numerical radius and zero pattern of matrices abstract let a be an n n complex matrix and r be the maximum size of its principal submatrices with no off diagonal zero entries suppose a has zero main diagonal and x is a unit n vector then letting a be the frobenius norm of a we show that a x x 2 1 1 2 r 1 2 n a 2 this inequality is tight within an additive term o r n 2 if the matrix a is hermitian then a x x 2 1 1 r a 2 this inequality is sharp moreover it implies the turan theorem for graphs

a metric for performance evaluation of multi target tracking algorithms performance evaluation of multi target tracking algorithms is of great practical importance in the design parameter optimization and comparison of tracking systems the goal of performance evaluation is to measure the distance between two sets of tracks the ground truth tracks and the set of estimated tracks this paper proposes a mathematically rigorous metric for this purpose the basis of the proposed distance measure is the recently formulated consistent metric for performance evaluation of multi target filters referred to as the ospa metric multi target filters sequentially estimate the number of targets and their position in the state space the ospa metric is therefore defined on the space of finite sets of vectors the distinction between filtering and tracking is that tracking algorithms output tracks and a track represents a labeled temporal sequence of state estimates associated with the same target the metric proposed in this paper is therefore defined on the space of finite sets of tracks and incorporates the labeling error numerical examples demonstrate that the proposed metric behaves in a manner consistent with our expectations

on some matrix inequalities the arithmetic geometric mean inequality for singular values due to bhatia and kittaneh says that 2sj ab sj a a b b j 1 2 for any matrices a b we first give new proofs of this inequality and its equivalent form then we use it to prove the following trace inequality let a0 be a positive definite matrix and a1 ak be positive semidefinite matrices then tr j 1k i 0jai 2aj tra0 1

the cardinality balanced multi target multi bernoulli filter and its implementations it is shown analytically that the multitarget multibernoulli member recursion proposed by mahler has a significant bias in the number of targets to reduce the cardinality bias a novel multibernoulli approximation to the multi target bayes recursion is derived under the same assumptions as the member recursion the proposed recursion is unbiased in addition a sequential monte carlo smc implementation for generic models and a gaussian mixture gm implementation for linear gaussian models are proposed the latter is also extended to accommodate mildly nonlinear models by linearization and the unscented transform

a consistent metric for performance evaluation of multi object filters the concept of a miss distance or error between a reference quantity and its estimated controlled value plays a fundamental role in any filtering control problem yet there is no satisfactory notion of a miss distance in the well established field of multi object filtering in this paper we outline the inconsistencies of existing metrics in the context of multi object miss distances for performance evaluation we then propose a new mathematically and intuitively consistent metric that addresses the drawbacks of current multi object performance evaluation metrics

set jpda filter for multitarget tracking in this article we show that when targets are closely spaced traditional tracking algorithms can be adjusted to perform better under a performance measure that disregards identity more specifically we propose an adjusted version of the joint probabilistic data association jpda filter which we call set jpda sjpda through examples and theory we motivate the new approach and show its possibilities to decrease the computational requirements we further show that the sjpda filter can be formulated as a continuous optimization problem which is fairly easy to handle optimal approximations are also discussed and an algorithm kullback leibler sjpda klsjpda which provides optimal gaussian approximations in the kullback leibler sense is derived finally we evaluate the sjpda filter on two scenarios with closely spaced targets and compare the performance in terms of the mean optimal subpattern assignment mospa measure with the jpda filter and also with the gaussian mixture cardinalized probability hypothesis density gm cphd filter the results show that the sjpda filter performs substantially better than the jpda filter and almost as well as the more complex gm cphd filter

a newborn track detection and state estimation algorithm using bernoulli random finite sets in multi target tracking mtt problems there are many important issues that affect performance including statistical filtering measurement target association and estimating the number of targets while newborn target detection and state estimation should also be considered as important factors in mtt only a few studies have addressed these topics in this paper a novel newborn track detection and state estimation method is proposed using the concept of bernoulli random finite sets the posterior finite set statistical probability density function fisst pdf of a newborn target is analytically derived and a tractable implementation scheme is proposed using importance sampling finally the validity of the proposed method is demonstrated via integration with a gaussian mixture probability hypothesis density gm phd filter and subsequent application to mtt problems

