a constrained min max algorithm for rival models of the same economic system there are well established rival theories about the economy these have in turn led to the development of rival models purporting to represent the economic system the models are large systems of discrete time nonlinear dynamic equations observed data of the real system does not in general provide sufficient information for statistical methods to invalidate all but one of the rival models in such a circumstance there is uncertainty about which model to use in the formulation of policy prudent policy design would suggest that a model based policy should take into account all the rival models this is achieved as a pooling of the models the pooling that yields the policy which is robust to model choice is formulated as a constrained min max problem the minimization is over the decision variables and the maximization is over the rival models only equality constraints are considered

an sqp augmented lagrangian bfgs algorithm for constrained optimization in this research an effective algorithm for nonlinearly constrained optimization using the structured augmented lagrangian secant update recently proposed by tapia is presented the algorithm is globally defined and uses a new and reliable method for choosing the lagrangian augmentation parameter that does not require prior knowledge of the true hessian considerable numerical experimentation with this algorithm both embedded in a merit function line search sqp framework and without line search is presented the algorithm is compared to the widely used damped bfgs secant update of powell which like the one in this paper was designed to circumvent the lack of positive definiteness in the hessian of the lagrangian it is also established that when the algorithm converges it converges r superlinearly which is a strong result in that it makes no assumptions on the approximate hessian or the augmentation parameter an immediate corollary is a new result in unconstrained optimization whenever the uncons

on secant updates for use in general constrained optimization in this paper we present two new classes of successive quadratic programming sqp secant methods for the equality constrained optimization problem one class of methods uses the sqp augmented lagrangian formulation while the other class uses the sqp lagrangian formulation we demonstrate under the standard assumptions that in both cases the bfgs and dfp versions of the algorithm are locally q superlinearly convergent to our knowledge this is the first time that either local or q superlinear convergence has been established for an sqp lagrangian secant method which uses either the bfgs or dfp updating philosophy and assumes no more than the standard assumptions since the standard assumptions do not require positive definiteness of the hessian of the lagrangian at the solution it is no surprise that our bfgs and dfp updates possess the hereditary positive definiteness property only on a proper subspace

a robust comparison of biological shapes summary localized differences in the form of two related animal skeletons are more effectively determined when resistant fitting techniques are used rather than least squares the repeated median resistant fitting algorithm is introduced the methods are tested by comparing hominid skulls with those of other primates and the male ostracode with the female in each case the resistant fit is superior to the least squares fit in that the difierences are more readily identified and agl ee more closely with the structural differences perceived on biological grounds le llltroductioll studies of allometric growth and of morphological change often run into difficulties in identifying and measuring the location and extent of relative shape deformations a reference system is needed for measuring these deformations and in the absence of an inherent fixed reference system one must be estimated the problem of estimating shape change in animal morphology provides an elegant setting for the comparison of least squares with recently developed robust and resistant statistical techniques being robust these new methods are protected against departures from the assumptions of analysis such as independent identically and normally distributed errors being resistant they are also protected against the potentially strong influences of atypical or incorrect data values two shapes can rarely be superimposed perfectly different fitting criteria will generally yield different results by allowing regions with large deformations to have a large impact on the fit least squares methods can minimize true shape differences and thereby obscure them a resistant technique however limits the influence of large deformations and the resulting fit is close in similar regions and not close in relatively deformed regions in this way resistant techniques can help to identify similarities and diflerences in form more effectively than least squares methods when sneath 1967 used least squares as a basis for some of the first quantitative estimates of relative deformation and so extended the fundamental descriptive work of thompson 1917 useful robust methods were not generally available sneath did in fact consider using the available robust methods but concluded that alternative methods that are less affected by wild values are not attractive in practice since then however the development of new theory and methods as reported by huber 1972 1981 hampel

on certain linear mappings between inner product and squared distance matrices abstract we obtain the spectral decomposition of four linear mappings the first is a mapping of the linear hull of all centered inner product matrices onto the linear hull of all the induced squared distance matrices it is based on the natural generalization of the cosine law of elementary euclidean geometry the other three mappings studied are 1 the adjoint and 1 extensions and applications particularly to multidimensional scaling are discussed in some detail

subspace procrustes analysis procrustes analysis pa has been a popular technique to align and build 2 d statistical models of shapes given a set of 2 d shapes pa is applied to remove rigid transformations later a non rigid 2 d model is computed by modeling the residual e g pca although pa has been widely used it has several limitations for modeling 2 d shapes occluded landmarks and missing data can result in local minima solutions and there is no guarantee that the 2 d shapes provide a uniform sampling of the 3 d space of rotations for the object to address previous issues this paper proposes subspace pa spa given several instances of a 3 d object spa computes the mean and a 2 d subspace that can model rigid and non rigid deformations of the 3 d object we propose a discrete dspa and continuous cspa formulation for spa assuming that 3 d samples of an object are provided dspa extends the traditional pa and produces unbiased 2 d models by uniformly sampling different views of the 3 d object cspa provides a continuous approach to uniformly sample the space of 3 d rotations being more efficient in space and time we illustrate the benefits of spa in two different applications first spa is used to learn 2 d face and body models from 3 d datasets experiments on the facewarehouse and cmu motion capture mocap datasets show the benefits of our 2 d models against the state of the art pa approaches and conventional 3 d models second spa learns an unbiased 2 d model from cmu mocap dataset and it is used to estimate the human pose on the leeds sports dataset

