1. INTRODUCTION

If we are imaging an ensemble of objects, then it is natural to describe this ensemble as a random field on some support region in ${\mathbb{R}}^q$. (Other terms used for this concept are random function and stochastic process.) For example, in medical imaging, the object is described by a function of three variables that varies unpredictably from one patient to the next. In x-ray computed tomography, this function gives the x-ray attenuation at each point, while in position emission tomography and single photon emission computed tomography imaging, this function gives the activity distribution inside the body. While there are anatomical structures that are common to all patients, there are also unpredictable variations in these structures throughout the patient population. As another example, for the x-ray imaging of luggage, there is an almost infinite variety of objects that can be packed in a bag and also infinite variation in the arrangement of these objects in the bag. The corresponding attenuation maps then vary in an unpredictable way from one bag to the next. In each of these examples, any given attenuation map or activity distribution can be regarded as a realization of a random field defined on a region of space. In medical imaging, this support region can be thought of as a cylinder large enough to contain all of the patients being imaged. In a luggage scanner, the support set may be a box shape large enough to contain all of the bags being scanned. In all imaging situations, there will be unpredictable variations in the function being imaged. If there were no such variations, then the object function would be known already and there would be no need to create an image. Therefore, we will consider the object functions being imaged to be realizations of a random field defined on a support set.

Describing the statistical properties of a random field can be difficult since we are dealing with essentially a random vector in an infinite-dimensional space. The concept of a probability density function (PDF), which is used to describe the statistics of continuous finite-dimensional random vectors, is not defined in general for a random field. For finite-dimensional random vectors, an equivalent description of the statistics is given by the characteristic function, which is the Fourier transform of the PDF. Indeed, many concepts in probability theory and multivariate statistics are easier to understand if we use the characteristic function as our description of the random vector. Fortunately, the concept of the characteristic function generalizes easily to random fields and is referred to as the characteristic functional in this context.

After a brief discussion of probability models and random variables in Section 2 to establish notation, we introduce the idea of a random field in Section 3. In Section 4, we define the characteristic functional for a random field and discuss some of its properties. Sections 5 and 6 introduce generalized random fields and their characteristic functionals; these ideas are needed to describe point processes, for example. In Section 7, we show how certain mathematical operations with random fields translate to relations among the corresponding characteristic functionals. Section 8 consists of a myriad of examples of random fields and their characteristic functionals that are relevant to imaging applications. We then show how characteristic functionals are propagated through noisy imaging systems in Section 9. This discussion includes both binned and list-mode imaging systems. Finally, in Section 10, we show how characteristic functionals and image data can be used to estimate population parameters, classify populations of objects, and compute a Fisher information matrix for the estimation of population parameters. In all of these applications, an analytic expression for the characteristic functional is essential to carrying out the task. Fortunately, in Section 8, we have a long list of such analytic expressions. We hope that these results will convince readers that modeling object populations as random fields and describing their statistical properties with characteristic functionals is both useful and important for the analysis of imaging systems and the data generated by them.

Since this paper is intended as a tutorial on the use of characteristic functionals in image science, most of the results have been derived elsewhere. Exceptions to this rule are Subsection 8.L on dynamically evolving random fields and Sections 10 and 11 on using characteristic functionals to generate mathematical observers and figures of merit for imaging systems.

2. PROBABILITY MODELS AND RANDOM VARIABLES

To define the characteristic functional of a random field, we must first discuss the definition of a random field. This definition in turn necessitates a brief discussion of probability spaces and random variables.

To define a probability space [1–3] we start with a set $\mathrm{\Omega }$, sometimes called the sample space, the space of outcomes, or the set of elementary events. This set may be finite. For example, if $N$ photons strike a detector and $n$ of them are detected, then $\mathrm{\Omega }=\{0,1,2,\dots ,N\}$ because these are the possible outcomes from this experiment. The outcome space may be infinite but discrete. An example of this situation is operating a photon-counting detector being illuminated by a source for a fixed amount of time. In this case, it is standard practice to take $\mathrm{\Omega }=\{0,1,2,\dots ,\}$. If an imaging system performs $M$ real-valued measurements of an object, then $\mathrm{\Omega }={\mathbb{R}}^M$. In the theory of random fields, we have to allow for the possibility that $\mathrm{\Omega }$ is an infinite-dimensional space. The set of all real-valued functions of $q$ variables is an example of such a space. In general, we will use the symbol $\omega$ to represent a point in $\mathrm{\Omega }$. Thus, $\omega$ may represent an integer, a real number, a finite-dimensional vector, a function, or some other more abstract way of representing outcomes.