generalized recursive track before detect with proposal partitioning for tracking varying number of multiple targets in low snr track before detect tbd algorithms incorporate unthresholded measurements to track targets under low signal to noise ratio snr conditions in this paper we generalize a single target recursive tbd tracking algorithm to track a time varying number of targets in low snr and cluttered environments this algorithm is important for example in tracking scenes involving multiple closely spaced targets moving along the same direction such as a convoy of low observable vehicles moving through a forest or multiple targets moving in a crisscross fashion the proposed multi mode multi target tbd mm mm tbd algorithm is based on estimating the probabilities of all possible combinations of target existence scenarios to obtain the joint multi target posterior probability density function in a recursive bayesian framework we implement the algorithm recursively using particle filtering to also incorporate nonlinear non gaussian tracking models as the number of target existence combinations dynamically increases with the number of targets we also propose an efficient proposal density function through partitioning of the multiple target space in order to decrease the number of particles and thus improve the approximation accuracy of the particle filter we employ a heuristic decision directed based approach to keep the computational complexity as a linear function of the maximum number of possible targets by exploiting the information obtained from the estimated mode probabilities

commutators and accretive operators abstract in this paper singular values of commutators of hilbert space operators are estimated to this aim the accretivity of a transform of the operators is applied some recent results of kittaneh f kittaneh singular value inequalities for commutators of hilbert space operators linear algebra appl 430 2009 2362 2367 are extended

some comments on six inequalities associated with the inefficiency of ordinary least squares with one regressor abstract the study of the inefficiency of the ordinary least squares estimator olse with one regressor by watson 1951 required a lower bound for the efficiency defined as the ratio of the variance of the best linear unbiased estimator blue to the variance of the olse such a lower bound was provided by the cassels inequality 1951 which we note is closely related to five other inequalities including the well known inequality usually attributed to kantorovich 1948 but which was established already by frucht 1943 the main purpose in this paper is to show how these six inequalities are related with a historical perspective we present some proofs and conclude that all six inequalities are essentially equivalent in the sense that any one inequality implies the other five we identify conditions for equality in each inequality and present the six continuous integral analogues we end the paper with english translations of the seminal papers by frucht 1943 and schweitzer 1914 respectively from the spanish and hungarian and a fairly extensive bibliography

multiobject tracking for generic observation model using labeled random finite sets this paper presents an exact bayesian filtering solution for the multiobject tracking problem with the generic observation model the proposed solution is designed in the labeled random finite set framework using the product styled representation of labeled multiobject densities with the standard multiobject transition kernel and no particular simplifying assumptions on the multiobject likelihood computationally tractable solutions are also devised by applying a principled approximation involving the replacement of the full multiobject density with a labeled multi bernoulli density that minimizes the kullback leibler divergence and preserves the first order moment to achieve the fast performance a dynamic grouping procedure based implementation is presented with a step by step algorithm the performance of the proposed filter and its tractable implementations are verified and compared with the state of the art in numerical experiments

brief paper sensor control for multi object state space estimation using random finite sets the problem addressed in this paper is information theoretic sensor control for recursive bayesian multi object state space estimation using random finite sets the proposed algorithm is formulated in the framework of partially observed markov decision processes where the reward function associated with different sensor actions is computed via the renyi or alpha divergence between the multi object prior and the multi object posterior densities the proposed algorithm in implemented via the sequential monte carlo method the paper then presents a case study where the problem is to localise an unknown number of sources using a controllable moving sensor which provides range only detections four sensor control reward functions are compared in the study and the proposed scheme is found to perform the best

design of a 3 3 multivariable feedback system with large plant uncertainty a recent paper presented a quantitative synthesis theory based on schauder s fixed point theorem for uncertain multiple input output feedback systems this paper is devoted to practical design execution by means of a detailed 3 3 problem with large plant uncertainty including open loop instability and some non minimum phase plant elements the outstanding features of the theory are 1 the quantitative nature of the assigned tolerances on the nine closed loop system response functions in this problem four are specified basically non interacting and the other five interacting 2 the design problem is translated into separate quantitative single loop problems each with plant uncertainty external disturbances and closed loop tolerances derived from the original problem the solutions for these single loop problems are guaranteed to solve the original multivariable problem whose system characteristic equation need never be examined 3 there is developed a systematic means of a optimization of the

singular value inequalities for commutators of hilbert space operators abstract we prove several singular value inequalities for commutators of hilbert space operators it is shown among other inequalities that if a b and x are operators on a complex separable hilbert space such that a and b are positive and x is compact then the singular values of ax xb are dominated by those of max a b x x where is the usual operator norm

generalized labeled multi bernoulli approximation of multi object densities in multiobject inference the multiobject probability density captures the uncertainty in the number and the states of the objects as well as the statistical dependence between the objects exact computation of the multiobject density is generally intractable and tractable implementations usually require statistical independence assumptions between objects in this paper we propose a tractable multiobject density approximation that can capture statistical dependence between objects in particular we derive a tractable generalized labeled multi bernoulli glmb density that matches the cardinality distribution and the first moment of the labeled multiobject distribution of interest it is also shown that the proposed approximation minimizes the kullback leibler divergence over a special tractable class of glmb densities based on the proposed glmb approximation we further demonstrate a tractable multiobject tracking algorithm for generic measurement models simulation results for a multiobject track before detect example using radar measurements in low signal to noise ratio snr scenarios verify the applicability of the proposed approach

