a new method for computing the closed loop eigenvalues of a discrete time algebraic riccati equation abstract we present a fast method for computing the closed loop eigenvalues of a discrete time algebraic riccati equation the method is close to the symplectic method for finding all the eigenvalues of a hamiltonian matrix and is based on a g orthogonal transformation which preserves structure and has desirable numerical properties the algorithm requires about one fourth the number of floating point operations and one half the storage of the qz algorithm

convergence cones near bifurcation the performance of predictor solver continuation methods near bifurcation points is considered a general theory is developed for the convergence of two commonly used solvers namely newton s method and the chord method near singular points of a nonlinear equation this theory provides a uniform treatment of continuation for simple bifurcation multiple bifurcation and multiple limit point bifurcation problems for these types of bifurcation it is shown that there are conical domains of attraction for the above iterative methods centred on solution branches which correspond to isolated roots of the algebraic bifurcation equations

the effect of spatial information characterization on 3d local feature descriptors designing local feature descriptors for 3d objects is a fundamental yet challenging task in 3d computer vision both geometry and spatial information descriptions are critical for a 3d local descriptor while most previous studies concentrate on the former one this paper investigates on how the characterization of spatial information would affect a 3d local descriptor in terms of descriptiveness robustness compactness and efficiency the evaluation process is deployed as follows first based on the analysis of representative spatial information characterization methods of existing local shape descriptors six typical characterization methods with different spatial dimensions and partition principles of spatial information are presented second three geometric attributes i e normal deviation local depth and shape index are respectively assigned to each point in the local surface for local geometry description creating a total of 18 different feature descriptors then a quantitative analysis of performance i e descriptiveness robustness compactness and efficiency for these descriptors is carried out on three benchmark datasets grounded on the experimental outcomes the traits merits and demerits of each spatial information encoding approach are eventually summarized this study reveals that different spatial information encoding approaches would bring significant effect on a local shape descriptor with respect to its discriminative power stability compactness and efficiency highlightsthe effect of spatial information encoding sie on 3d local descriptors is emphasized eighteen types of 3d local descriptors with different sie approaches are proposed quantitative evaluation of sie approaches is performed on three standard datasets the merits demerits and suitable applications of different sie ways are summarized instructive guidance is concluded for the design of a 3d local descriptor

rotational projection statistics for 3d local surface description and object recognition recognizing 3d objects in the presence of noise varying mesh resolution occlusion and clutter is a very challenging task this paper presents a novel method named rotational projection statistics rops it has three major modules local reference frame lrf definition rops feature description and 3d object recognition we propose a novel technique to define the lrf by calculating the scatter matrix of all points lying on the local surface rops feature descriptors are obtained by rotationally projecting the neighboring points of a feature point onto 2d planes and calculating a set of statistics including low order central moments and entropy of the distribution of these projected points using the proposed lrf and rops descriptor we present a hierarchical 3d object recognition algorithm the performance of the proposed lrf rops descriptor and object recognition algorithm was rigorously tested on a number of popular and publicly available datasets our proposed techniques exhibited superior performance compared to existing techniques we also showed that our method is robust with respect to noise and varying mesh resolution our rops based algorithm achieved recognition rates of 100 98 9 95 4 and 96 0 respectively when tested on the bologna uwa queen s and ca foscari venezia datasets

a method for registration of 3 d shapes the authors describe a general purpose representation independent method for the accurate and computationally efficient registration of 3 d shapes including free form curves and surfaces the method handles the full six degrees of freedom and is based on the iterative closest point icp algorithm which requires only a procedure to find the closest point on a geometric entity to a given point the icp algorithm always converges monotonically to the nearest local minimum of a mean square distance metric and the rate of convergence is rapid during the first few iterations therefore given an adequate set of initial rotations and translations for a particular class of objects with a certain level of shape complexity one can globally minimize the mean square distance metric over all six degrees of freedom by testing each initial registration one important application of this method is to register sensed data from unfixtured rigid objects with an ideal geometric model prior to shape inspection experimental results show the capabilities of the registration algorithm on point sets curves and surfaces

some perspectives on the eigenvalue problem this expository paper explores the relationships among a number of algorithms for solving eigenvalue problems including the power method subspace iteration the qr algorithm and the arnoldi and symmetric lanczos algorithms the symmetric lanczos algorithm is shown to be identical to the three term recursion stieltjes procedure for computing orthogonal polynomials with respect to a measure on the real line the connection between measures on the line and symmetric tridiagonal jacobi matrices is investigated if such a matrix is transformed by a step of the qr algorithm there is a corresponding transformation in the measure the tridiagonal matrices are also exploited for the construction of gaussian quadrature formulas for measures on the line the developments on the real line are replicated with suitable modifications on the unit circle via lanczos like procedures for unitary operators the best known procedure of this type is the recursion of szego for computing orthogonal polynomials on the

computation of singularities in large nonlinear systems a general approach is proposed for the numerical computation of one parameter singularities in the sense of golubitsky and schaeffer singularities and groups in bifurcation theory vol i springer verlag berlin 1985 the singularities of codimension three or less are organized in a new way our scheme is close to the numerics and in some cases gives explicit expressions for the defining equations obtained inductively where no such expressions were previously available the numerical computation by the lyapunov schmidt reduction summarizes work by many authors during the last 20 years treating all cases in a unified way a numerical example is provided where singularities with codimension up to three are detected and computed in a parameterized boundary value problem

large scale discrete time algebraic riccati equations doubling algorithm and error analysis we consider the numerical solution of large scale discrete time algebraic riccati equations the doubling algorithm is adapted with the iterates for a not computed explicitly but recursively the resulting algorithm is efficient with computational complexity and memory requirement proportional to the size of the problem and essentially converges quadratically an error analysis on the truncation of iterates and some numerical results are presented

