Estimation for Random Sets

Abstract

The random set (RS) concept generalizes that of a random vector. It permits the mathematical modeling of random systems that can be interpreted as random patterns. Algorithms based on RSs have been extensively employed in image processing. More recently, they have found application in multitarget detection and tracking and in the modeling and processing of human-mediated information sources. The purpose of this entry is to briefly summarize the concepts, theory, and practical application of RSs.

Introduction

In ordinary signal processing, one models physical phenomena as “sources,” which generate “signals” obscured by random “noise.” The sources are to be extracted from the noise using optimal-estimation algorithms. Random set (RS) theory was devised about 40 years ago by mathematicians who also wanted to construct optimal-estimation algorithms. The “signals” and “noise” that they had in mind, however, were geometric patterns in images. The resulting theory, stochastic geometry, is the basis of the “morphological operators” commonly employed today in image-processing applications. It is also the basis for the theory of RSs. An important special case of RS theory, the theory of random finite sets (RFSs), addresses problems in which the patterns of interest consist of a finite number of points. It is the theoretical basis of many modern medical and other image-processing algorithms. In recent years, RFS theory has found application to the problem of detecting, localizing, and tracking unknown numbers of unknown, evasive point targets. Most recently and perhaps most surprisingly, RS theory provides a theoretically rigorous way of addressing “signals” that are human-mediated, such as natural-language statements and inference rules. The breadth of RS theory is suggested in the various chapters of Goutsias et al. (1997).

The purpose of this entry is to summarize the RS and RFS theories and their applications. It is divided in to the following sections: A Simple Example, Mathematics of Random Sets, Random Sets and Image Processing, Random Sets and Multitarget Processing, Random Sets and Human-Mediated Data, Summary and Future Directions, Cross-References, and Recommended Reading.

A Simple Example

To illustrate the concept of a RS, let us begin by examining a simple example: locating stars in the nighttime sky. We will proceed in successively more illustrative steps:

Locating a single non-dim star (estimating a random point). When we try to locate a star, we are trying to estimate its actual position – its “state” x = (α₀, θ₀) – in terms of its azimuth angle α₀ and elevation angle θ₀. When the star is dim but not too dim, its apparent position will vary slightly. We can estimate its position by averaging many measurements – i.e., by applying a point estimator.

Locating a very dim star (estimating an RS with at most one element). Assume that the star is so dim that, when we see it, it might be just a momentary visual illusion. Before we can estimate its position, we mustfirst estimate whether or not it exists. We must record not only its apparent position z = (α, θ) (if we see it) but its apparent existence$\varepsilon$, with $\varepsilon = 1$ (we saw it) or $\varepsilon = 0$ (we did not). Averaging $\varepsilon$ over many observations, we get a number q between 0 and 1. If $q> \frac{1} {4}$ (say), we could declare that the star probably actually is a star; and then we could average the non-null observations to estimate its position.

Locating multiple stars (estimating an RFS). Suppose that we are trying to locate all of the stars in some patch of sky. In some cases, two dim stars may be so close that they are difficult to distinguish. We will then collect three kinds of measurements from them: Z = ∅ (did not see either star), Z = { (α, θ)} (we saw one or the other), or $Z =\{ (\alpha _{1},\theta _{1}),(\alpha _{2},\theta _{2})\}$ (saw both). The total collected measurement in the patch of sky is a finite set $Z =\{ \mathbf{z}_{1},\ldots,\mathbf{z}_{m}\}$ of point measurements with $\mathbf{z}_{j} = (\theta _{j},\alpha _{j})$, where each z _i is random, where m is random, and where m = 0 corresponds to the null measurement Z = ∅.

