linearized multidimensional earth mover s distance gradient flows this paper presents the first framework capable of performing active contour segmentation using earth mover s distance emd to measure dissimilarity between multidimensional feature distributions emd is the best known and understood cross bin histogram distance measure and as such it allows for meaningful comparisons between distributions unlike bin to bin measures that only account for discrepancies on a bin to bin basis because emd is obtained with linear programming techniques its differential structure with respect to variations in bin weights as the active contour evolves is expressed through sensitivity analysis euler lagrange equations are then derived from the computed sensitivity at every iteration to produce gradient descent flows we validate our approach with color image segmentation in comparison with state of the art bhattacharyya bin to bin and 1d emd cross bin active contours some unique advantages of cross bin comparison are highlighted in our segmentation results better perceptual value and increased robustness with respect to the initialization

a rank one reduction formula and its applications to matrix factorizations let a in r m times n denote an arbitrary matrix if x in r n and y in r m are vectors such that omega y t ax ne 0 then the matrix b a omega 1 axy t a a has rank exactly one less than the rank of a this wedderburn rank one reduction formula is easy to prove yet the idea is so powerful that perhaps all matrix factorizations can be derived from it the formula also appears in places such as the positive definite secant updates bfgs and dfp as well as the abs methods by repeatedly applying the formula to reduce ranks a biconjugation process analogous to the gram schmidt process with oblique projections can be developed this process provides a mechanism for constructing factorizations such as text ldm t qr and svd under a common framework of a general biconjugate decomposition v t au omega that is diagonal and nonsingular two characterizations of biconjugation provide new insight into the lanczos method and its breakdown one characterization shows that

on the continuity of the map ϑ ϑ from the predual of a w algebra abstract let be elements in the predual of a w algebra for their absolute value parts the estimate 2 1 2 is obtained

extended rank reduction formulas containing wedderburn and abaffy broyden spedicato rank reducing processes abstract the wedderburn rank reduction formula and the abaffy broyden spedicato abs algorithms are powerful methods for developing matrix factorizations and many fundamental numerical linear algebra processes such as gram schmidt conjugate direction and lanczos methods we present a rank reduction formula for transforming the rows and columns of a extending the wedderburn rank reduction formula and the abs approach by repeatedly applying the formula to reduce the rank an extended rank reducing process is derived the biconjugation process associated with the wedderburn rank reduction process and the scaled extended abs class of algorithms are shown to be in our proposed rank reducing process while the process is more general to produce several other effective reduction algorithms to compute various structured factorizations the process provides a general finite iterative approach for constructing factorizations of a and a t under a common framework of a general decomposition v t a p we also show that the biconjugation process associated with the wedderburn rank reduction process can be derived from the scaled abs class of algorithms applied to a or a t finally we provide a list of some well known reduction procedures as special cases of our extended rank reducing process the approach is general enough to produce various structured decompositions as well

perturbation bounds for triangular and full rank factorizations we give componentwise bounds for the perturbations of the lu and ldu factorizations these bounds are valid for all perturbations which keeps nonsingularity and lu factorizability it is shown that the new perturbation bounds are sharper than the earlier results the perturbation bounds are then applied to full rank factorizations produced by the rank reduction procedure the result indicates that these full rank factorizations are stable if the ldu factorization of a certain matrix is also stable

response surface approximation of pareto optimal front in multi objective optimization a systematic approach is presented to approximate the pareto optimal front pof by a response surface approximation the data for the pof is obtained by multi objective evolutionary algorithm improvements to address drift in the pof are also presented the approximated pof can help visualize and quantify trade offs among objectives to select compromise designs the bounds of this approximate pof are obtained using multiple convex hulls the proposed approach is applied to study trade offs among objectives of a rocket injector design problem where performance and life objectives compete the pof is approximated using a quintic polynomial the compromise region quantifies trade offs among objectives 2006 elsevier b v all rights reserved

active contours without edges we propose a new model for active contours to detect objects in a given image based on techniques of curve evolution mumford shah 1989 functional for segmentation and level sets our model can detect objects whose boundaries are not necessarily defined by the gradient we minimize an energy which can be seen as a particular case of the minimal partition problem in the level set formulation the problem becomes a mean curvature flow like evolving the active contour which will stop on the desired boundary however the stopping term does not depend on the gradient of the image as in the classical active contour models but is instead related to a particular segmentation of the image we give a numerical algorithm using finite differences finally we present various experimental results and in particular some examples for which the classical snakes methods based on the gradient are not applicable also the initial curve can be anywhere in the image and interior contours are automatically detected

alternative characterizations of the extended wedderburn guttman theorem let a be a u by v matrix of rank a and let m and n be u by p and v by q matrices respectively where p is not necessarily equal to q or rank m an min p q takane and yanai y takane h yanai on the wedderburn guttman theorem linear algebra appl 410 2005 267 278 investigated the conditions under which rank a an m an m a rank a rank an m an m a this is called the extended wedderburn guttman theorem in this paper we give alternative characterizations of these conditions using the product singular value decomposition psvd of matrix triplets