the numerical solution of nonlinear equations having several parameters part iii equations with z 2 symmetry the computation of symmetry breaking bifurcation points of nonlinear multiparameter problems with z 2 reflectional symmetry is considered the numerical approach is based on recent work in singularity theory which is used to construct systems of equations and inequalities characterising various types of symmetry breaking bifurcation points numerical continuation methods are then used to follow paths of symmetry breaking bifurcations and hence compute regions in parameter space for which a problem has qualitatively similar bifurcation diagrams the power of the numerical approach is illustrated by computations of axisymmetric flows in the finite taylor problem

on the repeatability and quality of keypoints for local feature based 3d object retrieval from cluttered scenes 3d object recognition from local features is robust to occlusions and clutter however local features must be extracted from a small set of feature rich keypoints to avoid computational complexity and ambiguous features we present an algorithm for the detection of such keypoints on 3d models and partial views of objects the keypoints are highly repeatable between partial views of an object and its complete 3d model we also propose a quality measure to rank the keypoints and select the best ones for extracting local features keypoints are identified at locations where a unique local 3d coordinate basis can be derived from the underlying surface in order to extract invariant features we also propose an automatic scale selection technique for extracting multi scale and scale invariant features to match objects at different unknown scales features are projected to a pca subspace and matched to find correspondences between a database and query object each pair of matching features gives a transformation that aligns the query and database object these transformations are clustered and the biggest cluster is used to identify the query object experiments on a public database revealed that the proposed quality measure relates correctly to the repeatability of keypoints and the multi scale features have a recognition rate of over 95 for up to 80 occluded objects

accelerated projection and iterated projection methods with applications to nonlinear integral equations a new approach is developed to numerically approximate fixed points for a class of nonlinear operators by projection and iterated projection methods it is shown that isolated solutions can be determined to any desired accuracy for a fixed choice of the projection neither the structure nor the rank of the projection is altered to achieve increased accuracy instead the fixed point is decomposed by the given projection into a finite and an infinite dimensional component successive approximations to the infinite dimensional component are employed to construct a sequence of accelerated finite dimensional systems their solutions define a sequence of higher order approximations to the finite dimensional component and consequently to the fixed point error expressions are obtained and several examples of solving nonlinear integral equations are given

on the calculation of limit and bifurcation points in stability problems of elastic shells abstract numerical methods for the computation of singular points of nonlinear equations g u 0 are discussed where and are real parameters simple and double limit points are treated in some detail and numerical algorithms are presented and applied to elastic shell stability problems the case of simple symmetry breaking bifurcation points is also treated with applications to nonsymmetric bifurcation from axisymmetric states of deformation of shells of revolution

tracking by detection of 3d human shapes from surfaces to volumes 3d human shape tracking consists in fitting a template model to temporal sequences of visual observations it usually comprises an association step that finds correspondences between the model and the input data and a deformation step that fits the model to the observations given correspondences most current approaches follow the iterative closest point icp paradigm where the association step is carried out by searching for the nearest neighbors it fails when large deformations occur and errors in the association tend to propagate over time in this paper we propose a discriminative alternative for the association that leverages random forests to infer correspondences in one shot regardless the choice of shape parameterizations being surface or volumetric meshes we convert 3d shapes to volumetric distance fields and thereby design features to train the forest we investigate two ways to draw volumetric samples voxels of regular grids and cells from centroidal voronoi tessellation cvt while the former consumes considerable memory and in turn limits us to learn only subject specific correspondences the latter yields much less memory footprint by compactly tessellating the interior space of a shape with optimal discretization this facilitates the use of larger cross subject training databases generalizes to different human subjects and hence results in less overfitting and better detection the discriminative correspondences are successfully integrated to both surface and volumetric deformation frameworks that recover human shape poses which we refer to as tracking by detection of 3d human shapes it allows for large deformations and prevents tracking errors from being accumulated when combined with icp for refinement it proves to yield better accuracy in registration and more stability when tracking over time evaluations on existing datasets demonstrate the benefits with respect to the state of the art

a novel 3d vorticity based approach for automatic registration of low resolution range images this paper tackles the problem of feature matching and range image registration our approach is based on a novel set of discriminating three dimensional 3d local features named 3d vor vorticity in contrast to conventional local feature representation techniques which use the vector field i e surface normals to just construct their local reference frames the proposed feature representation exploits the vorticity of the vector field computed at each point of the local surface to capture the distinctive characteristics at each point of the underlying 3d surface the 3d vor descriptors of two range images are then matched using a fully automatic feature matching algorithm which identifies correspondences between the two range images correspondences are verified in a local validation step of the proposed algorithm and used for the pairwise registration of the range images quantitative results on low resolution kinect 3d data washington rgb d dataset show that our proposed automatic registration algorithm is accurate and computationally efficient the performance evaluation of the proposed descriptor was also carried out on the challenging low resolution washington rgb d kinect object dataset for the tasks of automatic range image registration reported experimental results show that the proposed local surface descriptor is robust to resolution noise and more accurate than state of the art techniques it achieves 90 registration accuracy compared to 50 69 2 and 52 for spin image 3d surf and sisi ld sift descriptors respectively highlightsa novel local surface descriptor 3d vor is proposed for surface representation the proposed 3d vor exploits the vector field s vorticity a novel pairwise registration algorithm is also proposed 3d vor is tested on low resolution dataset for range image registration 3d vor based registration achieves 90 accuracy on low resolution data and outperforms state of the art techniques

a novel local surface feature for 3d object recognition under clutter and occlusion this paper presents a highly distinctive local surface feature called the trisi feature for recognizing 3d objects in the presence of clutter and occlusion for a feature point we first construct a unique and repeatable local reference frame lrf using the implicit geometrical information of neighboring triangular faces we then generate three signatures from the three orthogonal coordinate axes of the lrf these signatures are concatenated and then compressed into a trisi feature finally we propose an effective 3d object recognition algorithm based on hierarchical feature matching we tested our trisi feature on two popular datasets rigorous experimental results show that the trisi feature was highly descriptive and outperformed existing algorithms under all levels of gaussian noise laplacian noise shot noise varying mesh resolutions occlusion and clutter moreover we tested our trisi based 3d object recognition algorithm on four standard datasets the experimental results show that our algorithm achieved the best overall recognition results on these datasets

