This article examines performance evaluation criteria for basic vision tasks involving sets of objects namely, object detection, instance-level segmentation and multi-object tracking. The rankings of algorithms by a criterion can fluctuate with different choices of parameters, e.g. Intersection over Union (IoU) threshold, making their evaluations unreliable. More importantly, there is no means to verify whether we can trust the evaluations of a criterion. This work suggests a notion of trustworthiness for performance criteria, which requires (i) robustness to parameters for reliability, (ii) contextual meaningfulness in sanity tests, and (iii) consistency with mathematical requirements such as the metric properties. We observe that these requirements were overlooked by many widely-used criteria, and explore alternative criteria using metrics for sets of shapes. We also assess all these criteria based on the suggested requirements for trustworthiness.

This paper proposes an online multi-camera multi-object tracker that only requires monocular detector training, independent of the multi-camera configurations, allowing seamless extension/deletion of cameras without retraining effort. The proposed algorithm has a linear complexity in the total number of detections across the cameras, and hence scales gracefully with the number of cameras. It operates in the 3D world frame, and provides 3D trajectory estimates of the objects. The key innovation is a high fidelity yet tractable 3D occlusion model, amenable to optimal Bayesian multi-view multi-object filtering, which seamlessly integrates, into a single Bayesian recursion, the sub-tasks of track management, state estimation, clutter rejection, and occlusion/misdetection handling. The proposed algorithm is evaluated on the latest WILDTRACKS dataset, and demonstrated to work in very crowded scenes on a new dataset.

Meeting or conference assistance is a popular application that typically requires compact configurations of co-located audio and visual sensors. This paper proposes a novel solution for online separation of an unknown and time-varying number of moving sources using only a single microphone array co-located with a single visual device. The approach exploits the complementary nature of simultaneous audio and visual measurements, accomplished by a model-centric 3-stage process of detection, tracking, and (spatial) filtering, which performs separation in a block-wise or recursive fashion. Fusing the measurements requires solving the multi-modal space-time permutation problem, since the audio and visual measurements reside in different observation spaces, but also are unidentified or unlabeled (with respect to the unknown and time-varying number of sources), and are subject to noise, extraneous measurements and missing measurements. A labeled random finite set tracking filter is applied to resolve the permutation problem and recursively estimate the source identities and trajectories. A time-varying set of generalized side-lobe cancellers is constructed based on the tracking estimates to perform online separation. Evaluations are undertaken with live human speakers.

We consider the challenging problem of online planning for a team of agents to autonomously search and track a time-varying number of mobile objects under the practical constraint of detection range limited onboard sensors. A standard POMDP with a value function that either encourages discovery or accurate tracking of mobile objects is inadequate to simultaneously meet the conflicting goals of searching for undiscovered mobile objects whilst keeping track of discovered objects. The planning problem is further complicated by misdetections or false detections of objects caused by range limited sensors and noise inherent to sensor measurements. We formulate a novel multi-objective POMDP based on information theoretic criteria, and an online multi-object tracking filter for the problem. Since controlling multi-agent is a well known combinatorial optimization problem, assigning control actions to agents necessitates a greedy algorithm. We prove that our proposed multi-objective value function is a monotone submodular set function; consequently, the greedy algorithm can achieve a (1-1/e) approximation for maximizing the submodular multi-objective function.

The traditional method of applying the optimal subpattern assignment (OSPA) metric cannot fully evaluate multitarget tracking performance, as it does not account for phenomena such as track label switching, and track fragmentation. The OSPA(2)has been proposed as a technique for applying the OSPA distance in a way that captures these effects, while retaining the properties of a true metric. In this paper, we demonstrate the behaviour of the OSPA(2)on some numerical examples, discuss some of its advantages and limitations, and show that it is capable of being applied to performance evaluation of large-scale scenarios in the order of a thousand targets.

This paper extends the generalized labeled multi-Bernoulli (GLMB) tracking filter to a batch multi-target tracker. In a labeled random finite set formulation, a multi-target tracking filter propagates the labeled multi-target filtering density while a batch multi-target tracker propagates the labeled multi-target posterior density. The GLMB filter is an analytic solution to the labeled multi-target filtering recursion. In this work, we show that the GLMB filter can be extended to an analytic multi-object posterior recursion.

