# Survey of Deep Learning-Based Methods for FMCW Radar Odometry and Ego-Localization
## 2 Problem Formulation  
In this section, we present fundamental concepts and the sequential stages involved in odometry estimation and ego-localization within radar-based SLAM. Additionally, we highlight key methods employed in classical radar SLAM. SLAM constitutes an estimation challenge wherein the objective is to refine the collection of all ego poses \(\mathbf{X}_{1:t}=\{\mathbf{X}_{1},\dots,\mathbf{X}_{t}\}\) at times 1 through \(t\) and \(n\) landmark positions \(\mathbf{M}=\mathbf{m}_{1:n}=\{\mathbf{m}_{1},\dots,\mathbf{m}_{n}\}\) recorded in a global map \(\mathbf{M}\). This refinement is accomplished by utilizing observed data encompassing perceptions \(\mathbf{Z}_{1:t}=\{\mathbf{Z}_{1},\dots,\mathbf{Z}_{t}\}\), gathered odometry readings \(\mathbf{U}_{1:t}=\{\mathbf{U}_{1},\dots,\mathbf{U}_{t}\}\), and an initial starting point \(\mathbf{X}_{0}\). Mathematically, the probabilistic formulation is articulated as defined by [PERSON] et al. [31] as:  
\[p(\mathbf{X}_{1:t},\mathbf{M}|\mathbf{Z}_{1:t},\mathbf{U}_{1:t},\mathbf{X}_{0}) \tag{1}\]  
For convenience, the starting position \(\mathbf{X}_{0}\) from Equation (1) will be omitted from now on. The ego-position and odometry measurements are commonly depicted using 2D  
Figure 1: Overview of the proposed survey paper. or 3D special Euclidean group transformation matrices \(\mathbf{T}\in SE(2)\) or \(\mathbf{T}\in SE(3)\). Such transformation matrices consist of a translation vector \(t\) and a rotation matrix \(R\) which can be represented with homogeneous coordinates in Equation (2) as  
\[\mathbf{T}= \begin{bmatrix}R&t\\ 0&1\end{bmatrix}  \tag{2}\]  
The fundamental objective of any SLAM methodology is to determine the Maximum A Posteriori (MAP) state for trajectories \(\mathbf{X}_{1:t}\) and the landmark map \(\mathbf{M}\). This objective can be formally expressed as:  
\[\{\mathbf{X}^{*},\mathbf{M}^{*}\}=\underset{\{\mathbf{X},\mathbf{M}\}}{\operatorname{argmax}} \ p(\mathbf{X}_{1:t},\mathbf{M}|\mathbf{Z}_{1:t},\mathbf{U}_{1:t}) \tag{3}\]  
A commonly employed approach to solving Equation (3) involves making an assumption of the static global map and leveraging the Markov property. This allows the SLAM problem to be conceptualized as a dynamic Bayesian network (DBN) [31]. Within this framework, the initial pose \(\mathbf{X}_{0}\), observations \(\mathbf{Z}_{1:t}\) and odometry estimations \(\mathbf{U}_{1:t}\) are considered as observable variables. Meanwhile, the latent variables of the model encompass \(\mathbf{X}_{1:t}\) (sequential poses) and the map \(\mathbf{M}\). The structure of the DBN is characterized by two main components: a state transition model denoted as \(p(\mathbf{X}_{t}|\mathbf{X}_{t-1},\mathbf{U}_{t})\) which signifies that a new pose is contingent on the previous pose and the current odometry measurement. The second component is an observation model denoted as \(p(\mathbf{Z}_{t}|\mathbf{X}_{t},\mathbf{M})\), indicating that observations are dependent on the current pose and the overall map configuration. A DBN can also be portrayed using a graphical representation, such as that presented in [32], where nodes correspond to poses \(\mathbf{X}_{t}\). Within this graph, edges serve as spatial constraints connecting poses. These constraints can be established based on odometry measurements \(\mathbf{U}_{t}\) between poses or through the alignment of two distinct observations \(\mathbf{Z}_{t}\) taken at different poses. In practical implementations, the observation edges play a crucial role in the graphical representation by functioning as regulators. Their purpose is to rectify any potential accumulation of errors or drift that might arise from the utilization of odometry edges alone. Graph optimization is a prevailing technique employed to determine the optimal global solution, considering multiple outcomes from scan matching. Depending on the specific characteristics of a SLAM algorithm, the edges accumulated through odometry and observations might not align seamlessly.