a note on rounding error analysis of cholesky factorization abstract an almost sharp overall a priori bound is given for a ll t f where l is the computed cholesky factor of matrix a

mixed and componentwise condition numbers for matrix decompositions abstract we present normwise and componentwise perturbation bounds for the lu the cholesky the l d l t and the qr decompositions by using a new approach the explicit expressions of mixed and componentwise condition numbers for these matrix decompositions are derived the condition numbers improve known results of the normwise and componentwise cases and reveal the characterizations of the structured perturbations the exact explicit perturbation expressions are derived for the factors l and u of the lu decomposition and the rigorous normwise and componentwise perturbation bounds are presented for the lu decomposition

additive preconditioning eigenspaces and the inverse iteration we incorporate our recent preconditioning techniques into the classical inverse power rayleigh quotient iteration for computing matrix eigenvectors every loop of this iteration essentially amounts to solving an ill conditioned linear system of equations due to our modification we solve a well conditioned linear system instead we prove that this modification preserves local quadratic convergence show experimentally that fast global convergence is preserved as well and yield similar results for higher order inverse iteration covering the cases of multiple and clustered eigenvalues

normalized cuts and image segmentation we propose a novel approach for solving the perceptual grouping problem in vision rather than focusing on local features and their consistencies in the image data our approach aims at extracting the global impression of an image we treat image segmentation as a graph partitioning problem and propose a novel global criterion the normalized cut for segmenting the graph the normalized cut criterion measures both the total dissimilarity between the different groups as well as the total similarity within the groups we show that an efficient computational technique based on a generalized eigenvalue problem can be used to optimize this criterion we applied this approach to segmenting static images as well as motion sequences and found the results to be very encouraging

existence and regularity of minimizers of a functional for unsupervised multiphase segmentation abstract we consider a variational model for image segmentation proposed in sandberg et al 2010 12 in such a model the image domain is partitioned into a finite collection of subsets denoted as phases the segmentation is unsupervised i e the model finds automatically an optimal number of phases which are not required to be connected subsets unsupervised segmentation is obtained by minimizing a functional of the mumford shah type mumford and shah 1989 1 but modifying the geometric part of the mumford shah energy with the introduction of a suitable scale term the results of computer experiments discussed in 12 show that the resulting variational model has several properties which are relevant for applications in this paper we investigate the theoretical properties of the model we study the existence of minimizers of the corresponding functional first looking for a weak solution in a class of phases constituted by sets of finite perimeter then we find various regularity properties of such minimizers particularly we study the structure of triple junctions by determining their optimal angles

rounding error and perturbation bounds for the indefinite qr factorization abstract indefinite qr factorization is a generalization of the well known qr factorization where q is a unitary matrix with respect to the given indefinite inner product matrix j this factorization can be used for accurate computation of eigenvalues of the hermitian matrix a g jg where g and j are initially given or naturally formed from initial data the classical example of such a matrix is a b b c c with given b and c in this paper we present the rounding error and perturbation bounds for the so called triangular case of the indefinite qr factorization these bounds fit well into the relative perturbation theory for hermitian matrices given in factorized form

preconditioning sparse nonsymmetric linear systems with the sherman morrison formula let ax b be a large sparse nonsymmetric system of linear equations a new sparse approximate inverse preconditioning technique for such a class of systems is proposed we show how the matrix a0 1 a 1 where a0 is a nonsingular matrix whose inverse is known or easy to compute can be factorized in the form u omega v t using the sherman morrison formula when this factorization process is done incompletely an approximate factorization may be obtained and used as a preconditioner for krylov iterative methods for a0 sin where in is the identity matrix and s is a positive scalar the existence of the preconditioner for m matrices is proved in addition some numerical experiments obtained for a representative set of matrices are presented results show that our approach is comparable with other existing approximate inverse techniques

on some perturbation inequalities for operators abstract an estimate for the norm of the solution to the equation ax xb s obtained by r bhatia c davis and a mcintosh for normal operators a and b is shown to be valid for a larger class some other inequalities in the same spirit are obtained including a sin theorem for singular vectors some inequalities concerning the continuity of the map a ita obtained recently by kittaneh and kosaki are extended using these ideas

a multiphase level set framework for image segmentation using the mumford and shah model we propose a new multiphase level set framework for image segmentation using the mumford and shah model for piecewise constant and piecewise smooth optimal approximations the proposed method is also a generalization of an active contour model without edges based 2 phase segmentation developed by the authors earlier in t chan and l vese 1999 in scale space 99 m nilsen et al eds lncs vol 1682 pp 141 151 and t chan and l vese 2001 ieee ip 10 2 266 277 the multiphase level set formulation is new and of interest on its own by construction it automatically avoids the problems of vacuum and overlaps it needs only log n level set functions for n phases in the piecewise constant cases it can represent boundaries with complex topologies including triple junctionss in the piecewise smooth case only two level set functions formally suffice to represent any partition based on the four color theorem finally we validate the proposed models by numerical results for signal and image denoising and segmentation implemented using the osher and sethian level set method