This article proposes a framework for model-based point pattern learning using point process theory. Likelihood functions for point pattern data derived from point process theory enable principled yet conceptually transparent extensions of learning tasks, such as classification, novelty detection and clustering, to point pattern data. Furthermore, tractable point pattern models as well as solutions for learning and decision making from point pattern data are developed.

This letter proposes analytical expressions for the fusion of certain classes of labeled multiobject densities via Kullback-Leibler averaging. Specifically, we provide analytical fusion rules for the labeled multi-Bernoulli and marginalized δ-generalized labeled multi-Bernoulli families of labeled multiobject densities. Information fusion via Kullback-Leibler averaging ensures immunity to double counting of information and is essential to the development of effective multiagent multiobject estimation.

This paper considers a discrete-time sequential latent model for point pattern data, specifically a hidden Markov model (HMM) where each observation is an instantiation of a random finite set (RFS). This so-called RFS-HMM is worthy of investigation since point pattern data are ubiquitous in artificial intelligence and data science. We address the three basic problems typically encountered in such a sequential latent model, namely likelihood computation, hidden state inference, and parameter estimation. Moreover, we develop algorithms for solving these problems including forward-backward smoothing for likelihood computation and hidden state inference, and expectation-maximisation for parameter estimation. Simulation studies are used to demonstrate key properties of RFS-HMM, whilst real data in the domain of human dynamics are used to demonstrate its applicability.

This paper proposes an efficient implementation of the multi-sensor generalized labeled multi-Bernoulli (GLMB) filter. The solution exploits the GLMB joint prediction and update together with a new technique for truncating the GLMB filtering density based on Gibbs sampling. The resulting algorithm has a complexity in the order of the product of the number of measurements from each sensor, and quadratic in the number of hypothesized objects.

Clustering is one of the most common unsupervised learning tasks in machine learning and data mining. Clustering algorithms have been used in a plethora of applications across several scientific fields. However, there has been limited research in the clustering of point patterns - sets or multi-sets of unordered elements - that are found in numerous applications and data sources. In this paper, we propose two approaches for clustering point patterns. The first is a non-parametric method based on novel distances for sets. The second is a model-based approach, formulated via random finite set theory, and solved by the Expectation-Maximization algorithm. Numerical experiments show that the proposed methods perform well on both simulated and real data.

Point patterns are sets or multi-sets of unordered elements that can be found in numerous data sources. However, in data analysis tasks such as classification and novelty detection, appropriate statistical models for point pattern data have not received much attention. This paper proposes the modelling of point pattern data via random finite sets (RFS). In particular, we propose appropriate likelihood functions, and a maximum likelihood estimator for learning a tractable family of RFS models. In novelty detection, we propose novel ranking functions based on RFS models, which substantially improve performance.

In this paper, we extend the notion of Cauchy-Schwarz divergence to point processes and establish that the Cauchy-Schwarz divergence between the probability densities of two Poisson point processes is half the squared L2-distance between their intensity functions. Extension of this result to mixtures of Poisson point processes and, in the case where the intensity functions are Gaussian mixtures, closed form expressions for the Cauchy-Schwarz divergence are presented. Our result also implies that the Bhattacharyya distance between the probability distributions of two Poisson point processes is equal to the square of the Hellinger distance between their intensity measures. We illustrate the result via a sensor management application where the system states are modeled as point processes.

Quantitative analysis of the dynamics of tiny cellular and sub-cellular structures, known as particles, in time-lapse cell microscopy sequences requires the development of a reliable multi-target tracking method capable of tracking numerous similar targets in the presence of high levels of noise, high target density, complex motion patterns and intricate interactions. In this paper, we propose a framework for tracking these structures based on the random finite set Bayesian filtering framework. We focus on challenging biological applications where image characteristics such as noise and background intensity change during the acquisition process. Under these conditions, detection methods usually fail to detect all particles and are often followed by missed detections and many spurious measurements with unknown and time-varying rates. To deal with this, we propose a bootstrap filter composed of an estimator and a tracker. The estimator adaptively estimates the required meta parameters for the tracker such as clutter rate and the detection probability of the targets, while the tracker estimates the state of the targets. Our results show that the proposed approach can outperform state-of-the-art particle trackers on both synthetic and real data in this regime.