large sample theory of intrinsic and extrinsic sample means on manifolds ii this article develops nonparametric inference procedures for estimation and testing problems for means on manifolds a central limit theorem for frechet sample means is derived leading to an asymptotic distribution theory of intrinsic sample means on riemannian manifolds central limit theorems are also obtained for extrinsic sample means w r t an arbitrary embedding of a differentiable manifold in a euclidean space bootstrap methods particularly suitable for these problems are presented applications are given to distributions on the sphere s d directional spaces real projective space rp n 1 axial spaces complex projective space cp k 2 planar shape spaces w r t veronese whitney embeddings and a three dimensional shape space 4 3

a strategy for global convergence in a sequential quadratic programming algorithm n a previous work p boggs and j tolle siam j numer anal 21 1984 pp 1146 1161 the authors introduced a merit function for use with the sequential quadratic programming sqp algorithm for solving nonlinear programming problems here further theoretical justification including a global convergence theorem is provided in addition modifications are suggested that allow the efficient implementation of the merit function while maintaining the important convergence properties numerical results are presented demonstrating the effectiveness of the procedure

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

matching by linear programming and successive convexification we present a novel convex programming scheme to solve matching problems focusing on the challenging problem of matching in a large search range and with cluttered background matching is formulated as metric labeling with l1 regularization terms for which we propose a novel linear programming relaxation method and an efficient successive convexification implementation the unique feature of the proposed relaxation scheme is that a much smaller set of basis labels is used to represent the original label space this greatly reduces the size of the searching space a successive convexification scheme solves the labeling problem in a coarse to fine manner importantly the original cost function is reconvexified at each stage in the new focus region only and the focus region is updated so as to refine the searching result this makes the method well suited for large label set matching experiments demonstrate successful applications of the proposed matching scheme in object detection motion estimation and tracking

robust control of markov decision processes with uncertain transition matrices optimal solutions to markov decision problems may be very sensitive with respect to the state transition probabilities in many practical problems the estimation of these probabilities is far from accurate hence estimation errors are limiting factors in applying markov decision processes to real world problems we consider a robust control problem for a finite state finite action markov decision process where uncertainty on the transition matrices is described in terms of possibly nonconvex sets we show that perfect duality holds for this problem and that as a consequence it can be solved with a variant of the classical dynamic programming algorithm the robust dynamic programming algorithm we show that a particular choice of the uncertainty sets involving likelihood regions or entropy bounds leads to both a statistically accurate representation of uncertainty and a complexity of the robust recursion that is almost the same as that of the classical recursion hence robustness can be added at practically no extra computing cost we derive similar results for other uncertainty sets including one with a finite number of possible values for the transition matrices we describe in a practical path planning example the benefits of using a robust strategy instead of the classical optimal strategy even if the uncertainty level is only crudely guessed the robust strategy yields a much better worst case expected travel time

maximum a posteriori estimation of linear shape variation with application to vertebra and cartilage modeling the estimation of covariance matrices is a crucial step in several statistical tasks especially when using few samples of a high dimensional representation of shapes the standard maximum likelihood estimation ml of the covariance matrix can be far from the truth is often rank deficient and may lead to unreliable results in this paper we discuss regularization by prior knowledge using maximum a posteriori map estimates we compare ml to map using a number of priors and to tikhonov regularization we evaluate the covariance estimates on both synthetic and real data and we analyze the estimates influence on a missing data reconstruction task where high resolution vertebra and cartilage models are reconstructed from incomplete and lower dimensional representations our results demonstrate that our methods outperform the traditional ml method and tikhonov regularization

robust multidimensional matched subspace classifiers based on weighted least squares we propose and design two classes of robust subspace classifiers for classification of multidimensional signals our classifiers are based on robust m estimators and the least median of squares principle and we show that they may be unified as iterated reweighted oblique subspace classifiers the performance of the proposed classifiers are demonstrated by two examples noncoherent detection of space time frequency shift keying signals and shape classification of partially occluded two dimensional 2 d objects in both cases the proposed robust subspace classifiers outperform the conventional subspace classifiers

stratified generalized procrustes analysis generalized procrustes analysis computes the best set of transformations that relate matched shape data in shape analysis the transformations are usually chosen as similarities while in general statistical data analysis other types of transformation groups such as the affine group may be used generalized procrustes analysis has a nonlinear and nonconvex formulation the classical approach alternates the computation of a so called reference shape and the computation of transformations relating this reference shape to each shape datum in turn we propose the stratified approach to generalized procrustes analysis it first uses the affine transformation group to analyze the data and then upgrades the solution to the sought after group whether euclidean or similarity we derive a convex formulation for each of these two steps and efficient practical algorithms that gracefully handle missing data incomplete shapes extensive experimental results show that our approaches perform well on simulated and real data in particular our closed form solution gives very accurate results for generalized procrustes analysis of euclidean data

