimation of means and variances of the manifold valued parameters using two popular sampling methods independent and importance sampling using euclidean embeddings we specify a notion of extrinsic means employ monte carlo methods to estimate these means and utilize large sample asymptotics to approximate the estimator covariances experimental results are presented for target pose estimation orthogonal groups and signal subspace estimation grassmann manifolds asymptotic covariances are utilized to construct confidence regions to compare estimators and to determine the sample size for mc sampling

hierarchical lung field segmentation with joint shape and appearance sparse learning lung field segmentation in the posterior anterior pa chest radiograph is important for pulmonary disease diagnosis and hemodialysis treatment due to high shape variation and boundary ambiguity accurate lung field segmentation from chest radiograph is still a challenging task to tackle these challenges we propose a joint shape and appearance sparse learning method for robust and accurate lung field segmentation the main contributions of this paper are 1 a robust shape initialization method is designed to achieve an initial shape that is close to the lung boundary under segmentation 2 a set of local sparse shape composition models are built based on local lung shape segments to overcome the high shape variations 3 a set of local appearance models are similarly adopted by using sparse representation to capture the appearance characteristics in local lung boundary segments thus effectively dealing with the lung boundary ambiguity 4 a hierarchical deformable segmentation framework is proposed to integrate the scale dependent shape and appearance information together for robust and accurate segmentation our method is evaluated on 247 pa chest radiographs in a public dataset the experimental results show that the proposed local shape and appearance models outperform the conventional shape and appearance models compared with most of the state of the art lung field segmentation methods under comparison our method also shows a higher accuracy which is comparable to the inter observer annotation variation

a multi domain fourier pseudospectral time domain method for the linearized euler equations the fourier pseudospectral time domain f pstd method is computationally one of the most cost efficient methods for solving the linearized euler equations for wave propagation through a medium with smoothly varying spatial inhomogeneities in the presence of rigid boundaries as the method utilizes an equidistant discretization local fine scale effects of geometry or medium inhomogeneities require a refinement of the whole grid which significantly reduces the computational efficiency for this reason a multi domain f pstd methodology is presented with a coarse grid covering the complete domain and fine grids acting as a subgrid resolution of the coarse grid near local fine scale effects data transfer between coarse and fine grids takes place utilizing spectral interpolation with super gaussian window functions to impose spatial periodicity local time stepping is employed without intermediate interpolation the errors introduced by the window functions and the multi domain implementation are quantified and compared to errors related to the initial conditions and from the time iteration scheme it is concluded that the multi domain methodology does not introduce significant errors compared to the single domain method examples of scattering from small scale density scatters sound reflecting from a slitted rigid object and sound propagation through a jet are accurately modelled by the proposed methodology for problems that can be solved by f pstd the presented methodology can lead to a significant gain in computational efficiency

non conservative sandpile cellular automaton on the bethe lattice the non conservative noisy critical height sandpile cellular automaton with open boundary conditions is studied analytically on the bethe lattice using the modified method of dhar and majumdar 1990 the single site probabilities pair probabilities and the avalanche size distributions for the three versions of the automaton with different amount of dissipated particles are calculated

the spatial asymmetric two dimensional continuous abelian sandpile model we insert some asymmetries into the continuous abelian sandpile models such as directedness and ellipticity we analyze probability distribution of different heights and also find the corresponding field theories also we find the fields associated with some height variables

morphometric tools for landmark data geometry and biology 1 introduction 2 preliminaries 3 landmarks 4 distance measures 5 shape coordinates 6 principal axes of shape change for triangles 7 features of shape comparison 8 retrospect and prospect

probability convexity and harmonic maps with small image i uniqueness and fine existence this paper uses probability theory to provide an approach to the dirichlet problem for harmonic maps the probabilistic tools used are those of manifold valued brownian motion and martingales probabilistic proofs are given of uniqueness continuous dependence on data and existence of generalized solutions finely harmonic maps for target domains which have convex geometry this class of domain includes all regular geodesic balls links are made with new results in other fields the martingale dirichlet problem riemannian centres of mass

riemann manifold langevin and hamiltonian monte carlo methods the paper proposes metropolis adjusted langevin and hamiltonian monte carlo sampling methods defined on the riemann manifold to resolve the shortcomings of existing monte carlo algorithms when sampling from target densities that may be high dimensional and exhibit strong correlations the methods provide fully automated adaptation mechanisms that circumvent the costly pilot runs that are required to tune proposal densities for metropolis hastings or indeed hamiltonian monte carlo and metropolis adjusted langevin algorithms this allows for highly efficient sampling even in very high dimensions where different scalings may be required for the transient and stationary phases of the markov chain the methodology proposed exploits the riemann geometry of the parameter space of statistical models and thus automatically adapts to the local structure when simulating paths across this manifold providing highly efficient convergence and exploration of the target density the performance of these riemann manifold monte carlo methods is rigorously assessed by performing inference on logistic regression models log gaussian cox point processes stochastic volatility models and bayesian estimation of dynamic systems described by non linear differential equations substantial improvements in the time normalized effective sample size are reported when compared with alternative sampling approaches matlab code that is available from http www ucl ac uk statistics research rmhmc allows replication of all the results reported

the abelian sandpile and related models abstract the abelian sandpile model is the simplest analytically tractable model of self organized criticality this paper presents a brief review of known results about the model the abelian group structure of the algebra of operators allows an exact calculation of many of its properties in particular when there is a preferred direction one can calculate all the critical exponents characterizing the distribution of avalanche sizes in all dimensions for the undirected case the model is related to q 0 state potts model this enables exact calculation of some exponents in two dimensions and there are some conjectures about others we also discuss a generalization of the model to a network of communicating reactive processors this includes sandpile models with stochastic toppling rules as a special case we also consider a nondashabelian stochastic variant which lies in a different universality class related to directed percolation

a multi dimensional scaling approach to shape analysis summary we propose an alternative to kendall s shape space for reflection shapes of configurations in rm with k labelled vertices where reflection shape consists of all the geometric information that is invariant under compositions of similarity and reflection transformations the proposed approach embeds the space of such shapes into the space p k 1 of k 1 x k 1 real symmetric positive semidefinite matrices which is the closure of an open subset of a euclidean space and defines mean shape as the natural projection of euclidean means in p k 1 on to the embedded copy of the shape space this approach has strong connections with multi dimensional scaling and the mean shape so defined gives good approximations to other commonly used definitions of mean shape we also use standard perturbation arguments for eigenvalues and eigenvectors to obtain a central limit theorem which then enables the application of standard statistical techniques to shape analysis in two or more dimensions

a family of low dispersive and low dissipative explicit schemes for flow and noise computations explicit numerical methods for spatial derivation filtering and time integration are proposed they are developed with the aim of computing flow and noise with high accuracy and fidelity all the methods are constructed in the same way by minimizing the dispersion and the dissipation errors in the wavenumber space up to k x 2 corresponding to four points per wavelength they are shown to be more accurate and also more efficient numerically than most of the standard explicit high order methods for uniform and slowly non uniform grids two problems involving long range sound propagation are resolved to illustrate their respective precisions remarks about their practical applications are then made especially about the connection with the boundary conditions finally their relevance for the simulation of turbulent flows is emphasized

higher order and boundary scaling fields in the abelian sandpile model the abelian sandpile model asm is a paradigm of self organized criticality soc which is related to c 2 conformal field theory the conformal fields corresponding to some height clusters have been suggested before here we derive the first corrections to such fields in a field theoretical approach when the lattice parameter is non vanishing and consider them in the presence of a boundary