Although simultaneous localization and mapping (SLAM) algorithms are widely appreciated in mobile robot navigation, they can be further improved to suit practical applications in dynamic environmental conditions. One such important improvement is the detection and tracking of moving objects present in the sensor field of view (FOV). In this paper we propose to extend our recently introduced Collaborative Multi-vehicle SLAM (CMSLAM) solution based on the random finite set (RFS) representation of the feature map and measurements, by tracking both static and dynamic features. We represent static features observed during the SLAMprocess, along with dynamic features present in the current sensor FOV, as an augmented RFS. The corresponding probability density is propagated using a Bayes recursion, from which the static feature map and the estimates of dynamic feature locations can be obtained. Measurement update in the CMSLAM process is carried out only using the static feature map to take advantage of obvious accuracy improvements.

It was recently demonstrated that the Gaussian Mixture Cardinalised Probability Hypothesis Density (GMCPHD) filter can be used when the clutter density is unknown. Here we examine the performance of this filter, and as one would expect, it does not do as well as the conventional GMCPHD with matched clutter density. To improve the performance, we propose a bootstrap filtering scheme, and demonstrate by simulations on a bearings-only multitarget filtering scenario, that it is capable of performing almost as well as the matched GMCPHD filter.

In this paper we present a novel solution to the Multi-Vehicle SLAM (MVSLAM) problem by extending the random finite set (RFS) based SLAM filter framework using two recently developed multi-sensor information fusion approaches. Our solution is based on the modelling of the measurements and the landmark map as RFSs and factorizing the MVSLAM posterior into a product of the joint vehicle trajectories posterior and the landmark map posterior conditioned the vehicle trajectories. The joint vehicle trajectories posterior is propagated using a particle filter while the landmark map posterior conditioned on the vehicle trajectories is propagated using a Gaussian Mixture (GM) implementation of the probability hypothesis density (PHD) filter.

A new method is presented for integration of audio and visual information in multiple target tracking applications. The proposed approach uses a Bayesian filtering formulation and exploits multi-Bernoulli random finite set approximations. The work presented in this paper is the first principled Bayesian estimation approach to solve the sensor fusion problems that involve intermittent sensory data (e.g. audio data for a person who occasionally speaks.) We have examined our method with case studies from the SPEVI database. The results show nearly perfect tracking of people not only when they are silent but also when they are not visible to the camera (but speaking).

This paper proposes an integrated Bayesian frame work for feature-based simultaneous localization and map building (SLAM) in the general case of uncertain feature number and data association. By modeling the measurements and feature map as random finite sets (RFSs), a formulation of the feature-based SLAM problem is presented that jointly estimates the number and location of the features, as well as the vehicle trajectory. More concisely, the joint posterior distribution of the set-valued map and vehicle trajectory is propagated forward in time as measurements arrive, thereby incorporating both data association and feature management into a single recursion. Furthermore, the Bayes optimality of the proposed approach is established. A first-order solution, which is coined as the probability hypothesis density (PHD) SLAM filter, is derived, which jointly propagates the posterior PHD of the map and the posterior distribution of the vehicle trajectory. A Rao-Blackwellized (RB) implementation of the PHD-SLAM filter is proposed based on the Gaussian-mixture PHD filter (for the map) and a particle filter (for the vehicle trajectory). Simulated and experimental results demonstrate the merits of the proposed approach, particularly in situations of high clutter and data association ambiguity.

Almost every single-view visual multi-target tracking method presented in the literature includes a detection routine that maps the image data to point measurements relevant to the target states. These measurements are commonly further processed by a filter to estimate the number of targets and their states. This paper presents a novel visual tracking technique based on a multi-object filtering algorithm that operates directly on the image observations without the need for any detection. Experimental results on tracking sport players show that our proposed method can automatically track numerous interacting targets and quickly finds players entering or leaving the scene.

This paper proposes a tractable solution to feature-based (FB) SLAM in the presence of data association uncertainty and uncertainty in the number of features. By modeling the feature map as a random finite set (RFS), a rigorous Bayesian formulation of the FB-SLAM problem that accounts for uncertainty in the number of features and data association is presented. As such, the joint posterior distribution of the set-valued map and vehicle trajectory is propagated forward in time as measurements arrive. A first order solution, coined the PHD-SLAM filter, is derived, which jointly propagates the posterior PHD or intensity function of the map and the posterior distribution of the trajectory of the vehicle. A Rao-Blackwellised implementation of the PHD-SLAM filter is proposed based on the Gaussian mixture PHD filter for the map and a particle filter for the vehicle trajectory. Simulated results demonstrate the merits of the proposed approach, particularly in situations of high clutter and data association ambiguity.