policy approximation in policy iteration approximate dynamic programming for discrete time nonlinear systems policy iteration approximate dynamic programming dp is an important algorithm for solving optimal decision and control problems in this paper we focus on the problem associated with policy approximation in policy iteration approximate dp for discrete time nonlinear systems using infinite horizon undiscounted value functions taking policy approximation error into account we demonstrate asymptotic stability of the control policy under our problem setting show boundedness of the value function during each policy iteration step and introduce a new sufficient condition for the value function to converge to a bounded neighborhood of the optimal value function aiming for practical implementation of an approximate policy we consider using volterra series which has been extensively covered in controls literature for its good theoretical properties and for its success in practical applications we illustrate the effectiveness of the main ideas developed in this paper using several examples including a practical problem of excitation control of a hydrogenerator

approximate robust policy iteration using multilayer perceptron neural networks for discounted infinite horizon markov decision processes with uncertain correlated transition matrices we study finite state finite action discounted infinite horizon markov decision processes with uncertain correlated transition matrices in deterministic policy spaces existing robust dynamic programming methods cannot be extended to solving this class of general problems in this paper based on a robust optimality criterion an approximate robust policy iteration using a multilayer perceptron neural network is proposed it is proven that the proposed algorithm converges in finite iterations and it converges to a stationary optimal or near optimal policy in a probability sense in addition we point out that sometimes even a direct enumeration may not be applicable to addressing this class of problems however a direct enumeration based on our proposed maximum value approximation over the parameter space is a feasible approach we provide further analysis to show that our proposed algorithm is more efficient than such an enumeration method for various scenarios

a weighted procrustes criterion the procrustes criterion is a common measure for the distance between two matricesx andy and can be interpreted as the sum of squares of the euclidean distances between their respective column vectors often a weighted procrustes criterion using for example a weighted sum of the squared distances between the column vectors is called for this paper describes and analyzes the performance of an algorithm for rotating a matrixx such that the column weighted procrustes distance toy is minimized the problem of rotatingx intoy such that an aggregate measure of tucker s coefficient of congruence is maximized is also discussed

statistical shape analysis clustering learning and testing using a differential geometric treatment of planar shapes we present tools for 1 hierarchical clustering of imaged objects according to the shapes of their boundaries 2 learning of probability models for clusters of shapes and 3 testing of newly observed shapes under competing probability models clustering at any level of hierarchy is performed using a minimum variance type criterion and a markov process statistical means of clusters provide shapes to be clustered at the next higher level thus building a hierarchy of shapes using finite dimensional approximations of spaces tangent to the shape space at sample means we implicitly impose probability models on the shape space and results are illustrated via random sampling and classification hypothesis testing together hierarchical clustering and hypothesis testing provide an efficient framework for shape retrieval examples are presented using shapes and images from eth surrey and amcom databases

an algorithm for the inequality constrained discrete min max problem in this paper we discuss an approach using an augmented lagrangian formulation to directly solve the inequality constrained min max problem the algorithm involves a sequential quadratic programming subproblem an adaptive penalty parameter selection rule and a stepsize strategy convergent to unit steps that ensures progress toward optimality and feasibility of the inequality constraints it is shown that the penalty parameter does not grow indefinitely the convergence of the algorithm is established and its numerical effectiveness is demonstrated with test examples

eigenshape analysis of a cut grow mapping for triangles and its application to phyllotaxis in plants phyllotaxis the regular arrangement of leaves in plants has been extensively studied geometrically and through theories of mechanism recent observations of the biophysics of cell growth and division at the plant apex during leaf formation lead to a cut and grow algorithm in which a newly formed leaf is first cut off from the meristem and then grows rapidly deforming the adjacent tissue to a new configuration the properties of the sequence of shapes constructed by a cut grow map of triangles are discussed the cut grow map excises a triangular segment of a triangle and dilates the newly formed edge in shape space the cut grow map is equivalent to a reflection followed by inversion each map has a conjugate pair of eigenshapes in the limiting case when the eigenshapes are equal the shapes converge with successive iterations at rate 1 n however an arbitrarily small perturbation from the eigenshape can initialize a divergent sequence of arbitrary length before the shapes begin to converge

error bounds of adaptive dynamic programming algorithms for solving undiscounted optimal control problems in this paper we establish error bounds of adaptive dynamic programming algorithms for solving undiscounted infinite horizon optimal control problems of discrete time deterministic nonlinear systems we consider approximation errors in the update equations of both value function and control policy we utilize a new assumption instead of the contraction assumption in discounted optimal control problems we establish the error bounds for approximate value iteration based on a new error condition furthermore we also establish the error bounds for approximate policy iteration and approximate optimistic policy iteration algorithms it is shown that the iterative approximate value function can converge to a finite neighborhood of the optimal value function under some conditions to implement the developed algorithms critic and action neural networks are used to approximate the value function and control policy respectively finally a simulation example is given to demonstrate the effectiveness of the developed algorithms