componentwise perturbation analyses for the qr factorization this paper gives componentwise perturbation analyses for q and r in the qr factorization a qr q mathrm t q i r upper triangular for a given real m times n matrix a of rank n such specific analyses are important for example when the columns of a are badly scaled first order perturbation bounds are given for both q and r the analyses more accurately reflect the sensitivity of the problem than previous such results the condition number for r is bounded for a fixed n when the standard column pivoting strategy is used this strategy also tends to improve the condition of q so usually the computed q and r will both have higher accuracy when we use the standard column pivoting strategy practical condition estimators are derived the assumptions on the form of the perturbation delta a are explained and extended weaker rigorous bounds are also given

convergence rate of weak local linearization schemes for stochastic differential equations with additive noise there exists a diversity of weak local linearization ll schemes for the integration of stochastic differential equations with additive noise which differ in the algorithms employed for the numerical implementation of the weak local linear discretizations despite convergence results for these discretizations have been already developed the convergence of the weak ll schemes has not been considered up to date in this work a general result concerning the convergence rate of the weak ll schemes is presented as well as specificities for a number of particular schemes as an application the convergence of weak ll schemes for equations driven by poisson processes is presented in addition

image segmentation and selective smoothing by using mumford shah model recently chan and vese developed an active contour model for image segmentation and smoothing by using piecewise constant and smooth representation of an image tsai et al also independently developed a segmentation and smoothing method similar to the chan and vese piecewise smooth approach these models are active contours based on the mumford shah variational approach and the level set method in this paper we develop a new hierarchical method which has many advantages compared to the chan and vese multiphase active contour models first unlike previous works the curve evolution partial differential equations pdes for different level set functions are decoupled each curve evolution pde is the equation of motion of just one level set function and different level set equations of motion are solved in a hierarchy this decoupling of the motion equations of the level set functions speeds up the segmentation process significantly second because of the coupling of the curve evolution equations associated with different level set functions the initialization of the level sets in chan and vese s method is difficult to handle in fact different initial conditions may produce completely different results the hierarchical method proposed in this paper can avoid the problem due to the choice of initial conditions third in this paper we use the diffusion equation for denoising this method therefore can deal with very noisy images in general our method is fast flexible not sensitive to the choice of initial conditions and produces very good results

a symplectic qr like algorithm for the solution of the real algebraic riccati equation a method is presented to solve the real algebraic riccati equation xnx xa a t x k 0 where k k t and n n t the solution for the corresponding eigenvalue problem mx lambda x where m is a hamiltonian matrix is computed by an algorithm similar to the qr algorithm special symplectic matrices are used for the transformation of m such that the hamiltonian form is preserved during the computations

stochastic relaxation gibbs distributions and the bayesian restoration of images we make an analogy between images and statistical mechanics systems pixel gray levels and the presence and orientation of edges are viewed as states of atoms or molecules in a lattice like physical system the assignment of an energy function in the physical system determines its gibbs distribution because of the gibbs distribution markov random field mrf equivalence this assignment also determines an mrf image model the energy function is a more convenient and natural mechanism for embodying picture attributes than are the local characteristics of the mrf for a range of degradation mechanisms including blurring nonlinear deformations and multiplicative or additive noise the posterior distribution is an mrf with a structure akin to the image model by the analogy the posterior distribution defines another imaginary physical system gradual temperature reduction in the physical system isolates low energy states annealing or what is the same thing the most probable states under the gibbs distribution the analogous operation under the posterior distribution yields the maximum a posteriori map estimate of the image given the degraded observations the result is a highly parallel relaxation algorithm for map estimation we establish convergence properties of the algorithm and we experiment with some simple pictures for which good restorations are obtained at low signal to noise ratios

semi nonnegative rank for real matrices and its connection to the usual rank abstract the present paper introduces the semi nonnegative rank for real matrices as an alternative to the usual rank it is shown that the semi nonnegative rank takes two possible values which are simple functions of the usual rank several characterizations of matrices for which the two ranks coincide are given

rounding error and perturbation bounds for the cholesky and ldl t factorizations abstract componentwise rounding error and perturbation bounds for the cholesky and ldl t factorizations are derived the precision of the computed cholesky and ldl t factorizations is given

structured condition numbers for some matrix factorizations of structured matrices abstract using the modified matrix vector approach and the differential calculus we study the structured condition numbers for lu cholesky and qr factorizations of some structured matrices that can be represented by sets of parameters the obtained explicit expressions of these structured condition numbers are very general which are applicable to most of linear and non linear structured matrices and include the popular normwise mixed and componentwise condition numbers as special cases more specific explicit expressions of the structured condition numbers for linear structured matrices are also provided we compare the structured condition numbers with the corresponding unstructured ones in theory and experiment numerical results show that for non linear structured matrices the structured condition numbers can be much smaller than the unstructured ones in addition we also test the applications of structured condition numbers in estimating the first order perturbation bounds of matrix factorizations using numerical examples

