optimization algorithms on subspaces revisiting missing data problem in low rank matrix low rank matrix approximation has applications in many fields such as 3d reconstruction from an image sequence and 2d filter design in this paper one issue with low rank matrix approximation is re investigated the missing data problem much effort was devoted to this problem and the wiberg algorithm or the damped newton algorithm were recommended in previous studies however the wiberg or damped newton algorithms do not suit for large especially long matrices because one needs to solve a large linear system in every iteration in this paper we revitalize the usage of the levenberg marquardt algorithm for solving the missing data problem by utilizing the property that low rank approximation is a minimization problem on subspaces in two proposed implementations of the levenberg marquardt algorithm one only needs to solve a much smaller linear system in every iteration especially for long matrices simulations and experiments on real data show the superiority of the proposed algorithms though the proposed algorithms achieve a high success rate in estimating the optimal solution by random initialization as illustrated by real examples it still remains an open issue how to properly do the initialization in a severe situation that is a large amount of data is missing and with high level noise

3d trajectory reconstruction under perspective projection we present an algorithm to reconstruct the 3d trajectory of a moving point from its correspondence in a collection of temporally non coincidental 2d perspective images given the time of capture that produced each image and the relative camera poses at each time instant triangulation based solutions do not apply as multiple views of the point may not exist at each time instant we represent a 3d trajectory using a linear combination of compact trajectory basis vectors such as the discrete cosine transform basis that have been shown to approximate object independence we note that such basis vectors are also coordinate independent which allows us to directly use camera poses estimated from stationary areas in the scene in contrast to nonrigid structure from motion techniques where cameras are simultaneously estimated this reduces the reconstruction optimization to a linear least squares problem allowing us to robustly handle missing data that often occur due to motion blur texture deformation and self occlusion we present an algorithm to determine the number of trajectory basis vectors individually for each trajectory via a cross validation scheme and refine the solution by minimizing the geometric error the relationship between point and camera motion can cause degeneracies to occur we geometrically analyze the problem by studying the relationship of the camera motion point motion and trajectory basis vectors we define the reconstructability of a 3d trajectory under projection and show that the estimate approaches the ground truth when reconstructability approaches infinity this analysis enables us to precisely characterize cases when accurate reconstruction is achievable we present qualitative results for the reconstruction of several real world scenes from a series of 2d projections where high reconstructability can be guaranteed and report quantitative results on motion capture sequences

comments on design of optimal dead beat controllers it is shown that the system considered in the numerical example of the above paper belongs to a class of systems with n states and r inputs for which the set of all minimum time dead beat mtdb controllers is parameterized by an arbitrary matrix of order lambda 1 r n x n lambda r where is the greatest integer less than the ratio n r since the example treats the case of n 3 r 2 the mdtb controller is parameterized by a single scalar element

dead beat control of linear discrete time systems the paper considers the problem of determining the various classes of state feed back dead beat controllers for linear multivariable discrete time systems unlike earlier work attention is not restricted to the special case of minimum time dead beat controllers a general dead beat control problem is posed and solved the solution is based principally on an approach of closed loop eigenstructure assignment the minimum of free parameters in each class are specified and used in an explicit parametric form for the feedback gain matrix the concept of the greatest minimum of free parameters is introduced finally the method is illustrated by a numerical example

linearization techniques for singular initial value problems of ordinary differential equations linearization methods for singular initial value problems in second order ordinary differential equations are presented these methods result in linear constant coefficients ordinary differential equations which can be integrated analytically thus yielding piecewise analytical solutions and globally smooth solutions the accuracy of these methods is assessed by comparisons with exact and asymptotic solutions of homogeneous and non homogeneous linear and nonlinear lane emden equations it is shown that linearization methods provide accurate solutions even near the singularity or the zeros of the solution in fact it is shown that linearization methods provide more accurate solutions than methods based on perturbation methods it is also shown that the accuracy of these techniques depends on the nonlinearity of the ordinary differential equations and may not be a monotonic function of the step size

defect correction finite element method for the stationary incompressible magnetohydrodynamics equation in this study we give the defect correction finite element method for the stationary incompressible mhd equation firstly the nonlinear mhd equation is solved with an artificial viscosity term then the numerical solution is improved on the same grid by a linearized defect correction technique then we give the numerical analysis including stability analysis and error analysis the numerical analysis proves that our method is stable and has an optimal convergence rate then we give some numerical results from the numerical results we can see that our method is efficient for solving the mhd equations