analytic implementations of the cardinalized probability hypothesis density filter the probability hypothesis density phd recursion propagates the posterior intensity of the random finite set rfs of targets in time the cardinalized phd cphd recursion is a generalization of the phd recursion which jointly propagates the posterior intensity and the posterior cardinality distribution in general the cphd recursion is computationally intractable this paper proposes a closed form solution to the cphd recursion under linear gaussian assumptions on the target dynamics and birth process based on this solution an effective multitarget tracking algorithm is developed extensions of the proposed closed form recursion to accommodate nonlinear models are also given using linearization and unscented transform techniques the proposed cphd implementations not only sidestep the need to perform data association found in traditional methods but also dramatically improve the accuracy of individual state estimates as well as the variance of the estimated number of targets when compared to the standard phd filter our implementations only have a cubic complexity but simulations suggest favorable performance compared to the standard joint probabilistic data association jpda filter which has a nonpolynomial complexity

principal gains and principal phases in the analysis of linear multivariable feedback systems the concepts of principal gain and principal phase are introduced for linear multivariable systems and their use in the analysis of feedback behavior is demonstrated a sufficient nyquist type stability criterion is presented in terms of these quantities and is used to characterize the robustness of the closed loop stability property when the system model is subjected to a linear perturbation either multiplicative or additive at any point in the feedback configuration the results presented are less conservative than those obtained via the small gain theorem

bernoulli particle box particle filters for detection and tracking in the presence of triple measurement uncertainty this work presents sequential bayesian detection and estimation methods for nonlinear dynamic stochastic systems using measurements affected by three sources of uncertainty stochastic set theoretic and data association uncertainty following mahler s framework for information fusion the paper develops the optimal bayes filter for this problem in the form of the bernoulli filter for interval measurements two numerical implementations of the optimal filter are developed the first is the bernoulli particle filter pf which turns out to require a large number of particles in order to achieve a satisfactory performance for the sake of reduction in the number of particles the paper also develops an implementation based on box particles referred to as the bernoulli box pf a box particle is a random sample that occupies a small and controllable rectangular region of nonzero volume in the target state space manipulation of boxes utilizes the methods of interval analysis the two implementations are compared numerically and found to perform remarkably well the target is reliably detected and the posterior probability density function of the target state is estimated accurately the bernoulli box pf however when designed carefully is computationally more efficient

matrix versions of some classical inequalities abstract some natural inequalities related to rearrangement in matrix products can also be regarded as extensions of classical inequalities for sequences or integrals in particular we show matrix versions of chebyshev and kantorovich type inequalities the matrix approach may also provide simplified proofs and new results for classical inequalities for instance we show a link between cassel s inequality and the basic rearrangement inequality for sequences of hardy littlewood polya and we state a reverse inequality to the hardy littlewood polya inequality in which matrix technics are essential

a generalization of gruss s inequality in inner product spaces and applications a generalization of the classical gruss s integral inequality in inner product spaces is given some applications for positive linear functionals and integrals are also pointed out

joint detection and estimation of multiple objects from image observations the problem of jointly detecting multiple objects and estimating their states from image observations is formulated in a bayesian framework by modeling the collection of states as a random finite set analytic characterizations of the posterior distribution of this random finite set are derived for various prior distributions under the assumption that the regions of the observation influenced by individual objects do not overlap these results provide tractable means to jointly estimate the number of states and their values from image observations as an application we develop a multi object filter suitable for image observations with low signal to noise ratio snr a particle implementation of the multi object filter is proposed and demonstrated via simulations

accretive operators and cassels inequality abstract let a b 0 and let z m n r such that z lies into the operator ball of diameter ai bi then for all positive definite a m n r tr az a b 2 ab tr az several related reverse inequalities are considered extending some known results

a metric on the space of finite sets of trajectories for evaluation of multi target tracking algorithms in this article we propose a metric on the space of finite sets of trajectories for assessing multi target tracking algorithms in a mathematically sound way the main use of the metric is to compare estimates of trajectories from different algorithms with the ground truth of trajectories the proposed metric includes intuitive costs associated to localization error for properly detected targets missed and false targets and track switches at each time step the metric computation is based on solving a multi dimensional assignment problem we also propose a lower bound for the metric which is also a metric for sets of trajectories and is computable in polynomial time using linear programming we also extend the proposed metrics on sets of trajectories to random finite sets of trajectories

