on computing structural changes in evolving surfaces and their appearance as a surface undergoes a one parameter family of deformations its shape and its appearance change smoothly except at certain critical parameter values where abrupt structural changes occur this paper considers the case of surfaces defined as the zero set of smooth density functions undergoing a gaussian diffusion process and addresses the problem of computing the critical parameter values corresponding to structural changes in the parabolic curves of a surface and in its aspect graph an algorithm based on homotopy continuation and curve tracing is proposed in the case of polynomial density functions whose zero set is an algebraic surface it has been implemented and examples are presented

a similarity based aspect graph approach to 3d object recognition this paper describes a view based method for recognizing 3d objects from 2d images we employ an aspect graph structure where the aspects are not based on the singularities of visual mapping but are instead formed using a notion of shape similarity between views specifically the viewing sphere is endowed with a metric of dis similarity for each pair of views and the problem of aspect generation is viewed as a segmentation of the viewing sphere into homogeneous regions the viewing sphere is sampled at regular 5 degree intervals and the similarity metric is used in an iterative procedure to combine views into aspects with a prototype representing each aspect this is done in a region growing regime which stands in contrast to the usual edge detection styles to computing the aspect graph the aspect growth is constrained such that two aspects of an object remain distinct under the given similarity metric once the database of 3d objects is organized as a set of aspects and prototypes for these aspects for each object unknown views of database objects are compared with the prototypes and the results are ordered by similarity we use two similarity metrics for shape one based on curve matching and the other based on matching shock graphs which for a database of 64 objects and unknown views of objects from the database give a recall rate of 90 3 74 2 59 7 and 95 2 69 0 57 5 respectively for the top three matchess cumulative recall rate based on the top three matches is 98 and 100 respectively the result of indexing unknown views of objects not in the database also produce intuitive matches we also develop a hierarchical indexing scheme to prune unlikely objects at an early stage to improve the efficiency of indexing resulting in savings of 35 at the top level and of 55 at the next level cumulatively

structure ranks of matrices abstract the structure rank of a matrix i e the maximum order of a nonsingular submatrix all of whose entries are located in a given structure a subset of m n where m n are the row and column index sets respectively were previously studied for the off diagonal part here results on more general ranks e g off block diagonal strictly upper triangular and strictly upper block triangular are presented in particular it is shown that for a nonsingular matrix a in each of the last mentioned three cases the ranks of a and a 1 coincide

bounds on polynomial roots using intercyclic companion matrices abstract the frobenius companion matrix and more recently the fiedler companion matrices have been used to provide lower and upper bounds on the modulus of any root of a polynomial p x in this paper we explore new bounds obtained from taking the 1 norm and norm of a matrix in the wider class of intercyclic companion matrices as is the case with fiedler matrices we observe that the new bounds from intercyclic companion matrices can improve those from the frobenius matrix by at most a factor of two by using the hessenberg form of an intercyclic companion matrix we describe how to determine the best upper bound when restricted to fiedler companion matrices using the norm we also obtain a new general bound by considering the polynomial x q p x for q 0 we end by considering upper bounds obtained from inverses of monic reversal polynomials of intercyclic companion matrices noting that these can make more significant improvements on the bounds from a frobenius companion matrix for certain polynomials

computing the aspect graph for line drawings of polyhedral objects an algorithm for computing the aspect graph for polyhedral objects is described the aspects graph is a representation of three dimensional objects by a set of two dimensional views the set of viewpoints on the gaussian sphere is partitioned into regions such that in each region the qualitative structure of the line drawing remains the same at the boundaries between adjacent regions are the accidental viewpoints where the structure for the line drawing changes it is shown that for polyhedral objects there are two fundamental visual events 1 the projections of an edge and a vertex coincide and 2 the projections of three nonadjacent edges intersect at a point the geometry of the object is reflected in the locus of the accidental viewpoints the algorithm computes the partition together with a representative view for each region of the partition

structured strong linearizations from fiedler pencils with repetition ii abstract in this paper we give strong linearizations of a matrix polynomial p preserving the skew symmetry or t alternating structure of p the linearizations obtained are of the form s l where l is a block symmetric fiedler pencil with repetition and s is a direct sum of blocks of the form i or i with i the identity matrix this paper is a continuation of 2 where the corresponding problem for p with a symmetric structure was studied and as a consequence the block symmetric fiedler pencils with repetition were characterized

fast qr iterations for unitary plus low rank matrices some fast algorithms for computing the eigenvalues of a block companion matrix have recently appeared in the literature in this paper we generalize the approach to encompass unitary plus low rank matrices of the form a u xy h where u is a general unitary matrix three important cases for applications are u unitary diagonal u unitary block hessenberg and u unitary in block cmv form our extension exploits the properties of a larger matrix hat a obtained by a certain embedding of the hessenberg reduction of a suitable to maintain its structural properties we show that hat a can be factored as product of lower and upper unitary hessenberg matrices possibly perturbed in the first k rows and moreover such a data sparse representation is well suited for the design of fast eigensolvers based on the qr iteration the resulting eigenvalue algorithm is fast and backward stable

the visual hull of smooth curved objects the visual hull is a geometric entity that relates the shape of an object to its silhouettes or shadows this paper develops the theory of the visual hull of generic smooth objects we show that the visual hull can be constructed using surfaces which partition the viewpoint space of the aspect graph of the object the surfaces are those generated by the visual events tangent crossing and triple point an analysis based on the shape of the object at the tangency points of these surfaces allows pruning away many surfaces and patches not relevant to the construction an algorithm for computing the visual hull is outlined

a note on companion matrices abstract we show that the usual companion matrix of a polynomial of degree n can be factored into a product of n matrices n 1 of them being the identity matrix in which a 2 2 identity submatrix in two consecutive rows and columns is replaced by an appropriate 2 2 matrix the remaining being the identity matrix with the last entry replaced by possibly different entry by a certain similarity transformation we obtain a simple new companion matrix in a pentadiagonal form some generalizations are also possible

the scale space aspect graph currently the aspect graph is computed from the theoretical standpoint of perfect resolution in object shape the viewpoint and the projected image this means that the aspect graph may include details that an observer could never see in practice introducing the notion of scale into the aspect graph framework provides a mechanism for selecting a level of detail that is large enough to merit explicit representation this effectively allows control over the number of nodes retained in the aspect graph this paper introduces the concept of the scale space aspect graph defines three different interpretations of the scale dimension and presents a detailed example for a simple class of objects with scale defined in terms of the spatial extent of features in the image

a quasiseparable approach to five diagonal cmv and fiedler matrices abstract recent work in the characterization of structured matrices in terms of characteristic polynomials of principal submatrices is furthered in this paper some classical classes of matrices with quasiseparable structure include tridiagonal related to real orthogonal polynomials and banded matrices unitary hessenberg matrices related to szego polynomials and semiseparable matrices as well as others hence working with the class of quasiseparable matrices provides new results which generalize and unify classical results previous work has focused on characterizing h 1 quasiseparable matrices matrices with order one quasiseparable structure that are also upper hessenberg in this paper the authors introduce the concept of a twist transformation and use such transformations to explain the relationship between h 1 quasiseparable matrices and the subclass of 1 1 quasiseparable matrices without the upper hessenberg restriction which are related to the same systems of polynomials these results generalize the discoveries of cantero fiedler kimura moral and velazquez of five diagonal matrices related to horner and szego polynomials in the context of quasiseparable matrices