minimum gain minimum time deadbeat controllers abstract this paper considers the design of minimum gain minimum time deadbeat controllers mgmtdc for linear discrete time systems a new direct method for constructing the mgmtdc of minimum frobenius norm is developed compared with many existing techniques the proposed method gives analytic expressions for the constructed mgmtdc requires less computational effort and does not require that the transition matrix be nonsingular

construction and parameterization of a class of minimum time deadbeat controllers abstract a method for constructing and parametrizing a class of minimum time deadbeat controllers mtdc for a discrete time system is presented the number of construction steps is equal to the controllability index of the controllable part of the system the method reduces the considered problem to that of solving at most r linear equations at each step where r is the input dimension the minimum number of free parameters that characterizes the constructed class of mtdc is specified and arranged in a new compact linear parametric form of mtdc contrary to many existing methods the presented method does not require the system to be controllable

minimal basis subspace representation a unified framework for rigid and non rigid motion segmentation motion segmentation and non rigid structure from motion are two challenging computer vision problems that have attracted numerous research interests while the previous works handle these two problems separately we present a general motion segmentation framework in this paper for solving these two seemingly different problems in a unified manner at the heart of our general motion segmentation framework is a model selection mechanism based on finding the minimal basis subspace representation by seeking the joint sparse representation of the data matrix however such formulation is np hard and we solve the convex proxy instead unlike other compressive sensing related works this convex proxy solution is insufficient for our problem the convex relaxation artefacts and noise yield multiple subspace representations making identification of the exact number of motion subspaces challenging we solve for the right number of subspaces by transforming this problem into a facility location problem with global cost and solve the factor graph formulation using max product belief propagation message passing

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

the monotone iterative method and zeros of bessel functions for nonlinear singular derivative dependent bvp in the presence of upper and lower solutions abstract in this paper we consider a class of nonlinear singular boundary value problems x y x x f x y x x y x 0 0 x 1 y 0 y 1 0 for 1 we assume that the source function f x y x y is lipschitz in x y and one sided lipschitz in y the initial approximations are an upper solution u 0 x and a lower solution v 0 x which can be ordered in one way v 0 x u 0 x or the other u 0 x v 0 x we propose an iterative scheme and establish the existence of solutions bounded by v 0 and u 0 and allow f y to take both positive and negative values the method is constructive in nature and can be used to generate solutions of the nonlinear singular boundary value problems

optimal adaptive solution of piecewise regular systems of ivps with unknown switching hypersurface we study the solution of piecewise regular systems of ivps with a switching hypersurface defined by an unknown event function our aim is to design an algorithm that is not based on heuristic steps it does not use information on the event function and preserves the order known for regular systems the algorithm is based on the adaptive mesh which differs from the equidistant mesh only by a single point we provide a rigorous that is independent of heuristic arguments analysis of its error and cost

the convergence of shooting methods for singular boundary value problems we investigate the convergence properties of single and multiple shooting when applied to singular boundary value problems particular attention is paid to the well posedness of the process it is shown that boundary value problems can be solved efficiently when a high order integrator for the associated singular initial value problems is available moreover convergence results for a perturbed newton iteration are discussed

nonrigid structure from motion estimating shape and motion with hierarchical priors this paper describes methods for recovering time varying shape and motion of nonrigid 3d objects from uncalibrated 2d point tracks for example given a video recording of a talking person we would like to estimate the 3d shape of the face at each instant and learn a model of facial deformation time varying shape is modeled as a rigid transformation combined with a nonrigid deformation reconstruction is ill posed if arbitrary deformations are allowed and thus additional assumptions about deformations are required we first suggest restricting shapes to lie within a low dimensional subspace and describe estimation algorithms however this restriction alone is insufficient to constrain reconstruction to address these problems we propose a reconstruction method using a probabilistic principal components analysis ppca shape model and an estimation algorithm that simultaneously estimates 3d shape and motion for each instant learns the ppca model parameters and robustly fills in missing data points we then extend the model to represent temporal dynamics in object shape allowing the algorithm to robustly handle severe cases of missing data