distributed fusion with multi bernoulli filter based on generalized covariance intersection in this paper we propose a distributed multiobject tracking algorithm through the use of multi bernoulli mb filter based on generalized covariance intersection g ci our analyses show that the g ci fusion with two mb posterior distributions does not admit an accurate closed form expression to solve this problem we first approximate the fused posterior as the unlabeled version of delta generalized labeled mb distribution referred to as generalized mb gmb distribution then to allow the subsequent fusion with another mb posterior distribution e g fusion with a third sensor node in the sensor network or fusion in the feedback working mode we further approximate the fused gmb posterior distribution as an mb distribution which matches its first order statistical moment the proposed fusion algorithm is implemented using sequential monte carlo technique and its performance is highlighted by numerical results

uncertain multiple input multiple output systems with internal variable feedback there is given a linear time invariant multiple input multiple output mimo plant p with internal variables xi and outputs yi available for feedback due to uncertainty p is known only to be in a set p there are given tolerances on the overall system response functions tuv system output yu in response to command input ru to be satisfied pep it is shown how this mimo uncertainty problem may be replaced by a number of single input single output siso multiple loop uncertainty problems a suitable fixed point theorem guarantees that the solution of the siso problems solves the original mimo problem plant elements which appear singly in the mimo structure also appear singly in the siso structures a design example demonstrates how inner loop feedback may enormously decrease the effect of sensor noise at the mimo plant inputs extensions to disturbance attenuation under uncertainty and to uncertain time varying and non linear mimo plants are discussed

inequalities for commutators of positive operators abstract it is shown that if a b and x are operators on a complex separable hilbert space such that a and b are compact and positive then the singular values of the generalized commutator a x x b are dominated by those of x a b where is the usual operator norm consequently for every unitarily invariant norm we have a x x b x a b it is also shown that if a and b are positive and x is compact then a x x b max a b x for every unitarily invariant norm moreover if x is positive then the singular values of the commutator a x x a are dominated by those of 1 2 a x x consequently a x x a 1 2 a x x for every unitarily invariant norm for the usual operator norm these norm inequalities hold without the compactness conditions and in this case the first two norm inequalities are the same our inequalities include and improve upon earlier inequalities proved in this context and they seem natural enough and applicable to be widely useful

polynomial time algorithms for the exact mmospa estimate of a multi object probability density represented by particles in multi object estimation the traditional minimum mean squared error mmse objective is unsuitable a simple permutation of object identities can turn a very good estimate into what is apparently a very bad one fortunately a criterion tailored to sets minimization of the mean optimal sub pattern assignment mmospa has recently evolved aside from special cases exact mmospa estimates have seemed difficult to compute but in this work we present the first exact polynomial time algorithms for calculating the mmospa estimate for probability densities that are represented by particles the key insight is that the mmospa estimate can be found by means of enumerating the cells of a hyperplane arrangement which is a traditional problem from computational geometry although the runtime complexity is still high for the general case efficient algorithms are obtained for two special cases i e i two targets with arbitrary state dimensions and ii an arbitrary number of one dimensional targets

inequalities for sums and direct sums of hilbert space operators abstract we prove several singular value inequalities and norm inequalities involving sums and direct sums of hilbert space operators it is shown among other inequalities that if x and y are compact operators then the singular values of x y 2 are dominated by those of x y applications of these inequalities are also given

robustness results in linear quadratic gaussian based multivariable control designs the robustness of control systems with respect to model uncertainty is considered using simple frequency domain criteria available and new results are derived under a common framework in which the minimum singular value of the return differences transfer matrix is the key quantity in particular robustness results associated with multivariable control systems designed on the basis of linear quadratic lq and the linear quadratic gaussian lqg design methodologies are presented

a novel joint multitarget estimator for multi bernoulli models in this paper the joint multitarget jom estimator proposed for the joint target detection and tracking jott filter is reformulated for the gaussian mixture gm implementations of the multitarget multi bernoulli member filters for this purpose a mode finding algorithm is employed to search for the most significant mode of a gm density thus the maximum a posterior map estimates of bernoulli targets are determined in addition the multi bernoulli versions of the two conflicting objective functions for the pareto optimal value of the unknown jom estimation constant are derived simulations compare the performance of the proposed jom estimator with that of the marginal multitarget mam estimator in a multitarget tracking scenario where the probability of target detection is a function of target states the simulation results demonstrate that the proposed jom estimator outperforms the mam estimator under moderately low observable conditions this is because the incomplete cost function of the mam estimator is not adequate to obtain accurate cardinality estimates of targets without considering how well targets are localized nevertheless the proposed jom estimator may suffer from track termination latency more than the mam estimator due to the definition of its cost function

