# Multi-Player Tracking for Multi-View Sports Videos with Improved K-Shortest Path Algorithm
## 2 Related Work  
As a detailed review stated [4], the objective of MOT is to find optimal sequential states of all the objects, which can be generally modeled by performing MAP (maximal a posteriori) estimation from the conditional distribution of the sequential states given all the observations:  
\[\widehat{\mathbf{S}}_{1:t}=\operatorname*{arg\,max}_{\mathbf{S}_{1:t}}\;P( \mathbf{S}_{1:t}|\mathbf{O}_{1:t}), \tag{1}\]  
where \(\mathbf{S}_{1:t}=\{\mathbf{S}_{1},\mathbf{S}_{2},...,\mathbf{S}_{t}\}\) denotes all the sequential states of all the objects from the first frame to the \(t\)-th frame, and \(\mathbf{O}_{1:t}=\{\mathbf{O}_{1},\mathbf{O}_{2},...,\mathbf{O}_{t}\}\) represents all the collected sequential observations of all the objects throughout the same frames. Different MOT algorithms from previous works can be thought as designing different approaches to solving the above MAP problem, either from a probabilistic inference perspective or a deterministic optimization perspective. The former approach usually solves the MAP problem using a two-step iterative procedure: step one is a prediction process with a dynamic model and the other step is an update process using an observation model, such as some traditional tracking methods; e.g., mean-shift [13], Kalman filter [14; 15], particle filter [16; 17], and their variations [18; 19; 20]. Once the up-to-time observations are given, these kinds of dynamic approaches can predict the current observations with gradually extend existing trajectories, which make them being very suitable for the online tracking. However, they would easily miss a tracking object when occluded. Not to mention the terrible occlusions in the sports matches. As for the MOT methods based on deterministic optimization, they solve the above problem by maximizing a likelihood function or conversely minimizing an energy function, which is very popular nowadays. This is because they can obtain a global optimal solution, and thus are usually more robust at dealing with false positives and occlusions. One main disadvantage of these approaches is that there are some delays in outputting final results, as they need to process a batch of consecutive frames. Also, they must depend on the detecting results for each frame. Fortunately, the state-of-the-art detectors, such as Faster RCNN, YOLO et al. [21], can be used in real time. So the tracking output only needs to wait for a short time for processing a batch of frames, the process being able to be controlled within 5 or 10 s, which is acceptable in most applications, even a sports live broadcast. In the team sports analysis field, the global optimization methods are obviously more popular. Earlier in 2007, people formulated the MOT problem as an integer linear program to obtain a nearly optimal solution [22; 23]. But the number of targets in the scene needs to be fixed a-priori, which is a bit confusing. Also, there are some studies that consider multiple-player tracking as a network flow problem, which can be solved in polynomial time [24; 25]. Other global optimal methods [26] applied in MOT include graph-based approaches [27; 28], and quadratic and linear objective optimization [29]. Even though most of the above global approaches improve a lot compared with the earlier dynamic methods, they are still difficult to optimize and sometimes become trapped in local minima. In practice, issues manifest in tracking errors such as fragmented trajectories and identity switches. Compared with the above methods, the MOT based on KSP algorithm [9] is very fast and easily obtains the global optimal solution. However, identities are also easily be switched when two players are close to each other. To address this problem of identity switches, people have explored some appearance models [30] or motion models [31]. For example, [PERSON] et al. [32] defined a set of game context features to describe the current state of a match.