a numerical investigation of the schaeffer homotopy in the problem of taylor couette flows numerical methods are used to study the 4 6 cell exchange process in the taylor vortex problem with particular reference to the homotopy devised by schaeffer the homotopy describes a transformation between two models one incorporating periodic boundary conditions and so referring to flows in an infinite annulus the other with realistic boundary conditions our calculations indicate that the former model is more complicated than previously suspected and lead to a better understanding of the consequences of schaeffer s device

semilinear elliptic equations with near critical growth rates on etudie les solutions symetriques de l equation elliptique semi lineaire u u u u p 1 0 u b 0 ou b est la boule unite dans r 3 on prend p proche de 5 on obtient une description asymptotique des solutions avec une grande norme

the symplectic eigenvalue problem the butterfly form the sr algorithm and the lanczos method abstract we discuss some aspects of the recently proposed symplectic butterfly form which is a condensed form for symplectic matrices any 2n 2n symplectic matrix can be reduced to this condensed form which contains 8n 4 nonzero entries and is determined by 4n 1 parameters the symplectic eigenvalue problem can be solved using the sr algorithm based on this condensed form the sr algorithm preserves this form and can be modified to work only with the 4n 1 parameters instead of the 4n2 matrix elements the reduction of symplectic matrices to symplectic butterfly form has a close analogy to the reduction of arbitrary matrices to hessenberg form a lanczos like algorithm for reducing a symplectic matrix to butterfly form is also presented

a general procedure for the direct computation of turning and bifurcation points we address the formulation and numerical implementation of general purpose algorithme suitable for the direct computation of stability points in non linear solid mechanics the effectiveness of the resulting algorithm is illustrated by numerical simulation that include bifurcation and loss of stability in elastoplasticity non linear rods and non linear three dimensional shells

toldi an effective and robust approach for 3d local shape description abstract feature description for the 3d local shape in the presence of noise varying mesh resolutions clutter and occlusion is a quite challenging task in 3d computer vision this paper tackles the problem by proposing a new local reference frame lrf together with a novel triple orthogonal local depth images toldi representation forming the toldi method for local shape description compared with previous methods toldi manages to perform efficient distinctive and robust description for the 3d local surface simultaneously under various feature matching contexts the proposed lrf differs from many prior ones in its calculation of the z axis and x axis the z axis is calculated using the normal of the keypoint and the x axis is computed by aggregating the weighted projection vectors of the radius neighbors toldi feature descriptors are then obtained by concatenating three local depth images ldi captured from three orthogonal view planes in the lrf into feature vectors the performance of our toldi approach is rigorously evaluated on several public datasets which contain three major surface matching scenarios namely shape retrieval object recognition and 3d registration experimental results and comparisons with the state of the arts validate the effectiveness robustness high efficiency and overall superiority of our method our method is also applied to aligning 3d object and indoor scene point clouds obtained by different devices i e lidar and kinect the accurate outcomes further confirm the effectiveness of our method

a comprehensive performance evaluation of 3d local feature descriptors a number of 3d local feature descriptors have been proposed in the literature it is however unclear which descriptors are more appropriate for a particular application a good descriptor should be descriptive compact and robust to a set of nuisances this paper compares ten popular local feature descriptors in the contexts of 3d object recognition 3d shape retrieval and 3d modeling we first evaluate the descriptiveness of these descriptors on eight popular datasets which were acquired using different techniques we then analyze their compactness using the recall of feature matching per each float value in the descriptor we also test the robustness of the selected descriptors with respect to support radius variations gaussian noise shot noise varying mesh resolution distance to the mesh boundary keypoint localization error occlusion clutter and dataset size moreover we present the performance results of these descriptors when combined with different 3d keypoint detection methods we finally analyze the computational efficiency for generating each descriptor

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

on hamiltonian and symplectic lanczos processes abstract large sparse hamiltonian eigenvalue problems arise in a variety of contexts these problems can be attacked directly or they can first be transformed to problems having some related structure such as symplectic or skew hamiltonian in the interest of efficiency stability and accuracy such problems should be solved by methods that preserve the structure whether it be hamiltonian skew hamiltonian or symplectic the present work outlines krylov subspace methods for computing partial eigensystems of skew hamiltonian hamiltonian and symplectic matrices and records some of the relationships between them the ordinary unsymmetric lanczos process is a structure preserving method for skew hamiltonian matrices the hamiltonian and symplectic lanczos processes developed here are condensed versions of the processes that have been published previously the condensed hamiltonian lanczos process applied to h is equivalent to the unsymmetric lanczos process applied to the skew hamiltonian h 2 but costs half as much to execute the condensed symplectic lanczos process applied to s is equivalent to the unsymmetric lanczos process applied to the skew hamiltonian matrix s s 1 but also costs half as much to execute implicit restarts of the hamiltonian and symplectic lanczos processes can be effected by the sr algorithm because of the known relationship between the sr and hr algorithms the much simpler hr algorithm can be used to restart the condensed symplectic and hamiltonian lanczos processes each of these restart procedures is equivalent to restarting the unsymmetric lanczos process using the hr algorithm the hr algorithm is most effectively implemented as an implicit bulge chasing hz algorithm on a symmetric tridiagonal diagonal pencil t d