norm inequalities for commutators of positive operators and applications let x y and z be operators on a hilbert space such that x and z are positive it is shown that left xy yz right leqslant rm max left left x right left z right right left y right applications of this commutator inequality are given

bounds for singular values using traces abstract bounds for the singular values ratios of singular values and rank of a square matrix a involving tr a tr a 2 and tr a h a are presented some of the results are analogous to the well known bounds for the eigenvalues ratios of eigenvalues and rank of a involving tr a and tr a 2

a joint multitarget estimator for the joint target detection and tracking filter abstract this paper proposes a joint multitarget jom estimator for the joint target detection and tracking jott filter an efficient choice to the unknown jom estimation constant i e hypervolume around target state estimate is proposed as a pareto optimal solution to a multi objective nonlinear convex optimization problem the multi objective function is formulated as two convex objective functions in conflict the first objective function is the information theoretic part of the problem and aims for entropy maximization while the second one arises from the constraint in the definition of the jom estimator and aims to improve the accuracy of the jom estimates the pareto optimal solution is obtained using the weighted sum method where objective weights are determined as linear predictions from autoregressive models in contrast to the marginal multitarget mam estimator the target present decision from the jom estimator depends on the spatial information as well as the cardinality information in the finite set statistics fisst density the simulation results demonstrate that the jom estimator achieves better track management performance in terms of track confirmation latency and track maintenance than the mam estimator for different values of detection probability however the proposed jom estimator suffers from track termination latency more than the mam estimator since the localization performance of the jott filter does deteriorate gradually after target termination

singular values norms and commutators abstract let a i b i and x i i 1 n be bounded linear operators on a separable hilbert space such that x i is compact for i 1 n it is shown that the singular values of i 1 n a i x i b i are dominated by those of i 1 n a i b i i 1 n x i where is the usual operator norm among other applications of this inequality we prove that if a and b are self adjoint operators such that a 1 a a 2 and b 1 b b 2 for some real numbers a 1 a 2 b 1 and b 2 and if x is compact then the singular values of the generalized commutator ax xb are dominated by those of max b 2 a 1 a 2 b 1 x x this inequality proves a recent conjecture concerning the singular values of commutators several inequalities for norms of commutators are also given

generalized matrix version of reverse holder inequality abstract a generalized matrix version of reverse cauchy schwarz holder inequality is proved this includes the recent results proved by bourin fujii lee niezgoda and seo

norm inequalities for sums and differences of positive operators abstract it is shown that if a and b are positive operators on a separable complex hilbert space and if is any unitarily invariant norm then 2 a b 0 0 a b a b 0 0 a a b b a 1 2 b 1 2 a 1 2 b 1 2 a 1 2 b 1 2 a 1 2 b 1 2 when specialized to the usual operator norm this inequality reduces to max a b a 1 2 b 1 2 a b related inequalities for sums and differences of positive operators are obtained and applications of these inequalities to norms of self commutators are also considered

convergence results for the particle phd filter bayesian single target tracking techniques can be extended to a multiple target environment by viewing the multiple target state as a random finite set but evaluating the multiple target posterior distribution is currently computationally intractable for real time applications a practical alternative to the optimal bayes multitarget filter is the probability hypothesis density phd filter which propagates the first order moment of the multitarget posterior instead of the posterior distribution itself it has been shown that the phd is the best fit approximation of the multitarget posterior in an information theoretic sense the method avoids the need for explicit data association as the target states are viewed as a single global target state and the identities of the targets are not part of the tracking framework sequential monte carlo approximations of the phd using particle filter techniques have been implemented showing the potential of this technique for real time tracking applications this paper presents mathematical proofs of convergence for the particle filtering algorithm and gives bounds for the mean square error

a second order phd filter with mean and variance in target number the probability hypothesis density phd and cardinalized phd cphd filters are popular solutions to the multitarget tracking problem due to their low complexity and ability to estimate the number and states of targets in cluttered environments the phd filter propagates the first order moment i e mean of the number of targets while the cphd propagates the cardinality distribution in the number of targets albeit for a greater computational cost introducing the panjer point process this paper proposes a second order phd so phd filter propagating the second order moment i e variance of the number of targets alongside its mean the resulting algorithm is more versatile in the modeling choices than the phd filter and its computational cost is significantly lower compared to the cphd filter this paper compares the three filters in statistical simulations which demonstrate that the proposed filter reacts more quickly to changes in the number of targets i e target births and target deaths than the cphd filter in addition a new statistic for multiobject filters is introduced in order to study the correlation between the estimated number of targets in different regions of the state space and propose a quantitative analysis of the spooky effect for the three filters