rigid structure from motion from a blind source separation perspective we present an information theoretic approach to define the problem of structure from motion sfm as a blind source separation one given that for almost all practical joint densities of shape points the marginal densities are non gaussian we show how higher order statistics can be used to provide improvements in shape estimates over the methods of factorization via singular value decomposition svd bundle adjustment and bayesian approaches previous techniques have either explicitly or implicitly used only second order statistics in models of shape or noise a further advantage of viewing sfm as a blind source problem is that it easily allows for the inclusion of noise and shape models resulting in maximum likelihood ml or maximum a posteriori map shape and motion estimates a key result is that the blind source separation approach has the ability to recover the motion and shape matrices without the need to explicitly know the motion or shape pdf we demonstrate that it suffices to know whether the pdf is sub or super gaussian i e semi parametric estimation and derive a simple formulation to determine this from the data we provide extensive experimental results on synthetic and real tracked points in order to quantify the improvement obtained from this technique

a note on the sturmian theorem for singular boundary value problems in this paper we study properties of the eigenfunctions of singular boundary value problems the main results show that for a class of singular boundary value problems the sturmian oscillation theorem holds it is a generalization of the standard case to the singular case

a simple prior free method for non rigid structure from motion factorization this paper proposes a simple prior free method for solving the non rigid structure from motion nrsfm factorization problem other than using the fundamental low order linear combination model assumption our method does not assume any extra prior knowledge either about the non rigid structure or about the camera motions yet it works effectively and reliably producing optimal results and not suffering from the inherent basis ambiguity issue which plagued most conventional nrsfm factorization methods our method is very simple to implement which involves solving a very small sdp semi definite programming of fixed size and a nuclear norm minimization problem we also present theoretical analysis on the uniqueness and the relaxation gap of our solutions extensive experiments on both synthetic and real motion capture data assuming following the low order linear combination model are conducted which demonstrate that our method indeed outperforms most of the existing non rigid factorization methods this work offers not only new theoretical insight but also a practical everyday solution to nrsfm

singular free boundary problems and local bifurcation theory a constructive method applicable to the solution of a wide class of free boundary problems is presented a solution dependent transformation technique is introduced by considering a singular limit of the transformation a related problem to which local bifurcation theory may be applied is derived by inverting the near singular mapping between the two problems an expression for solutions of the original problem is obtained the method is illustrated by the study of a singularly perturbed elliptic equation approximate solutions are constructed and the validity of the approximations established by means of the contraction mapping theorem

kronecker markov prior for dynamic 3d reconstruction recovering dynamic 3d structures from 2d image observations is highly under constrained because of projection and missing data motivating the use of strong priors to constrain shape deformation in this paper we empirically show that the spatiotemporal covariance of natural deformations is dominated by a kronecker pattern we demonstrate that this pattern arises as the limit of a spatiotemporal autoregressive process and derive a kronecker markov random field as a prior distribution over dynamic structures this distribution unifies shape and trajectory models of prior art and has the individual models as its marginals the key assumption of the kronecker mrf is that the spatiotemporal covariance is separable into the product of a temporal and a shape covariance and can therefore be modeled using the matrix normal distribution analysis on motion capture data validates that this distribution is an accurate approximation with significantly fewer free parameters using the trace norm we present a convex method to estimate missing data from a single sequence when the marginal shape distribution is unknown the kronecker markov distribution fit to a single sequence outperforms state of the art methods at inferring missing 3d data and additionally provides covariance estimates of the uncertainty

recursive eigenstructure assignment in linear systems in arbel and rath 1985 an iterative eigenvalue assignment algorithm was presented this algorithm has the shortcomings of clustering most of the closed loop eigenvalues on the real axis a general method is given here for recursive eigenstructure assignment in linear systems the method eliminates the shortcomings of the algorithm of arbel and rath it is shown that the right and left eigenvectors of the closed loop system matrix can be determined in terms of those of a small dimension matrix a c it is also shown that the results of arbel and tse 1980 are special cases of those of the proposed method moreover the arbitrary parameters beyond eigenvalue assignment are shown to be embedded in the choice of a certain arbitrary invertible matrix s furthermore computer oriented steps are outlined for recursive eigenstructure assignment in large scale systems numerical examples are worked out to illustrate the generality and feasibility of the proposed method