Locating multiple stars in a quantized sky (estimation using imprecise measurements). Suppose that, for computational reasons, the patch of sky must be quantized into a finite number of hexagonal-shaped cells, c₁, …, c _M . Then, the measurement from any star is not a specific point z, but instead the cell c that contains z. The measurement c is imprecise – a randomly varying hexagonal cell c. There are two ways of thinking about the total measurement collection. First, it is a finite set $Z =\{ c_{1}^{{\prime}},\ldots,c_{m}^{{\prime}}\}\subseteq \{ c_{1},\ldots,c_{M}\}$ of cells. Second, it is the union $Z = c_{1}^{{\prime}}\cup \ldots \cup c_{m}^{{\prime}}$ of all of the observed cells – i.e., it is a geometrical pattern.

Locating multiple stars over an extended period of time (estimating multiple moving targets). As the night progresses, we must continually redetermine the existence and positions of each star – a process called multitarget tracking. We must also account for appearances and disappearances of the stars in the patch – i.e., for target death and birth.

Mathematics of Random Sets

The purpose of this section is to sketch the elements of the theory of random sets. It is organized as follows: General Theory of Random Sets, Random Finite Sets (Random Point Processes), and Stochastic Geometry. Of necessity, the material is less elementary than in later sections.

General Theory of Random Sets

Let $\mathfrak{Y}$ be a topological space – for example, an N-dimensional Euclidean space $\mathbb{R}^{N}$. The power set$2^{\mathfrak{Y}}$ of $\mathfrak{Y}$ is the class of all possible subsets $S \subseteq \mathfrak{Y}$. Any subclass of $2^{\mathfrak{Y}}$ is called a “hyperspace.” The “elements” or “points” of a hyperspace are thus actually subsets of some other space. For a hyperspace to be of interest, one must extend the topology on $\mathfrak{Y}$ to it. There are many possible topologies for hyperspaces (Michael 1950). The most well studied is the Fell-Matheron topology, also called the “hit-and-miss” topology (Matheron 1975). It is applicable when $\mathfrak{Y}$ is Hausdorff, locally compact, and completely separable. It topologizes only the hyperspace $\mathfrak{c}(2^{\mathfrak{Y}})$ of all closed subsets C of $\mathfrak{Y}$. In this case, a random (closed)set$\Theta$ is a measurable mapping from some probability space into $\mathfrak{c}(2^{\mathfrak{Y}})$.

The Fell-Matheron topology’s major strength is its relative simplicity. Let “$\Pr (\mathcal{E})$” denote the probability of a probabilistic event $\mathcal{E}$. Then, normally, the probability law of $\Theta$ would be described by a very abstract probability measure $p_{\Theta }(O) =\Pr (\Theta \in O)$. This measure must be defined on the Borel-measurable subsets $O \subseteq \mathfrak{c}(2^{\mathfrak{Y}})$, with respect to the Fell-Matheron topology, where O is itself a class of subsets of $\mathfrak{Y}$. However, define the Choquet capacity functional by $c_{\Theta }(G) =\Pr (\Theta \cap G\neq \emptyset )$ for all open subsets $G \subseteq \mathfrak{Y}$.  Then, the Choquet-Matheron theorem states that the probability law of $\Theta$ is completely described by the simpler, albeit nonadditive, measure $c_{\Theta }(G)$.

The theory of random sets has evolved into a substantial subgenre of statistical theory (Molchanov 2005). For estimation theory, the concept of the expected value$\mathbb{E}$[$\Theta$] of a random set $\Theta$ is of particular interest. Most definitions of $\mathbb{E}$[$\Theta$] are very abstract (Molchanov 2005, Chap. 2). In certain circumstances, however, more conventional-looking definitions are possible. Suppose that $\mathfrak{Y}$ is a Euclidean space and that $\mathfrak{c}(2^{\mathfrak{Y}})$ is restricted to $\mathfrak{K}(2^{\mathfrak{Y}})$, the bounded, convex, closed subsets of $\mathfrak{Y}$. If C, C^′ are two such subsets, their Minkowski sum is $C + C^{{\prime}} =\{ c + c^{{\prime}}\vert \;c \in C,c^{{\prime}}\in C^{{\prime}}\}$.  Endowed with this definition of addition, $\mathfrak{K}(2^{\mathfrak{Y}})$ can be homeomorphically and homomorphically embedded into a certain space of functions (Molchanov 2005, pp. 199–200). Denote this embedding by $C\longmapsto \phi _{C}$. Then, the expected value $\mathbb{E}$[$\Theta$] of $\Theta$, defined in terms of Minkowski addition, corresponds to the conventional expected value $\mathbb{E}$[$\phi _{\Theta }$]  of the random function $\phi _{\Theta }$.