beyond map estimation with the track oriented multiple hypothesis tracker the track oriented multiple hypothesis tracker tomht is a popular algorithm for tracking multiple targets in a cluttered environment in tracking parlance it is known as a multi scan maximum a posteriori map estimator multi scan because it enumerates possible data associations jointly over several scans and map because it seeks the most likely data association conditioned on the observations this paper extends the tomht building on its internal representation to support probabilistic queries other than map estimation specifically by summing over the tomht s pruned space of data association hypotheses one can compute marginal probabilities of individual tracks since this summation is generally intractable any practical implementation must replace it with an approximation we introduce a factor graph representation of the tomht s data association posterior and use variational message passing to approximate track marginals in an empirical evaluation we show that marginal estimates computed through message passing compare favorably to those computed through explicit summation over the k best hypotheses especially as the number of possible hypotheses increases we also show that track marginals enable parameter estimation in the tomht via a natural extension of the expectation maximization algorithm used in single target tracking in our experiments online em updates using approximate marginals significantly increased tracker robustness to poor initial parameter specification

the recursive form of error bounds for rfs state and observation with p_d this paper presents recursive performance bounds for dynamic estimation and tracking problems in both single and multiple target cases within the framework of finite set statistics because the targets can appear and disappear when the detection probability of a sensor is less than unity pd 1 we must determine the existence or nonexistence of the state as well as its value following the possible detection miss sequences within the framework of a random vector possible observation sets sequences are first defined based on these sequences the performance bounds can be represented by a multivariate function of some time based auxiliary elements recursive relations for all the auxiliary elements are derived rigorously for the multitarget case this recursive estimated bound can be applied without data association applications are analyzed through simulations to verify our theoretical results and show that our bounds are tighter than all other bounds for the case where detection probability pd 1

translation invariant maps and applications in this paper we study translation invariant maps on a linear space a method for estimating quantities induced by such maps is presented applications for differences of operators commutators g contractive maps and gruss type inequalities are also given

sensor management for multi target tracking via multi bernoulli filtering abstract in multi object stochastic systems the issue of sensor management is a theoretically and computationally challenging problem in this paper we present a novel random finite set rfs approach to the multi target sensor management problem within the partially observed markov decision process pomdp framework the multi target state is modelled as a multi bernoulli rfs and the multi bernoulli filter is used in conjunction with two different control objectives maximizing the expected renyi divergence between the predicted and updated densities and minimizing the expected posterior cardinality variance numerical studies are presented in two scenarios where a mobile sensor tracks five moving targets with different levels of observability

cphd filtering with unknown clutter rate and detection profile in bayesian multi target filtering we have to contend with two notable sources of uncertainty clutter and detection knowledge of parameters such as clutter rate and detection profile are of critical importance in multi target filters such as the probability hypothesis density phd and cardinalized phd cphd filters significant mismatches in clutter and detection model parameters result in biased estimates in practice these model parameters are often manually tuned or estimated offline from training data in this paper we propose phd cphd filters that can accommodate model mismatch in clutter rate and detection profile in particular we devise versions of the phd cphd filters that can adaptively learn the clutter rate and detection profile while filtering moreover closed form solutions to these filtering recursions are derived using beta and gaussian mixtures simulations are presented to verify the proposed solutions

new inequalities of the kantorovich type for bounded linear operators in hilbert spaces some new inequalities of the kantorovich type are established they hold for larger classes of operators and subsets of complex numbers than considered before in the literature and provide refinements of the classical results in the case when the involved operator satisfies the usual conditions several new reverse inequalities for the numerical radius of a bounded linear operator are obtained as well

optimized self localization for slam in dynamic scenes using probability hypothesis density filters in many applications sensors that map the positions of objects in unknown environments are installed on dynamic platforms as measurements are relative to the observer s sensors scene mapping requires accurate knowledge of the observer state however in practice observer reports are subject to positioning errors simultaneous localization and mapping addresses the joint estimation problem of observer localization and scene mapping state of the art approaches typically use visual or optical sensors and therefore rely on static beacons in the environment to anchor the observer estimate however many applications involving sensors that are not conventionally used for simultaneous localization and mapping slam are affected by highly dynamic scenes such that the static world assumption is invalid this paper proposes a novel approach for dynamic scenes called generalized motion gem slam based on probability hypothesis density filters the proposed approach probabilistically anchors the observer state by fusing observer information inferred from the scene with reports of the observer motion this paper derives the general theoretical framework for gem slam and shows that it generalizes existing probability hypothesis density phd based slam algorithms simulations for a model specific realization using range bearing sensors and multiple moving objects highlight that gem slam achieves significant improvements over three benchmark algorithms