recovering the missing components in a large noisy low rank matrix application to sfm in computer vision it is common to require operations on matrices with missing data for example because of occlusion or tracking failures in the structure from motion sfm problem such a problem can be tackled allowing the recovery of the missing values if the matrix should be of low rank when noise free the filling in of missing values is known as imputation imputation can also be applied in the various subspace techniques for face and shape classification online recommender systems and a wide variety of other applications however iterative imputation can lead to the recovery of data that is seriously in error in this paper we provide a method to recover the most reliable imputation in terms of deciding when the inclusion of extra rows or columns containing significant numbers of missing entries is likely to lead to poor recovery of the missing parts although the proposed approach can be equally applied to a wide range of imputation methods this paper addresses only the sfm problem the performance of the proposed method is compared with jacobs and shum s methods for sfm

parameterization of the class of deadbeat controllers this technical note deals with the problem of parameterizing the class of deadbeat controllers for a given system through the minimum of parameters it is shown that if n and m are dimensions of the state vector and control vector respectively and the i s are the controllability indexes then almost the whole class of deadbeat controllers is explicitly parameterizable through mn mu 1 3 mu 2 2m 1 mu m parameters

the application of runge kutta schemes to singular initial value problems a theory for explicit runge kutta schemes applied to the initial value problem for a first order system of differential equations with a singularity of the first kind is developed it is shown that in general the order of convergence is at most two but that the usual order up to a logarithmic term can be obtained for level three and four schemes applied to specific problems 1 introduction in this paper we examine the application of explicit runge kutta schemes to the numerical solution of the problem

parameterization of the class of deadbeat controllers via the theory of decoupling a novel approach is presented for parameterizing the class of minimum time deadbeat controllers mtdc through the minimum number of parameters the approach is based on the theory of decoupling and the properties of square decouplable systems the main result is a compact parametric form for the class of mtdc of a discrete time system contrary to many existing techniques no special assumption on the invertibility of the transition matrix is required

computing smooth time trajectories for camera and deformable shape in structure from motion with occlusion we address the classical computer vision problems of rigid and nonrigid structure from motion sfm with occlusion we assume that the columns of the input observation matrix w describe smooth 2d point trajectories over time we then derive a family of efficient methods that estimate the column space of w using compact parameterizations in the discrete cosine transform dct domain our methods tolerate high percentages of missing data and incorporate new models for the smooth time trajectories of 2d points affine and weak perspective cameras and 3d deformable shape we solve a rigid sfm problem by estimating the smooth time trajectory of a single camera moving around the structure of interest by considering a weak perspective camera model from the outset we directly compute euclidean 3d shape reconstructions without requiring postprocessing steps such as euclidean upgrade and bundle adjustment our results on real sfm data sets with high percentages of missing data compared positively to those in the literature in nonrigid sfm we propose a novel 3d shape trajectory approach that solves for the deformable structure as the smooth time trajectory of a single point in a linear shape space a key result shows that compared to state of the art algorithms our nonrigid sfm method can better model complex articulated deformation with higher frequency dct components while still maintaining the low rank factorization constraint finally we also offer an approach for nonrigid sfm when w is presented with missing data

a defect correction method for unsteady conduction convection problems ii time discretization in this paper a fully discrete defect correction mixed finite element method mfem for solving the non stationary conduction convection problems in two dimension which is leaded by combining the back euler time discretization with the two step defect correction in space is presented in this method we solve the nonlinear equations with an added artificial viscosity term on a finite element grid and correct these solutions on the same grid using a linearized defect correction technique the stability and the error analysis are derived the theory analysis shows that our method is stable and has a good convergence property some numerical results are also given which show that this method is highly efficient for the unsteady conduction convection problems

a closed form solution to non rigid shape and motion recovery recovery of three dimensional 3d shape and motion of non static scenes from a monocular video sequence is important for applications like robot navigation and human computer interaction if every point in the scene randomly moves it is impossible to recover the non rigid shapes in practice many non rigid objects e g the human face under various expressions deform with certain structures their shapes can be regarded as a weighted combination of certain shape bases shape and motion recovery under such situations has attracted much interest previous work on this problem bregler c hertzmann a and biermann h 2000 in proc int conf computer vision and pattern recognition brand m 2001 in proc int conf computer vision and pattern recognition torresani l yang d alexander g and bregler c 2001 in proc int conf computer vision and pattern recognition utilized only orthonormality constraints on the camera rotations rotation constraints this paper proves that using only the rotation constraints results in ambiguous and invalid solutions the ambiguity arises from the fact that the shape bases are not unique an arbitrary linear transformation of the bases produces another set of eligible bases to eliminate the ambiguity we propose a set of novel constraints basis constraints which uniquely determine the shape bases we prove that under the weak perspective projection model enforcing both the basis and the rotation constraints leads to a closed form solution to the problem of non rigid shape and motion recovery the accuracy and robustness of our closed form solution is evaluated quantitatively on synthetic data and qualitatively on real video sequences