Random Finite Sets (Random Point Processes)

Suppose that the $\mathfrak{c}(2^{\mathfrak{Y}})$ is restricted to $\mathfrak{f}(2^{\mathfrak{Y}})$, the class of finite subsets of $\mathfrak{Y}$. (In many formulations, $\mathfrak{f}(2^{\mathfrak{Y}})$  is taken to be the class of locally finite subsets of $\mathfrak{Y}$ – i.e., those whose intersection with compact subsets is finite.) A random finite set (RFS) is a measurable mapping from a probability space into $\mathfrak{f}(2^{\mathfrak{Y}})$. An example: the field of twinkling stars in some patch of a night sky. RFS theory is a particular mathematical formulation of point process theory (Daley and Vere-Jones 1998; Snyder and Miller 1991; Stoyan et al. 1995).

A Poisson RFS$\Psi$ is perhaps the simplest nontrivial example of a random point pattern. It is specified by a spatial distribution s(y) and an intensity μ. At any given instant, the probability that there will be n points in the pattern is $p(n) = e^{-\mu }\mu ^{n}/n!$ (the value of the Poisson distribution). The probability that one of these n points will be y is s(y). The function $D_{\Psi }(\mathbf{y}) =\mu \cdot s(\mathbf{y})$ is called the intensity function of $\Psi$.

At any moment, the point pattern produced by $\Psi$ is a finite set $Y =\{ \mathbf{y}_{1},\ldots,\mathbf{y}_{n}\}$ of points $\mathbf{y}_{1},\ldots,\mathbf{y}_{n}$ in $\mathfrak{Y}$, where n = 0, 1, … and where Y = ∅ if n = 0. If n = 0 then Y represents the hypothesis that no objects at all are present. If n = 1 then Y = {y₁} represents the hypothesis that a single object y₁ is present. If n = 2 then Y = {y₁, y₂} represents the hypothesis that there are two distinct objects y₁ ≠ y₂. And so on.

Theprobability distribution of $\Psi$ – i.e., the probability that $\Psi$ will have $Y =\{ \mathbf{y}_{1},\ldots,\mathbf{y}_{n}\}$ as an instantiation – is entirely determined by its intensity function $D_{\Psi }(\mathbf{y})$:

$$\displaystyle\begin{array}{rcl} f_{\Psi }(Y )& =& f_{\Psi }(\{\mathbf{y}_{1},\ldots,\mathbf{y}_{n}\}) {}\\ & =& e^{-mu} \cdot D_{ \Psi }(\mathbf{y}_{1})\cdots D_{\Psi }(\mathbf{y}_{n}) {}\\ \end{array}$$

Every suitably well-behaved RFS $\Psi$ has a probability distribution  $f_{\Psi }(Y )$ and an intensity function $D_{\Psi }(\mathbf{y})$ (a.k.a. first-moment density). A Poisson RFS is unique in that $f_{\Psi }(Y )$ is completely determined by $D_{\Psi }(\mathbf{y})$.

Conventional signal processing is often concerned with single-object random systems that have the form

$$\displaystyle{ \mathbf{Z} =\eta (\mathbf{x}) + \mathbf{V} }$$

where x is the state of the system; η(x)  is the “signal” generated by the system; the zero-mean random vector V is the random “noise” associated the sensor; and Z is the random measurement that is observed. The purpose of signal processing is to construct an estimate $\mathbf{\hat{x}}(\mathbf{z}_{1},\ldots.,\mathbf{z}_{k})$ of x, using the information contained in one or more draws z₁, …. , z _k from the random variable Z.

