the generalized eigenvalue problem for tridiagonal symmetric interval matrices an algorithm to compute the eigenintervals of the generalized eigenvalue problem for tridiagonal symmetric matrices a b with interval entries is developed the algorithm is based on the sturm sequences and requires twice as much computational effort as for point matrices the algorithm is applied to the problem of computing the natural frequencies of a vibrating beam where the manufacturing tolerance is taken into consideration

interval analysis of vibrating systems interval calculus is a tool to evaluate a mathematical expression for ranges of values of its parameters the basic mathematical operations are defined in the interval algebra vibrating systems having system parameters initial conditions or forcing functions defined as intervals rather than single valued quantities have been modelled using interval calculus the eigenvalue problem for determining natural frequencies and vibrating modes of a system in the case of system matrix elements given as a range of values cannot be solved by exhaustion due to the prohibitively large number of solutions of the point number eigenvalue problem eigenvalue solution with interval evaluation of the commonly used numerical techniques is not feasible because if they are applied directly the solution intervals diverge an optimization technique was used to obtain the minimum radius intervals of the solution for the eigenvalue sensitivity problem to assure monotonicity and absolute inclusion necessary for convergence to the exact interval a converging interval halving sequence was developed for finite width interval matrices for numerical tests of the method a monte carlo solution was developed the results showed that interval analysis can predict the range of the eigenvalues with sufficient accuracy the response of a linear system to general excitation for interval matrices of the system parameters cannot be found with interval evaluation of the commonly used numerical techniques because if they are applied directly the solution intervals diverge interval modal analysis and the interval solution of the eigenvalue problem were developed an application is presented for the dynamic response of a rotor with interval bearing properties

multi view aam fitting and construction active appearance models aams are generative parametric models that have been successfully used in the past to model deformable objects such as human faces the original aams formulation was 2d but they have recently been extended to include a 3d shape model a variety of single view algorithms exist for fitting and constructing 3d aams but one area that has not been studied is multi view algorithms in this paper we present multi view algorithms for both fitting and constructing 3d aams fitting an aam to an image consists of minimizing the error between the input image and the closest model instance i e solving a nonlinear optimization problem in the first part of the paper we describe an algorithm for fitting a single aam to multiple images captured simultaneously by cameras with arbitrary locations rotations and response functions this algorithm uses the scaled orthographic imaging model used by previous authors and in the process of fitting computes or calibrates the scaled orthographic camera matrices in the second part of the paper we describe an extension of this algorithm to calibrate weak perspective or full perspective camera models for each of the cameras in essence we use the human face as a non rigid calibration grid we demonstrate that the performance of this algorithm is roughly comparable to a standard algorithm using a calibration grid in the third part of the paper we show how camera calibration improves the performance of aam fitting a variety of non rigid structure from motion algorithms both single view and multi view have been proposed that can be used to construct the corresponding 3d non rigid shape models of a 2d aam in the final part of the paper we show that constructing a 3d face model using non rigid structure from motion suffers from the bas relief ambiguity and may result in a scaled stretched compressed model we outline a robust non rigid motion stereo algorithm for calibrated multi view 3d aam construction and show how using calibrated multi view motion stereo can eliminate the bas relief ambiguity and yield face models with higher 3d fidelity

solving systems of linear fuzzy equations by parametric functions an improved algorithm buckley and qu proposed a method to solve systems of linear fuzzy equations basically in their method the solutions of all systems of linear crisp equations formed by the a levels are calculated we propose a new method for solving systems of linear fuzzy equations based on a practical algorithm using parametric functions in which the variables are given by the fuzzy coefficients of the system by observing the monotonicity of the parametric functions in each variable i e each fuzzy coefficient in the system we improve the algorithm by calculating less parametric functions and less evaluations of these parametric functions we show that our algorithm is much more efficient than the method of buckley and qu

on optimal solution of interval linear equations for interval linear algebraic systems bf a x bf b we consider the problem of component wise evaluation of the set sigma exists exists bf a bf b a 1 b mid a in bf a b in bf b formed by all solutions of ax b when a and b vary independently in bf a and bf b respectively an iterative pss algorithm is introduced that computes optimal exact componentwise estimates of sigma exists exists and its convergence is proved under fairly general conditions on the interval system we introduce the concept of a sequentially guaranteeing algorithm as a reasonable compromise between the requirements for the interval result to be guaranteed and to be obtained in a practically acceptable time

