eigenvalues and degree deviation in graphs abstract let g be a graph with n vertices and m edges and let g 1 g n g be the eigenvalues of its adjacency matrix set s g u v g d u 2 m n we prove that s 2 g 2 n 2 2 m g 2 m n s g in addition we derive similar inequalities for bipartite g we also prove that the inequality k g n k 2 g 1 2 2 s g holds for every k 2 n finally we prove that for every graph g of order n n g n g 1 s 2 g 2 n 3 we show that these inequalities are tight up to a constant factor

conformal geometry and its applications on 3d shape matching recognition and stitching three dimensional shape matching is a fundamental issue in computer vision with many applications such as shape registration 3d object recognition and classification however shape matching with noise occlusion and clutter is a challenging problem in this paper we analyze a family of quasi conformal maps including harmonic maps conformal maps and least squares conformal maps with regards to 3d shape matching as a result we propose a novel and computationally efficient shape matching framework by using least squares conformal maps according to conformal geometry theory each 3d surface with disk topology can be mapped to a 2d domain through a global optimization and the resulting map is a diffeomorphism i e one to one and onto this allows us to simplify the 3d shape matching problem to a 2d image matching problem by comparing the resulting 2d parametric maps which are stable insensitive to resolution changes and robust to occlusion and noise therefore highly accurate and efficient 3d shape matching algorithms can be achieved by using the above three parametric maps finally the robustness of least squares conformal maps is evaluated and analyzed comprehensively in 3d shape matching with occlusion noise and resolution variation in order to further demonstrate the performance of our proposed method we also conduct a series of experiments on two computer vision applications i e 3d face recognition and 3d nonrigid surface alignment and stitching

gap metrics for time varying linear systems in a continuous time setting a recently proposed time varying generalisation of vinnicombe s v gap is shown to have the properties of a metric on a class of causal linear continuous time systems that admit normalised strong graph representations an alternative time varying gap metric is then defined in more general terms and a corresponding robust stability result is derived for linear feedback interconnections finally it is shown that the time varying gap and v gap metrics induce the same topology this being the coarsest topology under which feedback stability and closed loop sensitivity are both robust properties c 2014 elsevier b v all rights reserved

why do stabilizing controllers stabilize an alternative notion of a stabilizing controller is presented being defined in terms of the trajectories of plant and controller it is well suited for situations in which the controller s action is an interconnecting one in the second part of the paper a parametrization is given of the kernel representations of all controllers that give rise to an a priori specified closed loop behavior as a special case the parametrization yields all stabilizing controllers unlike the youla parametrization it also includes the so called singular stabilizing controllers

a behavioral approach to the h optimal control problem this paper considers the general h optimal control problem from a behavioral perspective a formalization of this problem is given that departs from the usual h control paradigm in the sense that system variables of the plant are treated in a symmetric way without distinguishing control inputs measurements exogenous inputs and to be controlled variables interconnection variables are introduced and controllers are allowed to constrain the interconnection variables of the plant necessary and sufficient conditions are given for the existence of controllers that achieve an h control objective for a given linear time invariant plant the set of all such controllers is parametrized by means of j spectral factorizations

observer synthesis in the behavioral approach analyzes the observer design problem in the behavioral context observability and detectability notions are first introduced and fully characterized necessary and sufficient conditions for the existence of an observer possibly an asymptotic or an exact one are introduced and a complete parameterization of all admissible observers is given the problem of obtaining observers endowed with a strictly proper transfer matrix and the design of observer based controllers are later addressed and solved finally the above issues are particularized to the case of state space systems thus showing they naturally generalize well known theorems of traditional system theory

analytical relationships between metric and centrality measures of a network and its dual the centrality and efficiency measures of a network g are strongly related to the respective measures on the dual g and the bipartite b g associated networks we show some relationships between the bonacich centralities c g c g and c b g and between the efficiencies e g and e g and we compute the behavior of these parameters in some examples

constraints on deformable models recovering 3d shape and nongrid motion abstract inferring the 3d structures of nonrigidly moving objects from images is a difficult yet basic problem in computational vision our approach makes use of dynamic elastically deformable object models that offer the geometric flexibility to satisfy a diversity of real world visual constraints we specialize these models to include intrinsic forces inducing a preference for axisymmetry image based constraints are applied as extrinsic forces that mold the symmetry seeking model into shapes consistent with image data we describe an extrinsic force that applies constraints derived from profiles of monocularly viewed objects we generalize this constraint force to incorporate profile information from multiple views and use it to exploit binocular image data for time varying images the force becomes dynamic and the model is able to infer not only depth but nonrigid motion as well we demonstrate the recovery of 3d shape and nonrigid motion from natural imagery

controller discretization a gap metric framework for analysis and synthesis although techniques for directly synthesising sampled data sd compensators are available in the literature feedback controller design is perhaps best understood in a purely continuous time setting as such a feedback controller is often designed in the continuous time domain and then discretized for digital implementation it is important for the discretization step involved to yield an sd approximation which captures the essential features of the original controller from the perspective of closed loop behavior in this note a gap metric framework is developed for studying the controller discretization problem for linear time invariant lti plants and controllers importantly knowledge of a gap metric distance between an lti controller and an sd approximation permits explicit characterization of the possible difference in closed loop performance with any lti plant for which the lti controller is known to work well accounting for intersample behavior the central result of the new framework gives rise to an algorithm for computing a gap metric measure of the distance between an lti controller and a given discretization and a technique for synthesising a sd approximation which is optimal with respect to this metric