RFS theory is analogously concerned with random systems that have the form

$$\displaystyle{ \Sigma = \Upsilon (X) \cup \Omega }$$

where a random finite point pattern $\Upsilon (X)$ is the “signal” generated by the point pattern  X (which is an instantiation of a random point pattern $\Xi$); $\Omega$ is a random finite point “noise” pattern; $\Sigma$ is the total random finite point pattern that has been observed; and “∪” denotes set-theoretic union. One goal of RFS theory is to devise algorithms that can construct an estimate $\hat{X}(Z_{1},\ldots.,Z_{k})$ of X, using multiple point patterns $\ Z_{1},\ldots.,Z_{k} \subseteq \mathfrak{Y}$ drawn from $\Sigma$. One approximate approach is that of estimating only the first-moment density $D_{\Xi }(\mathbf{x})$ of $\Xi$.

Stochastic Geometry

Stochastic geometry addresses more complicated random patterns. An example: the field of twinkling stars in a quantized patch of the night sky, in which case the measurement is the union c₁ ∪ … ∪ c _m of a finite number of hexagonally shaped cells.

This is one instance of a germ-grain process (Stoyan et al. 1995, pp. 59–64). Such a process is specified by two items: an RFS $\Psi$ and a function c _y that associates with each y in $\mathfrak{Y}$ a closed subset $c_{\mathbf{y}} \subseteq \mathfrak{Z}$. For example, if $\mathfrak{Y} = \mathbb{R}^{2}$ is the real-valued plane, then c _y could be the disk of radius r centered at y = (x, y). Let $Y =\{ \mathbf{y}_{1},\ldots,\mathbf{y}_{n}\}$ be a particular random draw from $\Psi$. The points $\mathbf{y}_{1},\ldots,\mathbf{y}_{n}$ are the “germs,” and $c_{\mathbf{y}_{1}},\ldots,c_{\mathbf{y}_{n}}$ are the “grains” of this random draw from the germ-grain process $\Theta$. The total pattern in $\mathfrak{Y}$ is the union $c_{\mathbf{y}_{1}} \cup \ldots \cup c_{\mathbf{y}_{n}}$ of the grains – a random draw from $\Theta$. Germ-grain processes can be used to model many kinds of natural processes. One example is the distribution of graphite particles in a two-dimensional section of a piece of iron, in which case the c _y could be chosen to be line segments rather than disks.

Stochastic geometry is concerned with random binary images that have observation structures such as

$$\displaystyle{ \Theta = (S \cap \Delta ) \cup \Omega }$$

where S is a “signal” pattern; $\Delta$ is a random pattern that models obscurations; $\Omega$ is a random pattern that models clutter; and $\Theta$ is the total pattern that has been observed. A common simplifying assumption is that $\Omega$ and $\Delta ^{c}$ are germ-grain processes. One goal of stochastic geometry is to devise algorithms that can construct an optimal estimate $\hat{S}(T_{1},\ldots.,T_{k})$ of S, using multiple patterns $T_{1},\ldots.,T_{k} \subseteq \mathfrak{Y}$ drawn from $\Theta$.

Random Sets and Image Processing

Both point process theory and stochastic geometry have found extensive application to image-processing applications. These are considered briefly in turn.

Stochastic Geometry and Image Processing. Stochastic geometry methods are based on the use of a “structuring element” B (a geometrical shape, such as a disk, sphere, or more complex structure) to modify an image.

The dilation of a set S by B is S ⊕ B where “⊕” is Minkowski addition (Stoyan et al. 1995). Dilation tends to fill in cavities and fissures in images. The erosion of S is S ⊖ B = (S^c ⊕ B^c)^c where “^c” indicates set-theoretic complement. Erosion tends to create and increase the size of cavities and fissures. Morphological filters are constructed from various combinations of dilation and erosion operators.

