# Multi-Agent-Supported Tracking Based on Hidden Markov Model with Weighted Entropy
## 3 Multi-Agent-Supported Tracking Based on Hidden Markov Model with Weighted Entropy
### Kalman Filter Reconstruction Through HMM  
With detections acquired by the object detector, establishing continuous trajectories through the matching of new detections with existing tracks constitutes a fundamental challenge. A critical issue arises from the fact that the trajectories generated at time \(t-1\) may not accurately represent the exact positions of moving objects at time \(t\), with the complexity of pedestrian motion further exacerbating this difficulty. To address this, the Kalman filter is employed for prediction. The Kalman filter is a method for predicting the future state of a system based on previous ones as a different approach to statistical prediction and filtering [14]. In this section, we present a reformulation of the Kalman filter within the framework of HMMs to better model the dual-phase movement patterns of pedestrians, as previously discussed. As the Kalman filter is rooted in principled probabilistic theory, it can indeed be systematically described within the paradigm of HMMs [15]. This integration aims to enhance the predictive capability for scenarios characterized by abrupt changesin pedestrian dynamics, thereby improving the robustness of trajectory association in complex environments. The state-space model is described as follows:  
* The sequence representing the states of objects is denoted as \((\mathbf{x}_{t})_{t=1,\ldots,n}\), with these states being non-observable and solely influenced by past states through the last observation. Mathematically, \(\mathbf{x}_{t}\) is expressed as a Markov process with \(\mathbb{P}(\mathbf{x}_{t}\mid\mathbf{x}_{1},\mathbf{x}_{2},\ldots,\mathbf{x}_ {t-1})=\mathbb{P}(\mathbf{x}_{t}\mid\mathbf{x}_{t-1})\);
* For each state, there exists a corresponding observable variable \(\mathbf{z}_{t}\), which is a manifestation of the underlying latent state \(\mathbf{x}_{t}\). To ensure computational tractability of the proposed framework, several constraints are integrated. * The space and the state exhibit linear dependence (this is shared by both the Kalman filter and HMMs): \[\mathbf{z}_{t}=\mathbf{H}_{t}\mathbf{x}_{t}+\mathbf{v}_{t}\] (6) \[\mathbf{v}_{t}\sim\mathcal{N}(0,\mathbf{R}_{t})\] (7) where \(\mathbf{v}_{t}\) represents the zero-mean Gaussian measurement noise, and \(\mathbf{H}\) denotes the observation matrix that projects the state space onto the measurement space. * The states at time \(t-1\) and \(t\) exhibit linear dependence (this is shared by both the Kalman filter and HMMs): \[\mathbf{x}_{t}=\mathbf{F}_{t}\mathbf{x}_{t-1}+\mathbf{B}_{t}\mathbf{u}_{t}+ \mathbf{w}_{t}\] (8) \[\mathbf{w}_{t}\sim\mathcal{N}(0,\mathbf{Q}_{t})\] (9) where \(\mathbf{w}_{t}\) represents the zero-mean Gaussian process noise, and \(\mathbf{F}\) is the state transition matrix. The term \(\mathbf{B}_{t}\mathbf{u}_{t}\) accounts for additional control inputs. * The initial state \(\mathbf{x}_{0}\) and noise vectors at each step are all mutually independent (this is shared by both the Kalman filter and HMMs). * The state variable \(\mathbf{x}_{t}\) adheres to a multi-dimensional normal distribution. This is specifically common to the Kalman filter. Meanwhile, for HMMs, the state variable is presumed to follow a multinomial distribution. Under these constraints, the original state-space model specializes to a more restricted formulation, the Linear-Gaussian State-Space Model (LG-SSM).