palindromic companion forms for matrix polynomials of odd degree the standard way to solve polynomial eigenvalue problems p x 0 is to convert the matrix polynomial p into a matrix pencil that preserves its spectral information a process known as linearization when p is palindromic the eigenvalues elementary divisors and minimal indices of p have certain symmetries that can be lost when using the classical first and second frobenius companion linearizations for numerical computations since these linearizations do not preserve the palindromic structure recently new families of pencils have been introduced with the goal of finding linearizations that retain whatever structure the original p might possess with particular attention to the preservation of palindromic structure however no general construction of palindromic linearizations valid for all palindromic polynomials has as yet been achieved in this paper we present a family of linearizations for odd degree polynomials p which are palindromic whenever p is and which are valid for all palindromic polynomials of odd degree we illustrate our construction with several examples in addition we establish a simple way to recover the minimal indices of the polynomial from those of the linearizations in the new family

integrally normalizable matrices and zero nonzero patterns abstract the problem of determining whether or not an n n integer matrix is diagonally similar to an integer matrix with n 1 off diagonal entries equal to 1 is studied such a matrix is called integrally normalizable and a zero nonzero pattern is integrally normalizable if each matrix with this zero nonzero pattern is integrally normalizable with respect to the same set of n 1 off diagonal entries matrices that are integrally normalizable with respect to a fixed spanning tree and integrally normalizable zero nonzero patterns are characterized the maximum number of nonzero entries in an n n integrally normalizable zero nonzero pattern is shown to be n 2 3 n 2 2 extensions to matrices over other integral domains are also presented

large vector spaces of block symmetric strong linearizations of matrix polynomials abstract given a matrix polynomial p i 0 k i a i of degree k where a i are n n matrices with entries in a field f the development of linearizations of p that preserve whatever structure p might posses has been a very active area of research in the last decade most of the structure preserving linearizations of p discovered so far are based on certain modifications of block symmetric linearizations the block symmetric linearizations of p available in the literature fall essentially into two classes linearizations based on the so called fiedler pencils with repetition which form a finite family and a vector space of dimension k of block symmetric pencils called dl p such that most of its pencils are linearizations one drawback of the pencils in dl p is that none of them is a linearization when p is singular in this paper we introduce new vector spaces of block symmetric pencils most of which are strong linearizations of p the dimensions of these spaces are o n 2 which for n k are much larger than the dimension of dl p when k is odd many of these vector spaces contain linearizations also when p is singular the coefficients of the block symmetric pencils in these new spaces can be easily constructed as k k block matrices whose n n blocks are of the form 0 i n a i or arbitrary n n matrices where is an arbitrary nonzero scalar

companion matrix patterns abstract companion matrices especially the frobenius companion matrices are used in algorithms for finding roots of polynomials and are also used to find bounds on eigenvalues of matrices in 2003 fiedler introduced a larger class of companion matrices that includes the frobenius companion matrices one property of the combinatorial pattern of these companion matrices is that up to diagonal similarity they uniquely realize every possible spectrum of a real matrix we characterize matrix patterns that have this property and consequently introduce more companion matrix patterns we observe that each fiedler companion matrix is permutationally similar to a unit hessenberg matrix we provide digraph characterizations of the classes of patterns described and in particular all sparse companion matrices noting that there are companion matrices that are not sparse

computing the generalized aspect graph for objects with moving parts algorithms for computing the aspect graph representation are generalized to include a larger more realistic domain of objects known as articulated assemblies those objects composed of rigid parts with articulated connections allowed between parts the generalization suggests two slightly different representations one that directly summarizes the possible general views of the object and another hierarchical form summarizing the possible general configurations and their respective views algorithms are outlined for computing both representations the generalized aspect graphs of assemblies formed using translational connections are examined

block kronecker ansatz spaces for matrix polynomials abstract in this paper we introduce a new family of equations for matrix pencils that may be utilized for the construction of strong linearizations for any square or rectangular matrix polynomial we provide a comprehensive characterization of the resulting vector spaces and show that almost every matrix pencil therein is a strong linearization regardless whether the matrix polynomial under consideration is regular or singular these novel ansatz spaces cover all block kronecker pencils as introduced in 6 as a subset and therefore contain all fiedler pencils modulo permutations the important case of square matrix polynomials is examined in greater depth we prove that the intersection of any number of block kronecker ansatz spaces is never empty and construct large subspaces of block symmetric matrix pencils among which still almost every pencil is a strong linearization moreover we show that the original ansatz spaces l 1 and l 2 may essentially be recovered from block kronecker ansatz spaces via pre and postmultiplication respectively of certain constant matrices

reprint of a quasiseparable approach to five diagonal cmv and fiedler matrices abstract recent work in the characterization of structured matrices in terms of characteristic polynomials of principal submatrices is furthered in this paper some classical classes of matrices with quasiseparable structure include tridiagonal related to real orthogonal polynomials and banded matrices unitary hessenberg matrices related to szego polynomials and semiseparable matrices as well as others hence working with the class of quasiseparable matrices provides new results which generalize and unify classical results previous work has focused on characterizing h 1 quasiseparable matrices matrices with order one quasiseparable structure that are also upper hessenberg in this paper the authors introduce the concept of a twist transformation and use such transformations to explain the relationship between h 1 quasiseparable matrices and the subclass of 1 1 quasiseparable matrices without the upper hessenberg restriction which are related to the same systems of polynomials these results generalize the discoveries of cantero fiedler kimura moral and velazquez of five diagonal matrices related to horner and szego polynomials in the context of quasiseparable matrices

the continuing influence of fiedler s work on companion matrices this is a reconstruction in article like form of a talk given at the minisymposium in honor of miroslav fiedler at the 17th ilas conference held at tu braunschweig germany on thurs 25 aug 2011

efficiently computing and representing aspect graphs of polyhedral objects an efficient algorithm and a data structure for computing and representing the aspect graph of polyhedral objects under orthographic projection are presented the aspect graph is an approach to representing 3 d objects by a set of 2 d views for the purpose of object recognition in this approach the viewpoint space is partitioned into regions such that in each region the qualitative structure of the line drawing does not change the viewing data of an object is the partition of the viewpoint space together with a representative view in each region the algorithm computes the viewing data for line drawings of polyhedral objects under orthographic projection

computing exact aspect graphs of curved objects solid of revolution this paper introduces a new approach to computing the exact orthographic aspect graph of curved objects curves corresponding to various visual events partition the view sphere into regions where the image structure is stable a catalog of these events for piecewise smooth objects is available from singularity theory for a solid of revolution whose generator is an algebraic curve each visual event is characterized by a system of polynomial equations whose roots can be computed by continuation methods within each region the stable image structure is characterized by a variation of cylindrical algebraic decomposition and ray tracing this approach has been implemented and several examples are presented

counting fiedler pencils with repetitions abstract we introduce a new notation based on diagrams to deal with fiedler pencils with repetitions fpr and use it to solve several counting problems in particular we give explicit recurrences to count the number of fprs of a given degree d the number of symmetric palindromic and antipalindromic ones where the latter two structures are intended in the sense of 5 we relate these structures to the presence of symmetries in the associated diagrams