Suppose that a binary image $\Sigma = S$ has been degraded by some measurement process – for example, the process $\Theta = (S \cap \Delta ) \cup \Omega$. Then, image restoration refers to the construction of an estimate $\hat{S}(T)$ of the original image S from a single degraded image $\Theta = T$. The restoration operator $\hat{S}(T)$ is optimal if it can be shown to be optimally close to S, given some concept of closeness. The symmetric difference

$$\displaystyle{ T_{1} \sqcup T_{2} = (T_{1} \cup T_{2}) - (T_{1} \cap T_{2}) }$$

is a commonly used method for measuring the dissimilarity of binary images. It can be used to construct measures of distance between random images. One such distance is

$$\displaystyle{ d(\Theta _{1},\Theta _{2}) = \mathbb{E}\left [\vert \Theta _{1} \sqcup \Theta _{2}\vert \right ] }$$

where | S | denotes the size of the set S and $\mathbb{E}$[A] is the expected value of the random number A. Other distances require some definition of the expected value $\mathbb{E}$[$\Theta$]  of a random set $\Theta$. It has been shown that, under certain circumstances, certain morphological operators can be viewed as consistent maximum a posteriori (MAP) estimators of S (Goutsias et al. 1997, p. 97).

RFS Theory and Image Processing. Positron-emission tomography (PET) is one example of the application of RFS theory. In PET, tissues of interest are suffused with a positron-emitting radioactive isotope. When a positron annihilates an electron in a suitable fashion, two photons are emitted in opposite directions. These photons are detected by sensors in a ring surrounding the radiating tissue. The location of the annihilation on the line can be estimated by calculating time difference of arrival.

Because of the physics of radioactive decay, the annihilations can be accurately modeled as a Poisson RFS $\Psi$. Since a Poisson RFS is completely determined by its intensity function $D_{\Psi }(\mathbf{x})$, it is natural to try to estimate $D_{\Psi }(\mathbf{x})$. This yields the spatial distribution $s_{\Psi }(\mathbf{y})$ of annihilations – which, in turn, is the basis of the PET image (Snyder and Miller 1991, pp. 115–119).

Random Sets and Multitarget Processing

The purpose of this section is to summarize the application of RFS theory to multitarget detection, tracking, and localization. An example: tracking the positions of stars in the night sky over an extended period of time.

Suppose that at time t _k there are an unknown number n of targets with unknown states $\mathbf{x}_{1},\ldots,\mathbf{x}_{n}$. The state of the entire multitarget system is a finite set $X =\{ \mathbf{x}_{1},\ldots,\mathbf{x}_{n}\}$ with n ≥ 0. When interrogating a scene, many sensors (such as radars) produce a measurement of the form $Z =\{ \mathbf{z}_{1},\ldots,\mathbf{z}_{m}\}$ – i.e., a finite set of measurements. Some of these measurements are generated by background clutter $\Omega _{k}$. Others are generated by the targets, with some targets possibly not having generated any. Mathematically speaking, Z is a random draw from an RFS $\Sigma _{k}$ that can be decomposed as $\Sigma _{k} = \Upsilon (X_{k}) \cup \Omega _{k}$, where $\Upsilon (X_{k})$ is the set of target-generated measurements.

Conventional Multitarget Detection and Tracking. This is based on a “divide and conquer” strategy with three basic steps: time update, data association, and measurement update. At time t _k we have n “tracks” $\tau _{1},\ldots,\tau _{n}$ (hypothesized targets). In the time update, an extended Kalman filter (EKF) is used to time-predict the tracks τ _i to predicted tracks$\tau _{i}^{+}$ at the time $t_{k+1}$ of the next measurement set $Z_{k+1} =\{ \mathbf{z}_{1},\ldots,\mathbf{z}_{m}\}$.

Given Z_k+1, we can construct the following data-association hypothesis H: for each i = 1, …, n, the predicted track τ _i ⁺ generated the detection $\mathbf{z}_{j_{i}}$, for some index j _i , or, alternatively, this track was not detected at all. If we remove from Z_k+1 all of the $\mathbf{z}_{j_{1}},\ldots,\mathbf{z}_{j_{n}}$, the remaining measurements are interpreted either as being clutter or as having been generated by new targets. Enumerating all possible association hypotheses (which is a combinatorily complex procedure), we end up with a “hypothesis table” H₁, …, H _ν .