This paper presents an alternative formulation for the Bayesian feature-based simultaneous localisation and mapping (SLAM) problem, using a random finite set approach. For a feature based map, SLAM requires the joint estimation of the vehicle location and the map. The map itself involves the joint estimation of both the number of features and their states (typically in a 2D Euclidean space), as an a priori unknown map is completely unknown in both landmark location and number. In most feature based SLAM algorithms, so-called dasiafeature managementpsila algorithms as well as data association hypotheses along with extended Kalman filters are used to generate the joint posterior estimate. This paper, however, presents a recursive filtering algorithm which jointly propagates both the estimate of the number of landmarks, their corresponding states, and the vehicle pose state, without the need for explicit feature management and data association algorithms. Using a finite set-valued joint vehicle-map state and set-valued measurements, the first order statistic of the set, called the intensity, is propagated via the probability hypothesis density (PHD) filter, from which estimates of the map and vehicle can be jointly extracted. Assuming a mildly non-linear Gaussian system, an extended-Kalman Gaussian Mixture implementation of the recursion is then tested for both feature-based robotic mapping (known location) and SLAM. Results from the experiments show promising performance for the proposed SLAM framework, especially in environments of high spurious measurements.

Exploiting channel state information at the transmitter and receiver to design an optimal linear precoder and decoder for a multiple-input multiple-output (MIMO) communication system is an active research area. The design is often based on the information rate criterion, that is to design the precoder and decoder such that the system capacity is maximized, subject to the average transmit power constraint. Although such an optimization problem has been considered intensively and there have been numerous proposals so far, they are not rigorously correct. We propose a mathematically rigorous framework for solving this optimization problem. Our proposed solution is applicable to both MIMO flat fading and frequency selective fading channels. Simulations verify the theoretical analysis.

The Gaussian mixture probability hypothesis density filter (GM-PHD Filter) was proposed recently for jointly estimating the time-varying number of targets and their states from a noisy sequence of sets of measurements which may have missed detections and false alarms. The initial implementation of the GM-PHD filter provided estimates for the set of target states at each point in time but did not ensure continuity of the individual target tracks. It is shown here that the trajectories of the targets can be determined directly from the evolution of the Gaussian mixture and that single Gaussians within this mixture accurately track the correct targets. Furthermore, the technique is demonstrated to be successful in estimating the correct number of targets and their trajectories in high clutter density and shows better performance than the MHT filter

In recent years there has been much interest in the probability hypothesis density (PHD) filtering approach, an attractive alternative to tracking unknown numbers of targets and their states in the presence of data association uncertainty, clutter, noise, and miss-detection. In particular, it has been discovered that the PHD filter has a closed form solution under linear Gaussian assumptions on the target dynamics and birth. This finding opens up a new direction where the PHD filter can be practically implemented in an effective and reliable fashion. However, the previous work is not general enough to handle jump Markov systems (JMS), a popular approach to modeling maneuvering targets. In this paper, a closed form solution for the PHD filter with linear JMS is derived. Our simulations demonstrate that the proposed PHD filtering algorithm provides promising performance. In particular, the algorithm is capable of tracking multiple maneuvering targets that cross each other

The probability hypothesis density (PHD) recursion is a first moment approximation to the multi-target Bayes filter which propagates the posterior intensity of the random finite set 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. Using the recently developed closed form solutions to both the PHD and CPHD recursions for linear Gaussian multi-target models, we present a comparative study of their performances

In this paper, a new technique for designing linear phase 2-D filter based on semi-definite programming (SDP) is proposed. This approach allows the design of 2-D filters with accurate cut-off frequency, subject to hard bounds on the frequency response to be achieved on a standard computer. Using the notion of 2-D trigonometric curves, we generalize the 2-D trigonometric Markov-Lukacs theorem to identify the pass-band and the stop-band in the region of support. The 2-D filter specifications are expressed as linear matrix inequalities. We also exploit convex duality to derive SDP formulations of reduced dimensions. Numerical examples illustrating the advantages of our method are also presented

