Before applying Bayes’ theorem, a significant effort must be spent on modeling the dynamics of the processes which describe the problem. In object tracking this cor-responds to developing object dynamical models and sensor measurement models.
Object motion is usually described by characteristics such as position, velocity, acceleration and other kinematic components, like angular velocity and its rate of change. These kinematic components constitute the object state. Using simple definitions like the rate of change of position is equal to velocity, it is possible to relate one component of object state to the other. Such models can be elegantly derived using the state space approach in either discrete or continuous time. The ideas and concepts in this book all refer to discrete-time. For example, in the case of the single-object tracking problem, we may consider the state Sk to represent only the position xkof a single object. In the state-space approach, the components of the state are grouped as a single vector called the state vector xk. An example state vector of the object is
xk= [position, velocity, acceleration]Tk.
The dimension of the object state is equal to the number of components of the vector and in this case it is 3. Another example of the object state vector is
xk = [position, velocity, acceleration, angular velocity, angular accelaration]Tk, where the dimension of the state vector is 5. Thus the object state is a finite-dimensional vector of those kinematic components of the object motion that explain its motion.
Many physical processes, such as the motion of an object, which is subject to random disturbances, and whose state can be represented by a finite-dimensional
1.3 Recursive Bayesian solution for object tracking 17 vector, can be modeled using a vector difference equation. Suppose Sk is an n-dimensional state vector at time k and vk is the m-dimensional vector of random disturbances (m ≤ n). We can write a difference equation
Sk= g(Sk−1, vk), (1.6)
where g is a real, in general non-linear, n-vector function, which, we suppose, is a twice continuously differentiable function of its arguments. The disturbance vkis often called the random noise input to the system.
1.3.2 The generalized sensor measurement equation
Sensors are devices which observe aspects of the object state and the measure-ments made by sensors are used to make inferences about the entire object state.
Measurement or sensor models enable one to determine the likelihood function p(yk|Sk, yk−1). Hence one of the primary properties of the sensor model is that it must be a function of the object state. Moreover, as presented earlier, the Bayes’
recursion is based on another sensor measurement characteristic, i.e., the mea-surements originated from the object state at a particular time are conditionally independent of measurements from other times. Sensor models of the form
yk = l(Sk, wk), (1.7)
where wkis a white noise (measurement error), satisfy the conditions needed for a recursive Bayesian solution.
1.3.3 Generalized object state prediction and conditional densities In the case where the state and measurement equations do satisfy (1.6) and (1.7), and under certain regularity conditions the solution (1.5) can be further expanded to obtain the object state prediction and conditional densities. The state prediction density is p(Sk|yk−1) at time k. It represents knowledge about the state Sk given all measurements up to time k− 1. The state conditional density, or posterior pdf of Sk, is p(Sk|yk), and represents the updated knowledge at time k after receiving the measurement yk. In complex tracking problems the state dynamics equation may not be captured in an equation of the form of (1.6), or it may not satisfy some regularity conditions. Similarly, the measurement equation in some track-ing problems may not be in the form of (1.7) with the regularity conditions met.
Even in these cases, the solution to the object tracking problem is derived from the recursive solutions (1.4) and (1.5). The derivation of the solution differs for each problem.
1.3.4 Generalized object state prediction and update The generalized object state transition density
Using the object dynamics model of (1.6) we derive the transition density p(Sk|Sk−1) needed in the Bayes’ solution (1.5). If the random input is absent from (1.6), we have an ordinary difference equation and we speak of Sk as its solution.
However, in the presence of the random input term, we refer to (1.6) as a stochastic difference equation (SDE). In using the SDE, it is the pdf of Skwhich is of interest.
If the probability law of vkis arbitrary, little can be said about the dynamic system (1.6). However, by modeling vk as a white noise sequence, we can make the fol-lowing observation. Since Sk−1, Sk depends only on vk, which is independent of Sk−2, . . . , S0, the solution of (1.6) is a Markov sequence. It is well known that the Markov sequence is defined using its transition density and an initial condition for its states. In the context of the SDE in (1.6), the transition density is p(Sk|Sk−1) and the initial condition is p(Sk−1).
Assume that the SDE in (1.6) can be solved for vk. In other words, for a given Sk−1, g(Sk−1, ·) has an inverse g−1 which is continuously differentiable. Then given Sk−1, the pdf of Sk, the transition density p(Sk|Sk−1) is
p(Sk|Sk−1) = pvk(g−1(Sk, Sk−1)) ∂g−1
∂Sk
. (1.8)
Here it is assumed that g−1(·) exists (i.e., it is assumed m = n). If m < n, the above direct derivation of the transition density is not possible. However, by partitioning Skinto
STk = [S(1)Tk S(2)Tk ],
and with the aid of the implicit function theorem (Kudryavtsev, 2001), (1.6) can be put in the form of S(1)k = g(1)(Sk−1, wk) and S(2)k = g(2)(Sk−1, S(1)k ) and the transition density can be derived as follows:
p(Sk|Sk−1) = p(S(1)k , S(2)k |Sk−1)
= p(S(2)k |S(1)k , Sk−1)p(S(1)k |Sk−1).