a shuffled complex evolution metropolis algorithm for optimization and uncertainty assessment of hydrologic model parameters author s vrugt ja gupta hv bouten w sorooshian s abstract markov chain monte carlo mcmc methods have become increasingly popular for estimating the posterior probability distribution of parameters in hydrologic models however mcmc methods require the a priori definition of a proposal or sampling distribution which determines the explorative capabilities and efficiency of the sampler and therefore the statistical properties of the markov chain and its rate of convergence in this paper we present an mcmc sampler entitled the shuffled complex evolution metropolis algorithm scem ua which is well suited to infer the posterior distribution of hydrologic model parameters the scem ua algorithm is a modified version of the original sce ua global optimization algorithm developed by duan et al 1992 the scem ua algorithm operates by merging the strengths of the metropolis algorithm controlled random search competitive evolution and complex shuffling in order to continuously update the proposal distribution and evolve the sampler to the posterior target distribution three case studies demonstrate that the adaptive capability of the scem ua algorithm significantly reduces the number of model simulations needed to infer the posterior distribution of the parameters when compared with the traditional metropolis hastings samplers

the factorability of symmetric matrices and some implications for statistical linear models abstract general conditions where a symmetric matrix is factorable by cholesky decomposition are described while numerical stability is a remaining issue whenever the cholesky decomposition is used to factor indefinite matrices the existence of such factors is demonstrated for matrix structures that are commonly found in statistics kalman filtering for example is rediscovered in the cholesky decomposition of an indefinite matrix moreover the cholesky decomposition uniquely defines the likelihood function in linear statistical models and this includes situations when the variance matrix is singular or when the cholesky decomposition does not run to completion alternative methods of likelihood evaluation which may involve for example the bunch parlett factorization are available only when the cholesky decomposition exists suggestions are made for computing an adaptive precision cholesky decomposition when numerical stability is an issue

a binary level set model and some applications to mumford shah image segmentation in this paper we propose a pde based level set method traditionally interfaces are represented by the zero level set of continuous level set functions instead we let the interfaces be represented by discontinuities of piecewise constant level set functions each level set function can at convergence only take two values i e it can only be 1 or 1 thus our method is related to phase field methods some of the properties of standard level set methods are preserved in the proposed method while others are not using this new method for interface problems we need to minimize a smooth convex functional under a quadratic constraint the level set functions are discontinuous at convergence but the minimization functional is smooth we show numerical results using the method for segmentation of digital images

calibration of rainfall runoff models application of global optimization to the sacramento soil moisture accounting model conceptual rainfall runoff models are difficult to calibrate by means of automatic methods one major reason for this is the inability of conventional procedures to locate the globally optimal set of parameters this paper investigates the consistency with which two global optimization methods the shuffled complex evolution sce ua method developed by the authors and the multistart simplex msx method are able to find the optimal parameter set during calibration of the sacramento soil moisture accounting model sac sma of the national weather service river forecast system nwsrfs in the first phase of this study error free synthetic data are used to conduct a comparative evaluation of the algorithms under ideal conditions in 10 independent trials of each algorithm in which 13 parameters of the sac sma model were optimized simultaneously the sce ua method achieved a 100 success rate in locating the precise global optimum i e the true parameter values while the msx method failed in all trials even with more than twice the number of function evaluations in the second phase historical data from the leaf river watershed are used to conduct a comparative evaluation of the algorithms under real conditions using two different estimation criteria drms and hmle the sce ua algorithm obtained consistently lower function values and more closely grouped parameter estimates while using one third fewer function evaluations than the msx algorithm

additive preconditioning and aggregation in matrix computations we combine our novel svd free additive preconditioning with aggregation and other relevant techniques to facilitate the solution of a linear system of equations and other fundamental matrix computations our analysis and experiments show the power of our algorithms guide us in selecting most effective policies of preconditioning and aggregation and provide some new insights into these and related subjects compared to the popular svd based multiplicative preconditioners our additive preconditioners are generated more readily and for a much larger class of matrices furthermore they better preserve matrix structure and sparseness and have a wider range of applications e g they facilitate the solution of a consistent singular linear system of equations and of the eigenproblem

multiobjective calibration with pareto preference ordering an application to rainfall runoff model calibration 1 automatic calibration routines for hydrologic models with multiple objective capabilities are becoming increasingly popular due to advances in computational power population based optimization techniques and the recognition that a single performance measure such as the root mean square error is no longer sufficient to characterize the complex behavior of the catchment however as more objective functions are included in the calibration the number of pareto optimal solutions as well as the number of near pareto optimal parameter sets increases the calibration problem quickly becomes a decision making problem in the practical sense users of automatic calibration routines have to face the task of selecting a set of suitable model parameters from the numerous pareto optimal sets a new method of automatic calibration is proposed which combines an effective optimization routine based on multiobjective genetic algorithms and pareto preference ordering in this case pareto optimal points that are also pareto optimal in different subspace combinations of the objective functions space are preferred the proposed method is used to calibrate the mike11 nam rainfall runoff model for a danish catchment the results indicated that the method is able to sieve through the numerous pareto optimal solutions and select a small number of preferred solutions this is extremely useful to modelers who typically are required to provide the best estimated parameter sets with good overall model performance

convex non convex image segmentation a convex non convex variational model is proposed for multiphase image segmentation we consider a specially designed non convex regularization term which adapts spatially to the image structures for a better control of the segmentation boundary and an easy handling of the intensity inhomogeneities the nonlinear optimization problem is efficiently solved by an alternating directions methods of multipliers procedure we provide a convergence analysis and perform numerical experiments on several images showing the effectiveness of this procedure