a monotonically convergent algorithm for orthogonal congruence rotation brokken has proposed a method for orthogonal rotation of one matrix such that its columns have a maximal sum of congruences with the columns of a target matrix this method employs an algorithm for which convergence from every starting point is not guaranteed in the present paper an iterative majorization algorithm is proposed which is guaranteed to converge from every starting point specifically it is proven that the function value converges monotonically and that the difference between subsequent iterates converges to zero in addition to the better convergence properties another advantage of the present algorithm over brokken s one is that it is easier to program the algorithms are compared on 80 simulated data sets and it turned out that the new algorithm performed well in all cases whereas brokken s algorithm failed in almost half the cases the derivation of the algorithm is given in full detail because it involves a series of inequalities that can be of use to derive similar algorithms in different contexts

deformotion deforming motion shape average and the joint registration and approximation of structures in images what does it mean for a deforming object to be moving how can we separate the overall motion a finite dimensional group action from the more general deformation a diffeomorphism in this paper we propose a definition of motion for a deforming object and introduce a notion of shape average as the entity that separates the motion from the deformation our definition allows us to derive novel and efficient algorithms to register non identical shapes using region based methods and to simultaneously approximate and align structures in greyscale images we also extend the notion of shape average to that of a moving average in order to track moving and deforming objects through time the algorithms we propose extend prior work on landmark based matching to smooth curves and involve the numerical integration of partial differential equations which we address within the framework of level set methods

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

online selective kernel based temporal difference learning in this paper an online selective kernel based temporal difference osktd learning algorithm is proposed to deal with large scale and or continuous reinforcement learning problems osktd includes two online procedures online sparsification and parameter updating for the selective kernel based value function a new sparsification method i e a kernel distance based online sparsification method is proposed based on selective ensemble learning which is computationally less complex compared with other sparsification methods with the proposed sparsification method the sparsified dictionary of samples is constructed online by checking if a sample needs to be added to the sparsified dictionary in addition based on local validity a selective kernel based value function is proposed to select the best samples from the sample dictionary for the selective kernel based value function approximator the parameters of the selective kernel based value function are iteratively updated by using the temporal difference td learning algorithm combined with the gradient descent technique the complexity of the online sparsification procedure in the osktd algorithm is o n in addition two typical experiments maze and mountain car are used to compare with both traditional and up to date o n algorithms gtd gtd2 and tdc using the kernel based value function and the results demonstrate the effectiveness of our proposed algorithm in the maze problem osktd converges to an optimal policy and converges faster than both traditional and up to date algorithms in the mountain car problem osktd converges requires less computation time compared with other sparsification methods gets a better local optima than the traditional algorithms and converges much faster than the up to date algorithms in addition osktd can reach a competitive ultimate optima compared with the up to date algorithms

a least squares framework for component analysis over the last century component analysis ca methods such as principal component analysis pca linear discriminant analysis lda canonical correlation analysis cca locality preserving projections lpp and spectral clustering sc have been extensively used as a feature extraction step for modeling classification visualization and clustering ca techniques are appealing because many can be formulated as eigen problems offering great potential for learning linear and nonlinear representations of data in closed form however the eigen formulation often conceals important analytic and computational drawbacks of ca techniques such as solving generalized eigen problems with rank deficient matrices e g small sample size problem lacking intuitive interpretation of normalization factors and understanding commonalities and differences between ca methods this paper proposes a unified least squares framework to formulate many ca methods we show how pca lda cca lpp sc and its kernel and regularized extensions correspond to a particular instance of least squares weighted kernel reduced rank regression ls wkrrr the ls wkrrr formulation of ca methods has several benefits 1 provides a clean connection between many ca techniques and an intuitive framework to understand normalization factors 2 yields efficient numerical schemes to solve ca techniques 3 overcomes the small sample size problem 4 provides a framework to easily extend ca methods we derive weighted generalizations of pca lda sc and cca and several new ca techniques

monte carlo extrinsic estimators of manifold valued parameters monte carlo mc methods have become an important tool for inferences in non gaussian and non euclidean settings we study their use in those signal image processing scenarios where the parameter spaces are certain riemannian manifolds finite dimensional lie groups and their quotient sets we investigate the estimation of means and variances of the manifold valued parameters using two popular sampling methods independent and importance sampling using euclidean embeddings we specify a notion of extrinsic means employ monte carlo methods to estimate these means and utilize large sample asymptotics to approximate the estimator covariances experimental results are presented for target pose estimation orthogonal groups and signal subspace estimation grassmann manifolds asymptotic covariances are utilized to construct confidence regions to compare estimators and to determine the sample size for mc sampling