stability analysis for time varying systems via quadratic constraints abstract in this paper we introduce the concept of quadratic constraint qc to deal with various stability problems for infinite dimensional linear discrete time varying ltv systems first we derive a necessary and sufficient condition for the close loop stability based on quadratic constraints qcs an equivalent condition of this stability criterion presents the relationship between the stabilization of each finite dimensional truncation system and that of the whole system moreover applying this stability criterion and the complete finiteness of the nest algebra we show that the plant is stabilizable if and only if it has a single strong representation finally we use qcs to give a necessary and sufficient condition for the robust stabilization of connected uncertainty set defined by gap metric

a unifying framework for robust synchronization of heterogeneous networks via integral quadratic constraints a general framework for analysing robust synchronization in large scale heterogenous networks is proposed based on the theory of integral quadratic constraints iqcs dynamic agents are represented as linear time invariant single input single output systems the agents exchange information according to a sparse dynamical interconnection operator in order to achieve synchronization where their outputs are steered to the same possibly time varying signal the main technical hindrance to applying iqcs in this context lies with the presence of the marginally stable dynamics which define the trajectory to which the agents outputs synchronize it is shown that by working with conditions defined on modified signal spaces of interest and exploiting the graph structure underlying the connections between the dynamic systems iqc methods can be applied directly to synchronization analysis without recourse to loop transformations which may obscure the inherent structural properties of the multi agent networked systems decentralized and scalable conditions for synchronization are proposed within this setting the iqc framework is demonstrated to unify and generalize some of the existing results in the literature including certain nyquist type consensus certificates for time delay systems moreover it allows the role of feedback in robustness against uncertainty to be better manifested within the context of synchronization

robust stability of feedback systems a geometric approach using the gap metric a geometric framework for robust stabilization of infinite dimensional time varying linear systems is presented the uncertainty of a system is described by perturbations of its graph and is measured in the gap metric necessary and sufficient conditions for robust stability are generalized from the time invariant case an example is given to highlight an important difference between the obstructions which limit the size of a stabilizable gap ball in the time varying and time invariant cases several results on the gap metric and the gap topology are established that are central in a geometric treatment of the robust stabilizability problem in the gap in particular the concept of a graphable subspace is introduced in the paper subspaces that fail to be graphable are characterized by an index condition on a certain semi fredholm operator

on the control of finite support behaviors abstract in the paper the main properties of finite support behaviors are introduced and characterized in terms of the polynomial matrices involved in their representation necessary and sufficient conditions for the existence of an input output possibly non anticipating description of a finite support behavior are given the control problem is also addressed and properly restated in the behavior theory context as an interconnection problem conditions for the existence of controllers that make the resulting connected system observable as well as a complete description of the class of characteristic polynomials that the resulting controlled system possibly exhibits are given finally the problem of controlling a dynamical system in input output from by means of a controller with the same structure is investigated and solved

on shape of plane elastic curves we study shapes of planar arcs and closed contours modeled on elastic curves obtained by bending stretching or compressing line segments non uniformly along their extensions shapes are represented as elements of a quotient space of curves obtained by identifying those that differ by shape preserving transformations the elastic properties of the curves are encoded in riemannian metrics on these spaces geodesics in shape spaces are used to quantify shape divergence and to develop morphing techniques the shape spaces and metrics constructed are novel and offer an environment for the study of shape statistics elasticity leads to shape correspondences and deformations that are more natural and intuitive than those obtained in several existing models applications of shape geodesics to the definition and calculation of mean shapes and to the development of shape clustering techniques are also investigated

nasal similarity measure of 3d faces based on curve shape space abstract we propose a novel method for measuring the nasal similarity among 3d faces firstly we construct a representation for the nose shape which is composed of a set of geodesic curves each crosses the bridge of the nose next using these geodesic curves we formulate a similarity measure to compare among noses in the curve shape space under the riemannian framework the shape space is a quotient space for which the scaling translation and rotation are removed since the nose similarity measure is based on the shape comparison the proposed method has the following advantages 1 the similarity measure is robust to facial expressions since the nose is not affected by facial expressions 2 the geometric features of the nose shape match well with the human perception 3 the similarity measure is independent of the mesh grid because the chosen nose curves are not sensitive to the triangular mesh model we construct a nasal hierarchical structure for noses organization which is based on nose similarity measure results in our experiments we evaluate the performance of the proposed method and compare it with competing methods on three public face databases namely frgc2 0 texas3d and bosphorusdb the results show superiority of the proposed method in terms of both the speed and the accuracy when the nasal measurements are processed in the nasal hierarchical structure and the nasal samples with low sampling rate 5 25 of original point cloud

paradigms and puzzles in the theory of dynamical systems a self contained exposition is given of an approach to mathematical models in particular to the theory of dynamical systems the basic ingredients form a triptych with the behavior of a system in the center and behavioral equations with latent variables as side panels the author discusses a variety of representation and parametrization problems in particular questions related to input output and state models the proposed concept of a dynamical system leads to a new view of the notions of controllability and observability and of the interconnection of systems in particular to what constitutes a feedback control law the final issue addressed is that of system identification it is argued that exact system identification leads to the question of computing the most powerful unfalsified model

stabilisation and control design by partial output feedback and by partial interconnection we study stabilisation and control design by partial output feedback on the one hand and by partial interconnection according to willems on the other hand this article is based on the paper design parametrization and pole placement of stabilizing output feedback compensators via injective cogenerator quotient signal modules by i blumthaler and u oberst and the same technique is used both discrete and continuous behaviours and various notions of stability are treated simultaneously in the case of partial output feedback we require that both the compensator and the feedback behaviour have proper transfer matrices without presuming properness of the plant we obtain conditions ensuring the existence of such stabilising compensators which perform additional control tasks like for instance tracking and an algorithm for constructing a large class of such compensators in the case of stabilisation by interconnection no input output structures of plant or compensator are assumed but the controlled be