Since S(2)k depends only on S(1)k and Sk, the first term in the above equation is given by the following dirac delta function:
p(S(2)k |S(1)k , Sk−1) = δ(S(2)k − g(2)(Sk−1, S(1)k )).
1.3 Recursive Bayesian solution for object tracking 19 The second term depends on wk and is given by
p(S(1)k |Sk−1) = pvk(g(1)−1(Sk, Sk−1))
∂g(1)−1
∂S(1)k . Thus the transition density when m < n is given by
p(Sk|Sk−1) = δ(S(2)k − g(2)(Sk−1, S(1)k ))pvk(g(1)−1(Sk, Sk−1))
∂g(1)−1
∂S(1)k
. (1.9)
The generalized object state prediction density
Substituting the state transition density p(Sk|Sk−1) from either (1.8) or (1.9), we obtain the state prediction density
p(Sk|yk−1) =
Sk−1
p(Sk|Sk−1)p(Sk−1|yk−1)dSk−1,
where p(Sk−1|yk−1) is the prior pdf of the object state, or the state conditional density at time k− 1. It is assumed known at time k − 1, having been derived recursively, based on all measurements yk−1up to time k− 1.
1.3.5 Generalized object state filtering The likelihood function
The sensor model (1.7) satisfies the two measurement related assumptions required by (1.5). The whiteness property of wk is what gives the measurements, yk, the property of conditional independence and the fact that l(·) is a function of Sk
enforces the property that the measurements at time k depend on the object state at time k. To infer about Sk using yk, the function l(·) must be invertible. Under these assumptions the likelihood function in (1.5) can be derived by treating (1.7) as a transformation of the random variable wk:
p(yk|Sk) = pwk(l−1(yk, Sk)) ∂l−1
∂yk
. (1.10)
The normalization factor The normalization factor is given by:
p(yk|yk−1) =
Sk
p(yk|Sk)p(Sk|yk−1)dSk.
Desired conditional density p(Sk|yk)
Expand measurement history p(Sk|yk, yk−1)
Invoke Bayes’ rule
1
δp(yk|Sk, yk−1)p(Sk|yk−1) Simplifying
assumptions
Simplifying assumptions
Likelihood p(yk|Sk)
Prediction p(Sk|yk−1)
Normalization δ = p(yk|Sk)p(Sk|yk−1)dSk
Find approximate solution for p(Sk|yk) = 1δp(yk|Sk)p(Sk|yk−1)
Recursive Bayesian solution
Summary of the recursive Bayesian framework for object tracking.
Substituting for the likelihood function p(yk|Sk) from (1.10) and for the predicted density p(Sk|yk−1), the normalization factor becomes
p(yk|yk−1) =
Sk
pwk(l−1(yk, Sk)) ∂l−1
∂yk
p(Sk|yk−1)dSk. (1.11)
The generalized object state conditional density
The conditional density of the general single-object tracking problem can be obtained by substituting the transition density of (1.8) or (1.9), the likelihood
1.4 Summary 21 function of (1.10) and the normalizing factor of (1.11) into the recursive Bayes’
formula in (1.5), and simplifying it.
1.3.6 Generalized object state estimates
In object tracking, the expected value ˆSkof the object state Skis often used as an estimate of the object state at time k. To assess the accuracy of this estimate, the covariance matrix Pk|k of Sk is computed. These are the first two moments of the conditional density (1.5) of the object state Sk at time k,
p(Sk|yk) = 1
p(yk|yk−1)p(yk|Sk)
Sk−1
p(Sk|Sk−1)p(Sk−1|yk−1)dSk−1. p(Sk|yk) is obtained after all the terms have been computed in the above equation, using (1.8) or (1.9), (1.10) and (1.11). The track estimate is ˆSk= E(Sk|yk),
ˆSk =
Sk
Skp(Sk|yk)dSk.
The covariance matrix is Pk|k = E((Sk− ˆSk)(Sk− ˆSk)T|yk), Pk|k =
Sk
(Sk− ˆSk)(Sk− ˆSk)Tp(Sk|yk)dSk.
1.4 Summary
This chapter introduced the various important areas where object tracking forms the core part. It has discussed the application of Bayesian reasoning in object track-ing problems. The recursive form of the Bayesian formulation is presented as a useful format that can be applied to any object tracking problem and will be used in later chapters.