structured strong linearizations from fiedler pencils with repetition i abstract in many applications the polynomial eigenvalue problem p x 0 arises with p being a structured matrix polynomial for example palindromic symmetric skew symmetric in order to solve a structured polynomial eigenvalue problem it is convenient to use strong linearizations with the same structure as p to ensure that the symmetries in the eigenvalues due to that structure are preserved in numerical computations in this paper we characterize all the pencils in the family of the fiedler pencils with repetition introduced by vologiannidis and antoniou 25 associated with a square matrix polynomial p that are block symmetric for every matrix polynomial p we show that this family of pencils is precisely the set of all fiedler pencils with repetition that are symmetric when p is when some generic nonsingularity conditions are satisfied these pencils are strong linearizations of p in particular our family strictly contains the standard basis for dl p a k dimensional vector space of symmetric pencils introduced by mackey mackey mehl and mehrmann 20

linearizations of matrix polynomials in newton bases we discuss matrix polynomials expressed in a newton basis and the associated polynomial eigenvalue problems properties of the generalized ansatz spaces associated with such polynomials are proved directly by utilizing a novel representation of pencils in these spaces also we show how the family of fiedler pencils can be adapted to matrix polynomials expressed in a newton basis these new newton fiedler pencils are shown to be strong linearizations and some computational aspects related to them are discussed

on vector spaces of linearizations for matrix polynomials in orthogonal bases abstract regular and singular matrix polynomials p i 0 k p i i p i r n n given in an orthogonal basis 0 1 k are considered following the ideas in 9 the vector spaces called m 1 p m 2 p and dm p of potential linearizations for p are analyzed all pencils in m 1 p are characterized concisely moreover several easy to check criteria whether a pencil in m 1 p is a strong linearization of p are given the equivalence of some of them to the z rank condition 9 is pointed out results on the vector space dimensions the genericity of linearizations in m 1 p and the form of block symmetric pencils are derived in a new way on a basic algebraic level moreover an extension of these results to degree graded bases is presented throughout the paper structural resemblances between the matrix pencils in l 1 i e the results obtained in 9 and their generalized versions are pointed out

fiedler companion linearizations for rectangular matrix polynomials the development of new classes of linearizations of square matrix polynomials that generalize the classical first and second frobenius companion forms has attracted much attention in the last decade research in this area has two main goals finding linearizations that retain whatever structure the original polynomial might possess and improving properties that are essential for accurate numerical computation such as eigenvalue condition numbers and backward errors however all recent progress on linearizations has been restricted to square matrix polynomials since rectangular polynomials arise in many applications it is natural to investigate if the new classes of linearizations can be extended to rectangular polynomials in this paper the family of fiedler linearizations is extended from square to rectangular matrix polynomials and it is shown that minimal indices and bases of polynomials can be recovered from those of any linearization in this class via the same simple procedures developed previously for square polynomials fiedler linearizations are one of the most important classes of linearizations introduced in recent years but their generalization to rectangular polynomials is nontrivial and requires a completely different approach to the one used in the square case to the best of our knowledge this is the first class of new linearizations that has been generalized to rectangular polynomials

invariant properties of straight homogeneous generalized cylinders and their contours a fundamental group in computer vision is the recovery of three dimensional shape from image data while this problem is in general underconstrained the authors show that it can be simplified in the case where the objects being viewed are generalized cylinders they consider the class of straight homogeneous generalized cylinders shgcs without any further assumption on the viewing direction or the precise shape of these objects they present a rigorous mathematical study of the geometry of shgcs and characterize their gaussian curvature and occluding contours and use these results to prove several new invariant properties of the contours of shgcs these properties are in turn used in two implemented algorithms for recovering shgc descriptions from image contours several examples of segmentation of real images are given other applications are also discussed

a note on permanents and generalized complementary basic matrices abstract we say that the product of a row vector and a column vector is intrinsic if there is at most one non zero product of corresponding coordinates analogously we speak about intrinsic product of two or more matrices as well as about intrinsic factorizations of matrices since all entries of the intrinsic product are products of entries of the multiplied matrices there is no addition the class of complementary basic matrices cb matrices was recently introduced as matrices if of order n a g i 1 g i 2 g i n 1 where i 1 i 2 i n 1 is a permutation of 1 2 n 1 and the matrices g k k 1 n 1 have the form g k i k 1 c k i n k 1 for some 2 2 matrices c k it was observed that 1 independently of the permutation all such matrices with given c i s have the same spectrum though they do not form a similarity class 2 the classical companion matrix belongs to the class of cb matrices 3 the multiplication of the g i s is intrinsic connections between the 2 2 matrices c k and the resulting cb matrix g k have been explored in particular the properties which are inherited from the c k to the g k two situations were considered for the ordinary real cb matrices and for the corresponding sign pattern matrices in this paper we consider generalized complementary basic matrices where the c k matrices are replaced by square matrices of arbitrary sizes some interesting facts and results about the permanents and permanental polynomials of such matrices are obtained

comments on efficiently computing and representing aspect graphs of polyhedral objects this correspondence is originated by the definition of aspect given in the above paper by gigus canny seidel ibid vol 13 no 6 1991 on the computation of aspect graphs of polyhedra simple examples are presented which show that the data stored according to this definition can be unsuitable for identifying an object or its attitude through a topological match a definition is suggested which does not incur the problems pointed out

inertia laws and localization of real eigenvalues for generalized indefinite eigenvalue problems abstract sylvester s law of inertia states that the number of positive negative and zero eigenvalues of hermitian matrices is preserved under congruence transformations the same is true of generalized hermitian definite eigenvalue problems in which the two matrices are allowed to undergo different congruence transformations but not for the indefinite case in this paper we investigate the possible change in inertia under congruence for generalized hermitian indefinite eigenproblems and derive sharp bounds that show the inertia of the two individual matrices often still provides useful information about the eigenvalues of the pencil especially when one of the matrices is almost definite a prominent application of the original sylvester s law is in finding the number of eigenvalues in an interval our results can be used for estimating the number of real eigenvalues in an interval for generalized indefinite and nonlinear eigenvalue problems

factorization of hessenberg matrices abstract the analysis of any hessenberg matrix as a product of a companion matrix and a triangular matrix is presented in this paper the factors are explicitly given on terms of the entries of the hessenberg matrix and further various factorizations are undertaken applications to comrade and confederate matrices as well as for the corresponding block cases are included

jordan structures of alternating matrix polynomials alternating matrix polynomials that is polynomials whose coefficients alternate between symmetric and skew symmetric matrices generalize the notions of even and odd scalar polynomials we investigate the smith forms of alternating matrix polynomials showing that each invariant factor is an even or odd scalar polynomial necessary and sufficient conditions are derived for a given smith form to be that of an alternating matrix polynomial these conditions allow a characterization of the possible jordan structures of alternating matrix polynomials and also lead to necessary and sufficient conditions for the existence of structure preserving strong linearizations most of the results are applicable to singular as well as regular matrix polynomials

wavelet multilayer taylor galerkin schemes for hyperbolic and parabolic problems in this study wavelet based high order taylor galerkin methods is introduced additional time layers are used to obtain high order temporal accuracy unattainable within a two step strategy two explicit model schemes are constructed and asymptotic stability of schemes are verified the compactly supported orthogonal wavelet bases developed by daubechies are used in galerkin scheme the behavior of the proposed schemes is illustrated through one and two dimensional problems