The next step in defining a probability model is to define the collection $\mathcal{F}$ of subsets of $\mathrm{\Omega }$ that will be assigned a probability. These subsets are usually called events. When $\mathrm{\Omega }$ is a discrete set, $\mathcal{F}$ is usually simply the collection of all possible subsets of $\mathrm{\Omega }$. Thus, in this case, we will assign a probability to any subset of outcomes. When $\mathrm{\Omega }$ is not a discrete set, then $\mathcal{F}$ normally does not consist of all possible subsets of $\mathrm{\Omega }$. In general, all that we require of the collection $\mathcal{F}$ is three conditions. The first condition is that $\mathrm{\Omega }$ itself is one of the subsets in the collection. The second condition is that, for any subset $A$ in the collection, the complement of $A$, which is the subset of points in $\mathrm{\Omega }$ that are not in $A$, is also in the collection. The third condition is that, if the subsets $A_n$ are in the collection for $n=1,2,\dots$, then the subset consisting of the union of these subsets,

The final ingredient for a probability model is a probability measure $Pr(A)$ that assigns a probability to each event $A$ in $\mathcal{F}$. We always have $0\le Pr(A)\le 1$. To state the only other condition on the probability measure, we begin with a sequence of subsets $A_n$ that are in $\mathcal{F}$ for $n=1,2,\dots$. We assume that the events are mutually disjoint, i.e., $A_n\cap A_m$ is empty for $n\ne m$. Then, if $A$ is the union event as just described, the probability measure must satisfy

A probability model is specified by the triple $(\mathrm{\Omega },\mathcal{F},Pr)$ consisting of a sample space, a $\sigma$-algebra of subsets of this space, and a probability measure on this collection of subsets. The concept of a probability model forms the foundation of probability theory, which in turn is essential for any discussion of random fields and characteristic functionals. For those not familiar with $\sigma$-algebras and probability measures, we will discuss a few examples.

In the first example, we will consider the photon-counting system where $\mathrm{\Omega }=\{0,1,2,\dots ,\}$ and an event $E$ will be any subset of $\mathrm{\Omega }$. Since the one-element set $\{n\}$ is an event, it has an associated probability $P_n=Pr(\{n\})$. For any event, we then have

In the second example, we will take $\mathrm{\Omega }$ to be the set $\mathbb{R}$ of all real numbers. If the outcome of an experiment is a measurement of a continuous quantity, then this is the appropriate outcome space. If the real-valued function $pr(x)$ on $\mathbb{R}$ satisfies $pr(x)\ge 0$ for all $x$ and

As a final example of a situation where we need to restrict the subsets to which we want to assign a probability, consider $\mathrm{\Omega }$ to be the space of all real-valued functions on $\mathbb{R}$. We may think of an experiment where the outcome is an oscilloscope trace. In a typical measurement, we want to assign a probability to an event of the form $E=\{f\text{:}{a}_n\le f(x_n)\le b_n\mathrm{for}n=1,2,\dots ,N\}$. This event corresponds to a finite number of measurements of the function $f(x)$ at the points $x_n$, with the results falling into the given intervals. We can then take $\mathcal{F}$ to be the $\sigma$-algebra generated by these subsets for our probability model. In this case, however, the subset $C=\{f\text{:}a\le f(x)\le b\mathrm{for}c<x<d\}$ is not a member of $\mathcal{F}$ and cannot be assigned a probability since it requires an uncountably infinite number of measurements to verify that a given function is in $C$. We are usually not bothered by this, since we cannot make an uncountably infinite number of measurements anyway. However, this example does show that the natural $\sigma$-algebra for a probability model may be very far from containing all of the subsets of $\mathrm{\Omega }$. In any case, we will only use the general concept of a probability model as just outlined to define what a random field is and what the characteristic functional of a random field is. Once that is accomplished, we will not have to worry very much about the details of the probability model used to define any given random field.

