Detecting and Tracking Unknown Number of Objects with Dirichlet Process Mixture Models and Markov Random Fields

Hidden Markov random field (HMRF) models are widely used for image segmentation, as they appear naturally in problems where a spatially constrained clustering scheme is asked for. A major limitation of HMRF models concerns the automatic selection of the proper number of their states, i.e., the number of region clusters derived by the image segmentation procedure. Existing methods, including likelihood- or entropy-based criteria, and reversible Markov chain Monte Carlo methods, usually tend to yield noisy model size estimates while imposing heavy computational requirements. Recently, Dirichlet process (DP, infinite) mixture models have emerged in the cornerstone of nonparametric Bayesian statistics as promising candidates for clustering applications where the number of clusters is unknown a priori; infinite mixture models based on the original DP or spatially constrained variants of it have been applied in unsupervised image segmentation applications showing promising results. Under this motivation, to resolve the aforementioned issues of HMRF models, in this paper, we introduce a nonparametric Bayesian formulation for the HMRF model, the infinite HMRF model, formulated on the basis of a joint Dirichlet process mixture (DPM) and Markov random field (MRF) construction. We derive an efficient variational Bayesian inference algorithm for the proposed model, and we experimentally demonstrate its advantages over competing methodologies.

In machine learning and data mining tasks, Clustering is considered to be one of the most important techniques. The same sorts of documents are grouped by performing clustering techniques. Similarity measuring is used to determine transaction relationships. Hierarchical clustering model generates tree structured results. Partitioned based clustering produces the result in grid format. Text documents are unstructured data values with high dimensional attributes. Document clustering group transforms unlabeled text documents into meaningful clusters. In the event of document grouping process, traditional clustering methods require cluster count (K). Clustering accuracy degrades drastically with reference to the unsuitable cluster count. It is observed that document features are automatically partitioned into two groups namely — discriminative words and non-discriminative words. In particular, discriminative words are only useful for grouping documents. The involvement of non-discriminative words confuses the clustering process and leads to poor clustering solution in return. A variation inference algorithm is used to infer the document collection structure and partition of document words simultaneously. Dirichlet Process Mixture (DPM) model is used to partition documents in a way utilizing both the data likelihood and the clustering property of the Dirichlet Process (DP). Dirichlet Process Mixture Model for Feature Extraction (DPMFE) is used to discover the latent cluster structure based on the DPM model and it is performed without involving the number of clusters as input.

The aim of this study was to implement a Dirichlet process mixture (DPM) model for automatic tumor edge identification on (18)F-fluorodeoxyglucose positron emission tomography ((18)F-FDG PET) images by optimizing the parameters on which the algorithm depends, to validate it experimentally, and to test its robustness. The DPM model belongs to the class of the Bayesian nonparametric models and uses the Dirichlet process prior for flexible nonparametric mixture modeling, without any preliminary choice of the number of mixture components. The DPM algorithm implemented in the statistical software package R was used in this work. The contouring accuracy was evaluated on several image data sets: on an IEC phantom (spherical inserts with diameter in the range 10-37 mm) acquired by a Philips Gemini Big Bore PET-CT scanner, using 9 different target-to-background ratios (TBRs) from 2.5 to 70; on a digital phantom simulating spherical/uniform lesions and tumors, irregular in shape and activity; and on 20 clinical cases (10 lung and 10 esophageal cancer patients). The influence of the DPM parameters on contour generation was studied in two steps. In the first one, only the IEC spheres having diameters of 22 and 37 mm and a sphere of the digital phantom (41.6 mm diameter) were studied by varying the main parameters until the diameter of the spheres was obtained within 0.2% of the true value. In the second step, the results obtained for this training set were applied to the entire data set to determine DPM based volumes of all available lesions. These volumes were compared to those obtained by applying already known algorithms (Gaussian mixture model and gradient-based) and to true values, when available. Only one parameter was found able to significantly influence segmentation accuracy (ANOVA test). This parameter was linearly connected to the uptake variance of the tested region of interest (ROI). In the first step of the study, a calibration curve was determined to automatically generate the optimal parameter from the variance of the ROI. This "calibration curve" was then applied to contour the whole data set. The accuracy (mean discrepancy between DPM model-based contours and reference contours) of volume estimation was below (1 ± 7)% on the whole data set (1 SD). The overlap between true and automatically segmented contours, measured by the Dice similarity coefficient, was 0.93 with a SD of 0.03. The proposed DPM model was able to accurately reproduce known volumes of FDG concentration, with high overlap between segmented and true volumes. For all the analyzed inserts of the IEC phantom, the algorithm proved to be robust to variations in radius and in TBR. The main advantage of this algorithm was that no setting of DPM parameters was required in advance, since the proper setting of the only parameter that could significantly influence the segmentation results was automatically related to the uptake variance of the chosen ROI. Furthermore, the algorithm did not need any preliminary choice of the optimum number of classes to describe the ROIs within PET images and no assumption about the shape of the lesion and the uptake heterogeneity of the tracer was required.