on the computation of the gap metric for ltv systems abstract in this paper we consider the robust stabilization problem for linear discrete time varying ltv systems using the gap metric in particular we show that the time varying tv directed gap reduces to an operator with a tv hankel plus toeplitz structure computation of the norm of such an operator can be carried out using an iterative scheme involving a tv hankel operator defined on a space of hilbert schmidt causal operators the infimization in the tv directed gap formula is shown to be in fact a minimum by using duality theory the latter holds as well in the time invariant case

on the laplace beltrami operator and brain surface flattening in this paper using certain conformal mappings from uniformization theory the authors give an explicit method for flattening the brain surface in a way which preserves angles from a triangulated surface representation of the cortex the authors indicate how the procedure may be implemented using finite elements further they show how the geometry of the brain surface may be studied using this approach

piecewise linear manifolds einstein metrics and ricci flows this article provides an attempt to extend concepts from the theory of riemannian manifolds to piecewise linear p l spaces in particular we propose an analogue of the ricci tensor which we give the name of an einstein vector field on a given set of p l spaces we define and discuss normalized einstein flows p l einstein metrics are defined and examples are provided criteria for flows to approach einstein metrics are formulated second variations of the total scalar curvature at a specific einstein space are calculated

teichmuller shape descriptor and its application to alzheimer s disease study we propose a novel method to apply teichmuller space theory to study the signature of a family of nonintersecting closed 3d curves on a general genus zero closed surface our algorithm provides an efficient method to encode both global surface and local contour shape information the signature teichmuller shape descriptor is computed by surface ricci flow method which is equivalent to solving an elliptic partial differential equation on surfaces and is numerically stable we propose to apply the new signature to analyze abnormalities in brain cortical morphometry experimental results with 3d mri data from alzheimer s disease neuroimaging initiative adni dataset 152 healthy control subjects versus 169 alzheimer s disease ad patients demonstrate the effectiveness of our method and illustrate its potential as a novel surface based cortical morphometry measurement in ad research

fast modes in the set of minimal dissipation trajectories abstract in this paper we study the set of trajectories satisfying both a given lti system s laws and also laws of the corresponding adjoint system in other words trajectories in the intersection of the system s behavior and that of the adjoint system this intersection has important system theoretic significance for example it is known that the trajectories in this intersection are the ones with minimal dissipation underlying the notion of adjoint is that of a power supply it is with respect to this supply rate that the trajectories in the intersection are known to be stationary in this paper we deal with half line solutions to the differential equations governing both the system and its adjoint analysis of half line solutions plays a central role for example in initial value problems and in well posedness studies of an interconnection we interpret the set of half line trajectories allowed by a system and its adjoint as an interconnection of these two systems and thus address issues about well posedness ill posedness of the interconnection we formulate necessary and sufficient conditions for this intersection to be autonomous for the case of an ill posed interconnection and resulting autonomous system we derive conditions for existence of initial conditions that lead to impulsive solutions in the states of the system we link our conditions with the strongly reachable and weakly unobservable subspaces of a state space system we show that absence of impulsive initial conditions is equivalent to the well known subspace iteration algorithms for these subspaces converging in one step

what can the canonical controller in principle tell us given a plant and a desired specification our goal is to construct a controller system which when interconnected with the plant yields a system that behaves like the desired specification we can always construct the canonical controller introduced in van der schaft 2003 10 for linear systems there exists a controller which when interconnected to the plant yields the desired behaviour if and only if the canonical controller is itself one such controller see vinjamoor and van der schaft 2011 4 in this paper we extend this result to nonlinear systems it turns out that one has to look at the canonical controller together with its subsystems we obtain necessary and sufficient conditions for the existence of a controller for a class of nonlinear systems we end with examples which show that in certain cases looking at subsystems of the canonical controller also does not suffice

three dimensional face recognition an expression invariant 3d face recognition approach is presented our basic assumption is that facial expressions can be modelled as isometries of the facial surface this allows to construct expression invariant representations of faces using the bending invariant canonical forms approach the result is an efficient and accurate face recognition algorithm robust to facial expressions that can distinguish between identical twins the first two authors we demonstrate a prototype system based on the proposed algorithm and compare its performance to classical face recognition methods the numerical methods employed by our approach do not require the facial surface explicitly the surface gradients field or the surface metric are sufficient for constructing the expression invariant representation of any given face it allows us to perform the 3d face recognition task while avoiding the surface reconstruction stage

brain surface conformal parameterization with the ricci flow in brain mapping research parameterized 3 d surface models are of great interest for statistical comparisons of anatomy surface based registration and signal processing here we introduce the theories of continuous and discrete surface ricci flow which can create riemannian metrics on surfaces with arbitrary topologies with user defined gaussian curvatures the resulting conformal parameterizations have no singularities and they are intrinsic and stable first we convert a cortical surface model into a multiple boundary surface by cutting along selected anatomical landmark curves secondly we conformally parameterize each cortical surface to a parameter domain with a user designed gaussian curvature arrangement in the parameter domain a shape index based on conformal invariants is computed and inter subject cortical surface matching is performed by solving a constrained harmonic map we illustrate various target curvature arrangements and demonstrate the stability of the method using longitudinal data to map statistical differences in cortical morphometry we studied brain asymmetry in 14 healthy control subjects we used a manifold version of hotelling s t2 test applied to the jacobian matrices of the surface parameterizations a permutation test along with the cumulative distribution of p values were used to estimate the overall statistical significance of differences the results show our algorithm s power to detect subtle group differences in cortical surfaces