point signatures a new representation for 3d object recognition few systems capable of recognizing complex objects with free form sculptured surfaces have been developed the apparent lack of success is mainly due to the lack of a competent modelling scheme for representing such complex objects in this paper a new form of point representation for describing 3d free form surfaces is proposed this representation which we call the point signature serves to describe the structural neighbourhood of a point in a more complete manner than just using the 3d coordinates of the point being invariant to rotation and translation the point signature can be used directly to hypothesize the correspondence to model points with similar signatures recognition is achieved by matching the signatures of data points representing the sensed surface to the signatures of data points representing the model surface the use of point signatures is not restricted to the recognition of a single object scene to a small library of models instead it can be extended naturally to the recognition of scenes containing multiple partially overlapping objects which may also be juxtaposed with each other against a large model library no preliminary phase of segmenting the scene into the component objects is required in searching for the appropriate candidate model recognition need not proceed in a linear order which can become prohibitive for a large model library for a given scene signatures are extracted at arbitrarily spaced seed points each of these signatures is used to vote for models that contain points having similar signatures inappropriate models with low votes can be rejected while the remaining candidate models are ordered according to the votes they received in this way efficient verification of the hypothesized candidates can proceed by testing the most likely model first experiments using real data obtained from a range finder have shown fast recognition from a library of fifteen models whose complexities vary from that of simple piecewise quadric shapes to complicated face masks results from the recognition of both single object and multiple object scenes are presented

on the computation of manifolds of foldpoints for parameter dependent problems for parameter dependent nonlinear equations in r n a new augmentation method for computing submanifolds of foldpoints is presented the underlying theory is based on a characterization developed earlier of foldpoints on solution manifolds of parameterized equations the method allows for a locally convergent iterative computation of foldpoints of a certain class for a continuation along paths of such foldpoints and for a computation of simplicial approximations of higher dimensional submanifolds of foldpoints of the same type

schur parameter pencils for the solution of the unitary eigenproblem abstract let u v be an n n pencil with unitary matrices u and v an algorithm is presented which reduces u and v simultaneously to unitary block diagonal matrices g o q h up and g e q h vp with block size at most two it is an o n 3 process using householder eliminations and it is backward stable in the special case v i the block diagonal matrices g o g h e can be normalized so that their entries are just the schur parameters of the hessenberg condensed form of u we call g o g e a schur parameter pencil it can also be derived from u v by a lanczos like process for the solution of the eigenvalue problem for g o g e a qr type algorithm can be developed based on this unitary reduction of a pencil u v to a schur parameter pencil the condensed form is preserved throughout the process each iteration step needs only o n operations this method of solving the unitary eigenvalue problem seems to be the closest possible analogy to the qr method for the hermitian eigenvalue problem

dense 3d face correspondence we present an algorithm that automatically establishes dense correspondences between a large number of 3d faces starting from automatically detected sparse correspondences on the outer boundary of 3d faces the algorithm triangulates existing correspondences and expands them iteratively by matching points of distinctive surface curvature along the triangle edges after exhausting keypoint matches further correspondences are established by generating evenly distributed points within triangles by evolving level set geodesic curves from the centroids of large triangles a deformable model k3dm is constructed from the dense corresponded faces and an algorithm is proposed for morphing the k3dm to fit unseen faces this algorithm iterates between rigid alignment of an unseen face followed by regularized morphing of the deformable model we have extensively evaluated the proposed algorithms on synthetic data and real 3d faces from the frgcv2 bosphorus bu3dfe and und ear databases using quantitative and qualitative benchmarks our algorithm achieved dense correspondences with a mean localisation error of 1 28 mm on synthetic faces and detected 14 anthropometric landmarks on unseen real faces from the frgcv2 database with 3 mm precision furthermore our deformable model fitting algorithm achieved 98 5 percent face recognition accuracy on the frgcv2 and 98 6 percent on bosphorus database our dense model is also able to generalize to unseen datasets

an accurate and efficient voting scheme for a maximally all inlier 3d correspondence set we present a highly accurate and efficient yet simple two stage voting scheme for distinguishing inlier 3d correspondences by densely assessing and ranking their local and global geometric consistencies the strength of the proposed method stems from both the novel idea of post validated voting set as well as single point superimposition transforms which are computationally cheap and avoid rotational ambiguities using a well known dataset consisting of various 3d models and numerous scenes that include different occlusion rates the proposed scheme is evaluated against state of the art 3d voting schemes in terms of both the correspondence pr precision recall auc area under curve and the execution time a total of 374 experiments were conducted for each method which involved a combination of four models 50 scenes and two down samplings the proposed scheme outperforms the state of the art 3d voting schemes in terms of both accuracy and speed quantitatively the proposed scheme scores 97 0 12 9 on the pr auc metric averaged over all of the experiments while the two state of the art schemes score 74 2 22 2 and 78 3 26 4 furthermore the proposed scheme requires only 24 1 6 0 of the time consumed by the fastest state of the art scheme the proposed voting scheme also demonstrates high robustness against occlusions and scarce inliers

a schur decomposition for hamiltonian matrices abstract a schur type decomposition for hamiltonian matrices is given that relies on unitary symplectic similarity transformations these transformations preserve the hamiltonian structure and are numerically stable making them ideal for analysis and computation using this decomposition and a special singular value decomposition for unitary symplectic matrices a canonical reduction of the algebraic riccati equation is obtained which sheds light on the sensitivity of the nonnegative definite solution after presenting some real decompositions for real hamiltonian matrices we look into the possibility of an orthogonal symplectic version of the qr algorithm suitable for hamiltonian matrices a finite step initial reduction to a hessenberg type canonical form is presented however no extension of the francis implicit shift technique was found and reasons for the difficulty are given

numerical bifurcation analysis of the bipedal spring mass model abstract the spring mass model and its numerous extensions are currently one of the best candidates for templates of human and animal locomotion however with increasing complexity their applications can become very time consuming in this paper we present an approach that is based on the calculation of bifurcations in the bipedal spring mass model for walking since the bifurcations limit the region of stable walking locomotion can be studied by computing the corresponding boundaries originally the model was implemented as a hybrid dynamical system our new approach consists of the transformation of the series of initial value problems on different intervals into a single boundary value problem using this technique discontinuities can be avoided and sophisticated numerical methods for studying parametrized nonlinear boundary value problems can be applied thus appropriate extended systems are used to compute transcritical and period doubling bifurcation points as well as turning points we show that the resulting boundary value problems can be solved by the simple shooting method with sufficient accuracy making the application of the more extensive multiple shooting superfluous the proposed approach is fast robust to numerical perturbations and allows determining complete manifolds of periodic solutions of the original problem