global bifurcation sets and stable projections of nonsingular algebraic surfaces the view graph of a surface is a planar graph whose nodes are the stable views projections of the surface and whose edges represent transitional views of codimension one the space of all directions of orthogonal projection can be identified with the projective plane the set of bad projection directions associated with the degenerate views of positive codimension forms a graph in the projective plane the view bifurcation set this graph is dual to the view graph and divides the projective plane into a certain number of connected regions whose representatives are the nodes of the view graph we assume that the projected surface is nonsingular and parameterized by polynomials of degree d we present an estimate for the number of nodes in the view graph in terms of d and describe symbolic algorithms for computing the bifurcation set and the view graph of a surface from a parametrization

computing exact aspect graphs of curved objects algebraic surfaces this article presents an algorithm for computing the exact aspect graph of an opaque solid bounded by a smooth algebraic surface orthographic projection is assumed the algorithm is based on a catalog of visual events available from singularity theory it uses curve tracing cell decomposition homotopy continuation and ray tracing to construct the regions of the view sphere delineated by visual event curves the algorithm has been fully implemented and examples are presented

modified gershgorin disks for companion matrices all the zeros of a polynomial are contained in the union of gershgorin disks derived from its companion matrix a consequence of gershgorin s theorem however this theorem does not exploit the structure of the companion matrix we will use this structure to obtain smaller zero inclusion regions thereby providing some nonstandard results to accompany and illustrate this frequently covered topic in numerical and matrix analysis

high order taylor galerkin and adaptive h p methods for second order hyperbolic systems application to elastodynamics a new family of high order taylor galerkin schemes is presented for the analysis of second order hyperbolic systems the use of such schemes within an adaptive h p finite element strategy for spatial approximation in two dimensional problems is described numerical results are presented to demonstrate performance of these methods

skew symmetric matrix polynomials and their smith forms we characterize the smith form of skew symmetric matrix polynomials over an arbitrary field f showing that all elementary divisors occur with even multiplicity restricting the class of equivalence transformations to unimodular congruences a smith like skew symmetric canonical form for skew symmetric matrix polynomials is also obtained these results are used to analyze the eigenvalue and elementary divisor structure of matrices expressible as products of two skew symmetric matrices as well as the existence of structured linearizations for skew symmetric matrix polynomials by contrast with other classes of structured matrix polynomials e g alternating or palindromic polynomials every regular skew symmetric matrix polynomial is shown to have a structured strong linearization while there are singular skew symmetric polynomials of even degree for which a structured linearization is impossible for each odd degree we develop a skew symmetric companion form that uniformly provides a structured linearization for every regular and singular skew symmetric polynomial of that degree finally the results are applied to the construction of minimal symmetric factorizations of skew symmetric rational matrices

visibility occlusion and the aspect graph this chapter studies the ways in which the topology of the image of a polyhedron changes with changing viewpoint we catalog the ways that the topological appearance or aspect can change this enables us to find maximal regions of viewpoints of the same aspect we use these techniques to construct the viewpoint space partition vsp a partition of viewpoint space into maximal regions of constant aspect and its dual the aspect graph here we present tight bounds on the maximum size of the vsp and the aspect graph and give algorithms for their construction first in the convex case and then in the general case in particular we give bounds on the maximum size of n 2 and n 6 under an orthographic projection viewing model and of n 3 and n 9 under a perspective viewing model the algorithms make use of a new representation of the appearance of polyhedra from all viewpoints called the aspect representation or asp we believe that this representation is one of the significant contributions of this paper

a computational model of view degeneracy we quantify the observation by kender and freudenstein 1987 that degenerate views occupy a significant fraction of the viewing sphere surrounding an object for a perspective camera geometry we introduce a computational model that can be used to estimate the probability that a view degeneracy will occur in a random view of a polyhedral object for a typical recognition system parameterization view degeneracies typically occur with probabilities of 20 percent and depending on the parameterization as high as 50 percent we discuss the impact of view degeneracy on the problem of object recognition and for a particular recognition framework relate the cost of object disambiguation to the probability of view degeneracy to reduce this cost we incorporate our model of view degeneracy in an active focal length control paradigm that balances the probability of view degeneracy with the camera field of view in order to validate both our view degeneracy model as well as our active focal length control model a set of experiments are reported using a real recognition system operating on real images

a structural approach to representation of curved objects abstract representation of complex curved objects is still a challenging and difficult problem in this paper we present a scheme for multi level structural qualitative representation of object surfaces the analysis begins with local face based descriptions of object surfaces surface primitive graphs and global high level structural and qualitative features are derived from these local descriptions the final descriptions global feature graphs are based on the quasi convexity concavity structures incorporating quantized directional features of closed boundaries of cross sections along several directions the global feature graphs have the following properties 1 they provide a characterization of global structural qualitative properties of the object 2 the graphs are composed of a few components nodes and edges with rich features and give compact descriptions of the objects 3 the global feature graph representations are robust and qualitative in that they are not affected by geometrical quantitative properties of the shape such as curvatures length and position 4 the representation has clear and rigorous properties furthermore we present some theoretical basis for object recognition using the global features graphs

visual events identification of solids of revolution from perspective views abstract an algorithm is presented for identifying the visual events of solids of revolution from a perspective point of view the object representation model comprises of these different visual events or aspects which can be applied for object identification algorithms have been developed to handle general polyhedral objects from either orthographic or perspective views or to compute the aspect graph of solids of revolution using the orthographic projection the orthographic view aims to find the aspects of the solid of revolution which are purely object oriented the theory of differential geometry can be applied sufficiently to find the aspects without referring to the distance of the viewer in the real case the distance effect that may influence the aspect dramatically when the viewer is close to the object has to be considered for instance to generate the same aspect the viewer may have to change his viewing directions when he is at a different distance from the object firstly this paper uses the constraint of contour generator to identify the aspects of solids of revolution under perspective projection the difference between our approach and the others is the distance effect the present method combines the object oriented and viewer oriented aspect finding secondly a method is developed to find the limb termination point of the non transparent object from a certain perspective viewpoint many aspect finding methods are developed for transparent objects i e objects made of glass of which the projection contour limb is continuous for non transparent objects the self occlusion may induce the limb termination phenomena

uncertainty minimization in the localization of polyhedral objects a straightforward method is presented for the determination of the optimal sensor exploration in the localization of a polyhedral object whose geometry is known optimality is intended in the sense of the a posteriori covariance matrix of the object position and orientation parameters the method consists in decomposing the problem into simpler subproblems each one relative to a single planar face of the object it requires reasonable processing time i e comparable with the sensor activation time

complementary basic matrices abstract we show that an n n matrix which has both subdiagonal and superdiagonal rank at most one even if we distribute the diagonal positions except the first and last completely between the subdiagonal and superdiagonal part then this matrix can be factorized into a product of n 1 matrices each consisting of a 2 2 principal submatrix in two consecutive rows and columns in all possible of the n 1 positions and completed by ones along the diagonal the converse is also true it is shown that the spectrum does not depend on the order of the factors