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

surface bounded growth modeling applied to human mandibles from a set of longitudinal three dimensional scans of the same anatomical structure the authors have accurately modeled the temporal shape and size changes using a linear shape model on a total of 31 computed tomography scans of the mandible from six patients 14 851 semilandmarks are found automatically using shape features and a new algorithm called geometry constrained diffusion the semilandmarks are mapped into procrustes space principal component analysis extracts a one dimensional subspace which is used to construct a linear growth model the worst case mean modeling error in a cross validation study is 3 7 mm

hierarchical lung field segmentation with joint shape and appearance sparse learning lung field segmentation in the posterior anterior pa chest radiograph is important for pulmonary disease diagnosis and hemodialysis treatment due to high shape variation and boundary ambiguity accurate lung field segmentation from chest radiograph is still a challenging task to tackle these challenges we propose a joint shape and appearance sparse learning method for robust and accurate lung field segmentation the main contributions of this paper are 1 a robust shape initialization method is designed to achieve an initial shape that is close to the lung boundary under segmentation 2 a set of local sparse shape composition models are built based on local lung shape segments to overcome the high shape variations 3 a set of local appearance models are similarly adopted by using sparse representation to capture the appearance characteristics in local lung boundary segments thus effectively dealing with the lung boundary ambiguity 4 a hierarchical deformable segmentation framework is proposed to integrate the scale dependent shape and appearance information together for robust and accurate segmentation our method is evaluated on 247 pa chest radiographs in a public dataset the experimental results show that the proposed local shape and appearance models outperform the conventional shape and appearance models compared with most of the state of the art lung field segmentation methods under comparison our method also shows a higher accuracy which is comparable to the inter observer annotation variation

perturbation and stability theory for markov control problems a unified approach to the asymptotic analysis of a markov decision process disturbed by an epsilon additive perturbation is proposed irrespective of whether the perturbation is regular or singular the underlying control problem that needs to be understood is the limit markov control problem the properties of this problem are studied

evolutionary policy iteration for solving markov decision processes we propose a novel algorithm called evolutionary policy iteration epi for solving infinite horizon discounted reward markov decision processes epi inherits the spirit of policy iteration but eliminates the need to maximize over the entire action space in the policy improvement step so it should be most effective for problems with very large action spaces epi iteratively generates a population or a set of policies such that the performance of the elite policy for a population monotonically improves with respect to a defined fitness function epi converges with probability one to a population whose elite policy is an optimal policy epi is naturally parallelizable and along this discussion a distributed variant of pi is also studied

landmark based shape encoding and sparse dictionary learning in the continuous domain we provide a generic framework to learn shape dictionaries of landmark based curves that are defined in the continuous domain we first present an unbiased alignment method that involves the construction of a mean shape as well as training sets whose elements are subspaces that contain all affine transformations of the training samples the alignment relies on orthogonal projection operators that have a closed form we then present algorithms to learn shape dictionaries according to the structure of the data that needs to be encoded 1 projection based functional principal component analysis for homogeneous data and 2 continuous domain sparse shape encoding to learn dictionaries that contain imbalanced data outliers or different types of shape structures through parametric spline curves we provide a detailed and exact implementation of our method we demonstrate that it requires fewer parameters than purely discrete methods and that it is computationally more efficient and accurate we illustrate the use of our framework for dictionary learning of structures in biomedical images as well as for shape analysis in bioimaging

markov decision processes with imprecise transition probabilities we present new numerical algorithms and bounds for the infinite horizon discrete stage finite state and action markov decision process with imprecise transition probabilities we assume that the transition probability mass vector for each state and action is described by a finite number of linear inequalities this model of imprecision appears to be well suited for describing statistically determined confidence limits and or natural language statements of likelihood the numerical procedures for calculating an optimal max min strategy are based on successive approximations reward revision and modified policy iteration the bounds that are determined are at least as tight as currently available bounds for the case where the transition probabilities are precise

bounded parameter markov decision process in this paper we introduce the notion of a em bounded parameter markov decision process as a generalization of the traditional em exact mdp a bounded parameter mdp is a set of exact mdps specified by giving upper and lower bounds on transition probabilities and rewards all the mdps in the set share the same state and action space bounded parameter mdps can be used to represent variation or uncertainty concerning the parameters of sequential decision problems bounded parameter mdps can also be used in aggregation schemes to represent the variation in the transition probabilities for different base states aggregated together in the same aggregate state we introduce em interval value functions as a natural extension of traditional value functions an interval value function assigns a closed real interval to each state representing the assertion that the value of that state falls within that interval an interval value function can be used to bound the performance of a policy over the set of exact mdps associated with a given bounded parameter mdp we describe an iterative dynamic programming algorithm called em interval policy evaluation which computes an interval value function for a given bounded parameter mdp and specified policy interval policy evaluation on a policy policy computes the most restrictive interval value function that is sound i e that bounds the value function for policy in every exact mdp in the set defined by the bounded parameter mdp a simple modification of interval policy evaluation results in a variant of value iteration bellman57 that we call em interval value iteration which computes a policy for an bounded parameter mdp that is optimal in a well defined sense