the achievable dynamics via control by interconnection we consider the problem of finding a controller such that when interconnected to the plant we obtain a system that is equivalent to a desired system here equivalence is formalized as bisimilarity we give necessary and sufficient conditions for the existence of such a controller the systems we consider are linear input state output systems a comparison is made to previously obtained results about achievable implementable behaviors in the behavioral approach to systems theory among the advantages of using the notion of bisimilarity is the fact that it directly applies to state space systems while the computations involved are operations on constant matrices

coordinate systems for conformal cerebellar flat maps methods quasi conformal flat maps of the human cerebellum 1 4 were created from a high resolution tl weighted mri volume 5 a topologically correct surface was produced from a cerebellar volume defined by a plane parallel to the posterior commisure obex line and orthogonal to a plane passing through the vermal midline our quasi conformal flattening procedure 2 3 was applied to this surface to produce flat maps the cortical surface was parcellated according to 6 and activations produced by a target interception task were imposed eleven readily identifiable anatomical landmarks e g the apex of the fourth ventricle and the rostral most tip of the lingula were defined for the hyperbolic coordinate system one landmark was used as the map center and a second to specify the map orientation we also used a polar coordinate system on the hyperbolic flat map with equiangular grid lines radiating from the map focus and equidistant circles surrounding the focus for the spherical coordinate system one landmark was used as the north pole one as the south pole and a third as a distinguished equatorial point we used the usual latitude and longitude coordinates similar to 7 results and conclusions a major benefit of quasi conformal flat maps is that they can be used to generate canonical surface based coordinate systems for a cortical surface since our fiat maps to the hyperbolic plane and the sphere are l l and onto the coordinate systems produced by these flat maps can be used to quantitatively localize structure and function furthermore these flat maps are mathematically unique different choices of landmarks and coordinate systems will increase the utility of flat maps for studying functional imaging dam we believe that the use of cerebellar flat maps and canonical coordinate systems based on reproducible anatomical landmarks will improve our ability to localize functional activity on the cerebellar cortex and to quantify anatomical and functional differences between individual subjects and groups of subjects

efficiency of scale free networks error and attack tolerance the concept of network efficiency recently proposed to characterize the properties of small world networks is here used to study the effects of errors and attacks on scale free networks two different kinds of scale free networks i e networks with power law p k are considered 1 scale free networks with no local clustering produced by the barabasi albert model and 2 scale free networks with high clustering properties as in the model by klemm and eguiluz and their properties are compared to the properties of random graphs exponential graphs by using as mathematical measures the global and the local efficiency we investigate the effects of errors and attacks both on the global and the local properties of the network we show that the global efficiency is a better measure than the characteristic path length to describe the response of complex networks to external factors we find that at variance with random graphs scale free networks display both on a global and on a local scale a high degree of error tolerance and an extreme vulnerability to attacks in fact the global and the local efficiency are unaffected by the failure of some randomly chosen nodes though they are extremely sensitive to the removal of the few nodes which play a crucial role in maintaining the network s connectivity

combinatorial ricci flows on surfaces we show that the analogue of hamilton s ricci flow in the combinatorial setting produces solutions which converge exponentially fast to thurston s circle packing on surfaces as a consequence a new proof of thurston s existence of circle packing theorem is obtained as another consequence ricci flow suggests a new algorithm to find circle packings

totally dissipative systems in a totally dissipative behavior all non trivial trajectories dissipate energy a characterization of such behaviors is given in terms of properties of the one and two polynomial matrices associated with the supply rate and with their kernel and image representation

stabilization pole placement and regular implementability in this paper we study control by interconnection of linear differential systems we give necessary and sufficient conditions for regular implementability of a given linear differential system we formulate the problems of stabilization and pole placement as problems of finding a suitable regularly implementable sub behavior of the manifest plant behavior the problem formulations and their resolutions are completely representation free and specified only in terms of the system dynamics control is viewed as regular interconnection a controller is a system that constrains the plant behavior through a distinguished set of variables namely the control variables the issue of implementation of a controller in the feedback configuration and its relation to regularity of interconnection is addressed freedom of disturbances in a plant and regular interconnection with a controller also turn out to be inter related

a robust control framework for linear time invariant spatially distributed systems a robust control framework for linear time invariant lti spatially distributed systems is outlined in this paper we adopt an input output approach which takes account of the spatially distributed nature of the input and output signals for such systems the approach is a generalization of h infty xspace control in the sense that the 2 norm in both time and space is used to quantify the size of signals it is shown that a frequency domain representation in the form of a graph symbol exists for every lti spatially distributed system under very mild assumptions the graph symbol gives rise to left and right coprime representations if the system is also stabilizable we investigate fundamental issues of feedback control such as feedback stability and robust stability to plant and or controller uncertainty quantified in the gap metric this includes a generalization of the sefton ober gap formula to the infinite dimensional operator case a design example in which an electrostatically destabilized membrane is feedback stabilized concludes the paper

on the α nonbacktracking centrality for complex networks existence and limit cases abstract nonbactracking centrality was introduced as an attempt to correct some deficiencies of eigenvector centrality in this work the nonbacktracking centrality is introduced as an extension that interpolates between the nonbacktracking centrality of the edges of a directed network and the eigenvector centrality of the corresponding directed line graph the existence of this new nonbacktracking centrality is proved in terms of the connectivity of the original network we prove that the limit of the nonbacktracking centrality when decreases to zero exists and is well defined moreover it coincides with the nonbacktracking centrality when this measure is defined with the same techniques we also prove the convergence of pagerank vectors to the eigenvector centrality vector when the damping factor tends to 1