A real random variable is a real-valued function $X=f(\omega )$ on the sample space $\mathrm{\Omega }$ that satisfies the following measurability condition. For any real number $a$, the subset of $\mathrm{\Omega }$ defined by $E_X(a)=\{\omega \text{:}f(\omega )\le a\}$ is in the collection $\mathcal{F}$ of events, and may thus be assigned a probability. A complex random variable $Z=f(\omega )$ is defined by insisting that $\mathrm{Re}Z$ and $\mathrm{Im}Z$ are both real random variables. We can also have real random vectors $\mathbf{X}=\mathbf{f}(\omega )$ of any dimension by insisting that each component $X_n$ of the vector is a real random variable. This definition is also valid for complex random vectors.

For a real random variable $X$, the PDF $p{r}_X(x)$ is defined by the condition

3. RANDOM FIELDS

Let $S$ be some $q$-dimensional support set for all functions in this section. This may be a subset of ${\mathbb{R}}^q$ or a $q$-dimensional manifold such as a sphere. The support set $S$ may even be a discrete set of points in ${\mathbb{R}}^q$, such as a regular grid in this space. A random field [1–3] is a function $f(\mathbf{r},\omega )$ of points $\mathbf{r}$ in $S$ and $\omega$ in $\mathrm{\Omega }$ such that $X_{\mathbf{r}}=f(\mathbf{r},\omega )$ is a random variable for each $\mathbf{r}$. We may also describe a random field as a collection of random variables $X_{\mathbf{r}}$ for each $\mathbf{r}$ in $S$. For a fixed sample point $\omega$, the function $f_{\omega }(\mathbf{r})=f(\mathbf{r},\omega )$ is called a realization of the random field. Sometimes further constraints are placed on the random field, such as assuming that all of the realizations are continuous functions of $\mathbf{r}$. In general, however, we cannot expect the realizations to be continuous. We will use the common notation $f(\mathbf{r})=X_{\mathbf{r}}$ for a random field. This notation has the potential for ambiguity, but it will be clear from the context whether we are referring to a random field or an ordinary function of $\mathbf{r}$.

Given an ordered set of points $\mathbf{R}=[{\mathbf{r}}_1,\dots ,{\mathbf{r}}_N]$ in $S$, the vector $\mathbf{f}=[f({\mathbf{r}}_1),\dots ,f({\mathbf{r}}_N)]$ is a random vector with a corresponding multivariate PDF $p{r}_{\mathbf{R}}(x_1,\dots ,x_N)$ on ${\mathbb{R}}^N$ or ${\mathbb{C}}^N$. The Kolmogorov extension theorem [3] provides two sets of consistency conditions that this collection of PDFs has to satisfy. The two sets of conditions arise from permutations of the points in $\mathbf{R}$ and marginalization over any of the variables $x_n$. Conversely, if the conditions in the theorem are satisfied, then there is a random field corresponding to the collection of PDFs. This theorem is often used to derive the existence of a random field from a suitable collection of PDFs.

For any realization of a random field, we will need to consider the inner product defined by

If we have two functions ${\varphi }_1(\mathbf{r})$ and ${\varphi }_2(\mathbf{r})$ that satisfy this condition and $\varphi (\mathbf{r})=c_1{\varphi }_1(\mathbf{r})+c_2{\varphi }_2(\mathbf{r})$, then $(\varphi ,f)$ is also a random variable. Thus, the set of functions $\varphi (\mathbf{r})$ such that $(\varphi ,f)$ is a well-defined random variable forms a vector space. The details of the problem of determining which functions are in this space depend on the random field itself, and we will not be discussing that issue in this work. This issue is discussed in detail elsewhere and involves a considerable amount of mathematics [4,5]. In all of the equations that follow, we will simply assume that $\varphi (\mathbf{r})$ is a function in this space.

4. CHARACTERISTIC FUNCTIONALS OF RANDOM FIELDS

Another way to think of a random field is as a map that assigns the function $f_{\omega }(\mathbf{r})$ to each $\omega$ in the sample space. Since the set of functions on $S$ forms a vector space, this is analogous to a random vector in an infinite-dimensional space. However, there is no easy way to generalize the concept of a PDF for a finite-dimensional random vector $\mathbf{X}$ to a PDF for an infinite-dimensional random vector. The main reason for this is that there is no way to easily generalize an integral of the form