strings variational deformable models of multivariate continuous boundary features we propose a new image segmentation technique called strings a string is a variational deformable model that is learned from a collection of example objects rather than built from a priori analytical or geometrical knowledge as opposed to existing approaches an object boundary is represented by a one dimensional multivariate curve in functional space a feature function rather than by a point in vector space in the learning phase feature functions are defined by extraction of multiple shape and image features along continuous object boundaries in a given learning set the feature functions are aligned then subjected to functional principal components analysis and functional principal regression to summarize the feature space and to model its content respectively also a mahalanobis distance model is constructed for evaluation of boundaries in terms of their feature functions taking into account the natural variations seen in the learning set in the segmentation phase an object boundary in a new image is searched for with help of a curve the curve gives rise to a feature function a string that is weighted by the regression model and evaluated by the mahalanobis model the curve is deformed in an iterative procedure to produce feature functions with minimal mahalanobis distance strings have been compared with active shape models on 145 vertebra images showing that strings produce better results when initialized close to the target boundary and comparable results otherwise

morphometric tools for landmark data geometry and biology 1 introduction 2 preliminaries 3 landmarks 4 distance measures 5 shape coordinates 6 principal axes of shape change for triangles 7 features of shape comparison 8 retrospect and prospect

probability convexity and harmonic maps with small image i uniqueness and fine existence this paper uses probability theory to provide an approach to the dirichlet problem for harmonic maps the probabilistic tools used are those of manifold valued brownian motion and martingales probabilistic proofs are given of uniqueness continuous dependence on data and existence of generalized solutions finely harmonic maps for target domains which have convex geometry this class of domain includes all regular geodesic balls links are made with new results in other fields the martingale dirichlet problem riemannian centres of mass

a global convergence theory for the celis dennis tapia trust region algorithm for constrained optimization a global convergence theory for a class of trust region algorithms for solving the equality constrained optimization problem is presented this theory is sufficiently general that it holds for any algorithm that generates steps giving at least a fraction of cauchy decrease in the quadratic model of the constraints and that uses the augmented lagrangian as a merit function this theory is used to establish global convergence of the 1984 celis dennis tapia algorithm with a different scheme for updating the penalty parameter the behavior of the penalty parameter is also discussed

a structured quasi newton algorithm for optimizing with incomplete hessian information we present a structured quasi newton algorithm for unconstrained optimization problems that have unavailable second order derivatives or hessian terms we provide a formal derivation of the well kn

parameter orthogonality and approximate conditional inference on propose une statistique du rapport de vraisemblance construite a partir de la distribution conditionnelle des observations etant donne les estimateurs du maximum de vraisemblance pour les parametres de nuisance

online optimal control of affine nonlinear discrete time systems with unknown internal dynamics by using time based policy update in this paper the hamilton jacobi bellman equation is solved forward in time for the optimal control of a class of general affine nonlinear discrete time systems without using value and policy iterations the proposed approach referred to as adaptive dynamic programming uses two neural networks nns to solve the infinite horizon optimal regulation control of affine nonlinear discrete time systems in the presence of unknown internal dynamics and a known control coefficient matrix one nn approximates the cost function and is referred to as the critic nn while the second nn generates the control input and is referred to as the action nn the cost function and policy are updated once at the sampling instant and thus the proposed approach can be referred to as time based adp novel update laws for tuning the unknown weights of the nns online are derived lyapunov techniques are used to show that all signals are uniformly ultimately bounded and that the approximated control signal approaches the optimal control input with small bounded error over time in the absence of disturbances an optimal control is demonstrated simulation results are included to show the effectiveness of the approach the end result is the systematic design of an optimal controller with guaranteed convergence that is suitable for hardware implementation

kronecker products unitary matrices and signal processing applications discrete unitary transforms are extensively used in many signal processing applications and in the development of fast algorithms kronecker products have proved quite useful in this semitutorial paper we briefly review properties of kronecker products and direct sums of matrices which provide a compact notation in treating patterned matrices a generalized matrix product which inherits some useful algebraic properties from the standard kronecker product and allows a large class of discrete unitary transforms to be generated from a single recursion formula is then introduced the notation is intimately related to sparse matrix factorizations and examples are included illustrating the utility of the new notation in signal processing applications finally some novel characteristics of hadamard transforms and polyadic permutations are derived in the framework of kronecker products

representation and detection of deformable shapes we describe some techniques that can be used to represent and detect deformable shapes in images the main difficulty with deformable template models is the very large or infinite number of possible nonrigid transformations of the templates this makes the problem of finding an optimal match of a deformable template to an image incredibly hard using a new representation for deformable shapes we show how to efficiently find a global optimal solution to the nonrigid matching problem the representation is based on the description of objects using triangulated polygons our matching algorithm can minimize a large class of energy functions making it applicable to a wide range of problems we present experimental results of detecting shapes in medical images and images of natural scenes we also consider the problem of learning a nonrigid shape model for a class of objects from examples we show how to learn good models while constraining them to be in the form required by the matching algorithm