using spin images for efficient object recognition in cluttered 3d scenes we present a 3d shape based object recognition system for simultaneous recognition of multiple objects in scenes containing clutter and occlusion recognition is based on matching surfaces by matching points using the spin image representation the spin image is a data level shape descriptor that is used to match surfaces represented as surface meshes we present a compression scheme for spin images that results in efficient multiple object recognition which we verify with results showing the simultaneous recognition of multiple objects from a library of 20 models furthermore we demonstrate the robust performance of recognition in the presence of clutter and occlusion through analysis of recognition trials on 100 scenes

asymmetric solutions of the reaction diffusion equation a class of new non symmetric solutions are computed for the reaction diffusion equation the bratu problem governed by the partial differential equation delta theta epsilon e theta 1 mu theta 0 in a two dimensional rectangular domain where mu is the arrhenius factor and epsilon is the frank kamenetskii parameter when mu 0 previously computed solutions that conform to the expected symmetries are confirmed in addition certain hitherto unknown solution branches are found these appear to be disconnected from the primary branch with in the parameter region investigated here each of these asymmetric solution branch contains an additional limit point these additional solutions do not conform to the expected symmetries and hence they occur with multiplicities of four grid refinement tests and extrapolation to zero grid size indicate that these non symmetric solutions remain distinct indicating that they are not spurious solutions of a poorly resolved grid next a perturbed bratu problem is examined with convection as an additional heat transport mechanism and rayleigh number ra and tilt angle phi as additional parameters this variation is examined using both a regular perturbation series expansion in ra around the conduction state and a full numerical solution all of the newly computed solutions at ra 0 remain smoothly connected as ra is increased which is viewed as an additional evidence in support of the claim that the non symmetric solutions are real even for extremely small convection ra 5 the quenching effect of convection is significant in both extending the range of stable solutions in epsilon and in reducing the mean temperature of the region the quenching effect of the convection in turn reduces the sharp temperature gradients thus making the limit points less sensitive to grid refinement in the presence of weak convection

an implementation of the look ahead lanczos algorithm for non hermitian matrices the nonsymmetric lanczos method can be used to compute eigenvalues of large sparse non hermitian matrices or to solve large sparse non hermitian linear systems however the original lanczos algorithm is susceptible to possible breakdowns and potential instabilities an implementation of a look ahead version of the lanczos algorithm is presented that except for the very special situation of an incurable breakdown overcomes these problems by skipping over those steps in which a breakdown or near breakdown would occur in the standard process the proposed algorithm can handle look ahead steps of any length and requires the same number of matrix vector products and inner products as the standard lanczos process without look ahead

the parameterized sr algorithm for symplectic butterfly matrices the sr algorithm is a structure preserving algorithm for computing the spectrum of symplectic matrices any symplectic matrix can be reduced to symplectic butterfly form a symplectic matrix b in butterfly form is uniquely determined by 4n 1 parameters using these 4n 1 parameters we show how one step of the symplectic sr algorithm for b can be carried out in o n arithmetic operations compared to o n 3 arithmetic operations when working on the actual symplectic matrix moreover the symplectic structure which will be destroyed in the numerical process due to roundoff errors when working with a symplectic butterfly matrix will be forced by working just with the parameters

equilibrium subsets for multi parametric structural analysis the paper discusses multi parametric formulations for structural equilibrium problems by seeking subsets of equilibrium states which also fulfil extra conditions more detailed information on structural behaviour can be obtained these conditions can be related to analyses of e g critical behaviour imperfections or optimisation the paper shows a general problem setting and some interesting augmentations a solution method for the special case of one dimensional equilibrium subsets is described the algorithm is a generalisation of a path following algorithm where new methods are developed for the evaluation of the path tangent and for the isolation of special solution points examples are used to show the properties and the possibilities of the ideas

parameter sensitivity of inelastic buckling and post buckling response this paper is concerned with recent sensitivity results for buckling and post buckling problems the article starts with an overview of the fem based approach to inelastic structural mechanics systems with singular points along the equilibrium path the existing sensitivity techniques are reviewed their inherent drawbacks are indicated and a new methodology is suggested the discretized formalism is mainly employed throughout but a continuum formulation appropriate for this class of problems is also given numerical studies illustrate the presentation

matrix factorizations for symplectic qr like methods abstract symplectic qr like methods use symplectic or unitary symplectic similarity transformations instead of the usual unitary similarity transformations in the qr process a fundamental problem for the development of such methods is the choice of a suitable type of decomposition a sr corresponding to the qr decomposition where s is symplectic or unitary symplectic decompositions of this type are studied with regard to their application in a qr like process

computation of takens bogdanov type bifurcations with arbitrary codimension let dot u f u alpha u 0 u 0 describe a dynamical system where u in r n is the state variable alpha in r k is a parameter variable and f u alpha takes values in r n here n can be large but k is usually small say leqslant 10 a solution u alpha to the equilibrium equations f u alpha 0 is a takens bogdanov type bifurcation of codimension m if the jacobian f u u alpha contains an m dimensional jordan block corresponding to a zero eigenvalue and has no other eigenvalues equal to zero this includes the cases of turning point bifurcation m 1 takens bogdanov bifurcation m 2 and triple point bifurcation m 3 a unified direct method is proposed based on ideas of a griewank and g w reddien in the case of takens bogdanov points for both the detection and computation of parameter values a at which thkens bogdanov type bifurcations of any given codimension occur as an application we compute tkens bogdanov type bifurcations with codi