The characteristic functional for a random field $f(\mathbf{r})$ is defined by

Suppose that $f(\mathbf{r})$ is a real random field. Since $X_{\mathbf{r}}=f(\mathbf{r})$ is a random variable, we may use its PDF to compute a mean value: $\overline{f}(\mathbf{r})=⟨f(\mathbf{r})⟩$. Similarly, $[f(\mathbf{r}),f({\mathbf{r}}^{\prime })]$ is a random vector and its PDF may be used to compute the correlation function $R(\mathbf{r},{\mathbf{r}}^{\prime })=⟨f(\mathbf{r})f({\mathbf{r}}^{\prime })⟩$. The covariance function for the random field is then given by $K(\mathbf{r},{\mathbf{r}}^{\prime })=R(\mathbf{r},{\mathbf{r}}^{\prime })-\overline{f}(\mathbf{r})\overline{f}({\mathbf{r}}^{\prime })$. For a real random field, we have ${\mathrm{\Psi }}_f(\varphi )=⟨\exp [-2\pi i(\varphi ,f)]⟩$ when $\varphi (\mathbf{r})$ is a real-valued function, which will be the assumption made in this paragraph. For a parameter $\tau$, we then have

For complex random fields, the correlation function is usually defined by $R(\mathbf{r},{\mathbf{r}}^{\prime })=⟨f^{*}(\mathbf{r})f({\mathbf{r}}^{\prime })⟩$, with the corresponding covariance function $K(\mathbf{r},{\mathbf{r}}^{\prime })=R(\mathbf{r},{\mathbf{r}}^{\prime })-{\overline{f}}^{*}(\mathbf{r})\overline{f}({\mathbf{r}}^{\prime })$. If we introduce the complex parameter $\tau ={\tau }_R+i{\tau }_I$, then this correlation function is given by

Note that $\tilde{\mathcal{R}}$ is, in general, not a Hermitian operator.

5. GENERALIZED RANDOM FIELDS

Some physically important random fields, such as Poisson point processes, are actually generalized random fields. For a generalized random field we do not insist that $X_{\mathbf{r}}=f(\mathbf{r})$ is a random variable since, for any given realization, $f_{\omega }(\mathbf{r})$ may not be well defined as an ordinary function. Instead, we restrict $\varphi (\mathbf{r})$ to be a test function and define

6. CHARACTERISTIC FUNCTIONALS OF GENERALIZED RANDOM FIELDS

Just as with ordinary random fields, extending the concept of a PDF to a generalized random field is difficult. We would need the PDF to be defined on the space of distributions on $S$, which is an infinite-dimensional space that includes, for example, the space of locally integrable functions on $S$. However, the characteristic functional for a generalized random field is easily defined by ${\mathrm{\Psi }}_f(\varphi )=⟨\exp [-2\pi i\mathrm{Re}(\varphi ,f)]⟩$, where the argument $\varphi (\mathbf{r})$ of the functional is taken to be a test function. The expectation in this expression is computed using the PDF $p{r}_{\varphi }(x)$ for the random variable $X_{\varphi }=(\varphi ,f)$. All of the statistical information about the generalized random field is contained in this functional.

A generalized random field is real-valued if $f(\varphi )$ is real whenever $\varphi (\mathbf{r})$ is a real-valued test function. In this case, we may define the mean of the generalized random field as a distribution by using $\overline{f}(\varphi )=⟨f(\varphi )⟩$. Since, for any given test function $\varphi (\mathbf{r})$, $f(\varphi )$ is a random variable, the expectation in this definition may be computed by using the PDF for that random variable. We may also write this definition as $(\varphi ,\overline{f})=⟨(\varphi ,f)⟩$. Similarly, for a pair of test functions $[\varphi ,{\varphi }^{\prime }]$, the random vector $[f(\varphi )f({\varphi }^{\prime })]$ has a PDF that can be used to compute the correlation operator via $⟨f(\varphi )f({\varphi }^{\prime })⟩=(\varphi ,\mathcal{R}{\varphi }^{\prime })$. Notice that we do not attempt to define a correlation function, which in many cases would be a generalized function itself. The covariance operator is then defined by $(\varphi ,\mathcal{R}{\varphi }^{\prime })-\overline{f}(\varphi )\overline{f}({\varphi }^{\prime })=(\varphi ,\mathcal{K}{\varphi }^{\prime })$.