vertex method for computing functions of fuzzy variables abstract this paper is to introduce a new approach the vertex method for computing functions of fuzzy variables the method is based on the cut concept and interval analysis the vertex method can avoid abnormality due to the discretization technique on the variables domain and the widening of the resulting function value set due to multi occurrence of variables in the functional expression by conventional interval analysis methods the algorithm is very easy to implement and can be applied to many practical problems some examples are given to illustrate the applications

solution of the scalar wave equation over very long distances using nonlinear solitary waves relation to finite difference methods the linear wave equation represents the basis of many linear electromagnetic and acoustic propagation problems features that a computational model must have to capture large scale realistic effects for over the horizon or oth radar communication for example include propagation of short waves with scattering and partial absorption by complex topography for these reasons it is not feasible to use green s function or any simple integral method which neglects these intermediate effects and requires a known propagation function between source and observer in this paper we describe a new method for propagating such short waves over long distances including intersecting scattered waves the new method appears to be much simpler than conventional high frequency schemes lagrangian particle based approaches such as ray tracing become very complex in 3 d especially for waves that may be expanding or even intersecting the other high frequency scheme in common use the eikonal also has difficulty with intersecting waves our approach based on nonlinear solitary waves concentrated about centroid surfaces of physical wave features is related to that of whitham 1 which involves solving wave fronts propagating on characteristics then the evolving electromagnetic or acoustic field can be approximated as a collection of propagating co dimension one surfaces for example 2 d surfaces in three dimensions this approach involves solving propagation equations discretely on an eulerian grid to approximate the linear wave equation however to propagate short waves over long distances conventional eulerian numerical methods which attempt to resolve the structure of each wave require far too many grid cells and are not feasible on current or foreseeable computers instead we employ an extended wave equation that captures the important features of the propagating waves this method is first formulated at the partial differential equation pde level as a wave equation with an added confining term that involves both a positive and a negative dissipation once we have the stable pde the discrete formulation is simply a multidimensional pde with stable perturbations caused by the discretization the resulting discrete solution can then be low order and very simple and yet remain stable over arbitrarily long times when discretized and solved on an eulerian grid this new method allows far coarser grids than required by conventional resolution considerations while still accounting for the effects of varying atmospheric and topographic features an important point is that the new method is in the same form as conventional discrete wave equation methods however the conventional solution eventually decays and only the intermediate asymptotic solution can be used simply by adding an extra term we show that a nontrivial true asymptotic solution can be obtained a similar solitary wave based approach has been used successfully in a different problem involving vorticity confinement for a number of years

fuzzy finite element method for frequency response function analysis of uncertain structures a concept is presented for incorporating fuzzy uncertainties in dynamic e nite element analyses of uncertain structures the objective is twofold the e rst goal is to clarify and extend the classical fuzzy e nite element ffe method as it was introduced for static analyses the shortcomings of the classical approach are described and an extension to a generalized approach is proposed this generalized approach is proven to be a more realistic and therefore more reliable concept for taking uncertainty into account the second goal is to illustrate the applicability of the method for dynamic analyses the classical and the generalized approach are compared using an eigenvalue analysis of a simple numerical example the ffe method is also applied to the calculation of the total envelope frequency response function frf using the modal superposition principle this method requires safe approximations of the individual mode envelope frequency response functions for this purpose a number of safe approximate optimization strategies are introduced the numerical example shows that useful results are obtained using this ffe approach for frf calculations

fuzzy weighted averages and implementation of the extension principle abstract this paper addresses the computational aspect of the extension principle when the principle is applied to algebraic mappings and in particular to weighted average operations in risk and decision analysis a computational algorithm based on the cut representation of fuzzy sets and interval analysis is described the method provides a discrete but exact solution to extended algebraic operations in a very efficient and simple manner examples are given to illustrate the method and its relation to other discrete methods and the exact approach by non linear programming the algorithm has been implemented in a computer program which can handle very general extended algebraic operations on fuzzy numbers