properties of euclidean and non euclidean distance matrices abstract a distance matrix d of order n is symmetric with elements 1 2 d ij 2 where dii 0 d is euclidean when the 1 2 n n 1 quantities dij can be generated as the distances between a set of n points x n p in a euclidean space of dimension p the dimensionality of d is defined as the least value of p rank x of any generating x in general p 1 and p 2 are also acceptable but may include imaginary coordinates even when d is euclidean basic properties of euclidean distance matrices are established in particular when rank d it is shown that depending on whether etd e is not or is zero the generating points lie in either p 1 dimensions in which case they lie on a hypersphere or in p 2 dimensions in which case they do not the notation e is used for a vector all of whose values are one when d is non euclidean its dimensionality p r s will comprise r real and s imaginary columns of x and r s are invariant for all generating x of minimal rank higher ranking representations can arise only from p 1 r 1 s or p 1 r s 1 or p 2 r 1 s 1 so that not only are r s invariant but they are both minimal for all admissible representations x

the statistical analysis of shape data summary the paper deals with the statistical shape analysis of landmark data bookstein 1986 introduced a plausible model and obtained a useful normal approximation for three landmarks we use exact distribution theory and a normal approximation to consider maximum likelihood estimation the fisher information matrix for the normal approximation is given and we consider the hypothesis of shape change using these results we analyse a practical example based on digitized shapes this example motivated the work and the general direction for future work is indicated

on the realization of the wolfe conditions in reduced quasi newton methods for equality constrained optimization this paper describes a reduced quasi newton method for solving equality constrained optimization problems a major difficulty encountered by this type of algorithm is the design of a consistent technique for maintaining the positive definiteness of the matrices approximating the reduced hessian of the lagrangian a new approach is proposed in this paper the idea is to search for the next iterate along a piecewise linear path the path is designed so that some generalized wolfe conditions can be satisfied these conditions allow the algorithm to sustain the positive definiteness of the matrices from iteration to iteration by a mechanism that has turned out to be efficient in unconstrained optimization

convergence theorems for least change secant update methods the purpose of this paper is to present a convergence analysis of least change secant methods in which part of the derivative matrix being approximated is computed by other means the theorems and proofs given here can be viewed as generalizations of those given by broyden dennis more j inst math appl 12 1973 pp 223 246 and by dennis more math comp 28 1974 pp 549 560 the analysis is done in the orthogonal projection setting of dennis schnabel siam rev 21 1980 pp 443 459 and many readers might feel that it is easier to understand the theorems here readily imply local and q superlinear convergence of all the standard methods in addition to proving these results for the first time for the sparse symmetric method of marwil and toint and the nonlinear least squares method of dennis gay welsch

nonmonotonic projected algorithm with both trust region and line search for constrained optimization in this paper we combine a reduced hessian method with a mixed strategy using both trust region and line search techniques for constrained optimization the adopted strategy switches to back tracking steps when a trial step produced by the trust region subproblem is unacceptable by using fletcher s penalty function as a merit function the resulting algorithm possesses global convergence while maintaining a superlinear local convergence rate under some reasonable conditions a nonmonotonic criterion is suggested which does not require the merit function to reduce its value after every iteration

on the local convergence of quasi newton methods for constrained optimization we consider the application of a general class of quasi newton methods to the solution of the classical equality constrained nonlinear optimization problem specifically we develop necessary and sufficient conditions for the q superlinear convergence of such methods and present a companion linear convergence theorem the essential conditions relate to the manner in which the hessian of the lagrangian function is approximated

continuous generalized procrustes analysis two dimensional shape models have been successfully applied to solve many problems in computer vision such as object tracking recognition and segmentation typically 2d shape models are learned from a discrete set of image landmarks corresponding to projection of 3d points of an object after applying generalized procustes analysis gpa to remove 2d rigid transformations however the standard gpa process suffers from three main limitations firstly the 2d training samples do not necessarily cover a uniform sampling of all the 3d transformations of an object this can bias the estimate of the shape model secondly it can be computationally expensive to learn the shape model by sampling 3d transformations thirdly standard gpa methods use only one reference shape which can might be insufficient to capture large structural variability of some objects to address these drawbacks this paper proposes continuous generalized procrustes analysis cgpa cgpa uses a continuous formulation that avoids the need to generate 2d projections from all the rigid 3d transformations it builds an efficient in space and time non biased 2d shape model from a set of 3d model of objects a major challenge in cgpa is the need to integrate over the space of 3d rotations especially when the rotations are parameterized with euler angles to address this problem we introduce the use of the haar measure finally we extended cgpa to incorporate several reference shapes experimental results on synthetic and real experiments show the benefits of cgpa over gpa highlightscontinuous formulation of the generalized procrustes analysis cgpa avoids the need to generate 2d projections from all 3d rigid transformations cgpa builds an efficient non biased 2d shape model from 3d objects cgpa uses the haar measure to integrate over the space of 3d rotations experimental results building 2d shape models for different public datasets