For complex generalized random fields, we have the Hermitian correlation operator $\mathcal{R}$ defined by $⟨f^{*}(\varphi )f({\varphi }^{\prime })⟩=(\varphi ,\mathcal{R}{\varphi }^{\prime })$, and the corresponding covariance operator $\mathcal{K}$ given by $(\varphi ,\mathcal{R}{\varphi }^{\prime })-{\overline{f}}^{*}(\varphi )\overline{f}({\varphi }^{\prime })=(\varphi ,\mathcal{K}{\varphi }^{\prime })$. As with ordinary random fields, to complete the description of the second-order statistics of a complex generalized random field, we need the operator $\tilde{\mathcal{R}}$ defined by $⟨f(\varphi )f({\varphi }^{\prime })⟩=(\varphi ,\tilde{\mathcal{R}}{\varphi }^{\prime })$. The corresponding covariance operator $\tilde{\mathcal{K}}$ is then given by $(\varphi ,\tilde{\mathcal{R}}{\varphi }^{\prime })-\overline{f}(\varphi )\overline{f}({\varphi }^{\prime })=(\varphi ,\tilde{\mathcal{K}}{\varphi }^{\prime })$. The relations between the characteristic functional and these first- and second-order statistics are the same as those for ordinary random fields given previously.

7. OPERATIONS WITH RANDOM FIELDS

Various mathematical operations with random fields or generalized random fields give rise to relations between characteristic functionals. If $f_1(\mathbf{r},{\omega }_1)$ and $f_2(\mathbf{r},{\omega }_2)$ are two random fields on $S$ with corresponding probability models $({\mathrm{\Omega }}_1,{\mathcal{F}}_1,P{r}_1)$ and $({\mathrm{\Omega }}_2,{\mathcal{F}}_2,P{r}_2)$, then there is a probability model on $\mathrm{\Omega }={\mathrm{\Omega }}_1\times {\mathrm{\Omega }}_2$ with a probability measure that satisfies $Pr(A_1\times A_2)=P{r}_1(A_1)P{r}_2(A_2)$. We can then define a random field on $S$ by $f(\mathbf{r},({\omega }_1,{\omega }_2))=f_1(\mathbf{r},{\omega }_1)+f_2(\mathbf{r},{\omega }_2)$. We usually write $f(\mathbf{r})=f_1(\mathbf{r})+f_2(\mathbf{r})$ in this case and say that the two random fields $f_1(\mathbf{r})$ and $f_2(\mathbf{r})$ are statistically independent. The characteristic functionals for these random fields are related by the equation ${\mathrm{\Psi }}_f(\varphi )={\mathrm{\Psi }}_{f_1}(\varphi ){\mathrm{\Psi }}_{f_2}(\varphi )$. On the other hand, we may also take this equation as defining when two random fields are statistically independent when they share a probability model $(\mathrm{\Omega },\mathcal{F},Pr)$. The obvious condition for this equation to be valid is that it must be permissible to use $\varphi (\mathbf{r})$ in the argument of all of the characteristic functionals involved.

In contrast to the technical issues for summing two random fields, multiplying a random field by a non-random function $\alpha (\mathbf{r})$ is straightforward and leads to the relation ${\mathrm{\Psi }}_{\alpha f}(\varphi )={\mathrm{\Psi }}_f({\alpha }^{*}\varphi )$. The one constraint here is that the function ${\alpha }^{*}(\mathbf{r})\varphi (\mathbf{r})$ must be in the space of functions that can be used in the argument of the functional ${\mathrm{\Psi }}_f$. For example, if $f(\mathbf{r})$ is a generalized random field and ${\alpha }^{*}(\mathbf{r})$ is infinitely differentiable, then $\alpha (\mathbf{r})f(\mathbf{r})$ is also a generalized random field. If $\varphi (\mathbf{r})$ is now a test function, then ${\alpha }^{*}(\mathbf{r})\varphi (\mathbf{r})$ is also a test function and the equality is valid.