on the parametrization of all regularly implementing and stabilizing controllers in this paper we deal with problems of controller parametrization in the context of behavioral systems given a full plant behavior a subbehavior of the manifest plant behavior is called regularly implementable if it can be achieved as the controlled behavior resulting from the interconnection of the full plant behavior with a suitable controller behavior in such a way that the controller does not impose restrictions that are already present in the plant we establish a parametrization of all controllers that regularly implement a given behavior we also obtain a parametrization of all stabilizing controllers

optimal mass transport for shape matching and comparison surface based 3d shape analysis plays a fundamental role in computer vision and medical imaging this work proposes to use optimal mass transport map for shape matching and comparison focusing on two important applications including surface registration and shape space the computation of the optimal mass transport map is based on monge brenier theory in comparison to the conventional method based on monge kantorovich theory this method significantly improves the efficiency by reducing computational complexity from o n 2 to o n for surface registration problem one commonly used approach is to use conformal map to convert the shapes into some canonical space although conformal mappings have small angle distortions they may introduce large area distortions which are likely to cause numerical instability thus resulting failures of shape analysis this work proposes to compose the conformal map with the optimal mass transport map to get the unique area preserving map which is intrinsic to the riemannian metric unique and diffeomorphic for shape space study this work introduces a novel riemannian framework conformal wasserstein shape space by combing conformal geometry and optimal mass transport theory in our work all metric surfaces with the disk topology are mapped to the unit planar disk by a conformal mapping which pushes the area element on the surface to a probability measure on the disk the optimal mass transport provides a map from the shape space of all topological disks with metrics to the wasserstein space of the disk and the pullback wasserstein metric equips the shape space with a riemannian metric we validate our work by numerous experiments and comparisons with prior approaches and the experimental results demonstrate the efficiency and efficacy of our proposed approach

analysis and synthesis on a generalized stability region the generalization of the spl nu gap metric to a finitely connected stability region is derived the results hold for the class of all linear time invariant systems which admit a continuous coprime factorization including the nonrational case a limited extension of the h sub spl infin loopshaping synthesis procedure is obtained via conformal mapping an example is given where enlargement of the stability region can be used to transform the problem into one which is solvable via uniform stability margin optimization

a two operator approach to robust stabilization of linear systems on robust non fragile feedback stabilization is studied for causal linear time invariant lti input output systems defined in an l 2 signal space setup or equivalently in an l 2 j setup in the frequency domain two operator descriptions for the plant and the controller are used with bounded operators defined on the full space l 2 basic properties of causal lti operators on weighted l 2 spaces l 2 w which are relevant to stabilization theory are also established for different choices of the weighting function w weighted l 2 spaces allow the inclusion of persistent signals such as steps and thus provide the potential of important generalizations of the robust stabilization theory on l 2 in particular it is shown that there is a large class of weights w for which the l 2 w setup leads to h optimization this implies that robust h design can be applied to a large class of weighted l 2 spaces on problems of closability of unstable causal lti convolution operators are discussed t

graphs closability and causality of linear time invariant discrete time systems in this paper a linear time invariant discrete time system is considered by regarding it as an unbounded operator on 2 n 2 z cn or cz the graphs of closed systems are characterized and properties such as causality and closability of systems are studied it is shown that the closure of a causal closable system on 2 n cn and cz is again causal whereas this is in general not the case for systems on 2 z necessary and sufficient conditions which guarantee that this property hold for systems on 2 z are given many of the results extend without change to the continuous time case

shape analysis of planar multiply connected objects using conformal welding shape analysis is a central problem in the field of computer vision in 2d shape analysis classification and recognition of objects from their observed silhouettes are extremely crucial but difficult it usually involves an efficient representation of 2d shape space with a metric so that its mathematical structure can be used for further analysis although the study of 2d simply connected shapes has been subject to a corpus of literatures the analysis of multiply connected shapes is comparatively less studied in this work we propose a representation for general 2d multiply connected domains with arbitrary topologies using conformal welding a metric can be defined on the proposed representation space which gives a metric to measure dissimilarities between objects the main idea is to map the exterior and interior of the domain conformally to unit disks and circle domains unit disk with several inner disks removed using holomorphic 1 forms a set of diffeomorphisms of the unit circle mathbb s 1 can be obtained which together with the conformal modules are used to define the shape signature a shape distance between shape signatures can be defined to measure dissimilarities between shapes we prove theoretically that the proposed shape signature uniquely determines the multiply connected objects under suitable normalization we also introduce a reconstruction algorithm to obtain shapes from their signatures this completes our framework and allows us to move back and forth between shapes and signatures with that a morphing algorithm between shapes can be developed through the interpolation of the beltrami coefficients associated with the signatures experiments have been carried out on shapes extracted from real images results demonstrate the efficacy of our proposed algorithm as a stable shape representation scheme

the ricci flow on the 2 sphere the classical uniformization theorem interpreted differential geomet rically states that any riemannian metric on a 2 dimensional surface ispointwise conformal to a constant curvature metric thus one can con sider the question of whether there is a natural evolution equation whichconformally deforms any metric on a surface to a constant curvature met ric the primary interest in this question is not so much to give a newproof of the uniformization theorem but rather to understand nonlinearparabolic equations better especially those arising in differential geome try a sufficiently deep understanding of parabolic equations should yieldimportant new results in riemannian geometry the question in the preceding paragraph has been studied by richardhamilton 3 and brad osgood ralph phillips and peter sarnak 6 in 3 hamilton studied the following equation which we refer to as