a particle multi target tracker for superpositional measurements using labeled random finite sets in this paper we present a general solution for multi target tracking with superpositional measurements measurements that are functions of the sum of the contributions of the targets present in the surveillance area are called superpositional measurements we base our modelling on labeled random finite set rfs in order to jointly estimate the number of targets and their trajectories this modelling leads to a labeled version of mahler s multi target bayes filter however a straightforward implementation of this tracker using sequential monte carlo smc methods is not feasible due to the difficulties of sampling in high dimensional spaces we propose an efficient multi target sampling strategy based on superpositional approximate cphd sa cphd filter and the recently introduced labeled multi bernoulli lmb and vo vo densities the applicability of the proposed approach is verified through simulation in a challenging radar application with closely spaced targets and low signal to noise ratio

characterizations and lower bounds for the spread of a normal matrix abstract the spread of an n n matrix a with eigenvalues 1 n is defined by spr a max j k j k we prove that if a is normal then spr a max x ax y ay x y c n x y 1 max x ay y ax x y c n x y 1 re x y 0 max x ay y ax x y c n x y 1 x y 0 max spr z a z a 2 z c z 1 2 max x a 2 x x ax 2 x a ax x ax 2 1 2 x c n x 1 we also present several lower bounds for spr a given by these characterizations

a scalable algorithm for tracking an unknown number of targets using multiple sensors we propose an algorithm for tracking an unknown number of targets based on measurements provided by multiple sensors our algorithm achieves low computational complexity and excellent scalability by running belief propagation on a suitably devised factor graph a redundant formulation of data association uncertainty and the use of augmented target states including binary target indicators make it possible to exploit statistical independencies for a drastic reduction of complexity an increase in the number of targets sensors or measurements leads to additional variable nodes in the factor graph but not to higher dimensions of the messages as a consequence the complexity of our method scales only quadratically in the number of targets linearly in the number of sensors and linearly in the number of measurements per sensor the performance of the method compares well with that of previously proposed methods including methods with a less favorable scaling behavior in particular our method can outperform multisensor versions of the probability hypothesis density phd filter the cardinalized phd filter and the multi bernoulli filter

a relationship between sensitivity and stability of multivariable feedback systems both comparison sensitivity and robust stability analysis attempt to extend concepts which are well understood in scalar feedback systems to multivariable control system design although the two bodies of work have developed about a decade apart the basic philosophies are similar both attempt to maintain a system property sensitivity reduction versus stability in the presence of plant modeling errors this paper demonstrates how the similarity in philosophy translates into a similarity in results and interpretations

relating random vector and random finite set estimation in navigation mapping and tracking navigation mapping and tracking are state estimation problems relevant to a wide range of applications these problems have traditionally been formulated using random vectors in stochastic filtering smoothing or optimization based approaches alternatively the problems can be formulated using random finite sets which offer a more robust solution in poor detection conditions i e low probabilities of detection and high clutter intensity this paper mathematically shows that the two estimation frameworks are related and equivalences can be determined under a set of ideal detection conditions the findings provide important insights into some of the limitations of each approach these are validated using simulations with varying detection statistics along with a real experimental dataset

bayesian sequential track formation this paper presents a theoretical framework for track building in multiple target scenarios from the bayesian point of view it is assumed that the number of targets is fixed and known we propose two optimal methods for building tracks sequentially the first one uses the labelling of the current multitarget state estimate that minimizes the mean square labeled optimal subpattern assignment error this method requires knowledge of the posterior density of the vector valued state the second assigns the labeling that maximizes the probability that the current multi target state estimate is optimally linked with the available tracks at the previous time step in this case we only require knowledge of the random finite set posterior density without labels

an efficient variational approximation of the best fitting multi bernoulli filter the joint probabilistic data association jpda filter is a popular tracking methodology for problems involving well spaced targets but it is rarely applied in problems with closely spaced targets due to its complexity in these cases and due to the well known phenomenon of coalescence this paper addresses these difficulties using random finite sets rfss and variational inference deriving a highly tractable approximate method for obtaining the multi bernoulli distribution that minimizes the set kullback leibler kl divergence from the true posterior working within the rfs framework to incorporate uncertainty in target existence the derivation is interpreted as an application of expectation maximization em where the missing data is the correspondence of bernoulli components i e tracks under each data association hypothesis the missing data is shown to play an identical role to the selection of an ordered distribution in the same ordered family in the set jpda algorithm subsequently a special case of the proposed method is utilized to provide an efficient approximation of the minimum mean optimal subpattern assignment estimator the performance of the proposed methods is demonstrated in challenging scenarios in which up to twenty targets come into close proximity

inequalities for the numerical radius the norm and the maximum of the real part of bounded linear operators in hilbert spaces some inequalities for the numerical radius the operator norm and the maximum of the real part of bounded linear operators in hilbert spaces under suitable assumptions for the involved operator are given