Now suppose that $\mathcal{H}$ is a linear integral operator given by

Now suppose that we have a set of random fields $f_n(\mathbf{r},\omega )$ on $S$ such that they are all defined with the same probability model $(\mathrm{\Omega },\mathcal{F},Pr)$ for $n=1,\dots ,N$. Also assume that we have a set of probabilities $P{r}_n$. We can create a new probability model $(\tilde{\mathrm{\Omega }},\tilde{\mathcal{F}},\tilde{P}r)$ such that the samples are pairs $\tilde{\omega }=(\omega ,n)$ and $\tilde{P}r(E,n)=Pr(E)P{r}_n$. The mixture random field for this situation is defined by $f(\mathbf{r},\tilde{\omega })=f_n(\mathbf{r},\omega )$. We can think of the realizations of this random field as being equal to $f_n(\mathbf{r},\omega )$ with probability $P{r}_n$. We then have that the characteristic functional for the mixture random field is given by the convex combination

8. EXAMPLES

We now look at some examples of characteristic functionals that are relevant for imaging. In most cases, we do not present a derivation of the characteristic functionals since they are readily available elsewhere.

A. Real Gaussian

A random field on $S$ is Gaussian or normal if its characteristic functional has the form ${\mathrm{\Psi }}_f(\varphi )=\exp [-2\pi i(\varphi ,\overline{f})]\exp [-2{\pi }^2(\varphi ,\mathcal{K}\varphi )]$ [1 (page 410),2]. In this equation, $\overline{f}(\mathbf{r})$ is the mean of the random field and $\mathcal{K}$ is the covariance operator as described above. If the mean is zero and the covariance operator is a multiple of the identity operator, then we have a zero-mean white noise Gaussian random field. This noise field is often used in the description of Brownian motion. Note that a white noise Gaussian random field is actually a generalized random field since the kernel of the covariance operator is given by $K(\mathbf{r},{\mathbf{r}}^{\prime })=\delta (\mathbf{r}-{\mathbf{r}}^{\prime })$, which is a generalized function of two variables. Due to the central limit theorem, Gaussian random fields often occur in applications as an approximation for a random field that is a sum of many independent random fields.

Suppose that we have a random field $p(\mathbf{r})$ that satisfies $p(\mathbf{r})>0$ for all realizations and all $\mathbf{r}$ in $S$. If the random field $f(\mathbf{r})=\ln p(\mathbf{r})$ is a Gaussian random field, then we say that $p(\mathbf{r})$ is a log-normal random field [1 (page 1462),7]. There is no analytic form for the characteristic functional of a log-normal random field. However, in this case, the functional

B. Circular Gaussian

A complex random field on $S$ is a circular Gaussian if its characteristic functional is given by ${\mathrm{\Psi }}_f(\varphi )=\exp [-{\pi }^2(\varphi ,\mathcal{K}\varphi )]$, where the covariance operator $\mathcal{K}$ is a Hermitian operator. It is often assumed that the electric field in partially coherent light is a circular Gaussian random field. We can also have complex random fields that are Gaussian but not circular Gaussian. The characteristic functional in this case involves both covariance functions for the complex random field.

C. Poisson Random Field

A random point field is a generalized random field that satisfies

For a Poisson random point field, we have the Poisson distribution for $N$:

If we define the function $\tilde{\varphi }(\mathbf{r})$ by $2\pi i\tilde{\varphi }(\mathbf{r})=1-\exp [2\pi i\varphi (\mathbf{r})]$, then the characteristic functional of a Poisson random field is given by ${\mathrm{\Psi }}_f(\varphi )=\exp [-2\pi i(\tilde{\varphi },\overline{f})]$. Poisson random fields are often used to describe the distribution of photon locations on a photon-counting detector. In this case the mean of the field is called the photon fluence, and it determines the statistics of the field completely. Poisson random fields are also used to describe photon-emitting objects that are being imaged.

D. Random Fields Related to Poisson Random Fields

Suppose that the mean of a Poisson random field $f(\mathbf{r})$ is itself a random field. Then we have immediately that the characteristic functional of $f(\mathbf{r})$ is given by ${\mathrm{\Psi }}_f(\varphi )={\mathrm{\Psi }}_{\overline{f}}(\tilde{\varphi })$. In this case, we say that $f(\mathbf{r})$ is a doubly stochastic Poisson random field [1 (page 659),8]. If we are imaging an ensemble of objects with a photon-counting detector, then a doubly stochastic random field is the appropriate model of the statistics of the emitted or detected photon locations.