constructing membership functions using statistical data abstract determination of the membership functions is vital in practical applications of the fuzzy set theory this paper presents a guideline to construct the membership functions for fuzzy sets whose elements have a defining feature with a known probability density function pdf in the universe of discourse the method finds the smallest fuzzy set which assigns high average membership values to those objects with the defining features distributed according to the given pdf it is show that for any pdf the method is capable of generating membership functions in accordance with the possibility probability consistency principle membership functions derived from some of the well known pdfs and an application in solving noise contaminated linear system of equations are presented

a bayesian collocation method for static analysis of structures with unknown but bounded uncertainties abstract this paper proposes a bayesian collocation method bcm for static analysis of structures with unknown but bounded uncertainties or interval uncertainties unlike the existing collocation methods which construct surrogate model using preselected samples at fixed points the bcm chooses samples sequentially and adaptively based on bayesian inference theorem the bcm constructs a crude gaussian process surrogate model of structural response with selected samples then two acquisition functions named upper expected improvement ei up function and lower expected improvement ei lo function are defined simultaneously to guide the selection of next samples the samples will be collocated to the place where extrema of structural response are most likely to be found the procedure will repeat until the convergence criterion is met the extrema of the selected samples can be regarded as structural response bounds fewer samples are needed compared with existing collocation methods two numerical examples are used to demonstrate the accuracy and efficiency of the proposed method the results are compared with those obtained by existing methods the application of the method is performed by an engineering example the results show the validity of the proposed method

multidimensional morphable models a framework for representing and matching object classes we describe a flexible model for representing images of objects of a certain class known a priori such as faces and introduce a new algorithm for matching it to a novel image and thereby perform image analysis the flexible model known as a multidimensional morphable model is learned from example images of objects of a class in this paper we introduce an effective stochastic gradient descent algorithm that automatically matches a model to a novel image several experiments demonstrate the robustness and the broad range of applicability of morphable models our approach can provide novel solutions to several vision tasks including the computation of image correspondence object verification and image compression

long distance wave computation using nonlinear solitary waves a recently developed method is described to propagate short wave equation pulses over indefinite distances and through regions of varying indices of refraction including multiple reflections the method wave confinement utilizes a newly developed nonlinear partial differential equation pde that propagates basis functions according to the wave equation these basis functions are generated as stable solitary waves where the discretized equation can be solved without any numerical dissipation the method can also be used to solve for harmonic waves in the high frequency eikonal limit including multiple arrivals the solution involves discretizing the wave equation on a uniform eulerian grid and adding a simple nonlinear confinement term this term does not change the amplitude integrated through each point on the pulse surface or the propagation velocity or arrival time and yet results in capturing the waves as thin surfaces that propagate as thin nonlinear solitary waves and remain 2 3 grid cells in thickness indefinitely with no numerical spreading a new feature described in this paper involves computing scattering of short pulses from complex objects such as complete aircraft a simple immersed surface approach is used that utilizes the same uniform grid as the propagation and avoids complex body fitted or adaptive grid schemes the new method should be useful in areas of wave propagation from radar scattering and long distance communications to cell phone transmission

interval field model and interval finite element analysis uncertain parameters with inherent spatial variability are commonly encountered in engineering modeling of this kind of spatial uncertainty plays a fundamental role in structural uncertainty analysis which provides a necessary basis for subsequent uncertainty propagation through the system an interval field model for quantification of spatial uncertain parameters is proposed by which only the upper and lower bounds of the spatial uncertain parameters rather than their precise probability distributions are required the dependency can be fully considered by the proposed interval field model with the information of dependency an interval k l expansion is presented as a combination of deterministic functions with uncorrelated standard interval variables through which the continuous spatial interval field with dependency can be expressed only by very limited intervals necessary mathematical illustrations are provided for the proposed interval field model and the interval k l expansion the sampling method for the interval field model is given providing a robust numerical analysis basis for subsequent structural uncertainty analysis when the interval field model is applied in finite element analysis of structures with spatial uncertain parameters the non deterministic equilibrium equations with interval factors is then formulated the mcs method and the perturbation method are developed for solution of the derived interval equilibrium equations the feasibility and validity of the proposed interval field model and corresponding interval finite element methods are verified by numerical examples where the upper and lower bounds of the responses such as the displacements and the stresses of structures with spatial uncertain parameters are computed and compared