comments on eigenstructure assignment in linear multivariable systems a parametric solution the approaches to eigenstructure assignment in 1 3 and the above paper are shown to be equivalent

a difference method for a singular boundary value problem of second order the standard three point discretization applied to the numerical solution of linear boundary value problems for second order systems with a singularity at the origin is investigated a number of numerical examples illustrating the theoretical results are presented

sequential non rigid structure from motion using physical priors we propose a new approach to simultaneously recover camera pose and 3d shape of non rigid and potentially extensible surfaces from a monocular image sequence for this purpose we make use of the extended kalman filter based simultaneous localization and mapping ekf slam formulation a bayesian optimization framework traditionally used in mobile robotics for estimating camera pose and reconstructing rigid scenarios in order to extend the problem to a deformable domain we represent the object s surface mechanics by means of navier s equations which are solved using a finite element method fem with these main ingredients we can further model the material s stretching allowing us to go a step further than most of current techniques typically constrained to surfaces undergoing isometric deformations we extensively validate our approach in both real and synthetic experiments and demonstrate its advantages with respect to competing methods more specifically we show that besides simultaneously retrieving camera pose and non rigid shape our approach is adequate for both isometric and extensible surfaces does not require neither batch processing all the frames nor tracking points over the whole sequence and runs at several frames per second

low rank matrix fitting based on subspace perturbation analysis with applications to structure from motion the task of finding a low rank r matrix that best fits an original data matrix of higher rank is a recurring problem in science and engineering the problem becomes especially difficult when the original data matrix has some missing entries and contains an unknown additive noise term in the remaining elements the former problem can be solved by concatenating a set of r column matrices that share a common single r dimensional solution space unfortunately the number of possible submatrices is generally very large and hence the results obtained with one set of r column matrices will generally be different from that captured by a different set ideally we would like to find that solution that is least affected by noise this requires that we determine which of the r column matrices i e which of the original feature points are less influenced by the unknown noise term this paper presents a criterion to successfully carry out such a selection our key result is to formally prove that the more distinct the r vectors of the r column matrices are the less they are swayed by noise this key result is then combined with the use of a noise model to derive an upper bound for the effect that noise and occlusions have on each of the r column matrices it is shown how this criterion can be effectively used to recover the noise free matrix of rank r finally we derive the affine and projective structure from motion sfm algorithms using the proposed criterion extensive validation on synthetic and real data sets shows the superiority of the proposed approach over the state of the art

iterated defect correction for the solution of singular initial value problems we investigate the convergence properties of the iterated defect correction idec method based on the implicit euler rule for the solution of singular initial value problems with a singularity of the first kind we show that the method retains its classical order of convergence which means that the sequence of approximations obtained during the iteration shows gradually growing order of convergence limited by the smoothness of the data and technical details of the procedure

procrustean regression a flexible alignment based framework for nonrigid structure estimation non rigid structure from motion nrsfm is a fundamental problem of computer vision recently it has been shown that incorporating shape alignment in nrsfm can improve the performance significantly compared with the other algorithms which do not consider shape alignment however realizing this idea was at a cost of a heavy complicated process which limits its usefulness and possible extensions in this paper we propose a novel regression framework for nrsfm of which the variables 3d shapes are regularized based on their aligned shapes we show that this can be casted into an unconstrained problem or a problem with simple bound constraints which can be efficiently solved by existing solvers this framework can be easily integrated with numerous existing models and assumptions such as orthographic or perspective camera models occlusion low rank assumption smooth deformations and so on which makes it more practical for various real situations the experimental results show that the proposed method gives competitive result to the state of the art methods for orthographic projection with much less time complexity and memory requirement and outperforms the existing methods for perspective projection