If we convolve a Poisson random field with a continuous function $h(\mathbf{r})$, then we have a filtered Poisson random field [1 (page 662)]:

We can have a doubly stochastic filtered Poisson random field also. In this case, the characteristic functional is given by ${\mathrm{\Psi }}_f(\varphi )={\mathrm{\Psi }}_{\overline{f}}({\tilde{\varphi }}_h)$. Other generalizations are also possible. For example, the filter could have the more general form $h(\mathbf{r},{\mathbf{r}}_n)$, i.e., a space-variant filter. It is straightforward to work out the characteristic functional in these cases also.

E. Partitioned Random Fields

We will say that the random field is partitioned if we can write the support set $S$ as a disjoint union

F. Square of a Real Gaussian Random Field

First we consider a finite-dimensional, zero-mean Gaussian random vector $\mathbf{f}$ with covariance matrix $\mathbf{K}$. Define a random vector $\mathbf{I}$ by $I_n=f_n^2$. If we use $\mathbf{D}(\mathit{\xi })$ to denote the diagonal matrix with the components ${\xi }_n$ along the main diagonal, then we may write the characteristic function of $\mathbf{I}$ as

Now consider a zero-mean Gaussian random field $f(\mathbf{r})$ with covariance operator $\mathcal{K}$. We define the corresponding intensity field $I(\mathbf{r})$ by $I(\mathbf{r})=f^2(\mathbf{r})$. This is called the intensity field in analogy with optical applications where $f(\mathbf{r})$ would be the (real) electric field from a finite-bandwidth source, and the detector responds to the intensity field. If we define the diagonal operator ${\mathcal{D}}_{\varphi }$ for $\varphi (\mathbf{r})$ acting on a function $\beta (\mathbf{r})$ by the equation ${\mathcal{D}}_{\varphi }\beta (\mathbf{r})=\varphi (\mathbf{r})\beta (\mathbf{r})$, then the obvious generalization of the finite-dimensional formula is

This determinant is well defined if $\mathcal{K}$ is a trace-class operator since, in this case, $2\pi i\mathcal{K}{\mathcal{D}}_{\varphi }$ is also a trace-class operator and the determinant can be defined as an infinite product of eigenvalues. However, just because this functional is well defined does not mean it is necessarily the characteristic functional of the random field $I(\mathbf{r})$.

To see why we might believe that the given expression for ${\mathrm{\Psi }}_I(\varphi )$ is correct, we consider the following special cases:

After substituting this $\beta (\mathbf{r})$ into the eigenvlaue equations, we arrive at

G. Square Magnitude of a Circular Gaussian

Now suppose that $f(\mathbf{r})$ is a circular Gaussian complex random field and $I(\mathbf{r})={|f(\mathbf{r})|}^2$. Using the results of the previous subsection, we have

For the $N=1$ case, the inverse 1D Fourier transform integral can be computed analytically to give a chi-square PDF for the random variable $I(\mathbf{r})$ at a fixed $\mathbf{r}$. The $N=2$ case can also be computed analytically in terms of the $I_0$ Bessel function [1 (page 1258),9].

H. Complex-Amplitude Poisson Random Field

An example of a complex-amplitude Poisson random field has the form

We define the function $m(\mathbf{r})=\overline{N}pr(\mathbf{r})$ and

If the propagation of the field through space or an imaging system is defined by an operator $\mathcal{P}$, then ${\mathrm{\Psi }}_{\mathcal{P}f}(\varphi )={\mathrm{\Psi }}_f({\mathcal{P}}^{†}\varphi )$. If the propagation is implemented by convolving with a point spread function $h(\mathbf{r})$ we define, as before, ${\varphi }_h=h*\varphi$ and have ${\mathrm{\Psi }}_f(\varphi )=\exp [-2\pi i({\tilde{\varphi }}_h,m)]$ for a fixed collection of scatterers and ${\mathrm{\Psi }}_f(\varphi )={\mathrm{\Psi }}_m({\tilde{\varphi }}_h)$ for an ensemble of objects.