traveltime and amplitude estimation using wavefront construction we have developed and implemented a new method for estimating traveltimes and amplitudes in a general smooth two dimensional 2 d model the basic idea of this wavefront wf construction approach is to use ray tracing to estimate a new wf from the old one the wf is defined as a curve in 2 d of constant traveltime from the source the ray direction and amplitude will then be a function of s the distance along the front to maintain a sufficiently small sampling distance along the wf it is scanned at every time step and new rays are interpolated whenever the distance between two rays becomes larger than a predefined limit as the wavefronts are constructed the data i e traveltimes amplitude coefficients etc are transferred to the receivers by interpolation within the ray cells advantages of the wf construction method are its flexibility robustness and accuracy first second and later arrivals may be found at any point in the model any shape of the initial wavefront is possible the drawb

a survey of non probabilistic uncertainty treatment in finite element analysis the objective of this paper is to critically review the emerging non probabilistic approaches for uncertainty treatment in finite element analysis the paper discusses general theoretical and practical aspects of both the interval and fuzzy finite element analysis first the applicability of the non probabilistic concepts for numerical uncertainty analysis is discussed from a theoretical viewpoint the necessary conditions for a useful application of the non probabilistic concepts are determined and are proven to be complementary rather than competitive to the classical probabilistic approach the second part of the paper focuses on numerical aspects of the interval finite element method it describes two principal strategies for the implementation i e the anti optimisation and the interval arithmetic approach and gives a state of the art of the interval finite element algorithms available from literature it is shown how the application of the interval arithmetic approach to the classical finite element procedure can result in a severe overestimation of the uncertainty on the output and the main sources of this conservatism are identified a numerical example in the final part of the paper illustrates the capabilities of the different strategies on an eigenfrequency analysis of a built up benchmark structure

active appearance models revisited active appearance models aams and the closely related concepts of morphable models and active blobs are generative models of a certain visual phenomenon although linear in both shape and appearance overall aams are nonlinear parametric models in terms of the pixel intensities fitting an aam to an image consists of minimising the error between the input image and the closest model instances i e solving a nonlinear optimisation problem we propose an efficient fitting algorithm for aams based on the inverse compositional image alignment algorithm we show that the effects of appearance variation during fitting can be precomputed projected out using this algorithm and how it can be extended to include a global shape normalising warp typically a 2d similarity transformation we evaluate our algorithm to determine which of its novel aspects improve aam fitting performance

the static displacement and the stress analysis of structures with bounded uncertainties using the vertex solution theorem abstract in practical engineering on account of physical imperfections model inaccuracies and system complexities almost all structures have physical and geometrical uncertainties in some degrees in this paper we focus on the static response problem and the stress distribution of structures with bounded uncertainties in terms of the vertex notation in interval analysis a new mathematical proof of the vertex solution theorem which is used to determine the supremum and the infimum of the static responses of structures with bounded uncertainties is given then we extend this theorem for calculating the interval stress and the deformation distributions which was only used to determine the supremum and the infimum of the static responses of structures with bounded uncertainties the basic idea of the vertex solution theorem is to convert the interval linear equations into systems of equivalent deterministic linear equations the resulting deterministic linear equations are then solved by using the familiar techniques like the gauss elimination method or the gauss seidel iteration method another advantage of this method is that the increase in number of the uncertain parameters in a structure element will not result in the increase in number of the uncertain elements in the stiffness matrix for practical engineering problem with bounded uncertainties the chief thing is to avoid the disadvantages of large calculation and much runtime of the vertex solution in this paper we present the parallel arithmetic of the vertex solution theorem which can be used in large scale computations in practical engineering to avoid much runtime on the other hand we can use the symmetrical characteristic of the stiffness matrix and the substructure method to reduce the computational effort in practical engineering the numerical results of the typical example presented by deif calculated by the proposed vertex solution theorem show that the width between the supremum and the infimum on the static response yielded by the vertex solution theorem is tighter than those produced by the kcn s method presented by koyluoglu cakmak and nielson the numerical examples of the two dimensional shell with an edge fixed and the two dimensional shell with surrounding fixed are used to illustrate the computational feasibility of the vertex solution theorem