reconstructing distances among objects from their discriminability we describe a principled way of imposing a metric representing dissimilarities on any discrete set of stimuli symbols handwritings consumer products x ray films etc given the probabilities with which they are discriminated from each other by a perceiving system such as an organism person group of experts neuronal structure technical device or even an abstract computational algorithm in this procedure one does not have to assume that discrimination probabilities are monotonically related to distances or that the distances belong to a predefined class of metrics such as minkowski discrimination probabilities do not have to be symmetric the probability of discriminating an object from itself need not be a constant and discrimination probabilities are allowed to be 0 s and 1 s the only requirement that has to be satisfied is regular minimality a principle we consider the defining property of discrimination for ordered stimulus pairs a b b is least frequently discriminated from a if and only if a is least frequently discriminated from b regular minimality generalizes one of the weak consequences of the assumption that discrimination probabilities are monotonically related to distances the probability of discriminating a from a should be less than that of discriminating a from any other object this special form of regular minimality also underlies such traditional analyses of discrimination probabilities as multidimensional scaling and cluster analysis

a new selection metric for multiobjective hydrologic model calibration a novel selection metric called convex hull contribution chc is introduced for solving multiobjective mo optimization problems with pareto fronts that can be accurately approximated by a convex curve the hydrologic model calibration literature shows that many biobjective calibration problems with a proper setup result in such pareto fronts the chc selection approach identifies a subset of archived nondominated solutions whose map in the objective space forms convex approximation of the pareto front the optimization algorithm can sample solely from these solutions to more accurately approximate the convex shape of the pareto front it is empirically demonstrated that chc improves the performance of pareto archived dynamically dimensioned search pa dds when solving mo problems with convex pareto fronts this conclusion is based on the results of several benchmark mathematical problems and several hydrologic model calibration problems with two or three objective functions the impact of chc on pa dds performance is most evident when the computational budget is somewhat limited it is also demonstrated that 1 000 solution evaluations limited budget in this study is sufficient for pa dds with chc based selection to achieve very high quality calibration results relative to the results achieved after 10 000 solution evaluations

region competition unifying snakes region growing and bayes mdl for multiband image segmentation we present a novel statistical and variational approach to image segmentation based on a new algorithm named region competition this algorithm is derived by minimizing a generalized bayes minimum description length mdl criterion using the variational principle the algorithm is guaranteed to converge to a local minimum and combines aspects of snakes balloons and region growing the classic snakes balloons and region growing algorithms can be directly derived from our approach we provide theoretical analysis of region competition including accuracy of boundary location criteria for initial conditions and the relationship to edge detection using filters it is straightforward to generalize the algorithm to multiband segmentation and we demonstrate it on gray level images color images and texture images the novel color model allows us to eliminate intensity gradients and shadows thereby obtaining segmentation based on the albedos of objects it also helps detect highlight regions

computing symmetric nonnegative rank factorizations an algorithm is described for the nonnegative rank factorization nrf of some completely positive cp matrices whose rank is equal to their cp rank the algorithm can compute the symmetric nrf of any nonnegative symmetric rank r matrix that contains a diagonal principal submatrix of that rank and size with leading cost o rm2 operations in the dense case the algorithm is based on geometric considerations and is easy to implement the implications for matrix graphs are also discussed

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

perturbation analysis of system hessenberg and hessenberg triangular forms abstract laub and linnemann have shown by numerous examples that the system hessenberg form of a single input controllable system can be extremely sensitive to perturbations however the question as to which quantities can be used to assess the condition of the system hessenberg form remains unanswered this paper gives an answer to this question perturbation bounds and condition numbers for the system hessenberg form of a single input controllable system are derived in various ways moreover this paper also derives perturbation bounds and condition numbers for the hessenberg triangular form of a more general single input controllable system

matrices with special reference to applications in psychometrics multidimensional scaling item response theory and factor analysis may be considered three major contributions of psychometricians to statistics matrix theory played an important role in early developments of these techniques unfortunately nonlinear models are currently very prevalent in these areas still one can identify several areas of psychometrics where matrix algebra plays a prominent role they include analysis of asymmetric square tables multiway data analysis reduced rank regression analysis and multiple set t set canonical correlation analysis among others in this article we review some of the important matrix results in these areas and suggest future studies

multiphase image segmentation via modica mortola phase transition we propose a novel multiphase segmentation model built upon the celebrated phase transition model of modica and mortola in material sciences and a properly synchronized fitting term that complements it the proposed sine sinc model outputs a single multiphase distribution from which each individual segment or phase can be easily extracted theoretical analysis is developed for the gamma convergence behavior of the proposed model and the existence of its minimizers since the model is not quadratic nor convex for computation we adopted the convex concave procedure cccp that has been developed in the literatures of both computational nonlinear pdes and neural computation numerical details and experiments on both synthetic and natural images are presented