scale invariant features for 3 d mesh models in this paper we present a framework for detecting interest points in 3 d meshes and computing their corresponding descriptors for that we propose an intrinsic scale detection scheme per interest point and utilize it to derive two scale invariant local features for mesh models first we present the scale invariant spin image local descriptor that is a scale invariant formulation of the spin image descriptor second we adapt the scale invariant feature transform feature to mesh data by representing the vicinity of each interest point as a depth map and estimating its dominant angle using the principal component analysis to achieve rotation invariance the proposed features were experimentally shown to be robust to scale changes and partial mesh matching and they were compared favorably with other local mesh features on the shrec 10 and shrec 11 testbeds we applied the proposed local features to mesh retrieval using the bag of features approach and achieved state of the art retrieval accuracy last we applied the proposed local features to register models to scanned depth scenes and achieved high registration accuracy

on hamiltonian and symplectic hessenberg forms abstract characterizations are given for the hamiltonian matrices that can be reduced to hamiltonian hessenberg form and the symplectic matrices that can be reduced to symplectic hessenberg form by orthogonal symplectic similarity transformations the reduction to these special hessenberg forms is the missing link in the solution of the open problem of constructing a stable structure preserving qr like method of complexity o n 3 for the computation of invariant subspaces of hamiltonian and symplectic matrices our considerations lead us to propose an approach to the computation of lagrangian invariant subspaces of a hamiltonian or symplectic matrix

sr and sz algorithms for the symplectic butterfly eigenproblem abstract sr and sz algorithms for the symplectic generalized eigenproblem that are based on the reduction of a symplectic matrix to symplectic butterfly form are discussed a 2 n 2 n symplectic butterfly matrix has 8 n 4 generically nonzero entries which are determined by 4 n 1 parameters while the sr algorithm operates directly on the matrix entries the sz algorithm works with the 4 n 1 parameters the algorithms are made more compact and efficient by using laurent polynomials instead of standard polynomials to drive the iterations

on computing the eigenvalues of a symplectic pencil this paper presents an algorithm for computing the eigenvalues of a symplectic pencil that arises in one of the commonly used approaches for solving the discrete time algebraic riccati equation the algorithm is numerically efficient and reliable in that it employs only orthogonal transformations and makes use of the structure of the symplectic pencil it requires about one fourth the number of floating point operations that the qz algorithm uses to compute the eigenvalues of the pencil directly the proposed method can be regarded as being analogous for the case of symplectic pencils to the method developed by van loan for computing the eigenvalues of hamiltonian matrices

the continuation method for rational and deficient systems deficient polynomial systems those with fewer than the full number of solutions cannot be solved efficiently with the usual continuation methods this paper describes a continuation technique which allows some deficient and rational systems of equations to be solved with little wasted computation several examples from the literature are discussed

keypoints based surface representation for 3d modeling and 3d object recognition the three dimensional 3d modeling and recognition of 3d objects have been traditionally performed using local features to represent the underlying 3d surface extraction of features requires cropping of several local surface patches around detected keypoints although an important step the extraction and representation of such local patches adds to the computational complexity of the algorithms this paper proposes a novel keypoints based surface representation ksr technique the proposed technique has the following two characteristics 1 it does not rely on the computation of features on a small surface patch cropped around a detected keypoint rather it exploits the geometrical relationship between the detected 3d keypoints for local surface representation 2 ksr is computationally efficient requiring only seconds to process 3d models with over 50 000 points with a matlab implementation experimental results on the uwa and stanford 3d models dataset suggest that it can accurately perform pairwise and multiview range image registration 3d modeling ksr was also tested for 3d object recognition with occluded scenes recognition results on the uwa dataset show that the proposed technique outperforms existing methods including 3d tensor vd lsd keypoint depth based feature spherical harmonics and spin image with a recognition rate of 95 9 the proposed approach also achieves a recognition rate of 93 5 on the challenging ca fascori dataset compared to 92 5 achieved by game theoretic the proposed method is computationally efficient compared to state of the art local feature methods highlightswe propose a novel technique called keypoint based surface representation ksr the proposed technique does not require local features around detected keypoints ksr exploits geometrical relationships between keypoints for surface representation ksr is tested on 3 popular datasets for 3d modeling and 3d object recognition ksr achieves superior 3d modeling and recognition results

large scale algebraic riccati equations with high rank constant terms abstract we consider the numerical solution of large scale algebraic riccati equations with high rank constant terms the solutions are not numerically low rank so the previously successful methods based on low rank representations are not directly applicable we modify the doubling algorithm making use of the low rank in the input matrix b we also solve the challenging problems in the estimation of residuals and relative errors convergence control and the output of the modified algorithm illustrative numerical examples are presented

the computation of symmetry breaking bifurcation points we consider symmetry breaking simple singular points of a parameter dependent nonlinear equation g lambda x 0 such points are intimately connected to pitchfork bifurcation points and we sh

a look ahead lanczos algorithm for unsymmetric matrices abstract the two sided lanczos algorithm is known to suffer from serious breakdowns these occur when the associated moment matrix does not permit triangular factorization we modify the algorithm slightly so that it corresponds to using a 2 x 2 pivot in triangular factorization whenever a 1 x 1 pivot would be dangerous the incidence of breakdown is greatly reduced the price paid is that the tridiagonal matrix produced by the algorithm now has bumps whenever a 2 x 2 pivot is used author