eigenstructure assignment via linear state feedback control in fahmy and o reilly 1982 a b a parametric solution for the linear state feedback eigenstructure assignment problem is developed this approach is here modified in such a way as to accommodate the case where the set of closed loop eigenvalues and the set of open loop eigenvalues have elements in common it is particularly shown that each common eigenvalue introduces a number of partially free design parameter vectors in addition to the original free ones the paper concludes with illustrative numerical examples

collocation for singular boundary value problems of second order the collocation methods for solving linear boundary value problems with a singularity of the first kind are investigated the stability and convergence results are established particular attention is paid to the superconvergence properties of the methods which are different as in the classical case when the singularity is not present the results are extended to the nonlinear case and illustrated by numerical examples

properties of numerical algorithms related to computing controllability the numerical properties of some methods for computing controllability are used in an expository way to motivate a wider understanding of numerical computations in particular the numerical rank of a matrix the numerical stability of algorithms the sensitivity of problems and the scaling of problems are discussed a numerically stable algorithm is given for computing controllability but it is pointed out that a measure of the distance of the given system from the nearest uncontrollable system would be more useful and this appears to be an open computational problem

spline finite difference methods for singular two point boundary value problems in this paper we discuss the construction of a spline function for a class of singular two point boundary value problemx x u f x u u 0 a u 1 b 0 1 or 1 2 the boundary conditions may also be of the formu 0 0 u 1 b three point finite difference methods using the above splines are obtained for the solution of the boundary value problem these methods are of second order and are illustrated by four numerical examples

shape and motion from image streams under orthography a factorization method inferring scene geometry and camera motion from a stream of images is possible in principle but is an ill conditioned problem when the objects are distant with respect to their size we have developed a factorization method that can overcome this difficulty by recovering shape and motion under orthography without computing depth as an intermediate step an image stream can be represented by the 2fxp measurement matrix of the image coordinates of p points tracked through f frames we show that under orthographic projection this matrix is of rank 3 based on this observation the factorization method uses the singular value decomposition technique to factor the measurement matrix into two matrices which represent object shape and camera rotation respectively two of the three translation components are computed in a preprocessing stage the method can also handle and obtain a full solution from a partially filled in measurement matrix that may result from occlusions or tracking failures the method gives accurate results and does not introduce smoothing in either shape or motion we demonstrate this with a series of experiments on laboratory and outdoor image streams with and without occlusions

affine approximation for direct batch recovery of euclidian structure and motion from sparse data we present a batch method for recovering euclidian camera motion from sparse image data the main purpose of the algorithm is to recover the motion parameters using as much of the available information and as few computational steps as possible the algorithm thus places itself in the gap between factorisation schemes which make use of all available information in the initial recovery step and sequential approaches which are able to handle sparseness in the image data euclidian camera matrices are approximated via the affine camera model thus making the recovery direct in the sense that no intermediate projective reconstruction is made using a little known closure constraint the fa closure we are able to formulate the camera coefficients linearly in the entries of the affine fundamental matrices the novelty of the presented work is twofold firstly the presented formulation allows for a particularly good conditioning of the estimation of the initial motion parameters but also for an unprecedented diversity in the choice of possible regularisation terms secondly the new autocalibration scheme presented here is in practice guaranteed to yield a least squares estimate of the calibration parameters as a bi product the affine camera model is rehabilitated as a useful model for most cameras and scene configurations e g wide angle lenses observing a scene at close range experiments on real and synthetic data demonstrate the ability to reconstruct scenes which are very problematic for previous structure from motion techniques due to local ambiguities and error accumulation

asymptotically correct error estimation for collocation methods applied to singular boundary value problems we discuss an a posteriori error estimate for collocation methods applied to boundary value problems in ordinary differential equations with a singularity of the first kind as an extension of previous results we show the asymptotical correctness of our error estimate for the most general class of singular problems where the coefficient matrix is allowed to have eigenvalues with positive real parts this requires a new representation of the global error for the numerical solution obtained by piecewise polynomial collocation when applied to our problem class