a class of preconditioners based on matrix splitting for nonsymmetric linear systems given a general matrix splitting a m n where m is nonsingular a new factorization scheme in terms of factorized and splitting matrices is given using the sherman morrison formula theoretical analysis shows that the factorization can give an ldu decomposition of a under some special choices we propose and implement a class of preconditioners based on this factorization combining with dropping rules a number of numerical experiments from discrete convection diffusion equation and some practical problems show that the new preconditioner is efficient and is comparable to existing preconditioners in term of storage requirement and computational cost

multi region active contours with a single level set function segmenting an image into an arbitrary number of coherent regions is at the core of image understanding many formulations of the segmentation problem have been suggested over the past years these formulations include among others axiomatic functionals which are hard to implement and analyze and graph based alternatives which impose a non geometric metric on the problem we propose a novel method for segmenting an image into an arbitrary number of regions using an axiomatic variational approach the proposed method allows to incorporate various generic region appearance models while avoiding metrication errors in the suggested framework the segmentation is performed by level set evolution yet contrarily to most existing methods here multiple regions are represented by a single non negative level set function the level set function evolution is efficiently executed through the voronoi implicit interface method for multi phase interface evolution the proposed approach is shown to obtain accurate segmentation results for various natural 2d and 3d images comparable to state of the art image segmentation algorithms

multiple region segmentation without supervision by adaptive global maximum clustering in this paper we propose a new method of segmenting an image into several sets of pixels with similar intensity values called regions a multiple region segmentation problem is unstable because the result considerably depends on the number of regions given a priori therefore one of the most important tasks in solving the problem is automatically finding the number of regions the method we propose is able to find the reasonable number of distinct regions not only for clean images but also for noisy ones our method is made up of two procedures first we develop the adaptive global maximum clustering in this procedure we deal with an image histogram and automatically obtain the number of significant local maxima of the histogram this number indicates the number of different regions in the image second we derive a simple and fast calculation to segment an image composed of distinct multiple regions then we split an image into multiple regions according to the previous procedure finally we show the efficiency of our method by comparing it with other previous methods

on the use of spatial regularization strategies to improve calibration of distributed watershed models 1 hydrologic models require the specification of unknown model parameters via calibration to historical input output data for spatially distributed models the large number of unknowns makes the calibration problem poorly conditioned spatial regularization can help to stabilize the problem by facilitating inclusion of additional information while a common regularization approach is to apply a scalar multiplier to the prior estimate of each parameter field this can cause problems by simultaneously changing both the mean and the variance of the distribution this paper explores a multiple criteria regularization approach that facilitates adjustment of the mean variance and shape of the parameter distribution using prior information to constrain the problem while providing sufficient degrees of freedom to enable model performance improvements we also test simple squashing functions to help in maintaining conceptually reasonable parameter values throughout the spatial domain we apply the method to three basins in the context of the distributed model intercomparison project dmip2 obtaining considerable performance improvements at the basin outlet however the prior parameter estimates are found to give much better performance at the interior points treated as ungauged suggesting that the spatial information has not been properly exploited the results also suggest that basin outlet hydrographs may not be particularly sensitive to spatial parameter variability and that an overall basin mean value may be sufficient for flow forecasting at the outlet although not at the interior points we discuss weaknesses in our study approach and suggest diagnostically more powerful strategies to be pursued

refined error analyses of cholesky factorization this paper concentrates on the practical cholesky process for solving positive definite symmetric systems of linear equations in standard floating point arithmetic two self contained round off analyses are presented which lead to novel a posteriors error bounds of practical use and to refined a priori results of more theoretical significance unmotivated assumptions are avoided with the aim of being constructive throughout

weak local linear discretizations for stochastic differential equations convergence and numerical schemes weak local linear wll discretizations are playing an increasing role in the construction of effective numerical integrators and inference methods for stochastic differential equations sdes with additive noise however due to limitations in the existing numerical implementations of wll discretizations the resulting integrators and inference methods have either been restricted to particular classes of autonomous sdes or showed low computational efficiency another limitation is the absence of a systematic theoretical study of the rate of convergence of the wll discretizations and numerical integratos this task is the main purpose of the present paper a second goal is introducing a new wll scheme that overcomes the numerical limitations mentioned above additionally a comparative analysis between the new wll scheme and some conventional weak integrators is also presented

a sparse approximate inverse preconditioner for nonsymmetric linear systems this paper is concerned with a new approach to preconditioning for large sparse linear systems a procedure for computing an incomplete factorization of the inverse of a nonsymmetric matrix is developed and the resulting factorized sparse approximate inverse is used as an explicit preconditioner for conjugate gradient type methods some theoretical properties of the preconditioner are discussed and numerical experiments on test matrices from the harwell boeing collection and from tim davis s collection are presented our results indicate that the new preconditioner is cheaper to construct than other approximate inverse preconditioners furthermore the new technique insures convergence rates of the preconditioned iteration which are comparable with those obtained with standard implicit preconditioners