bordered augmented linear systems in numerical continuation and bifurcation we present the algorithm gmbe to solve a block linear system begin gathered begin array 20 c n m 1 end array left begin array 20 c a 11 0 a 31 end array begin array 20 c a 12 a 22 a 32 end array begin array 20 c a 13 a 23 a 33 end array right left begin array 20 c x 1 x 2 x 3 end array right left begin array 20 c f 1 f 2 f 3 end array right begin array 20 c n m 1 end array hfill nm111 hfill end gathered or m z h which appears in numerical computation of folds and symmetry breaking bifurcation gmbe principally uses 1 solve witha 11 t anda 22 t each and 2 solves witha 11 anda 22 each m must be well conditioned buta 11 anda 22 may be arbitrarily ill conditioned in fact singular to machine precision the error analysis requires that the solvers fora 11 a 22 a 11 t anda 22 t are stable in the sense of backward projection of the errors and that the solvers fora 11 anda 22 are bounded in a sense to be made precise both properties are typically possessed in practice by solvers based on either direct or iterative methods gmbe is the first algorithm that solvesmz h by using the solvers fora 11 etc as black boxes and does not require prior computation of the left and right singular vectors ofa 11 anda 22

a method for the computation of nonsimple turning points corresponding to cusps abstract a direct method is described for the approximation of nonsimple turning points corresponding to cusp points of nonlinear operator equations depending on two parameters the procedure is based on the application of a special projection method to the computation of simple turning points of a suitable augmented system numerical examples illustrate the features of the proposed algorithm

folds in solutions of two parameter systems and their calculation part i this paper is concerned with paths of turning points in solutions of nonlinear systems having two parameters it is well known that these paths are solutions of a particular extended system of nonlinear equations in this paper both regular points and simple turning points in the extended system are related to the local geometry of the solution surface of the original nonlinear system a description is given of numerical methods both for solving the extended system and for calculating certain quantities which determine the local geometry of the solution surface applications to perturbed bifurcation to the formation of isolas and to the calculation of the multiplicity of solutions are also discussed numerical examples are given

a spherical representation for recognition of free form surfaces introduces a new surface representation for recognizing curved objects the authors approach begins by representing an object by a discrete mesh of points built from range data or from a geometric model of the object the mesh is computed from the data by deforming a standard shaped mesh for example an ellipsoid until it fits the surface of the object the authors define local regularity constraints that the mesh must satisfy the authors then define a canonical mapping between the mesh describing the object and a standard spherical mesh a surface curvature index that is pose invariant is stored at every node of the mesh the authors use this object representation for recognition by comparing the spherical model of a reference object with the model extracted from a new observed scene the authors show how the similarity between reference model and observed data can be evaluated and they show how the pose of the reference object in the observed scene can be easily computed using this representation the authors present results on real range images which show that this approach to modelling and recognizing 3d objects has three main advantages 1 it is applicable to complex curved surfaces that cannot be handled by conventional techniques 2 it reduces the recognition problem to the computation of similarity between spherical distributions in particular the recognition algorithm does not require any combinatorial search and 3 even though it is based on a spherical mapping the approach can handle occlusions and partial views

an implicitly restarted symplectic lanczos method for the hamiltonian eigenvalue problem abstract an implicitly restarted symplectic lanczos method for the hamiltonian eigenvalue problem is presented the lanczos vectors are constructed to form a symplectic basis the inherent numerical difficulties of the symplectic lanczos method are addressed by inexpensive implicit restarts the method is used to compute eigenvalues eigenvectors and invariant subspaces of large and sparse hamiltonian matrices and low rank approximations to the solution of continuous time algebraic riccati equations with large and sparse coefficient matrices

on numerical methods for discrete least squares approximation by trigonometric polynomials fast efficient and reliable algorithms for discrete least squares approximation of a real valued function given at arbitrary distinct nodes in 0 2 by trigonometric polynomials are presented the algorithms are based on schemes for the solution of inverse unitary eigenproblems and require only o mn arithmetic operations as compared to o mn 2 operations needed for algorithms that ignore the structure of the problem an algorithm which solves this problem with real valued data and real valued solution using only real arithmetic is given numerical examples are presented that show that the proposed algorithms produce consistently accurate results that are often better than those obtained by general qr decomposition methods for the least squares problem

two connections between the sr and hr eigenvalue algorithms abstract the sr and hr algorithms are members of the family of gr algorithms for calculating eigenvalues and invariant subspaces of matrices this paper makes two connections between the sr and hr algorithms 1 an iteration of the sr algorithm on a 2 n 2 n symplectic butterfly matrix using shifts i 1 i i 1 k is equivalent to an iteration of the hr algorithm on an n n tridiagonal sign symmetric matrix using shifts i 1 i i 1 k 2 an iteration of the sr algorithm on a 2 n 2 n j tridagonal hamiltonian matrix using shifts i i i 1 k is equivalent to an iteration of the hr algorithm on an n n tridiagonal sign symmetric matrix using shifts 2 i i 1 k

product eigenvalue problems many eigenvalue problems are most naturally viewed as product eigenvalue problems the eigenvalues of a matrix a are wanted but a is not given explicitly instead it is presented as a product of several factors a a k a k 1 cdots a 1 usually more accurate results are obtained by working with the factors rather than forming a explicitly for example if we want eigenvalues vectors of b t b it is better to work directly with b and not compute the product the intent of this paper is to demonstrate that the product eigenvalue problem is a powerful unifying concept diverse examples of eigenvalue problems are discussed and formulated as product eigenvalue problems for all but a couple of these examples it is shown that the standard algorithms for solving them are instances of a generic gr algorithm applied to a related cyclic matrix

on a condensed form for normal matrices under finite sequences of elementary unitary similarities abstract it is generally known that any hermitian matrix can be reduced to a tridiagonal form by a finite sequence of unitary similarities namely householder reflections recently a bunse gerstner and l elsner have found a condensed form to which any unitary matrix can be reduced again by a finite sequence of householder transformations this condensed form can be considered as a pentadiagonal or block tridiagonal matrix with some additional zeros inside the band we describe such a condensed form or more precisely a set of such forms for general normal matrices where the number of nonzero elements does not exceed o n 3 2 n being the order of the normal matrix given two approaches to constructing the condensed form are outlined the first approach is a geometrical lanczos type one where we use the so called generalized krylov sequences the second more constructive approach is an elimination process using householder reflections our condensed form can be thought of as a variable bandwidth form an interesting feature of it is that for normal matrices whose spectra lie on algebraic curves of low degree the bandwidth is much smaller

