## 3 Reducing the Uncertainty at the Output of Kalman Filter by Post Processing
### Data Association  
The goal of data association is to establish a correspondence between observations and existing tracks [18].  
Traditionally, multiple target tracking (MTT) algorithms have been designed for scenarios with multiple remote objects situated far away from the sensor. An object is not always detected by the sensor in such scenarios, and if it is detected, it occupies, at most, one sensor resolution cell. \"From traditional scenarios, specific assumptions on the mathematical model of MTT problems have evolved including the so-called \"small object\" assumptions\" [19]:  
* The evolution of objects is independent;
* The model of each object is a point at a given time;
* Each object produces at most a single measurement per scan/time frame.  
Due to: sensor noise, missed detections, clutter detections, measurement origin uncertainty and an unknown and time-varying number of targets, MTT based on the \"small object\" assumptions is a highly complex problem. Theoretically, each measurement can be associated with a different track. For an increased number of measurements, a very high number of associations is possible. To find the true data association, all these possible associations would need to be verified. This requires a tremendous volume of computation. To reduce this volume of computation, at the beginning of the data association procedure, a validation region (gate) is formed around the predicted track position, to eliminate unlikely observation-to-track pairing. The measurements observed in the interior of this gate are good candidates for the association with the considered track. An observation can be associated with a track and used to update the track filter if it is the single observation in the interior of the corresponding gate and if it is not in the interior of the gate of any other track. When an observation falls within the gates of multiple target tracks or when multiple observations fall within the gate of a target track in a dense target environment, additional logic is necessary.  
For multiple sensors and multiple targets, common association algorithms are:  
* Joint likelihood (JL): a variation of the multi-hypothesis track splitting algorithm extended to multiple tracks [20];
* Joint probabilistic data association (JPDA): for each track, this algorithm updates the filter based on a joint probability of association of the latest set of measurements and each track [19];
* Multiple hypothesis joint probabilistic (MHJP): a variation of the optimal Bayesian filter using joint probabilities among multiple track associations for multiple hypotheses [18];
* Random finite sets (RFS) approaches: rely on modelling the objects and the measurements as random sets.  
Inspired by [6,21], we use in this paper the Hungarian algorithm (Kuhn-Munkres algorithm) [17] for data association and the Kalman filter for tracking and realizing the tracking of multiple radar targets. The Hungarian algorithm deals with the problem of personnel assignment. It aims to find the best assignment of a set of persons to a set of jobs. In RADAR tracking, the necessity to correctly assign the position of a detected object to a track can be completed by computing a \"distance\" matrix between the track estimates at the current timestamp and the measurements. The Hungarian algorithm performs the optimal assignment between the tracks (rows of the matrix) to the measurements (columns). The result is a column vector that contains the assigned column number (measurement) in each row (track). The absence of measurements assigned at the current step for the respective track is marked with a zero value in this matrix.