A new design method for complex-valued two-channel FIR filter banks with both orthogonality and symmetry properties is developed. Based on a novel linear matrix inequality (LMI) characterization of trigonometric curves, the optimal design of the perfect reconstruction filter bank is reformulated as a semi-definite programme. The dimension of the resulting semi-definite programme is further reduced by exploiting the strong convex duality. Consequently, the globally optimal solution can be effectively found for any practical filter length and desired regularity order

The problem of designing a jointly optimal linear precoder and decoder for a multiple-input multiple-output (MIMO) channel has received much interest recently. However, most existing works only deal with white input signal. When the input signal is colored, pre-whitening and its inverse operation are often applied prior to precoding and after decoding, respectively. Consequently, the precoder and decoder are no longer optimal with respect to the original colored signal. In this paper, we propose a closed-form solution for optimal linear precoder and decoder for colored input signal. Our approach is based on minimizing the symbol mean squared error under an average output power constraint, and is applicable to both MIMO flat fading and frequency selective fading channels. Simulations show the advantage of our solution over prewhitening-based method.

We consider the problem of blind maximum-likelihood (ML) detection for the orthogonal space-time block code (OSTBC) scheme. Our previous work has shown that the problem can be simplified to a Boolean quadratic program (BQP). This sequel focuses on effective optimization methods for that BQP, which, from an optimization viewpoint, is still a computationally hard problem. First, we consider semidefinite relaxation (SDR), a high-precision BQP approximation algorithm with a computational cost that is polynomial in the problem size. We also propose a simple method that can significantly reduce the average complexity of the SDR technique. Second, we consider sphere decoding, an exact BQP solver that can be computationally expensive in the worst case, but generally incurs a reasonable average complexity particularly at high SNRs. Simulation results indicate that these two blind ML algorithms provide very similar bit error rate performance. Moreover, numerical studies show that SDR provides better complexity performance than sphere decoding in the worst-case sense, while sphere decoding provides better complexity performance in the average sense.

In this paper, we develop a new linear matrix inequality (LMI) technique, which is practical for solutions of the general trigonometric semi-infinite linear constraint (TSIC) of competitive orders. Based on the new full LMI characterization for the convex hull of a trigonometric curve, it is shown that the semi-infinite optimization problem involving TSIC can be solved by an LMI optimization problem with additional variables of dimension just n, the order of the trigonometric curve. Our solution method is very robust which allows us to address almost all practical filter design problems. Unlike most previous works involving several complex mathematical tools, our derivation arguments are based on simple results of the convex analysis and some formal elementary transforms. Furthermore, many filter/filterbank design problems can be reformulated as the optimization of linear/convex quadratic objectives over the TSIC. Based on this reformulation, these problems can be equivalently reduced to LMI optimization problems with the minimal size. Our examples of designing up to 1200-tap filters verifies the viability of our formulation.

Using time-difference-of-arrival (TDOA) measurements to perform speaker localization has received much interest recently. Motivated by the significant progress in TDOA single speaker localization, this paper presents a TDOA multi-speaker location tracking algorithm based on Bayesian particle filtering. The development is based on the random finite set framework, which provides an effective treatment to the problem of an unknown time-varying number of active speakers. The proposed method can be viewed as a generalization of the existing single-speaker particle filter. Using a simulated reverberant room, we demonstrate the tracking capability of the proposed particle filter.

Joint multi-target detection and tracking is a challenging problem due to several factors such as the number of targets being an unknown time-varying random parameter, and the target generated measurements being obscured by clutter. The theory of random finite sets permits an elegant formulation of the joint multi-target detection and tracking problem in a Bayesian framework, in particular, the random finite set formalism leads to the probability hypothesis density (PHD) filtering method, which realizes Bayesian joint detection and tracking in a suboptimal but numerically tractable manner. In this paper the PHD filter is applied to jointly detect and track multiple maneuvering targets. The tracking performance fidelity of the PHD filter is demonstrated by numerical results.

This paper addresses the blind symbol identifiability of the orthogonal space-time block code (OSTBC) scheme. That is, the conditions under which OSTBC symbols can be identified without ambiguity when channel state information is not available. In many space-time communication schemes, achieving unique blind symbol identification requires certain assumptions on the number of receiver antennas and the rank of the channel matrix. In this paper, we show that unique blind symbol identification of OSTBCs is possible for any number of receiver antennas and for any (nonzero) channel matrix. This attractive unique identifiability result is shown to be achieved by a class of OSTBCs that exhibit certain matrix non-rotational properties. Using these properties, we validate the identifiability of a number of commonly used OSTBCs.