cylindrical algebraic decomposition ii an adjacency algorithm for the plane in part i of the present paper we defined cylindrical algebraic decompositions cad s and described an algorithm for cad construction in part ii we give an algorithm that provides information about the topological structure of a cad of the plane informally two disjoint cells in e r r 1 are adjacent if they touch each other formally they are adjacent if their union is connected in a picture of a cad eg fig 2 of part i it is obvious to the eye which pairs of cells are adjacent however the cad algorithm of part i does not actually produce this information nor did the original version of that algorithm in collins 1975

topological recognition of polyhedral objects from multiple views abstract the aspect graph ag has been proposed as a viewer centered tool for object recognition although the aspects could be used for recognition from one image the full information stored in the ag aspects and visual events suggests a multiple views possibly active approach the topological nature of the ag also suggests a topological match of images of the unknown object and stored aspects in this paper we present a theoretical investigation on the use of the ag for topological recognition from multiple views of polyhedral objects first we discuss the topological matching process and give a suitable topological definition of aspect moreover since topological identification is approximate we tackle the problem of understanding the ability of the ag to topologically discriminate different objects more precisely we address the question how similar are two polyhedra with the same ag the isomorphism of polyhedra is chosen as the reference similarity condition the cases of general and convex polyhedra are discussed under perspective and parallel projections in practice it could be difficult to identify the visual events from the images of the unknown object thus we also compare isomorphism and the similarity induced by i a topologically reduced aspect graph ii the set of topologically different aspects neglecting visual events

hermitian matrix polynomials with real eigenvalues of definite type part i classification the spectral properties of hermitian matrix polynomials with real eigenvalues have been extensively studied through classes such as the definite or definitizable pencils definite hyperbolic or quasihyperbolic matrix polynomials and overdamped or gyroscopically stabilized quadratics we give a unified treatment of these and related classes that uses the eigenvalue type or sign characteristic as a common thread equivalent conditions are given for each class in a consistent format we show that these classes form a hierarchy all of which are contained in the new class of quasidefinite matrix polynomials as well as collecting and unifying existing results we make several new contributions we propose a new characterization of hyperbolicity in terms of the distribution of the eigenvalue types on the real line by analyzing their effect on eigenvalue type we show that homogeneous rotations allow results for matrix polynomials with nonsingular or definite leading coefficient to be translated into results with no such requirement on the leading coefficient which is important for treating definite and quasidefinite polynomials we also give a sufficient condition for a quasihyperbolic matrix polynomial to be diagonalizable by structure preserving congruence and show that this condition is always satisfied in the quadratic case and for any hyperbolic matrix polynomial thereby identifying an important new class of diagonalizable matrix polynomials

high order taylor galerkin methods for linear hyperbolic systems a new family of high order taylor galerkin schemes is presented for the analysis of first order linear hyperbolic systems the schemes are unconditionally stable which makes them very attractive to use in conjunction with adaptive hp finite element methods for spatial approximation

bounds for the zeros of polynomials from eigenvalues and singular values of some companion matrices abstract let a monic polynomial pn x xn a1xn 1 an aj plane1d 49e j 1 2 n be given bounds for the eigenvalues and the computation of the singular values of some special companion matrices give bounds for the zeros of the polynomial pn which improve classical and recently found ones also some new bounds for the zeros are given

characterization of a family of generalized companion matrices abstract matrices a of order n having entries in the field f x 1 x n of rational functions over a field f and characteristic polynomial det t i a t n x 1 t n 1 x n 1 t x n are studied it is known that such matrices are irreducible and have at least 2 n 1 nonzero entries such matrices with exactly 2 n 1 nonzero entries are called ma zhan matrices conditions are given that imply that a ma zhan matrix is similar via a monomial matrix to a generalized companion matrix that is a lower hessenberg matrix with ones on its superdiagonal and exactly one nonzero entry in each of its subdiagonals via the ax grothendieck theorem respectively its analog for the reals these conditions are shown to hold for a family of matrices whose entries are complex respectively real polynomials

an introduction to finite element methods for transient advection problems abstract an introduction is presented to time accurate finite element methods recently developed for solving unsteady problems governed by linear and nonlinear advection equations this includes methods making an explicit use of the characteristic curves methods based on the taylor series in time and least squares methods

finding the limbs and cusps of generalized cylinders this paper addresses the problem of finding analytically the limbs and cusps of generalized cylinders orthographic projections of generalized cylinders whose axis is straight and whose axis is an arbitrary 3d curve are considered in turn in both cases the general equations of the limbs and cusps are given they are solved for three classes of generalized cylinders solids of revolution straight homogeneous generalized cylinders whose scaling sweeping rule is a polynomial of degree less than or equal to 5 and generalized cylinders whose axis is an arbitrary 3d curve but the cross section is circular and constant examples of limbs and cusps found for each class are given applications and extensions to perspective projection and completely general straight generalized cylinders are discussed

a taylor galerkin method for convective transport problems a method is described to derive finite element schemes for the scalar convection equation in one or more space dimensions to produce accurate temporal differencing the method employs forward time taylor series expansions including time derivatives of second and third order which are evaluated from the governing partial differential equation this yields a generalized time discretized equation which is successively discretized in space by means of the standard bubnov galerkin finite element method the technique is illustrated first in one space dimension with linear elements and euler leap frog and crank nicolson time stepping several interesting relations with standard galerkin and recently developed petrov galerkin methods emerge and the new taylor galerkin schemes are found to exhibit particularly high phase accuracy with minimal numerical damping the method is successively extended to deal with variable coefficient problems and multi dimensional situations

a simplified approach to fiedler like pencils via block minimal bases pencils abstract the standard way of solving the polynomial eigenvalue problem associated with a matrix polynomial is to embed the matrix coefficients of the polynomial into a matrix pencil transforming the problem into an equivalent generalized eigenvalue problem such pencils are known as linearizations many of the families of linearizations for matrix polynomials available in the literature are extensions of the so called family of fiedler pencils these families are known as generalized fiedler pencils fiedler pencils with repetition and generalized fiedler pencils with repetition or fiedler like pencils for simplicity the goal of this work is to unify the fiedler like pencils approach with the more recent one based on strong block minimal bases pencils introduced in f m dopico et al 2017 17 to this end we introduce a family of pencils that we have named extended block kronecker pencils whose members are under some generic nonsingularity conditions strong block minimal bases pencils and show that with the exception of the non proper generalized fiedler pencils all fiedler like pencils belong to this family modulo permutations as a consequence of this result we obtain a much simpler theory for fiedler like pencils than the one available so far moreover we expect this simplification to allow for further developments in the theory of fiedler like pencils such as global or local backward error analyses and eigenvalue conditioning analyses of polynomial eigenvalue problems solved via fiedler like linearizations

on recognizing and positioning curved 3 d objects from image contours an approach for explicitly relating the shape of image contours to models of curved three dimensional objects is presented this relationship is used for object recognition and positioning object models consist of collections of parametric surface patches and their intersection curves this includes nearly all representations used in computer aided geometric design and computer vision the image contours considered are the projections of surface discontinuities and occluding contours elimination theory provides a method for constructing the implicit equation of these contours for an object observed under orthographic or perspective projection this equation is parameterized by the object s position and orientation with respect to the observer determining these parameters is reduced to a fitting problem between the theoretical contour and the observed data points the proposed approach readily extends to parameterized models it has been implemented for a simple world composed of various surfaces of revolution and tested on several real images