a multi dimensional scaling approach to shape analysis summary we propose an alternative to kendall s shape space for reflection shapes of configurations in rm with k labelled vertices where reflection shape consists of all the geometric information that is invariant under compositions of similarity and reflection transformations the proposed approach embeds the space of such shapes into the space p k 1 of k 1 x k 1 real symmetric positive semidefinite matrices which is the closure of an open subset of a euclidean space and defines mean shape as the natural projection of euclidean means in p k 1 on to the embedded copy of the shape space this approach has strong connections with multi dimensional scaling and the mean shape so defined gives good approximations to other commonly used definitions of mean shape we also use standard perturbation arguments for eigenvalues and eigenvectors to obtain a central limit theorem which then enables the application of standard statistical techniques to shape analysis in two or more dimensions

object matching using a locally affine invariant and linear programming techniques in this paper we introduce a new matching method based on a novel locally affine invariant geometric constraint and linear programming techniques to model and solve the matching problem in a linear programming formulation all geometric constraints should be able to be exactly or approximately reformulated into a linear form this is a major difficulty for this kind of matching algorithm we propose a novel locally affine invariant constraint which can be exactly linearized and requires a lot fewer auxiliary variables than other linear programming based methods do the key idea behind it is that each point in the template point set can be exactly represented by an affine combination of its neighboring points whose weights can be solved easily by least squares errors of reconstructing each matched point using such weights are used to penalize the disagreement of geometric relationships between the template points and the matched points the resulting overall objective function can be solved efficiently by linear programming techniques our experimental results on both rigid and nonrigid object matching show the effectiveness of the proposed algorithm

transformation invariant clustering using the em algorithm clustering is a simple effective way to derive useful representations of data such as images and videos clustering explains the input as one of several prototypes plus noise in situations where each input has been randomly transformed e g by translation rotation and shearing in images and videos clustering techniques tend to extract cluster centers that account for variations in the input due to transformations instead of more interesting and potentially useful structure for example if images from a video sequence of a person walking across a cluttered background are clustered it would be more useful for the different clusters to represent different poses and expressions instead of different positions of the person and different configurations of the background clutter we describe a way to add transformation invariance to mixture models by approximating the nonlinear transformation manifold by a discrete set of points we show how the expectation maximization algorithm can be used to jointly learn clusters while at the same time inferring the transformation associated with each input we compare this technique with other methods for filtering noisy images obtained from a scanning electron microscope clustering images from videos of faces into different categories of identification and pose and removing foreground obstructions from video we also demonstrate that the new technique is quite insensitive to initial conditions and works better than standard techniques even when the standard techniques are provided with extra data

shape appearance correlated active appearance model among the challenges faced by current active shape or appearance models facial feature localization in the wild with occlusion in a novel face image i e in a generic environment is regarded as one of the most difficult computer vision tasks in this paper we propose an active appearance model aam to tackle the problem of generic environment firstly a fast face model initialization scheme is proposed based on the idea that the local appearance of feature points can be accurately approximated with locality constraints nearest neighbors which have similar poses and textures to a test face are retrieved from a training set for constructing the initial face model to further improve the fitting of the initial model to the test face an orthogonal cca occa is employed to increase the correlation between shape features and appearance features represented by principal component analysis pca with these two contributions we propose a novel aam namely the shape appearance correlated aam sac aam and the optimization is solved by using the recently proposed fast simultaneous inverse compositional fast sic algorithm experiment results demonstrate a 5 10 improvement on controlled and semi controlled datasets and with around 10 improvement on wild face datasets in terms of fitting accuracy compared to other state of the art aam models a fast face model initialization scheme is proposed for aams orthogonal cca is employed to increase the correlation between shape and appearance a shape appearance correlated aam is proposed to tackle the generic problem of localizing feature points on unseen faces

perturbation theory for generalized procrustes analysis summary perturbation theory for procrustes statistics is developed when both configurations are subject to independent normal errors the total sum of squares is shown to partition into sums of squares for translation rotation reflection and scaling plus a residual these results are extended to generalized procrustes analysis this gives a theoretical justification to gower s analysis of variance summary of the results from a generalized procrustes analysis

the geometry of shape spaces kendall introduced shape spaces which classified n tuples of points in r k and explained their role for statistics the aim of this paper is to study such shape spaces and in particular to explore the corresponding spaces for points on spheres the ultimate purpose of the work is to be able to apply statistical techniques to observations where only the shape of the observations is significant the geometry of the shape space is critical in determining the type of statistical technique which one should use therefore the major part of this paper is devoted to an examination of the geometry and the statistics is only introduced in a simple case when considering random processes on the shape space

robust optimality for discounted infinite horizon markov decision processes with uncertain transition matrices we study finite state finite action discounted infinite horizon markov decision processes with uncertain transition matrices in the deterministic policy space the transition matrices are classified as either independent or correlated a generalized robust optimality criterion which can be degenerated to some popular optimality criteria is proposed under which an optimal or near optimal policy exists for any uncertain transition matrix theorems are developed to guarantee a stationary policy being optimal or near optimal in the deterministic policy space