active appearance models we describe a new method of matching statistical models of appearance to images a set of model parameters control modes of shape and gray level variation learned from a training set we construct an efficient iterative matching algorithm by learning the relationship between perturbations in the model parameters and the induced image errors

hybrid uncertainty propagation in structural acoustic systems based on the polynomial chaos expansion and dimension wise analysis abstract hybrid uncertainties are ubiquitous in the structural acoustic analysis and greatly affect the behaviors of the coupled system however requirements of small uncertainties and rewriting simulation codes for the numerical analysis are always necessary for current methods in this context a non intrusive dual layer analysis procedure i e a hybrid method of the polynomial chaos expansion and dimension wise analysis pce dw is proposed in this paper the pce procedure based on the sparse grid collocation strategy is utilized to handle the random uncertainty while the dw procedure is employed to evaluate the interval bounds of statistical characteristics of the system response the pce dw also applies to the structural acoustic analysis with only random or interval parameters the investigation of two structural acoustic systems with hybrid uncertainties finally demonstrates the accuracy efficiency and capability of the proposed method

shape ambiguities in structure from motion this paper examines the fundamental ambiguities and uncertainties inherent in recovering structure from motion by examining the eigenvectors associated with null or small eigenvalues of the hessian matrix we can quantify the exact nature of these ambiguities and predict how they affect the accuracy of the reconstructed shape our results for orthographic cameras show that the bas relief ambiguity is significant even with many images unless a large amount of rotation is present similar results for perspective cameras suggest that three or more frames and a large amount of rotation are required for metrically accurate reconstruction

a dual layer dimension wise fuzzy finite element method for structural analysis with epistemic uncertainties abstract epistemic uncertainty propagation is gaining momentum but remains a great challenge in the numerical analysis of a mechanical system a dual layer dimension wise analysis procedure for the fuzzy finite element method is proposed the core algorithm layer is to identify the minimal and maximal min max points of each slice of the response surface at the zero cut the data processing layer aims to filtrate the min max points of each response slice at each nonzero cut subsequently the min max input vectors at any alpha cut are dimension wisely assembled and the response interval at alpha cut is calculated by deterministic finite element analysis at the min max input vectors finally the fuzzy description of the system response is recomposed two examples are investigated to verify both the accuracy and efficiency of the proposed method

possibilistic kalman filtering for radar 2d tracking abstract standard kalman filter skf introduced by kalman in the 60s has gained a non estimated importance in control as well as in robotics community its importance arises from the obtained optimal result in the sense of variance minimization under stochastic gaussian and unbiased perturbations and when the state model as well as the measurement model are precisely known however when the last requirement is relaxed such that one or more parameters governing the models are ill defined and rather given in terms of interval evaluations chen et al ieee trans aerospace electr syst 33 1 1997 251 259 have proposed interval kalman filter ikf by extending the arithmetic operations to interval calculus in this paper we rather assume that the uncertainty pervading some parameters of the models are given in terms of possibility distributions 21 this leads to a formulation of possibilistic kalman filtering pkf which agrees with ikf the same example of 2d radar tracking is tackled comparisons with ikf are investigated as well the influence of the modelling process on the performance of the filter besides the proposal permits to capture certainty qualified information which cannot be obtained from ikf