impulse free interconnection of dynamical systems abstract in this paper we address the problem that impulses might occur when a to be controlled plant is connected to a suitable controller in the behavioral literature this issue is dealt when studying the so called regular feedback interconnection rfi of the plant and the controller behaviors we address the question of when a given controlled behavior satisfying desired specifications can be obtained by rfi of the plant and some suitable controller the paper presents an algorithm to construct a controller that yields the controlled behavior by rfi if such a controller exists and then parametrizes all such controllers we extend the results to the more general case when all the to be controlled variables might not be available for interconnection we show that any disturbances in the plant continue to remain free in the controlled behavior if and only if the interconnection is an rfi this is also equivalent to the well posedness of an interconnection this paper also makes concrete the intuitive relation between absence of inadmissibility of initial conditions and rfi a door closing mechanism is analysed in relation to the results in this paper

optimizing equiareality of nurbs surfaces using composite mobius transformations the equiareality of nurbs surfaces greatly affects the results of visualization and tessellation applications especially when dealing with extruding and intruding shapes to obtain more equiareal parameterizations of nurbs surfaces than the mobius based method an optimization algorithm is presented in this paper based on the more flexible composite reparameterization functions for fixed knots the optimal composite reparameterization can be obtained using the levenberg marquardt method for a given tolerance a uniform subdivision scheme is interleaved with the optimization procedure and this process finishes until the change of the equiareal deviation energy is less than the given tolerance examples are given to show the performance of our algorithm for visualization and tessellation applications

generalized gradients priors on minimization flows this paper tackles an important aspect of the variational problem underlying active contours optimization by gradient flows classically the definition of a gradient depends directly on the choice of an inner product structure this consideration is largely absent from the active contours literature most authors explicitely or implicitely assume that the space of admissible deformations is ruled by the canonical l 2 inner product the classical gradient flows reported in the literature are relative to this particular choice here we investigate the relevance of using i other inner products yielding other gradient descents and ii other minimizing flows not deriving from any inner product in particular we show how to induce different degrees of spatial consistency into the minimizing flow in order to decrease the probability of getting trapped into irrelevant local minima we report numerical experiments indicating that the sensitivity of the active contours method to initial conditions which seriously limits its applicability and efficiency is alleviated by our application specific spatially coherent minimizing flows we show that the choice of the inner product can be seen as a prior on the deformation fields and we present an extension of the definition of the gradient toward more general priors

trajectory control and interconnection of 1d and nd systems in this paper we examine the relationship between control viewed as concatenation of trajectories and control viewed as interconnection of systems we show that for one dimensional linear time invariant systems the ability to obtain a given subsystem by regular interconnection a prerequisite for any feedback type structure is equivalent to the ability to drive any trajectory into that subsystem however in the case of multidimensional systems the former is a stronger property than the latter trajectory controllability can however be expressed as a regular interconnection of behaviors in an extended variable space by introducing latent or auxiliary variables this leads as a by product to the notion of controlling a system by means of latent variables

robust input output stabilization on z for persistent signals robustness of feedback stabilization is studied for causal linear time invariant input output systems defined on the full time axis z the signal space setting used includes also certain unbounded signals this study has been possible due to the use of two operator plant models ay bu this avoids the intrinsic difficulties of unbounded causal input output plant mappings y gu on z the full time axis robust stabilization results obtained here are analogous to those on the semi infinite time axis n in the finite dimensional case however there are certain important differences between the full time axis results and the semi infinite time axis results for infinite dimensional systems even when two operator plant models are used

hyperbolic harmonic mapping for surface registration automatic computation of surface correspondence via harmonic map is an active research field in computer vision computer graphics and computational geometry it may help document and understand physical and biological phenomena and also has broad applications in biometrics medical imaging and motion capture industries although numerous studies have been devoted to harmonic map research limited progress has been made to compute a diffeomorphic harmonic map on general topology surfaces with landmark constraints this work conquers this problem by changing the riemannian metric on the target surface to a hyperbolic metric so that the harmonic mapping is guaranteed to be a diffeomorphism under landmark constraints the computational algorithms are based on ricci flow and nonlinear heat diffusion methods the approach is general and robust we employ our algorithm to study the constrained surface registration problem which applies to both computer vision and medical imaging applications experimental results demonstrate that by changing the riemannian metric the registrations are always diffeomorphic and achieve relatively high performance when evaluated with some popular surface registration evaluation standards

metric optimization for surface analysis in the laplace beltrami embedding space in this paper we present a novel approach for the intrinsic mapping of anatomical surfaces and its application in brain mapping research using the laplace beltrami eigen system we represent each surface with an isometry invariant embedding in a high dimensional space the key idea in our system is that we realize surface deformation in the embedding space via the iterative optimization of a conformal metric without explicitly perturbing the surface or its embedding by minimizing a distance measure in the embedding space with metric optimization our method generates a conformal map directly between surfaces with highly uniform metric distortion and the ability of aligning salient geometric features besides pairwise surface maps we also extend the metric optimization approach for group wise atlas construction and multi atlas cortical label fusion in experimental results we demonstrate the robustness and generality of our method by applying it to map both cortical and hippocampal surfaces in population studies for cortical labeling our method achieves excellent performance in a cross validation experiment with 40 manually labeled surfaces and successfully models localized brain development in a pediatric study of 80 subjects for hippocampal mapping our method produces much more significant results than two popular tools on a multiple sclerosis study of 109 subjects