analytical and numerical treatment of a singular initial value problem in avalanche modeling we discuss a leading edge model used in the computation of the run out length of dry flowing avalanches the model has the form of a singular initial value problem for a scalar ordinary differential equation describing the avalanche dynamics existence uniqueness and smoothness properties of the analytical solution are shown we also prove the existence of a unique root of the solution moreover we present a fortran 90 code for the numerical computation of the run out length the code is based on a solver for singular initial value problems which is an implementation of the acceleration technique known as iterated defect correction based on the implicit euler method

on the numerical solution of singular boundary value problems of second order by a difference method the standard three point discretization applied to the numerical solution of nonlinear boundary value problems for second order systems with a singularity of the first kind is investigated the results are extended to the boundary value problems arising in practical problems from mechanics and chemistry a number of numerical examples illustrating the theoretical results is presented

parametrization of minimum time output deadbeat controllers for all classes of invertible systems this paper considers the problem of parametrizing through the minimum number of parameters the minimum time output deadbeat controller mtodc for a given linear multivariable output controllable system s a b c e having n states m inputs p outputs and a transfer function matrix h z of full rank r the main result is a new compact linear parametric form for a class of mtodc that zeros the outputs of the system s in the minimum number of control iterations 1 where 1 i 1 2 r assumed to be decreasingly ordered are equal lo the negatives of the right factorization indices at infinity of h z the developed parametric class is a linear function in effective free parameters the principal benefits of this explicit parametrization of mtodcs lie in the ability to accommodate directly various different design criteria

on a singular boundary value problem arising in the theory of shallow membrane caps abstract we investigate the following singular boundary value problem which originates from the theory of shallow membrane caps t 3 u t t 3 1 8 u 2 t a 0 u t b 0 t 2 4 0 lim t 0 t 3 u t 0 u 1 0 where a 0 b 0 and are given constants we show the existence of a positive solution to the above problem by means of a generalized lower and upper functions method involving limiting processes we illustrate the theory by numerical experiments in which we used the new version of the matlab code sbvp based on polynomial collocation to approximate the solution of the membrane problem

optimal solution of a class of non autonomous initial value problems with unknown singularities in this paper we are interested in a rigorous analysis of the solution of a class of initial value problems with singularities we consider scalar non autonomous ivps with separated variables and one unknown singularity in each variable which leads to four unknown events in the two dimensional space many algorithms proposed in the literature for singular ivps are practically oriented they do not avoid heuristic arguments and are often checked for efficiency by numerical experiments we design an adaptive algorithm with no heuristic steps for solving the considered problems and provide rigorous bounds on the error we show that in spite of the presence of singularities the algorithm preserves the optimal error known for the regular case lower bounds on the error of any algorithm in the case of a larger number of singularities are also discussed

a novel approach to the numerical solution of boundary value problems on infinite intervals the classical numerical treatment of two point boundary value problems on infinite intervals is based on the introduction of a truncated boundary instead of infinity where appropriate boundary conditions are imposed then the truncated boundary allowing for a satisfactory accuracy is computed by trial motivated by several problems of interest in boundary layer theory here we consider boundary value problems on infinite intervals governed by a third order ordinary differential equation we highlight a novel approach to define the truncated boundary the main result is the convergence of the solution of our formulation to the solution of the original problem as a suitable parameter goes to zero in the proposed formulation the truncated boundary is an unknown free boundary and has to be determined as part of the solution for the numerical solution of the free boundary formulation a noniterative and an iterative transformation method are introduced furthermore we characterize the class of free boundary value problems that can be solved noniteratively a nonlinear flow problem involving two physical parameters and belonging to the characterized class of problems is then solved moreover the falkner skan equation with relevant boundary conditions is considered and representative results obtained by the iterative transformation method are listed for the homann flow all the obtained numerical results clearly indicate the effectiveness of our approach finally we discuss the possible extensions of the proposed approach and for the question of a priori error analysis

on the numerical solution of a class of singular two point boundary value problems abstract mono implicit runge kutta mirk formulae used in a deferred correction framework have proved to be a very effective way of solving first order systems of nonlinear two point boundary value problems we examine this general approach in detail and question whether mirk formulae are the most efficient class of runge kutta methods to use we also consider the extension of this deferred correction approach to deal with problems having a certain type of singularity at the end of the range of integration we consider the questions whether it is possible to derive mirk formulae which are applicable to such problems and whether it is in general possible to solve such problems automatically we will answer the first question in the affirmative however we will show that despite claims to the contrary appearing in the literature the answer to the second question is no