Tracking multiple speakers in an acoustic environment involves jointly estimating the number of speakers and their states. This important problem in signal processing is challenging in theory as well as implementation. The paper presents a novel and fundamentally well-grounded framework for tracking multiple speakers using random finite sets. Simulations are also presented to demonstrate the performance in tracking a randomly varying number of speakers in a reverberant room.

This paper studies a reference-assisted approach for interference canceling (IC) in microphone array systems. Conventionally, reference-assisted IC is based on the zero crosstalk assumption; i.e., when the desired source signal is absent in the reference microphones. In applications where crosstalk is inevitable, the conventional IC approach usually exhibits degraded performance due to cancellation of the desired signal. In this paper, we develop a crosstalk resilient IC method based on the Capon beamforming technique. The proposed beamformer deals with the uncertainty of crosstalk by applying a constraint on the worst-case crosstalk magnitude. The proposed beamformer not only performs IC, it also provides blind beamforming of the desired signal. We show that a blind beamformer based on the traditional minimum-mean-square-error (MMSE) IC method is a special case of the proposed beamformer. One key step of implementing the proposed Capon beamformer lies in solving a difficult nonconvex optimization problem, and we illustrate how the Capon optimal solution can be effectively approximated using the so-called semidefinite relaxation algorithm. Simulation results demonstrate that the proposed beamformer is more robust against crosstalk-induced signal cancellation than beamformers based on the MMSE-IC methods.

In the discrete-time envelope-constrained filtering problem, the gain of the filter is minimised subject to the constraint that the filter output to a prescribed input fits into a given envelope. A novel adaptive algorithm for solving this problem based on stochastic optimisation is presented. The algorithm is simple to implement on-line and convergence is demonstrated in numerical examples. Under mild regularity assumptions, convergence follows from standard stochastic approximation results.

In an envelope-constrained filtering problem with uncertain input the set of possible inputs and the set of permissible outputs are each defined by envelopes or masks. This paper considers a continuous-time filter which in structure is comprised of an A/D converter, an FIR filter, a D/A converter and an analog post-filter. The object is to design the digital component of the filter structure so as to minimize the noise enhancement whilst satisfying the constraint that every signal in the input envelope evokes a response which stays in the output envelope.

Consider a continuous-time filter the structure of which is comprised of an A/D converter, an FIR filter, a D/A converter and an analog post-filter. The envelope constrained (EC) filtering problem for this filter structure is to design the digital component so as to minimize the effect of the input noise whilst satisfying the constraint that the noiseless response of the filter to a specified excitation fits into a prescribed envelope. This problem is formulated as a quadratic programming (QP) problem with functional inequality constraints. Approximating this continuum of constraints by a finite set, the problem is solved by QP via an active set strategy.

The envelope constrained (EC) filtering problem is concerned with the design of a filter with minimum gain to white input noise and with a response to a given signal that fits into a prescribed envelope. Using the discrete-time orthonormal Laguerre network, the EC filtering problem is posed as a quadratic programming (QP) problem with affine inequality constraints. An iterative algorithm for solving this QP problem is proposed which serves as a basis for the derivation of adaptive algorithms for applications where the parameters of the underlying signal model are either not known or varying with time.

The discrete-time envelope constrained (EC) filtering problem can be formulated as a quadratic programming (QP) problem with linear inequality constraints. In this paper, the QP problem is approximated by an unconstrained minimization problem with two parameters. These parameters can be selected so that given an acceptable deviation from the norm of the optimal EC filter, the solution to the unconstrained problem satisfies both the deviation and envelope constraints. Newton's method with line search is applied to solve the unconstrained problem iteratively.

This paper presents new recursive procedures for designing optimum envelope-constrained (EC) filters. Using a constraint transcription technique, the inequality constrained quadratic programming problem associated with EC filtering can be approximated as an unconstrained minimization problem. Two types of optimization methods are developed to solve this unconstrained problem in a recursive adjusting manner. Simulations of the proposed recursive procedures are presented which support the theoretical predictions.