high dimensional model representation method for fuzzy structural dynamics abstract uncertainty propagation in multi parameter complex structures possess significant computational challenges this paper investigates the possibility of using the high dimensional model representation hdmr approach when uncertain system parameters are modeled using fuzzy variables in particular the application of hdmr is proposed for fuzzy finite element analysis of linear dynamical systems the hdmr expansion is an efficient formulation for high dimensional mapping in complex systems if the higher order variable correlations are weak thereby permitting the input output relationship behavior to be captured by the terms of low order the computational effort to determine the expansion functions using the cut method scales polynomically with the number of variables rather than exponentially this logic is based on the fundamental assumption underlying the hdmr representation that only low order correlations among the input variables are likely to have significant impacts upon the outputs for most high dimensional complex systems the proposed method is first illustrated for multi parameter nonlinear mathematical test functions with fuzzy variables the method is then integrated with a commercial finite element software adina modal analysis of a simplified aircraft wing with fuzzy parameters has been used to illustrate the generality of the proposed approach in the numerical examples triangular membership functions have been used and the results have been validated against direct monte carlo simulations it is shown that using the proposed hdmr approach the number of finite element function calls can be reduced without significantly compromising the accuracy

modal approaches for the stochastic finite element analysis of structures with material and geometric uncertainties this paper presents two efficient modal approaches as an alternative to direct formulations for the time harmonic dynamic analysis of structures with random material and shape parameters in both approaches the structural eigenproblem is solved and complemented by a sensitivity analysis to the random parameters the modal perturbation stochastic finite element method sfem then condenses the response sensitivities to assess the response variability the mixed perturbation monte carlo sfem assesses the response statistics by sampling the structural eigenmodes according to the perturbation estimation of their probability distribution functions pdfs geometric uncertainties are handled through an appropriate shape parameterisation and a shape design sensitivity analysis two numerical applications examine both approaches in terms of accuracy variability level and computational requirements the applications involve a plate bending problem with random young modulus or edge length and a plate with a random flatness default particular observations related to the influence of the parameter pdfs in simulation based methods are also provided

kink dynamics in the highly discrete sine gordon system abstract we study kink dynamics in a very discrete sine gordon system where the kink width is of the order of the lattice spacing numerical simulations exhibit new properties of kinks in this case they lose the memory of their initial velocity and propagate preferentially at well defined velocities which correspond to quasi steady states while a kink moving at other velocities suffers relatively high rates of radiation of small amplitude oscillations when a small external driving force is applied to the system the same velocities appear as plateus in the strongly nonlinear mobility of the kink the energy radiated by the kink is calculated for a simple model that preserves the discrete character of the system and the preferential velocities for the kink are obtained to good accuracy similar results may be expected to be valid for other discrete systems manifesting topological solitons the numerical simulations reveal also new stable multiple kink excitations which can propagate almost freely in extremely discrete systems where ordinary simple kinks are pinned to the lattice by discreteness the stability of the multiple kinks is discussed

lucas kanade 20 years on a unifying framework since the lucas kanade algorithm was proposed in 1981 image alignment has become one of the most widely used techniques in computer vision applications range from optical flow and tracking to layered motion mosaic construction and face coding numerous algorithms have been proposed and a wide variety of extensions have been made to the original formulation we present an overview of image alignment describing most of the algorithms and their extensions in a consistent framework we concentrate on the inverse compositional algorithm an efficient algorithm that we recently proposed we examine which of the extensions to lucas kanade can be used with the inverse compositional algorithm without any significant loss of efficiency and which cannot in this paper part 1 in a series of papers we cover the quantity approximated the warp update rule and the gradient descent approximation in future papers we will cover the choice of the error function how to allow linear appearance variation and how to impose priors on the parameters

fuzzy finite element approach for the analysis of imprecisely defined systems many engineering systems are too complex to be defined in precise mathematical terms they often contain information and features that are vague imprecise qualitative linguistic or incomplete the traditional deterministic and probabilistic techniques are not adequate to analyze such systems this paper aims at developing a fuzzy finite element approach for the analysis of imprecisely defined systems the development of the methodology starts from the basic concepts of fuzzy numbers of fuzzy arithmetic and implements suitably defined fuzzy calculus concepts such as differentiation and integration for the derivation manipulation and solution of the finite element equations simple stress analysis problems involving vaguely defined geometry material properties external loads and boundary conditions are solved to establish and to illustrate the new procedure the approach developed is applicable to systems that are described in linguistic terms as well as those that are described by incomplete information if complete data are known the method handles the information similar to that of a probabilistic approach the present approach represents a unique methodology that enables us to handle certain types of imprecisely known data more realistically compared with the existing procedures