This paper presents an empirical study of endogenous treatment effects in the presence of heterogeneous responses. We estimate the incentive and selection effects of having prescription drug coverage on total drug expenditures using a sample from the Medicare Current Beneficiary Survey (MCBS). A Dirichlet process mixture (DPM) model is used to model the heterogeneity in treatment effects. Rather than estimating a finite mixture model with a fixed number of components, we specify a DP prior on the parameters, thus allowing the data and prior knowledge to determine the number of components. Endogeneity is modeled by the covariance of the error terms of the selection and the outcome equations in a two-equation selection model. We find that there are strong incentive and advantageous selection effects, with the average treatment effect and the average treatment effect for the treated estimated at $1132 and $858, respectively. Substantial heterogeneity is found to exist in the selection effects, particularly for those having lower drug expenditures. Copyright © 2014 John Wiley & Sons, Ltd.

This paper presents a novel multilook SAR image segmentation algorithm with an unknown number of clusters. Firstly, the marginal probability distribution for a given SAR image is defined by a Gamma mixture model (GaMM), in which the number of components corresponds to the number of homogeneous regions needed to segment and the spatial relationship among neighboring pixels is characterized by a Markov Random Field (MRF) defined by the weighting coefficients of components in GaMM. During the algorithm iteration procedure, the number of clusters is gradually reduced by merging two components until they are equal to one. For each fixed number of clusters, the parameters of GaMM are estimated and the optimal segmentation result corresponding to the number is obtained by maximizing the marginal probability. Finally, the number of clusters with minimum global energy defined as the negative logarithm of marginal probability is indicated as the expected number of clusters with the homogeneous regions needed to be segmented, and the corresponding segmentation result is considered as the final optimal one. The experimental results from the proposed and comparing algorithms for simulated and real multilook SAR images show that the proposed algorithm can find the real number of clusters and obtain more accurate segmentation results simultaneously.

In this paper, an algorithm for tracking multiple maneuvering targets by Multiple Hypothesis Tracking (MHT) with nonlinear non-Gaussian Kalman filter is investigated. The main challenges in multiple maneuvering targets tracking are the nonlinearity and non -Gaussianity problems. The Multiple Hypothesis Tracking (MHT) is used to detect the multiple targets in maneuverable and non-maneuverable modes. The computational requirements increase exponentially with number of tracks, the backscan depth and this can be reduced by careful design and tuning of MHT. The 1-backscan MHT algorithm is a good compromise between the two conflicting requirements of good tracking performance and limitation of computation time. The nonlinear non-Gaussian Kalman filter is used to track the target with high maneuver rate. The nonlinear non-Gaussian Kalman filter is implemented in MHT to give less probability of missing the target. The 1-backscan MHT with nonlinear non-Gaussian Kalman filter is free from computational burden by using simple probability concepts. This method of tracking also shows the reduction in the overshoot of root mean square error (RMSE).

This paper discusses several theoretical issues related to the score function for the measurement-to-track association/assignment decision in the track oriented version of the multiple hypothesis tracker (MHT). This score function is the likelihood ratio: the ratio of the pdf of a measurement having originated from a track, to the pdf of this measurement having a different origin. The likelihood ratio score is derived rigorously starting from the fully Bayesian (hypothesis oriented) MHT, which is shown to be amenable under some (reasonable) assumptions to the track oriented MHT. The latter can be implemented efficiently using multidimensional assignment. The main feature of a likelihood ratio is the fact that it is a (physically) dimensionless quantity and, consequently, can be used for the association of different numbers of measurements and/or measurements of different dimension. The explicit forms of the likelihood ratio are discussed both for the commonly used Kalman tracking filter, as well as for the interacting multiple model estimator. The issues of measurements of different dimension and different coordinate systems are also discussed.

Hidden Markov Random Field (HMRF) models are important statistical tools for image segmentation. However, such models cannot automatically determine the number of clusters. In many practical applications, to determine the number of clusters in advance is impossible. The Dirichlet process mixture (DPM) models are nonparametric Bayesian models which can automatically determine the number of clusters for the clustering problem. A new HMRF model which combines the DPM model with Student's t-distribution is proposed in this paper. Compared to the Gaussian distribution, the Student's t-distribution with heavier-tailed can have more applications. This model can automatically determine the appropriate clustering number from the observed data. In addition, an efficient variational Bayesian inference algorithm is derived to the proposed model. Experimental results show that the proposed model outperforms other DPM-based models in image segmentation.

One of the central issues in statistics and machine learning is how to select an adequate model that can automatically adapt its complexity to the observed data. In the present paper, we focus on the issue of determining the structure of clustered data, both in terms of finding the appropriate number of clusters and of modelling the right dependence structure between the observations. Bayesian nonparametric (BNP) models, which do not impose an upper limit on the number of clusters, are appropriate to avoid the required guess on the number of clusters but have been mainly developed for independent data. In contrast, Markov random fields (MRF) have been extensively used to model dependencies in a tractable manner but usually reduce to finite cluster numbers when clustering tasks are addressed. Our main contribution is to propose a general scheme to design tractable BNP-MRF priors that combine both features: no commitment to an arbitrary number of clusters and a dependence modelling. A key ingredient in this construction is the availability of a stick-breaking representation which has the threefold advantage to allowing us to extend standard discrete MRFs to infinite state space, to design a tractable estimation algorithm using variational approximation and to derive theoretical properties on the predictive distribution and the number of clusters of the proposed model. This approach is illustrated on a challenging natural image segmentation task for which it shows good performance with respect to the literature.