two purposes for matrix factorization a historical appraisal matrix factorization in numerical linear algebra nla typically serves the purpose of restating some given problem in such a way that it can be solved more readily for example one major application is in the solution of a linear system of equations in contrast within applied statistics psychometrics as p a much more common use for matrix factorization is in presenting possibly spatially the structure that may be inherent in a given data matrix obtained on a collection of objects observed over a set of variables the actual components of a factorization are now of prime importance and not just as a mechanism for solving another problem we review some connections between nla and as p and their respective concerns with matrix factorization and the subsequent rank reduction of a matrix we note in particular that several results available for many decades in as p were more recently re discovered in the nla literature two other distinctions between nla and as p are also discussed briefly how a generalized singular value decomposition might be defined and the differing uses for the newer methods of optimization based on cyclic or iterative projections

on the wedderburn guttman theorem abstract let a be a u by v matrix of rank a and let m and n be u by g and v by g matrices respectively such that m an is nonsingular then rank a n m an 1m a a g where g rank an m an 1m a rank m an this is called wedderburn guttman theorem what happens if m an is rectangular and or singular in this paper we investigate conditions under which the regular inverse m an 1 can be replaced by a g inverse m an of some kind thereby extending the wedderburn guttman theorem the resultant conditions look similar to those arising in seemingly unrelated contexts namely cochran s and related theorems on distributions of quadratic forms involving a normal random vector

preconditioned iterative methods for solving linear least squares problems new preconditioning strategies for solving m times n overdetermined large and sparse linear least squares problems using the conjugate gradient for least squares cgls method are described first direct preconditioning of the normal equations by the balanced incomplete factorization bif for symmetric and positive definite matrices is studied and a new breakdown free strategy is proposed preconditioning based on the incomplete lu factors of an n times n submatrix of the system matrix is our second approach a new way to find this submatrix based on a specific weighted transversal problem is proposed numerical experiments demonstrate different algebraic and implementational features of the new approaches and put them into the context of current progress in preconditioning of cgls it is shown in particular that the robustness demonstrated earlier by the bif preconditioning strategy transfers into the linear least squares solvers and the use of the weighted transversal helps to improve the lu

tensor deflation for candecomp parafac part i alternating subspace update algorithm candecomp parafac cp approximates multiway data by sum of rank 1 tensors unlike matrix decomposition the procedure which estimates the best rank r tensor approximation through r sequential best rank 1 approximations does not work for tensors because the deflation does not always reduce the tensor rank in this paper we propose a novel deflation method for the problem when one factor matrix of a rank r cp decomposition is of full column rank the decomposition can be performed through r 1 rank 1 reductions at each deflation stage the residue tensor is constrained to have a reduced multilinear rank for decomposition of order 3 tensors of size r times r times r and rank r estimation of one rank 1 tensor has a computational cost of cal o r 3 per iteration which is lower than the cost cal o r 4 of the als algorithm for the overall cp decomposition the method can be extended to tracking one or a few rank one tensors of slow changes or inspect variations of common patterns in individual datasets

matrix factorizations and their perturbations there are several ways in which a matrix can be factorized as a product of two special matrices these factorizations are often used in numerical analysis and many perturbation bounds useful in such analysis have been proved by various authors in this paper a simple method which leads to these results and several new ones is discussed in the last section related results on the lipschitz continuity of the matrix absolute value are surveyed

effective and efficient algorithm for multiobjective optimization of hydrologic models 1 practical experience with the calibration of hydrologic models suggests that any single objective function no matter how carefully chosen is often inadequate to properly measure all of the characteristics of the observed data deemed to be important one strategy to circumvent this problem is to define several optimization criteria objective functions that measure different complementary aspects of the system behavior and to use multicriteria optimization to identify the set of nondominated efficient or pareto optimal solutions in this paper we present an efficient and effective markov chain monte carlo sampler entitled the multiobjective shuffled complex evolution metropolis moscem algorithm which is capable of solving the multiobjective optimization problem for hydrologic models moscem is an improvement over the shuffled complex evolution metropolis scem ua global optimization algorithm using the concept of pareto dominance rather than direct single objective function evaluation to evolve the initial population of points toward a set of solutions stemming from a stable distribution pareto set the efficacy of the moscem ua algorithm is compared with the original mocom ua algorithm for three hydrologic modeling case studies of increasing complexity

unsupervised multiphase segmentation a phase balancing model variational models have been studied for image segmentation application since the mumford shah functional was introduced in the late 1980s in this paper we focus on multiphase segmentation with a new regularization term that yields an unsupervised segmentation model we propose a functional that automatically chooses a favorable number of phases as it segments the image the primary driving force of the segmentation is the intensity fitting term while a phase scale measure complements the regularization term we propose a fast yet simple brute force numerical algorithm and present experimental results showing the robustness and stability of the proposed model

sensitivity analyses for factorizations of sparse or structured matrices abstract for a unique factorization of a matrix b the effect of sparsity or other structure on measuring the sensitivity of the factors of b to some change g in b is considered in particular norm based analyses of the qr and cholesky factorizations are examined if b is structured but g is not it is shown that the expressions for the condition numbers are identical to those when b is not structured but because of the structure the condition numbers may be easier to estimate if g is structured whether b is or not then the expressions for the condition numbers can change and it is shown how to derive the new expressions cases where b and g have the same sparsity structure occur often here for the qr factorization an example shows the value of the new expression can be arbitrarily smaller but for the cholesky factorization of a tridiagonal matrix and perturbation the value of the new expression cannot be significantly different from the value of the old one thus taking account of sparsity can show the condition is much better than would be suggested by ignoring it but only for some classes of problems and perhaps only for some types of factorization the generalization of these ideas to other factorizations is discussed