on the sign characteristic of hermitian linearizations in dl p abstract the computation of eigenvalues and eigenvectors of matrix polynomials is an important but difficult problem the standard approach to solve this problem is to use linearizations which are matrix polynomials of degree 1 that share the eigenvalues of p hermitian matrix polynomials and their real eigenvalues are of particular interest in applications attached to these eigenvalues is a set of signs called the sign characteristic from both a theoretical and a practical point of view it is important to be able to recover the sign characteristic of a hermitian linearization of p from the sign characteristic of p in this paper for a hermitian matrix polynomial p with nonsingular leading coefficient we describe in terms of the sign characteristic of p the sign characteristic of the hermitian linearizations in the vector space dl p mackey mackey mehl and mehrmann 2006 in particular we identify the hermitian linearizations in dl p that preserve the sign characteristic of p we also provide a description of the sign characteristic of the hermitian linearizations of p in the family of generalized fiedler pencils with repetition bueno dopico furtado and rychnovsky 2015

block minimal bases l ifications of matrix polynomials abstract the standard way of solving a polynomial eigenvalue problem associated with a matrix polynomial starts by embedding the matrix coefficients of the polynomial into a matrix pencil known as a strong linearization this process transforms the problem into an equivalent generalized eigenvalue problem however there are some situations in which is more convenient to replace linearizations by other low degree matrix polynomials this has motivated the idea of a strong l ification of a matrix polynomial which is a matrix polynomial of degree at most l having the same finite and infinite elementary divisors and the same numbers of left and right minimal indices as the original matrix polynomial we present in this work a novel method for constructing strong l ifications of matrix polynomials of size m n and grade d when l d and l divides nd or md this method is based on a family called strong block minimal bases matrix polynomials and relies heavily on properties of dual minimal bases we show how strong block minimal bases l ifications can be constructed from the coefficients of a given matrix polynomial p we also show that these l ifications satisfy many desirable properties for numerical applications they are strong l ifications regardless of whether p is regular or singular the minimal indices of the l ifications are related to those of p via constant uniform shifts and eigenvectors and minimal bases of p can be recovered from those of any of the strong block minimal bases l ifications in the special case where l divides d we introduce a subfamily of strong block minimal bases matrix polynomials named block kronecker matrix polynomials which is shown to be a fruitful source of companion l ifications

three dimensional weak visibility complexity and applications abstract in this paper we study the complexity of 3d weak visibility we obtain an o n 8 time and n 6 space algorithm to compute the weakly visible region of a triangle f from another triangle g among general scenes which are a set of n disjoint triangles we also consider the cases when the scenes are rectilinear objects and polyhedral terrains we show that in these special situations the weakly visible regions can be computed much faster in o n 6 time and o n 4 space with these results we obtain the first known polynomial time algorithm to decide whether or not a simple polyhedron is weakly internally or externally visible

computing occlusion free viewpoints this paper presents methods to compute the locus of all viewpoints from which features on known polyhedral objects can be viewed in their entirety without being occluded by anything in the environment convex and concave polyhedra with or without holes and the viewing model of perspective projection are employed in this work initially properties of the occlusion free and occluded loci of viewpoints are determined based on these properties two methods to construct these loci together with their complexity analysis are presented in the first method a boundary representation of the occlusion free locus is obtained in the second method the locus of occluded viewpoints is expressed in terms of a constructive solid geometry representation that consists of a union of component solids implementation results and comparison of the two methods are given

curve and surface duals and the recognition of curved 3d objects from their silhouettes this article addresses the problem of recognizing a solid bounded by a smooth surface in a single image the proposed approach is based on a new representation for two and three dimensional shapes called their signature that exploits the close relationship between the dual of a surface and the dual of its silhouette in weak perspective images objects are modeled by rotating them in front of a camera without any knowledge of or constraints on their motion the signatures of their silhouettes are concatenated into a single object signature to recognize an object from novel viewpoint other than those used during modeling the signature of the contours extracted from a test photograph is matched to the signatures of all modeled objects signatures this approach has been implemented and recognition examples are presented

root polynomials and their role in the theory of matrix polynomials abstract we develop a complete and rigorous theory of root polynomials of arbitrary matrix polynomials i e either regular or singular and study how these vector polynomials are related to the spectral properties of matrix polynomials we pay particular attention to the so called maximal sets of root polynomials and prove that they carry complete information about the eigenvalues finite or infinite of matrix polynomials and that they are related to the matrices that transform any matrix polynomial into its smith form in addition we describe clearly for the first time in the literature the extremality properties of such maximal sets and identify some of them whose vectors have minimal grade once the main theoretical properties of root polynomials have been established the interaction of root polynomials with three problems that have attracted considerable attention in the literature is analyzed more precisely we study the change of root polynomials under rational transformations or reparametrizations of matrix polynomials the recovery of the root polynomials of a matrix polynomial from those of its most important linearizations and the relationship between the root polynomials of two dual pencils we emphasize that for the case of regular matrix polynomials all the results in this paper can be translated into the language of jordan chains as a consequence of the well known relationship between root polynomials and jordan chains therefore a number of open problems are also solved for jordan chains of regular matrix polynomials we also briefly discuss how root polynomials can be used to define eigenvectors and eigenspaces for singular matrix polynomials

an analysis of time discretization in the finite element solution of hyperbolic problems abstract the problem of the time discretization of hyperbolic equations when finite elements are used to represent the spatial dependence is critically examined a modified equation analysis reveals that the classical second order accurate time stepping algorithms i e the lax wendroff leap frog and crank nicolson methods properly combine with piecewise linear finite elements in advection problems only for small values of the time step on the contrary as the courant number increases the numerical phase error does not decrease uniformly at all wavelengths so that the optimal stability limit and the unit cfl property are not achieved these fundamental numerical properties can however be recovered while still remaining in the standard galerkin finite element setting by increasing the order of accuracy of the time discretization this is accomplished by exploiting the taylor series expansion in the time increment up to the third order before performing the galerkin spatial discretization using piecewise linear interpolations as a result it appears that the proper finite element equivalents of second order finite difference schemes are implicit methods of incremental type having third and fourth order global accuracy on uniform meshes taylor galerkin methods numerical results for several linear examples are presented to illustrate the properties of the taylor galerkin schemes in one and two dimensional calculations

fast computation of the zeros of a polynomial via factorization of the companion matrix a new fast algorithm for computing the zeros of a polynomial in o n 2 time using o n memory is developed the eigenvalues of the frobenius companion matrix are computed by applying a nonunitary analogue of francis s implicitly shifted qr algorithm to a factored form of the matrix the algorithm achieves high speed and low memory use by preserving the factored form it also provides a residual and an error estimate for each root numerical tests confirm the high speed of the algorithm

matching 3 d models to 2 d images we consider the problem of analytically characterizing the set of all 2 d images that a group of 3 d features may produce and demonstrate that this is a useful thing to do our results apply for simple point features and point features with associated orientation vectors when we model projection as a 3 d to 2 d affine transformation we show how to represent the set of images that a group of 3 d points produces with two lines 1 d subspaces one in each of two orthogonal high dimensional spaces where a single image group corresponds to one point in each space the images of groups of oriented point features can be represented by a 2 d hyperbolic surface in a single high dimensional space the problem of matching an image to models is essentially reduced to the problem of matching a point to simple geometric structures moreover we show that these are the simplest and lowest dimensional representations possible for these cases we demonstrate the value of this way of approaching matching by applying our results to a variety of vision problems in particular we use this result to build a space efficient indexing system that performs 3 d to 2 d matching by table lookup this system is analytically built and accessed accounts for the effects of sensing error and is tested on real images we also derive new results concerning the existence of invariants and non accidental properties in this domain finally we show that oriented points present unexpected difficulties indexing requires fundamentally more space with oriented than with simple points we must use more images in a motion sequence to determine the affine structure of oriented points and the linear combinations result does not hold for oriented points