Hypothesis formation is a major computational burden for any multiple hypotheses tracking (MHT) method. In particular, a track-oriented MHT method defines compatible tracks to be tracks not sharing common observations and then re-forms hypotheses from compatible tracks after each new scan of data is received. The Cheap Joint Probabilistic Data Association (CJPDA) method provides an efficient means for computing approximate hypothesis probabilities. This paper presents a method of extending CJPDA calculations in order to eliminate low probability track branches in a track-oriented MHT method. The method is tested using IRST data. This approach reduces the number of tracks in a cluster and the resultant computations required for hypothesis formation. It is also suggested that the use of CJPDA methods can reduce assignment matrix sizes and resultant computations for the hypothesis-oriented (Reid’s algorithm) MHT implementation.

We present a people counting system that estimates the number of people in a scene by employing a clustering scheme based on Dirichlet Process Mixture Models (DP-MMs) which takes outputs of a person detector system as input. For each frame, we run a person detector on the frame, take its output as a set of detection areas and define a set of features based on spatial, color and temporal information for each detection. Then using these features, we cluster the detections using DPMMs and Gibbs sampling while having no restriction on the number of clusters, thus can estimate an arbitrary number of people or groups of people. We finally define a measure to calculate the actual number of people within each cluster to infer the final estimation of the number of people in the scene.

To date, Wald sequential probability ratio test (WSPRT) has been widely applied to track management of multiple hypothesis tracking (MHT). But in a real situation, if the false alarm spatial density is much larger than the new target spatial density, the original track score will be very close to the deletion threshold of the WSPRT. Consequently, all tracks, including target tracks, may easily be deleted, which means that the tracking performance is sensitive to the tracking environment. Meanwhile, if a target exists for a long time, its track will have a high score, which will make the track survive for a long time even after the target has disappeared. In this paper, to consider the relationship between the hypotheses of the test, we adopt the Shiryayev SPRT (SSPRT) for track management in MHT. By introducing a hypothesis transition probability, the original track score can increase faster, which solves the first problem. In addition, by setting an independent SSPRT for track deletion, the track score can decrease faster, which solves the second problem. The simulation results show that the proposed SSPRT-based MHT can achieve better tracking performance than MHT based on the WSPRT under a high false alarm spatial density.

In multi-target tracking, Multiple Hypothesis Tracking (MHT) can effectively solve the data association problem. However, traditional MHT can not make full use of motion information. In this work, we combine MHT with Interactive Multiple Model (IMM) estimator and feature fusion. New algorithm greatly improves the tracking performance due to the fact that IMM estimator provides better estimation and feature information enhances the accuracy of data association. The new algorithm is tested by tracking tropical fish in fish container. Experimental result shows that this algorithm can significantly reduce tracking lost rate and restrain the noises with higher computational effectiveness when compares with traditional MHT.

Interacting multiple model (IMM) filtering and multiple hypothesis tracking (MHT) represent the most accurate methods currently available for tracking multiple maneuvering targets in cluttered environments. Although these methods are complex, modern computational capabilities make their combined implementation feasible for modern tracking systems. This paper discusses alternative approaches for developing a combined IMM/MHT tracking system and describes specific implementations that have been developed for radar and IRST applications. Simulation results for both radar and IRST systems are presented. Track maintenance and accuracy performance for the IMM/MHT system is compared with that obtained from an MHT tracker using a conventional filtering approach. Results indicate that the improvements in data association derived from the use of IMM filtering with MHT may be comparable to the well known IMM improvements in track accuracy. The addition of multiple filter models to MHT data association has the potential to significantly increase computational requirements. Thus, several compromises, described in this paper, have been developed in order to assure computational feasibility. Preliminary estimates of computational requirements are given in order to demonstrate implementation feasibility.© (1997) COPYRIGHT SPIE--The International Society for Optical Engineering. Downloading of the abstract is permitted for personal use only.

We study the ambiguous data association problem confronting simultaneous localization and mapping (SLAM), specifically for the autonomous exploration of environments lacking rich features. In such environments, a single false positive assignment might lead to catastrophic failure, which even robust back-ends may be unable to resolve. Inspired by multiple hypothesis tracking, we present a novel approach to effectively manage multiple hypotheses (MH) of data association inherited from traditional joint compatibility branch and bound (JCBB), which entails the generation, ordering and elimination of hypotheses. We analyze the performance of MHJCBB in two particular situations, one applying it to SLAM over a predefined trajectory and the other showing its applicability in exploring unknown environments. Statistical results demonstrate that MHJCBB's maintenance of diverse hypotheses under ambiguous conditions significantly improves map accuracy.