[AMR] Finished summary for vision part

electives/others/amr/parts/04_vision/03_mapping.tex

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+\newpage
+\subsection{Mapping}
+\bi{Problem Formulations} \textit{Localisation} {\scriptsize (always static given map)}:\\
+$\vec{x}^*_{R, k} = \text{argmax}\; \P(x_{R, k} \divider \vec{x}_M, \vec{z}_{1 : k}, \vec{u}_{1 : k})$ (recursive)\\
+$\vec{x}^*_{R, 1:k} = \text{argmax}\; \P(x_{R, 1:k} \divider \vec{x}_M, \vec{z}_{1 : k}, \vec{u}_{1 : k})$ (batch)
+
+\textit{SLAM} $\{ \vec{x}^*_{R, k}, \vec{x}^*_M \} \! = \! \text{argmax}\; \P(\vec{x}_R, \vec{x}_M \divider \vec{z}_{1 : k}, \vec{u}_{1 : k})$ (as $\uparrow$)
+
+\textit{Mapp.}: $\vec{x}_M^* \! = \! \text{argmax}\; \P(\vec{x}_M \divider \vec{x}^*_{R, 1:k}, \vec{z}_{1 : k}, \vec{u}_{1 : k})$ with given poses $\vec{x}^*_{R, 1:k}$.
+Thus temp. model $\vec{u}_{1:k}$ doesn't matter.
+
+\bi{Prob. Occ. Grid} $\P(f_j) = \P(\neg o_j) = 1 - \P(o_j)$, with $\P(o_j)$ prob. cell $j$ occupied; pairwise independent.
+
+\bi{Occ. Map. w/ depth sensor} $\P(o_j \divider \vec{x}_{R, 1:k}) =$
+\[
+    \frac{
+        {\color{MidnightBlue} \P(o_j \divider \vec{x}_{R, k}, \vec{z}_k)}
+        \P(\vec{z}_k \divider \vec{x}_{R, k})
+        {\color{ForestGreen} \P(o_j \divider \vec{x}_{R, 1 : k - 1}, \vec{z}_{1 : k - 1})}
+    }{{\color{DarkOrchid} \P(o_j)} \P(\vec{z}_k \divider \vec{x}_{R, 1 : k}, \vec{z}_{1 : k - 1})}
+\]
+{\color{DarkOrchid} map prior}; {\color{ForestGreen} Prev. occ. est.}; {\color{MidnightBlue} Occ. based on curr. range meas.};
+
+\bi{Update function}:
+$l(a) := \log(\text{Odds}(a))$ with $\text{Odds}(a) = \frac{\P(a)}{1 - \P(a)}$ (inv. sensor model):
+$l(o_j | \vec{x}_{R, 1 : k}, \vec{z}_{1 : k})$ =
+\[
+    l(o_j | \vec{x}_{R, k}, \vec{z}_k) + l(o_j | \vec{x}_{R, 1 : k - 1}, \vec{z}_{1 : k - 1}) - l(o_j)
+\]
+
+\shade{gray}{In 3D} 3D voxel $j$ as signed dist. $s$ and weight $w$, update:
+$\displaystyle s_k = \frac{w_{k - 1} s_{k - 1} + \tilde{s}_k}{w_{k - 1} + 1}$ with $w_k = \min(w_{\max}, w_{k - 1} + 1)$
+
+\bi{Impl.} Using HashTables or octree