labeled random finite sets and the bayes multi target tracking filter an analytic solution to the multi target bayes recursion known as the delta generalized labeled multi bernoulli delta glmb filter has been recently proposed by vo and vo in labeled random finite sets and multi object conjugate priors ieee trans signal process vol 61 no 13 pp 3460 3475 2014 as a sequel to that paper the present paper details efficient implementations of the delta glmb multi target tracking filter each iteration of this filter involves an update operation and a prediction operation both of which result in weighted sums of multi target exponentials with intractably large number of terms to truncate these sums the ranked assignment and k th shortest path algorithms are used in the update and prediction respectively to determine the most significant terms without exhaustively computing all of the terms in addition using tools derived from the same framework such as probability hypothesis density filtering we present inexpensive relative to the delta glmb filter look ahead strategies to reduce the number of computations characterization of the l 1 error in the multi target density arising from the truncation is presented

the gaussian mixture probability hypothesis density filter a new recursive algorithm is proposed for jointly estimating the time varying number of targets and their states from a sequence of observation sets in the presence of data association uncertainty detection uncertainty noise and false alarms the approach involves modelling the respective collections of targets and measurements as random finite sets and applying the probability hypothesis density phd recursion to propagate the posterior intensity which is a first order statistic of the random finite set of targets in time at present there is no closed form solution to the phd recursion this paper shows that under linear gaussian assumptions on the target dynamics and birth process the posterior intensity at any time step is a gaussian mixture more importantly closed form recursions for propagating the means covariances and weights of the constituent gaussian components of the posterior intensity are derived the proposed algorithm combines these recursions with a strategy for managing the number of gaussian components to increase efficiency this algorithm is extended to accommodate mildly nonlinear target dynamics using approximation strategies from the extended and unscented kalman filters

feedback properties of multivariable systems the role and use of the return difference matrix for linear time invariant multivariable feedback systems the feedback properties of plant disturbance attenuation sensor noise response stability margins and sensitivity to plant and sensor variation are quantitatively related to the bode magnitude versus frequency plots of the singular values of the return difference matrix i l and of the associated inverse return difference matrix i l 1 implied fundamental limits of feedback performance are quantitatively described and design tradeoffs are discussed the penalty function in the stochastic linear quadratic gaussian lqg optimal control problem is found to be a weighted sum of the singular values with the weights determined by the quadratic cost and noise intensity matrices this enables systematic tuning of lqg cost and noise matrices so that the resulting optimal return difference and inversereturn difference meet inequality constraints derived from design specifications on feedback properties the theory has been used to synthesize a multivariable automatic controller for the longitudinal dynamics of an advanced fighter aircraft

the spline probability hypothesis density filter the probability hypothesis density phd filter is a multitarget tracker that can alleviate the computational intractability of the optimal multitarget bayes filter the phd filter recursively estimates the number of targets and their phd from a set of observations and works well in scenarios with false alarms and missed detections two distinct phd filter implementations are available in the literature the sequential monte carlo probability hypothesis density smc phd and the gaussian mixture probability hypothesis density gm phd filters while particle based phd implementations may suffer from degeneracy gm based methods may not be suitable for highly nonlinear non gaussian systems this paper proposes a b spline based spline probability hypothesis density sphd filter which has the capability to better approximate any arbitrary probability density function the resulting algorithm can handle linear non linear gaussian and non gaussian models the sphd filter can provide continuous estimates of the probability density function of the system state and it is immune to the degeneracy problem the sphd filter can maintain highly accurate tracks by taking advantage of dynamic knot movement but at the expense of higher computational complexity which makes it suitable for tracking a few high value targets under difficult conditions the sphd filter derivations and simulations are provided in this paper

lower bounds for the numerical radius abstract let a be a complex n n matrix we find lower bounds for its numerical radius r a max x ax x c n x x 1 first we choose x satisfying x x 1 and compute x ax we also improve the simple bound so obtained second we apply r a max z a z a 2 z c z 1 where denotes the largest eigenvalue

the labeled multi bernoulli filter this paper proposes a generalization of the multi bernoulli filter called the labeled multi bernoulli filter that outputs target tracks moreover the labeled multi bernoulli filter does not exhibit a cardinality bias due to a more accurate update approximation compared to the multi bernoulli filter by exploiting the conjugate prior form for labeled random finite sets the proposed filter can be interpreted as an efficient approximation of the delta generalized labeled multi bernoulli filter it inherits the advantages of the multi bernoulli filter in regards to particle implementation and state estimation it also inherits advantages of the delta generalized labeled multi bernoulli filter in that it outputs labeled target tracks and achieves better performance