Poisson Point Processes

The purpose of the book is to provide an accessible discussion of multidimensional nonhomogeneous Poisson point processes. While often overlooked in the literature, new applications are bringing them to greater prominence. One chapter is devoted to developing their basic properties in a constructive manner. The approach is the reverse of the usual abstract approach, and it greatly facilitates understanding of PPPs. Two chapters discuss intensity estimation algorithms, with special attention to Gaussian sums, and the Cramér-Rao bound on unbiased estimation error. Three chapters are devoted to applications from medical imaging, multitarget tracking, and distributed network sensing. A final chapter discusses non-Poisson point processes for modeling spatial correlation between the points of a process.

Properties of multidimensional Poisson point processes (PPPs) are discussed using a constructive approach readily accessible to a broad audience. The processes are defined in terms of a two-step simulation procedure, and their fundamental properties are derived from the simulation. This reverses the traditional exposition, but it enables those new to the subject to understand quickly what PPPs are about, and to see that general nonhomogeneous processes are little more conceptually difficult than homogeneous processes. After reviewing the basic concepts on continuous spaces, several important and useful operations that map PPPs into other PPPs are discussed—these include superposition, thinning, nonlinear transformation, and stochastic transformation. Following these topics is an amusingly provocative demonstration that PPPs are “inevitable.” The chapter closes with a discussion of PPPs whose points lie in discrete spaces and in discrete-continuous spaces. In contrast to PPPs on continuous spaces, realizations of PPPs in these spaces often sample the discrete points repeatedly. This is important in applications such as multitarget tracking.

PPPs are characterized, or parameterized, by their intensity function. When the intensity is not fully specified by application domain knowledge (e.g., the physics), it is necessary to estimate it from data. These inverse problems are addressed by the method of maximum likelihood (ML). The Expectation-Maximization method is used to obtain estimators for problems involving superposition of PPPs. Gaussian sums are developed as an important special case. Estimators are obtained for both sample and histogram data. Regularization methods are presented for stabilizing Gaussian sum algorithms.

The quality of unbiased estimators of intensity is analyzed in terms of the Cramér-Rao Bound on the smallest possible estimation variance. No estimator is needed to find the bound. The CRB for the intensity of a PPP takes a remarkably simple explicit form. The CRB is examined for several problems. One problem analyzes the effect of gating on estimating the mean of a Gaussian signal in the presence of clutter, or outliers. Another generalizes this to find the CRB of the parameters of a Gaussian sum.

PPP methods for tomographic imaging are presented in this chapter. The primary emphasis is on methods for emission tomography, but transmission tomography is also included. The famous Shepp-Vardi algorithm for positron emission tomography (PET) is obtained via the EM method for time-of-flight data. Single-photon emission computed tomography (SPECT) is used in practice much more often than PET. It differs from PET in many ways, yet the models and the mathematics of the two methods are similar. (Both PET and SPECT are also closely related to multitarget tracking problems discussed in Chapter 6.) Transmission tomography is the final topic discussed. The Lange-Carson algorithm is derived via the EM method. CRBs for unbiased estimators for emission and transmission tomography are discussed. Regularization and Grenander’s method of sieves are reviewed in the last section.

Multitarget tracking intensity filters are closely related to imaging problems, especially PET imaging. The intensity filter is obtained by three different methods. One is a Bayesian derivation involving target prediction and information updating. The second approach is a simple, compelling, and insightful intuitive argument. The third is a straightforward application of the Shepp-Vardi algorithm. The intensity filter is developed on an augmented target state space. The PHD filter is obtained from the intensity filter by substituting assumed known target birth and measurement clutter intensities for the intensity filter’s predicted target birth and clutter intensities. To accommodate heterogeneous targets and sensor measurement models, a parameterized intensity filter is developed using a marked PPP Gaussian sum model. Particle and Gaussian sum implementations of intensity filters are reviewed. Mean-shift algorithms are discussed as a way to extract target state estimates. Grenander’s method of sieves is discussed for regularization of the multitarget intensity filter estimates. Sources of error in the estimated target count are discussed. Finally, the multisensor intensity filter is developed using the same PPP target models as in the single sensor filter. It is closely related to the SPECT medical imaging problem. Both homogeneous and heterogeneous multisensor fields are discussed. Multisensor intensity filters reduce the variance of estimated target count by averaging.

PPPs make several important, albeit somewhat disjointed, contributions to distributed sensor network detection, tracking, and communication connectivity. The focus in this chapter is on detection and communication since tracking problems are specialized forms of the multisensor intensity filter presented in Section 6.5. Communication path lengths of a randomly distributed sensor field are characterized by distance distributions. Distance distributions are obtained for sensors located at the points of a nonhomogeneous PPP realization. Both sensor-to-target and sensor-to-sensor distances are discussed. Communication diversity, that is, the number of communication paths between sensors in a distributed sensor field is discussed as a threshold phenomenon a geometric random graph and related to the abrupt transition phenomenon of such graphs. Detection coverage is discussed for both stationary and drifting sensor fields using Boolean models. These problems relate to classical problems in stochastic geometry and geometric probability. The connection between stereology and distributed sensor fields is presented as a final topic.

Generalizations of PPPs are useful in a large variety of applications. A few of the better known of these point processes are presented here, with emphasis on the processes themselves, not the applications. Marked processes are relatively simple extensions of PPPs that model auxiliary phenomena related to the point distribution. Other processes model the point-to-point correlation that may exist between the otherwise random occurrences of points. These processes include hard core processes and cluster processes. Cox processes are briefly reviewed, along with two stochastic processes, namely, Markov modulated Poisson processes and filtered processes. Gibbs (or Markov) point processes are not straightforward generalizations of PPPs. Generating realizations of Gibbs processes is typically done using MCMC methods.

Several topics of interest not discussed elsewhere in this book are mentioned here.