geometric reasoning about mechanical assembly abstract in which order can a product be assembled or disassembled how many hands are required how many degrees of freedom what parts should be withdrawn to allow the removal of a specified subassembly to answer such questions automatically important theoretical issues in geometric reasoning must be addressed this paper investigates the planning of assembly algorithms specifying dis assembly operations on the components of a product and the ordering of these operations it also presents measures to evaluate the complexity of these algorithms and techniques to estimate the inherent complexity of a product the central concept underlying these planning and complexity evaluation techniques is that of a non directional blocking graph a qualitative representation of the internal structure of an assembly product this representation describes the combinatorial set of parts interactions in polynomial space it is obtained by identifying physical criticalities where geometric interferences among parts change it is generated from an input geometric description of the product the main application considered in the paper is the creation of smart environments to help designers create products that are easier to manufacture and service other possible applications include planning for rapid prototyping and autonomous robots

green s matrices abstract our goal is to identify and understand matrices a that share essential properties of the unitary hessenberg matrices m that are fundamental for szego s orthogonal polynomials those properties include i recurrence relations connect characteristic polynomials rk x of principal minors of a ii a is determined by generators parameters generalizing reflection coefficients of unitary hessenberg theory iii polynomials rk x correspond not only to a but also to a certain cmv like five diagonal matrix iv the five diagonal matrix factors into a product bc of block diagonal matrices with 2 2 blocks v submatrices above and below the main diagonal of a have rank 1 vi a is a multiplication operator in the appropriate basis of laurent polynomials vii eigenvectors of a can be expressed in terms of those polynomials conditions v connects our analysis to the study of quasi separable matrices but the factorization requirement iv narrows it to the subclass of green s matrices that share properties i vii the key tool is twist transformations that provide 2n matrices all sharing characteristic polynomials of principal minors with a one such twist transformation connects unitary hessenberg to cmv another twist transformation explains findings of fiedler who noticed that companion matrices give examples outside the unitary hessenberg framework we mention briefly the further example of a daubechies wavelet matrix infinite matrices are included

companion factorization in the general linear group gl n c and applications abstract the companion factorization for nonsingular matrices belonging to the general linear group gl n c is studied here the entries in the last row of the companion matrices are explicitly represented in terms of determinants of proper sub matrices of the matrix being factorized the computation of the inverse of a nonsingular matrix is given as an application of this factorization

block kronecker linearizations of matrix polynomials and their backward errors we introduce a new family of strong linearizations of matrix polynomials which we call block kronecker pencils and perform a backward stability analysis of complete polynomial eigenproblems these problems are solved by applying any backward stable algorithm to a block kronecker pencil such as the staircase algorithm for singular pencils or the qz algorithm for regular pencils this analysis allows us to identify those block kronecker pencils that yield a computed complete eigenstructure which is exactly that of a slightly perturbed matrix polynomial these favorable pencils include the famous fiedler linearizations which are just a very particular case of block kronecker pencils thus our analysis offers the first proof available in the literature of global backward stability for fiedler pencils in addition the theory developed for block kronecker pencils is much simpler than the theory available for fiedler pencils especially in the case of rectangular matrix polynomials the global backward error analysis in this work presents for the first time the following key properties it is a rigorous analysis valid for finite perturbations i e it is not a first order analysis it provides precise bounds it is valid simultaneously for a large class of linearizations and it establishes a framework that may be generalized to other classes of linearizations these features are related to the fact that block kronecker pencils are a particular case of the new family of strong block minimal bases pencils which include certain perturbations of block kronecker pencils this allows us to extend the results in this paper to other contexts

intrinsic products and factorizations of matrices we say that the product of a row vector and a column vector is intrinsic if there is at most one nonzero product of corresponding coordinates analogously we speak about intrinsic product of two or more matrices as well as about intrinsic factorizations of matrices since all entries of the intrinsic product are products of entries of the multiplied matrices there is no addition we present several examples together with important applications these applications include companion matrices and sign nonsingular matrices

condition numbers for inversion of fiedler companion matrices abstract the fiedler matrices of a monic polynomial p z of degree n are n n matrices with characteristic polynomial equal to p z and whose nonzero entries are either 1 or minus the coefficients of p z fiedler matrices include as particular cases the classical frobenius companion forms of p z frobenius companion matrices appear frequently in the literature on control and signal processing but it is well known that they posses many properties that are undesirable numerically which limit their use in applications in particular as n increases frobenius companion matrices are often nearly singular i e their condition numbers for inversion are very large therefore it is natural to investigate whether other fiedler matrices are better conditioned than the frobenius companion matrices or not in this paper we present explicit expressions for the condition numbers for inversion of all fiedler matrices with respect the frobenius norm i e a f ij a ij 2 this allows us to get a very simple criterion for ordering all fiedler matrices according to increasing condition numbers and to provide lower and upper bounds on the ratio of the condition numbers of any pair of fiedler matrices these results establish that if p 0 1 then the frobenius companion matrices have the largest condition number among all fiedler matrices of p z and that if p 0 1 then the frobenius companion matrices have the smallest condition number we also provide families of polynomials where the ratio of the condition numbers of pairs of fiedler matrices can be arbitrarily large and prove that this can only happen when both fiedler matrices are very ill conditioned we finally study some properties of the singular values of fiedler matrices and determine how many of the singular values of a fiedler matrix are equal to one

some inheritance properties for complementary basic matrices abstract we say that the product of a row vector and a column vector is intrinsic if there is at most one non zero product of corresponding coordinates analogously we speak about intrinsic product of two or more matrices as well as about intrinsic factorizations of matrices since all entries of the intrinsic product are products of entries of the multiplied matrices there is no addition the class of complementary basic matrices cb matrices was recently introduced as matrices if of order n a g i 1 g i 2 g i n 1 where i 1 i 2 i n 1 is a permutation of 1 2 n 1 and the matrices g k k 1 n 1 have the form g k i k 1 c k i n k 1 for some 2 2 matrices c k it was observed that 1 independently of the permutation all such matrices with given c i s have the same spectrum though they do not form a similarity class 2 the classical companion matrix belongs to the class of cb matrices m fiedler a note on companion matrices linear algebra appl 372 2003 325 331 9 3 the multiplication of the g i s is intrinsic we explore connections between the 2 2 matrices c k and the resulting cb matrix g k in particular which properties are inherited from the c k to the g k we consider two situations for the ordinary real cb matrices and for the corresponding sign pattern matrices