Given H _i , let $\mathbf{z}_{j_{i}}$ be the measurement that is hypothesized to have been generated by predicted track τ _i ⁺. Then, the measurement-update step of an EKF is used to construct a measurement-updated track $\tau _{i,j_{i}}$ from τ _i ⁺ and $\mathbf{z}_{j_{i}}$. Attached to each H _i is a hypothesis probability p _i – the probability that the particular hypothesis H _i is the correct one. The hypothesis with largest p _i yields the multitarget estimate $\hat{X} =\{ \mathbf{\hat{x}}_{1},\ldots,\mathbf{\hat{x}}_{\hat{n}}\}$.

RFS Multitarget Detection and Tracking. In the place of tracks and hypothesis tables, this uses multitarget state sets and multitarget probability distributions. In place of the conventional time update, data association, and measurement update, it uses a recursive Bayes filter. A random multitarget state set is an RFS $\Xi _{k\vert k}$ whose points are target states. A multitarget probability distribution is the probability distribution $f(X_{k}\vert Z_{1:k}) = f_{\Xi _{k\vert k}}(X)$ of the RFS $\Xi _{k\vert k}$, where $Z_{1:k} : Z_{1},\ldots,Z_{k}$ is the time sequence of measurement sets at time t _k .

RFS Time Update. The Bayes filter time-update step $f(X_{k}\vert Z_{1:k}) \rightarrow f(X_{k+1}\vert Z_{1:k})$ requires a multitarget Markov transition function f(X_k+1 | X _k ). It is the probability that the multitarget system will have multitarget state set X_k+1 at time t_k+1, if it had multitarget state set X _k at time t _k . It takes into account all pertinent characteristics of the targets: individual target motion, target appearance, target disappearance, environmental constraints, etc. It is explicitly constructed from an RFS multitarget motion model using a multitarget integrodifferential calculus.

RFS Measurement Update. The Bayes filter measurement-update step $f(X_{k+1}\vert Z_{1:k}) \rightarrow f(X_{k+1}\vert Z_{1:k+1})$ is just Bayes rule. It requires a multitarget likelihood function f_k+1(Z | X) – the likelihood that a measurement set Z will be generated, if a system of targets with state set X is present. It takes into account all pertinent characteristics of the sensor(s): sensor noise, fields of view and obscurations, probabilities of detection, false alarms, and/or clutter. It is explicitly constructed from an RFS measurement model using multitarget calculus.

RFS State Estimation. Determination of the number n and states x₁, …, x _n of the targets is accomplished using a Bayes-optimal multitarget state estimator. The idea is to determine the X_k+1 that maximizes $f(X_{k+1}\vert Z_{1:k+1})$ in some sense.

Approximate Multitarget RFS Filters. The multitarget Bayes filter is, in general, computationally intractable. Central to the RFS approach is a toolbox of techniques – including the multitarget calculus – designed to produce statistically principled approximate multitarget filters. The two most well studied are the probability hypothesis density (PHD) filter and its generalization the cardinalized PHD (CPHD) filter. In such filters, f(X _k | Z_{1: k}) is replaced by the first-moment density D(x _k | Z_{1: k}) of $\Xi _{k\vert k}$. These filters have been shown to be faster and perform better than conventional approaches in some applications.

Random Sets and Human-Mediated Data

Natural-language statements and inference rules have already been mentioned as examples of human-mediated information. Expert-systems theory was introduced in part to address situations – such as this – that involve uncertainties other than randomness. Expert-system methodologies includefuzzy set theory, the Dempster-Shafer (D-S)theory of uncertain evidence, andrule-based inference. RS theory provides solid Bayesian foundations for them and allows human-mediated data to be processed using standard Bayesian estimation techniques. The purpose of this section is to briefly summarize this aspect of the RS approach.

The relationships between expert-systems theory and random set theory were first established by researchers such as Orlov (1978), Höhle (1982), Nguyen (1978), and Goodman and Nguyen (1985). At a relatively early stage, it was recognized that random set theory provided a potential means of unifying much of expert-systems theory (Goodman and Nguyen 1985; Kruse et al. 1991).