a symplectic method for approximating all the eigenvalues of a hamiltonian matrix a fast method for computing all the eigenvalues of a hamiltonian matrix m is given the method relies on orthogonal symplectic similarity transformations which preserve structure and have desirable numerical properties the algorithm is about four times faster than the standard q r algorithm the computed eigenvalues are shown to be the exact eigenvalues of a matrix m e where vert e vert depends on the square root of the machine precision the accuracy of a computed eigenvalue depends on its condition and its magnitude larger eigenvalues typically being more accurate

cosmos a representation scheme for 3d free form objects we address the problem of representing and recognizing 3d free form objects when 1 the object viewpoint is arbitrary 2 the objects may vary in shape and complexity and 3 no restrictive assumptions are made about the types of surfaces on the object we assume that a range image of a scene is available containing a view of a rigid 3d object without occlusion we propose a new and general surface representation scheme for recognizing objects with free form sculpted surfaces in this scheme an object is described concisely in terms of maximal surface patches of constant shape index the maximal patches that represent the object are mapped onto the unit sphere via their orientations and aggregated via shape spectral functions properties such as surface area curvedness and connectivity which are required to capture local and global information are also built into the representation the scheme yields a meaningful and rich description useful for object recognition a novel concept the shape spectrum of an object is also introduced within the framework of cosmos for object view grouping and matching we demonstrate the generality and the effectiveness of our scheme using real range images of complex objects

three dimensional model based object recognition and segmentation in cluttered scenes viewpoint independent recognition of free form objects and their segmentation in the presence of clutter and occlusions is a challenging task we present a novel 3d model based algorithm which performs this task automatically and efficiently a 3d model of an object is automatically constructed offline from its multiple unordered range images views these views are converted into multidimensional table representations which we refer to as tensors correspondences are automatically established between these views by simultaneously matching the tensors of a view with those of the remaining views using a hash table based voting scheme this results in a graph of relative transformations used to register the views before they are integrated into a seamless 3d model these models and their tensor representations constitute the model library during online recognition a tensor from the scene is simultaneously matched with those in the library by casting votes similarity measures are calculated for the model tensors which receive the most votes the model with the highest similarity is transformed to the scene and if it aligns accurately with an object in the scene that object is declared as recognized and is segmented this process is repeated until the scene is completely segmented experiments were performed on real and synthetic data comprised of 55 models and 610 scenes and an overall recognition rate of 95 percent was achieved comparison with the spin images revealed that our algorithm is superior in terms of recognition rate and efficiency

keypoints and local descriptors of scalar functions on 2d manifolds this paper addresses the problem of describing surfaces using local features and descriptors while methods for the detection of interest points in images and their description based on local image features are very well understood their extension to discrete manifolds has not been well investigated we provide a methodological framework for analyzing real valued functions defined over a 2d manifold embedded in the 3d euclidean space e g photometric information local curvature etc our work is motivated by recent advancements in multiple camera reconstruction and image based rendering of 3d objects there is a growing need for describing object surfaces matching two surfaces or tracking them over time considering polygonal meshes we propose a new methodological framework for the scale space representations of scalar functions defined over such meshes we propose a local feature detector meshdog and region descriptor meshhog unlike the standard image features the proposed surface features capture both the local geometry of the underlying manifold and the scale space differential properties of the real valued function itself we provide a thorough experimental evaluation the repeatability of the feature detector and the robustness of feature descriptor are tested by applying a large number of deformations to the manifold or to the scalar function

generalized deflated block elimination a stable algorithm is presented to solve a nonsingular bordered system of the form left begin array 20 c a b c t d end array right left begin array 20 c x y end array right left begin array 20 c f g end array right where b and c are n by m matrices and the n by n matrix a could be nearly singular with at most mu small singular values the algorithm needs only a solver for a and the solution to an m mu by m mu dense linear system it is thus well suited for problems for which a has easily exploitable structures and m mu ll n such as in continuation methods bifurcation problems and constrained optimization

computation of cusp singularities for operator equations and their discretizations we discuss the direct calculation of cusp singularities as solutions of a minimally augmented defining system which is nonsingular under the canonical cusp conditions the underlying operator equation may have any finite fredholm index and bounds on the discretization error are derived for the case of projection methods

a novel representation and feature matching algorithm for automatic pairwise registration of range images automatic registration of range images is a fundamental problem in 3d modeling of free from objects various feature matching algorithms have been proposed for this purpose however these algorithms suffer from various limitations mainly related to their applicability efficiency robustness to resolution and the discriminating capability of the used feature representation we present a novel feature matching algorithm for automatic pairwise registration of range images which overcomes these limitations our algorithm uses a novel tensor representation which represents semi local 3d surface patches of a range image by third order tensors multiple tensors are used to represent each range image tensors of two range images are matched to identify correspondences between them correspondences are verified and then used for pairwise registration of the range images experimental results show that our algorithm is accurate and efficient moreover it is robust to the resolution of the range images the number of tensors per view the required amount of overlap and noise comparisons with the spin image representation revealed that our representation has more discriminating capabilities and performs better at a low resolution of the range images

convergence of algorithms of decomposition type for the eigenvalue problem we develop the theory of convergence of a generic gr algorithm for the matrix eigenvalue problem that includes the qr lr sr and other algorithms as special cases our formulation allows for shifts of origin and multiple gr steps the convergence theory is based on the idea that the gr algorithm performs nested subspace iteration with a change of coordinate system at each step thus the convergence of the gr algorithm depends on the convergence of certain sequences of subspaces it also depends on the quality of the coordinate transformation matrices as measured by their condition numbers we show that with a certain obvious shifting strategy the gr algorithm typically has a quadratic asymptotic convergence rate for matrices possessing certain special types of structure cubic convergence can be achieved