In this work a probabilistic graphical model is developed to solve the multiple object track- ing problem. While the literature on the MOT problem and traditional (non-PGM) algorithmic solutions is extensive, very few sources are available on probabilistic graphical models based solu- tions to the problem3. Furthermore, many of these models draw inspiration from the traditional,
non-PGM, algorithms. While this might be sensible, it is our view that the model functionality should arise naturally from the structure of a logically constructed PGM. This work therefore strives to develop a logically sound graphical model and to investigate the functionality that arises naturally from it.
Since all the functionality required to implement the proposed MOT PGM was not available in the EMDW library (or any other library to the best of our knowledge), the EMDW library needed to be expanded. The idea with this expansion was to write additional factor classes in a highly object-oriented manner that is consistent with the existing code base. The added classes are therefore suitable for general use in graphical models and not limited to the multiple object tracking problem. The functionality that was added to the EMDW library is as follows 4:
3
Of course, any probabilistic model can be framed as a PGM, but the details of such PGM interpretations are often somewhat obscured by more abstract algorithms and some of the advantages of the PGM approach are lost in the process.
4
The basic structure of some of the existing classes were used in the creation of these classes. Some of the code is therefore identical to that of the these existing classes. For the classes where a large portion of the code
• Gaussian class (adapted from the existing gausscanonical class) • Non-linear Gaussian class
• Gaussian mixture class
• Conditional non-linear Gaussian class • Factorised factor class
• Discrete log table class (adapted from the existing discretetable class)
The factorised factor class allows for the natural and efficient representation of products of factors. This functionality is especially useful when multiplication of independent factors is required during message passing. In this case it is unnecessary to explicitly multiply the factors, and the factorised factor class allows them to be kept in a factorised form while also supporting any subsequent operations on the (factorised) factor. This class supports Gaussian, Gaussian mixture, or non-linear Gaussian factors, or a combination of these as factorised components.
During this study and the development of the multiple object tracking model, a few valu- able insights were gained. These insights are presented in detail in the corresponding sections throughout this work and are summarised as follows:
• A logical, graphical model-based design process for the development of a graphical model for multiple object tracking.
• Interesting message passing characteristics:
– The effect of the means and variances when multiplying Gaussian distributions (see Section 5.3)
– How the association variable(s) causes the state posterior to be a superposition of the prior and measurement updated distribution (see sections 6.1 and 6.2). This explanation has not been found in any of the available literature.
Furthermore, some of the methods and concepts presented in this work are believed to be novel. These contributions are as follows:
• An alternative view on model selection that allows model selection to be performed efficiently during message passing inference in a single graph was formulated in this work. This effect was later found to be mentioned in the paper by Segal and Reid [11], but is not described in any detail and it has not been found to be mentioned in any other
remains unchanged, this is indicated by ”adapted from”. The other classes have little functional code in common with the classes that they were adapted from.
literature. In this work, the effect is described in detail. Here it is shown mathematically (see Section 7.1) that inference in graphs, with association type discrete variables, can correspond exactly to classical model selection. Furthermore, this type of model selection was implemented in the MOT PGM and shown to work in practice.
• The clutter classification graphical model developed as part of this work does not require the clutter density or detection probability to be known. This is in contrast to the majority of established MOT algorithms that are capable of tracking through clutter, where these parameters are required for accurate tracking. Although some models have been developed in the past that allow the clutter intensity and detection probabilities to be estimated [14], these approaches are very different to the approach presented in this work. The clutter classification model forms part of the larger MOT PGM and was shown to work very well in practice (see Section 10.3).
• An algorithm for performing approximate Gaussian mixture division was developed to allow for Gaussian mixtures to be used in the LBU message passing algorithm (where division operations are required). The quotient of Gaussian mixtures is not typically a Gaussian mixture but can often be approximated as one. The developed algorithm allows for quotients of Gaussian mixtures to be accurately approximated by Gaussian mixtures. Such an algorithm has not been found anywhere in the literature. Furthermore it has been shown that the use of Gaussian mixtures (as opposed to the single Gaussian approximation) can increase tracking accuracy under certain conditions (see Chapter 8).
In summary, while the use of PGMs has in the past (in a very small number of studies) been proposed to solve multiple object tracking, we believe that the model developed in this work is at least partially novel. Furthermore, this work contributes to the literature in terms of the detail of explanation and the approach that was taken to develop a multiple object tracking PGM.