on interconnections control and feedback the purpose of this paper is to study interconnections and control of dynamical systems in a behavioral context we start with an extensive physical example which serves to illustrate that the familiar input output feedback loop structure is not as universal as we have been taught to believe this leads to a formulation of control problems in terms of interconnections subsequently we study interconnections of linear time invariant systems from this vantage point let us mention two of the results obtained the first one states that any polynomial can be achieved as the characteristic polynomial of the interconnection with a given plant provided the plant is not autonomous the second result states that any subsystem of a controllable system can be implemented by means of a singular feedback control law these results yield pole placement and stabilization of controllable plants as a special case these ideas are finally applied to the stabilization of a nonlinear system around an operating point

on robustness in the gap metric and coprime factor uncertainty for ltv systems abstract in this paper we study the problem of robust stabilization for discrete linear time varying ltv systems subject to time varying normalized coprime factor uncertainty operator theoretic results which generalize similar results known to hold for linear time invariant infinite dimensional systems are developed in particular we compute an upper bound for the maximal achievable stability margin under tv normalized coprime factor uncertainty in terms of the norm of an operator with a time varying hankel structure we point to a necessary and sufficient condition which guarantees compactness of the tv hankel operator and in which case singular values and vectors can be used to compute the time varying stability margin and tv controller

on the regular feedback interconnection problem we give a necessary and sufficient condition for a subsystem of a plant to be a regular feedback interconnection of an admissible controller with the plant we also parameterize the set of controllers that give rise to a desired controlled system

computing quasiconformal maps using an auxiliary metric and discrete curvature flow surface mapping plays an important role in geometric processing which induces both area and angular distortions if the angular distortion is bounded the mapping is called a quasiconformal mapping qc mapping many surface mappings in our physical world are quasiconformal the angular distortion of a qc mapping can be represented by the beltrami differentials according to qc teichmuller theory there is a one to one correspondence between the set of beltrami differentials and the set of qc surface mappings under normalization conditions therefore every qc surface mapping can be fully determined by the beltrami differential and reconstructed by solving the so called beltrami equation in this work we propose an effective method to solve the beltrami equation on general riemann surfaces the solution is a qc mapping associated with the prescribed beltrami differential the main strategy is to define an auxiliary metric am on the domain surface such that the original qc mapping becomes conformal under the auxiliary metric the desired qc mapping can then be obtained by using the conventional conformal mapping method in this paper we first formulate a discrete analogue of qc mappings on triangular meshes then we propose an algorithm to compute discrete qc mappings using the discrete yamabe flow method to the best of our knowledge it is the first work to compute the discrete qc mappings for general riemann surfaces especially with different topologies numerically the discrete qc mapping converges to the continuous solution as the mesh grid size approaches to 0 we tested our algorithm on surfaces scanned from real life with different topologies experimental results demonstrate the generality and accuracy of our auxiliary metric method

robustness analysis of nonlinear feedback systems an input output approach this paper presents an approach to robustness analysis for nonlinear feedback systems we pursue a notion of model uncertainty based on the closeness of input output trajectories which is not tied to a particular uncertainty representation such as additive parametric structured etc the basic viewpoint is to regard systems as operators on signal spaces we present two versions of a global theory where stability is captured by induced norms or by gain functions we also develop local approaches over bounded signal sets and give a treatment for systems with potential for finite time escape we compute the relevant stability margin for several examples and demonstrate robustness of stability for some specific perturbations e g small time delays we also present examples of nonlinear control systems which have zero robustness margin and are destabilized by arbitrarily small gap perturbations the paper considers the case where uncertainty is present in the controller as well as the plant and the generalization of the approach to the case where uncertainty occurs in several subsystems in an arbitrary interconnection

ricci flow embedding for rectifying non euclidean dissimilarity data pairwise dissimilarity representations are frequently used as an alternative to feature vectors in pattern recognition one of the problems encountered in the analysis of such data is that the dissimilarities are rarely euclidean while statistical learning algorithms often rely on euclidean dissimilarities such non euclidean dissimilarities are often corrected or a consistent euclidean geometry is imposed on them via embedding this paper commences by reviewing the available algorithms for analysing non euclidean dissimilarity data the novel contribution is to show how the ricci flow can be used to embed and rectify non euclidean dissimilarity data according to our representation the data is distributed over a manifold consisting of patches each patch has a locally uniform curvature and this curvature is iteratively modified by the ricci flow the raw dissimilarities are the geodesic distances on the manifold rectified euclidean dissimilarities are obtained using the ricci flow to flatten the curved manifold by modifying the individual patch curvatures we use two algorithmic components to implement this idea firstly we apply the ricci flow independently to a set of surface patches that cover the manifold second we use curvature regularisation to impose consistency on the curvatures of the arrangement of different surface patches we perform experiments on three real world datasets and use these to determine the importance of the different algorithmic components i e ricci flow and curvature regularisation we conclude that curvature regularisation is an essential step needed to control the stability of the piecewise arrangement of patches under the ricci flow

focusr feature oriented correspondence using spectral regularization a method for precise surface matching existing methods for surface matching are limited by the tradeoff between precision and computational efficiency here we present an improved algorithm for dense vertex to vertex correspondence that uses direct matching of features defined on a surface and improves it by using spectral correspondence as a regularization this algorithm has the speed of both feature matching and spectral matching while exhibiting greatly improved precision distance errors of 1 4 percent the method focusr incorporates implicitly such additional features to calculate the correspondence and relies on the smoothness of the lowest frequency harmonics of a graph laplacian to spatially regularize the features in its simplest form focusr is an improved spectral correspondence method that nonrigidly deforms spectral embeddings we provide here a full realization of spectral correspondence where virtually any feature can be used as an additional information using weights on graph edges but also on graph nodes and as extra embedded coordinates as an example the full power of focusr is demonstrated in a real case scenario with the challenging task of brain surface matching across several individuals our results show that combining features and regularizing them in a spectral embedding greatly improves the matching precision to a submillimeter level while performing at much greater speed than existing methods