on the convergence of multiobjective evolutionary algorithms we consider the usage of evolutionary algorithms for multiobjective programming mop i e for decision problems with alternatives taken from a real valued vector space and evaluated according to a vector valued objective function selection mechanisms possibilities of temporary fitness deterioration and problems of unreachable alternatives for such multiobjective evolutionary algorithms moeas are studied theoretical properties of moeas such as stochastic convergence with probability 1 are analyzed

a variant of the level set method and applications to image segmentation in this paper we propose a variant of the level set formulation for identifying curves separating regions into different phases in classical level set approaches the sign of n level set functions are utilized to identify up to 2 phases the novelty in our approach is to introduce a piecewise constant level set function and use each constant value to represent a unique phase if 2 phases should be identified the level set function must approach 2 predetermined constants we just need one level set function to represent 2 unique phases and this gains in storage capacity further the reinitializing procedure requested in classical level set methods is superfluous using our approach the minimization functional for our approach is locally convex and differentiable and thus avoids some of the problems with the nondifferentiability of the delta and heaviside functions numerical examples are given and we also compare our method with related approaches

componentwise analysis of direct factorization of real symmetric and hermitian matrices abstract we derive componentwise error bound for the factorization h gjg t where h is a real symmetric matrix g has full column rank and j is diagonal with 1 s on the diagonal we also derive a componentwise forward error bound that is we bound the difference between the exact and the computed factor g in the cases where such a bound is possible we extend these results to the hermitian case and to the well known bunch parlett factorization finally we prove bounds for the scaled condition of the matrix g and show that the factorization can have the rank revealing property

wedderburn rank reduction and krylov subspace method for tensor approximation part 1 tucker case new algorithms are proposed for the tucker approximation of a 3 tensor accessed only through a tensor by vector by vector multiplication subroutine in the matrix case the krylov methods are methods of choice to approximate the dominant column and row subspaces of a sparse or structured matrix given through a matrix by vector operation using the wedderburn rank reduction formula we propose a matrix approximation algorithm that computes the krylov subspaces and can be generalized to 3 tensors the numerical experiments show that on quantum chemistry data the proposed tensor methods outperform the minimal krylov recursion of savas and elden

discrete region competition for unknown numbers of connected regions we present a discrete unsupervised multiregion competition algorithm for image segmentation over different energy functionals the number of regions present in an image does not need to be known a priori nor their photometric properties the algorithm jointly estimates the number of regions their photometries and their contours the required regularization is provided by defining a region as a connected set of pixels the evolving contours in the image are represented by computational particles that move as driven by an energy minimization algorithm we present an efficient discrete algorithm that allows minimizing a range of well known energy functionals under the topological constraint of regions being connected components the presented framework and algorithms are implemented in the open source insight toolkit image processing library

on a multivariate eigenvalue problem part i algebraic theory and a power method multivariate eigenvalue problems for symmetric and positive definite matrices arise from multivariate statistics theory where coefficients are to be determined so that the resulting linear combinations of sets of random variables are maximally correlated by using the method of lagrange multipliers such an optimization problem can be reduced to the multivariate eigenvalue problem for over 30 years an iterative method proposed by horst psychometrika 26 1961 pp 129 149j has been used for solving the multivariate eigenvalue problem yet the theory of convergence has never been complete the number of solutions to the multivariate eigenvalue problem also remains unknown this paper contains two new results by using the degree theory a closed form on the cardinality of solutions for the multivariate eigenvalue problem is first proved a convergence property of horst s method by forming it as a generalization of the so called power method is then proved the discussion leads to new formulations of nume

inequalities for the schatten p norm iv we present some inequalities for the schattenp norm of operators on a hilbert space it is shown among other things that ifa is an operator such that rea a 0 then for any operatorx ax xa p 2a x p also for any two operatorsa andb a b 2 2 a b 2 2 2 a b 2 2

a new energy framework with distribution descriptors for image segmentation segmentation of the target object s from images that have multiple complicated regions mixture intensity distributions or are corrupted by noise poses a challenge for the level set models in addition the conventional piecewise smooth level set models normally require prior knowledge about the number of image segments to address these problems we propose a novel segmentation energy function with two distribution descriptors to model the background and the target the single background descriptor models the heterogeneous background with multiple regions then the target descriptor takes into account the intensity distribution and incorporates local spatial constraint our descriptors which have more complete distribution information construct the unique energy function to differentiate the target from the background and are more tolerant of image noise we compare our approach to three other level set models 1 the chan vese 2 the multiphase level set and 3 the geodesic level set this comparison using 260 synthetic images with varying levels and types of image noise and medical images with more complicated backgrounds showed that our method outperforms these models for accuracy and immunity to noise on an additional set of 300 synthetic images our model is also less sensitive to the contour initialization as well as to different types and levels of noise