A conventional sensor measurement at time t _k is typically represented as $\mathbf{Z}_{k} =\eta (\mathbf{x}_{k}) + \mathbf{V}_{k}$ – equivalently formulated as a likelihood function f(z _k |x _k ). It is conventional to think of z _k as the actual “measurement” and of f(z _k |x _k ) as the full description of the uncertainty associated with it. In actuality, z _k is just a mathematical model$\mathbf{z}_{\zeta _{k}}$ of some real-world measurement ζ _k . Thus, the likelihood actually has the form $f(\zeta _{k}\mathbf{\vert x}_{k}) = f(\mathbf{z}_{\zeta _{k}}\mathbf{\vert x}_{k})$.

This observation assumes crucial importance when one considers human-mediated data. Consider the simple natural-language statement

$$\displaystyle{ \zeta = \text{"The target is near the tower" } }$$

where the tower is a landmark, located at a known position (x₀, y₀), and where the term “near” is assumed to have the following specific meaning: (x, y) is near (x₀, y₀) means that (x, y) ∈ T₅ where T₅ is a disk of radius 5 m, centered at (x₀, y₀). If z = (x, y) is the actual measurement of the target’s position, then ζ is equivalent to the formulaz ∈ T₅. Since z is just one possible draw from Z _k , we can say that ζ – or, equivalently, T₅ – is actually a constraint on the underlying measurement process: Z _k ∈ T₅.

Because the word “near” is rather vague, we could just as well say that z ∈ T₅ is the best choice, with confidence w₅ = 0. 7; that z ∈ T₄ is the next best choice, with confidence w₄ = 0. 2; and that z ∈ T₆ is the least best, with confidence w₆ = 0. 1. Let $\Theta$ be the random subset of $\mathfrak{Z}$ defined by $\Pr (\Theta = T_{i}) = w_{i}$ for i = 4, 5, 6. In this case, ζ is equivalent to the random constraint

$$\displaystyle{ \mathbf{Z}_{k} \in \Theta. }$$

The probability

$$\displaystyle\begin{array}{rcl} \rho _{k}(\Theta \vert \mathbf{x}_{k})& =& \Pr (\eta (\mathbf{x}_{k}) + \mathbf{V}_{k} \in \Theta ) {}\\ & =& \Pr (\mathbf{Z}_{k} \in \Theta \vert \mathbf{X}_{k} = \mathbf{x}_{k}) {}\\ \end{array}$$

is called a generalized likelihood function (GLF). GLFs can be constructed for more complex natural-language statements, for inference rules, and more. Using their GLF representations, such “nontraditional measurements” can be processed using single- and multi-object recursive Bayes filters and their approximations. As a consequence, it can be shown that fuzzy logic, the D-S theory, and rule-based inference can be subsumed within a single Bayesian-probabilistic paradigm.

Summary and Future Directions

In the engineering world, the theory of random sets has been associated primarily with certain specialized image-processing applications, such as morphological filters and tomographic imaging. It has more recently found application in fields such as multitarget tracking and in expert-systems theory. All of these fields of application remain areas of active research.

Cross-References

• Estimation, Survey on

• Extended Kalman Filters

• Nonlinear Filters

Recommended Reading

Molchanov (2005) provides a definitive exposition of the general theory of random sets. Two excellent references for stochastic geometry are Stoyan et al. (1995) and Barndorff-Nielsen and van Lieshout (1999). The books by Kingman (1993) and Daley and Vere-Jones (1998) are good introductions to point process theory. The application of point process theory and stochastic geometry to image processing is addressed in, respectively, Snyder and Miller (1991) and Stoyan et al. (1995). The application of RFSs to multitarget estimation is addressed in the tutorials Mahler (2004, 2013) and the book Mahler (2007). Introductions to the application of random sets to expert systems can be found in Kruse et al. (1991) and Mahler (2007), Chaps. 3–6.