linear feedback systems and the graph topology it is established for general linear systems that the gap metric induces the coarsest topology with respect to which both closed loop stability and closed loop performance are robust properties in earlier works similar topological results were obtained by exploiting the existence of particular coprime factor system representations not known to exist in general by contrast the development here does not rely on any specific system representations systems are simply characterized as subspaces of norm bounded input output pairs and the analysis hinges on the underlying geometric structure of the feedback stabilization problem unlike other work developed within such a framework fundamental issues concerning the causality of feedback interconnections are discussed explicitly the key result of this paper concerns the difference between linear feedback interconnections with identical controllers in terms of a performance robustness related closed loop mapping upper and lower bounds on the induced norm of this difference are derived allowing for possibly infinite dimensional input output spaces and time varying behavior the bounds are both proportional to the gap metric distance between the plants which clearly demonstrates the gap to be an appropriate measure of the difference between open loop systems from the perspective of closed loop behavior to conclude an example is presented to show that bounds of the form derived here for linear systems do not hold in a general nonlinear setting

stabilization lyapunov functions and dissipation for linear time invariant systems any stabilizing controller for a given plant can be associated with a supply rate with respect to which the plant in open loop is half line dissipative we also prove the equivalence between the stability of the interconnection of two systems and their half line dissipativity with respect to a supply rate and its negative

impulse controllability from descriptor systems to higher order daes impulsive solutions in lti dynamical systems have received ample attention but primarily for descriptor systems i e first order differential algebraic equations daes this paper focuses on the impulsive behavior of higher order dynamical systems and analyzes the causes of impulses in the context of interconnection of one or more dynamical systems we extend the definition of impulse controllability to the higher order case amongst the various nonequivalent notions of impulse controllability for first order systems available in the literature which mostly rely on the input output structure of the system our definition based on a so called state map obtained directly from the system equations generalizes many key first order results to the higher order case in particular we show that our higher order extension of the definition of impulse controllability generalizes the equivalence between impulse controllability and the ability to eliminate impulses in the closed loop by interconnecting with a suitable controller this requires an extension of the definition of regularity of interconnection from behaviors involving only smooth trajectories to behaviors on the positive half line involving impulsive smooth trajectories

ricci flow for 3d shape analysis ricci flow is a powerful curvature flow method which is invariant to rigid motion scaling isometric and conformal deformations we present the first application of surface ricci flow in computer vision previous methods based on conformal geometry which only handle 3d shapes with simple topology are subsumed by the ricci flow based method which handles surfaces with arbitrary topology we present a general framework for the computation of ricci flow which can design any riemannian metric by user defined curvature the solution to ricci flow is unique and robust to noise we provide implementation details for ricci flow on discrete surfaces of either euclidean or hyperbolic background geometry our ricci flow based method can convert all 3d problems into 2d domains and offers a general framework for 3d shape analysis we demonstrate the applicability of this intrinsic shape representation through standard shape analysis problems such as 3d shape matching and registration and shape indexing surfaces with large nonrigid anisotropic deformations can be registered using ricci flow with constraints of feature points and curves we show how conformal equivalence can be used to index shapes in a 3d surface shape space with the use of teichmuller space coordinates experimental results are shown on 3d face data sets with large expression deformations and on dynamic heart data

gap metric robustness analysis of linear periodically time varying feedback systems quantitative robust stability results are established in this paper for feedback systems that evolve in continuous time and exhibit linear periodically time varying lptv behavior the results presented are analogous to results known to hold for linear time invariant lti systems although they do not follow directly from these system uncertainty is measured using the gap metric which quantifies the distance between systems in terms of the aperture between their graphs the subspaces corresponding to all input output pairs for each system the main robustness result characterizes the largest gap ball of lptv plants stabilized by a nominal lptv feedback controller known to stabilize a nominal lptv plant at the center of this ball a key step in the proof of this result makes use of a formula derived for the directed gap between lptv systems this formula is essentially a generalization of georgiou s for lti systems importantly all of the results presented apply to a class of sampled data control systems as a special case

on graphs conic relations and input output stability of nonlinear feedback systems this paper presents necessary and sufficient conditions for the stability of general interconnected dynamical systems the results are expressed in terms of the graphs of the interconnection s subsystems conditions for robustness of stability for an interconnection with additive disturbances are also provided finally the paper gives sufficient conditions for stability using conic sector conditions that generalize earlier conditions and that incorporate a recent nonlinear small gain idea

output feedback pole placement with dynamic compensators derives a new rank condition which guarantees the arbitrary pole assignability of a given system by dynamic compensators of degree at most q by using this rank condition the authors establish several new sufficiency conditions which ensure the arbitrary pole assignability of a generic system the authors proofs also come with a concrete numerical procedure to construct a particular compensator which assigns a given set of closed loop poles

genus zero surface conformal mapping and its application to brain surface mapping we developed a general method for global conformal parameterizations based on the structure of the cohomology group of holomorphic one forms for surfaces with or without boundaries gu and yau 2002 gu and yau 2003 for genus zero surfaces our algorithm can find a unique mapping between any two genus zero manifolds by minimizing the harmonic energy of the map in this paper we apply the algorithm to the cortical surface matching problem we use a mesh structure to represent the brain surface further constraints are added to ensure that the conformal map is unique empirical tests on magnetic resonance imaging mri data show that the mappings preserve angular relationships are stable in mris acquired at different times and are robust to differences in data triangulation and resolution compared with other brain surface conformal mapping algorithms our algorithm is more stable and has good extensibility