automatic recovery of eigenvectors and minimal bases of matrix polynomials from generalized fiedler pencils with repetition abstract linearization is a widely used method for solving polynomial eigenproblems in which a matrix polynomial is transformed to a matrix pencil of larger size fiedler pencils generalized fiedler pencils fiedler pencils with repetition fprs and generalized fiedler pencils with repetition gfprs are well known classes of strong linearizations of matrix polynomials the class gfprs is also a rich source of structure preserving strong linearizations of structured matrix polynomials the recovery of eigenvectors minimal bases and minimal indices of matrix polynomials from those of the linearizations is an important task it is well known that eigenvectors and minimal bases of matrix polynomials can be easily recovered from those of the fiedler and generalized fiedler pencils by contrast barring a small subclass of fprs the recovery of eigenvectors and minimal bases of matrix polynomials from those of the fprs and gfprs is still an open problem the purpose of this paper is to fill this gap we consider the class of gfprs which subsumes the class of fprs and describe the recovery of eigenvectors minimal bases and minimal indices of matrix polynomials from those of the gfprs and show that the recovery is operation free

aspect graph generation for non convex polyhedra from perspective projection view abstract the aspect graph is basically a multiple view representation of the designated three dimensional 3d object in this paper a complete algorithm is presented for constructing the aspect graph of general polyhedral objects from the perspective point of view the aspect graph generation is based on a so called visual event these visual events will completely capture the way the appearance of an edge or a face can be occluded by other faces and edges in the model all the visual events can be found for polyhedra as facial ee and eee events the facial event indicates the visibility transition of the faces the ee event indicates the possible visibility variation of the occlusion of two edges and the eee event represents another visibility change the consecutive occlusion among three edges the system consists of the following five stages 1 find the viewing cell for each planar face ee event and eee event and then create the parcellation graph 2 employ the parcellation graph to create a face level enumeration array and edge level enumeration array 3 generate test points 4 examine plausible cells that enclose a volume of viewing space and 5 generate an explicit aspect graph structure finally nine non convex polyhedra are obtained in the experiments the aspect graphs of these non regular shaped objects can be easily generated by this system

spectral equivalence of matrix polynomials and the index sum theorem abstract the concept of linearization is fundamental for theory applications and spectral computations related to matrix polynomials however recent research on several important classes of structured matrix polynomials arising in applications has revealed that the strategy of using linearizations to develop structure preserving numerical algorithms that compute the eigenvalues of structured matrix polynomials can be too restrictive because some structured polynomials do not have any linearization with the same structure this phenomenon strongly suggests that linearizations should sometimes be replaced by other low degree matrix polynomials in applied numerical computations motivated by this fact we introduce equivalence relations that allow the possibility of matrix polynomials with coefficients in an arbitrary field to be equivalent with the same spectral structure but have different sizes and degrees these equivalence relations are directly modeled on the notion of linearization and consequently inherit the simplicity applicability and most relevant properties of linearizations simultaneously though they are much more flexible in the possible degrees of equivalent polynomials this flexibility allows us to define in a unified way the notions of quadratification and l ification to introduce the concept of companion form of arbitrary degree and to provide concrete and simple examples of these notions that generalize in a natural and smooth way the classical first and second frobenius companion forms the properties of l ifications are studied in depth in this process a fundamental result on matrix polynomials the index sum theorem is recovered and extended to arbitrary fields although this result is known in the systems theory literature for real matrix polynomials it has remained unnoticed by many researchers it establishes that the sum of the finite and infinite partial multiplicities together with the left and right minimal indices of any matrix polynomial is equal to the rank times the degree of the polynomial the index sum theorem turns out to be a key tool for obtaining a number of significant results on the possible sizes and degrees of l ifications and companion forms on the minimal index preservation properties of companion forms of arbitrary degree as well as on obstructions to the existence of structured companion forms for structured matrix polynomials of even degree this paper presents many new results blended together with results already known in the literature but extended here to the most general setting of matrix polynomials of arbitrary sizes and degrees over arbitrary fields therefore we have written the paper in an expository and self contained style that makes it accessible to a wide variety of readers

new bounds for roots of polynomials based on fiedler companion matrices abstract several matrix norms of the classical frobenius companion matrices of a monic polynomial p z have been used in the literature to obtain simple lower and upper bounds on the absolute values of the roots of p z recently m fiedler 2003 9 has introduced a new family of companion matrices of p z that has received considerable attention and it is natural to investigate if matrix norms of fiedler companion matrices may be used to obtain new and sharper lower and upper bounds on the development of such bounds requires first to know simple expressions for some relevant matrix norms of fiedler matrices and we obtain them in the case of the 1 and matrix norms with these expressions at hand we will show that norms of fiedler matrices produce many new bounds but that none of them improves significatively the classical bounds obtained from the frobenius companion matrices however we will prove that if the norms of the inverses of fiedler matrices are used then another family of new bounds on is obtained and some of the bounds in this family improve significatively the bounds coming from the frobenius companion matrices for certain polynomials

spectral embedding of graphs in this paper we explore how to embed symbolic relational graphs with unweighted edges in a pattern space we adopt a graph spectral approach we use the leading eigenvectors of the graph adjacency matrix to define eigenmodes of the adjacency matrix for each eigenmode we compute vectors of spectral properties these include the eigenmode perimeter eigenmode volume cheeger number inter mode adjacency matrices and intermode edge distance we embed these vectors in a pattern space using two contrasting approaches the first of these involves performing principal or independent components analysis on the covariance matrix for the spectral pattern vectors the second approach involves performing multidimensional scaling on the l2 norm for pairs of pattern vectors we illustrate the utility of the embedding methods on neighbourhood graphs representing the arrangement of corner features in 2d images of 3d polyhedral objects two problems are investigated the first of these is the clustering of graphs representing distinct objects viewed from different directions the second is the identification of characteristic views of single objects these two studies reveal that both embedding methods result in well structured view spaces for graph data extracted from 2d views of 3d objects

the visual hull concept for silhouette based image understanding many algorithms for both identifying and reconstructing a 3 d object are based on the 2 d silhouettes of the object in general identifying a nonconvex object using a silhouette based approach implies neglecting some features of its surface as identification clues the same features cannot be reconstructed by volume intersection techniques using multiple silhouettes of the object this paper addresses the problem of finding which parts of a nonconvex object are relevant for silhouette based image understanding for this purpose the geometric concept of visual hull of a 3 d object is introduced this is the closest approximation of object s that can be obtained with the volume intersection approach it is the maximal object silhouette equivalent to s i e which can be substituted for s without affecting any silhouette only the parts of the surface of s that also lie on the surface of the visual hull can be reconstructed or identified using silhouette based algorithms the visual hull depends not only on the object but also on the region allowed to the viewpoint two main viewing regions result in the external and internal visual hull in the former case the viewing region is related to the convex hull of s in the latter it is bounded by s the internal visual hull also admits an interpretation not related to silhouettes algorithms for computing visual hulls are presented and their complexity analyzed in general the visual hull of a 3 d planar face object turns out to be bounded by planar and curved patches

flandersʼ theorem for many matrices under commutativity assumptions we analyze the relationship between the jordan canonical form of products in different orders of k square matrices a 1 a k our results extend some classical results by h flanders motivated by a generalization of fiedler matrices we study permuted products of a 1 a k under the assumption that the graph of non commutativity relations of a 1 a k is a forest under this condition we show that the jordan structure of all nonzero eigenvalues is the same for all permuted products for the eigenvalue zero we obtain an upper bound on the dierence between the sizes of jordan blocks for any two permuted products and we show that this bound is attainable for k 3 we show that moreover the bound is exhaustive