I. Innovation Random Field

Sparsity of a random field is a property often assumed in compressed sensing that allows better object reconstructions. One model for a sparse random field $f(\mathbf{r})$ assumes that $f=\mathcal{L}w$ where $w(\mathbf{r})$ is a stationary white noise field, sometimes called an innovation field. We then have ${\mathrm{\Psi }}_f(\varphi )={\mathrm{\Psi }}_w({\mathcal{L}}^{†}\varphi )$. To describe the statistical properties of an innovation field, we introduce the translation operators ${\mathcal{T}}_{\mathbf{s}}\varphi (\mathbf{r})=\varphi (\mathbf{r}-\mathbf{s})$ and the random variables $({\mathcal{T}}_{\mathbf{s}}\varphi ,w)$. The stationarity assumption is that for any $N$ and functions $\{{\varphi }_1(\mathbf{r}),\dots ,{\varphi }_N(\mathbf{r})\}$, the $N$-dimensional PDF for the $N$-dimensional random vector with components $({\mathcal{T}}_{\mathbf{s}}{\varphi }_n,w)$ is independent of the translation vector $\mathbf{s}$. For a stationary random field, we necessarily have $S={\mathbb{R}}^q$. The white noise assumption is that whenever $\varphi (\mathbf{r}){\varphi }^{\prime }(\mathbf{r})\equiv 0$ for all $\mathbf{r}$ in $S$, the random variables $X_{\varphi }$ and $X_{{\varphi }^{\prime }}$ are statistically independent. In this case it can be shown that [4,5]

Therefore any random field that is the result of a linear operator acting on an innovation field must have a characteristic functional of the form ${\mathrm{\Psi }}_f(\varphi )={\mathrm{\Psi }}_w({\mathcal{L}}^{†}\varphi )$, where the functional ${\mathrm{\Psi }}_w$ is as just given. If the PDFs for the random variables $({\mathcal{L}}^{†}{\varphi }_n,w)=({\varphi }_n,f)$ for some basis sequence ${\varphi }_1(\mathbf{r}),{\varphi }_2(\mathbf{r}),\dots$ are highly kurtotic with long tails, then this sequence can be thought of as a sparsity basis for the random field $f(\mathbf{r})$. This requirement on these PDFs can then be related to properties of the Levy measure for the innovation process.

J. Plane-Wave Random Field

Suppose that the random field $f(\mathbf{r})$ is a superposition of a random number $N$ of plane waves with random amplitudes, phases, and spatial frequencies:

This would be the case, for example, if $f(\mathbf{r})$ was the inverse Fourier transform of a point process with complex amplitudes in frequency space. For a given $N$, we will assume that the amplitudes and spatial frequencies are independent of each other and are i.i.d. with PDFs $p{r}_A(A_n)$ and $p{r}_{\mathbf{k}}({\mathbf{k}}_n)$, respectively. If $\mathrm{\Phi }(\mathbf{k})$ is the Fourier transform of $\varphi (\mathbf{r})$ and $N$ is Poisson distributed with mean $\overline{N}$, then the characteristic functional of this random field is given by ${\mathrm{\Psi }}_f(\varphi )=\exp \{\overline{N}{⟨{\psi }_A[\mathrm{\Phi }(\mathbf{k})]⟩}_{\mathbf{k}}-\overline{N}\}$. If we assume that the spatial frequencies follow a circular Gaussian distribution, with variance ${\sigma }^2$, then

The single-point characteristic function for the random variable $X=f({\mathbf{r}}_0)$ is derived by setting $\varphi (\mathbf{r})=\xi \delta (\mathbf{r}-{\mathbf{r}}_0)$, which results in

K. Infinite-Series Random Field

Suppose that the function $F(\mathbf{n},\omega )$ describes a random field on a discrete set of integer vectors $\mathbf{n}$. A Markov random field [1 (page 428)] is an example of this situation. If the functions ${\mu }_{\mathbf{n}}(\mathbf{r})$ form a set of functions on $S$ indexed by these integer vectors, then we may define a random field $f(\mathbf{r},\omega )$ on $S$ via

If we define a vector $\mathit{\varphi }$ on the index set via ${[\mathit{\varphi }]}_{\mathbf{n}}=(\varphi ,{\mu }_{\mathbf{n}})$, then the characteristic functional for the random field $f(\mathbf{r})$ is given by ${\mathrm{\Psi }}_f(\varphi )={\mathrm{\Psi }}_F(\mathit{\varphi })$. This type of model can be used for compressed-sensing applications where the functions ${\mu }_{\mathbf{n}}(\mathbf{r})$ could form a wavelet basis or some similar basis or frame.

L. Dynamically Evolving Random Field

Suppose that we define a bounded spatial integral operator $\mathcal{A}$ on spatiotemporal functions via