[AMR] Finished summary for vision part

electives/others/amr/autonomous-mobile-robots-cheatsheet.tex

@@ -11,7 +11,6 @@
 \renewcommand{\subsectionnumbering}{section}
 \renewcommand{\numberingpreset}{off}
 \renewcommand{\definitionShortNamingEN}{Def}
-\noverticalspacing
 \fboxsep 1pt
 \fboxrule 0.1pt
 
@@ -23,6 +22,7 @@
 
 
 \begin{document}
+\noverticalspacing
 
 \vspace*{0mm}
 
@@ -75,6 +75,8 @@
 \input{parts/04_vision/00_keypoints.tex}
 \input{parts/04_vision/01_bootstrapping.tex}
 \input{parts/04_vision/02_place-recognition.tex}
+\input{parts/04_vision/03_mapping.tex}
+\input{parts/04_vision/04_dense-tracking_loop-closure.tex}
 % \input{parts/04_vision/}
 
 \end{document}

electives/others/amr/parts/04_vision/01_bootstrapping.tex

@@ -9,14 +9,12 @@ Find sol. for camera pose \textit{directly}
 Find good point in 3D. Fast sol: \bi{Midpoint Method}:
 
 \bi{1} Find p. along ray w/ min. dist (Lin. Least Squares)
-\rmvspace[0.7]
 \[
     \vec{\lambda}\! =\! [\lambda_1 \; \lambda_2]^\top\! = \! \argmin{} ||({_W}\vec{t}_{C_2} + \lambda_2 {_W}\vec{e}_2) - ({_W}\vec{t}_{C_1} + \lambda_1 {_W}\vec{e}_2)||^2
 \]
 
-\rmvspace[1]
+\rmvspace
 \bi{2} Solve normal equation $\mat{A} \vec{\lambda} = \vec{b}$ with $\vec{q} = -{_W}\vec{e}^\top_1 {_W}\vec{e}_2$:
-\rmvspace[0.7]
 \[
     \mat{A} = \begin{bmatrix}
         1       & \vec{q} \\
@@ -29,5 +27,4 @@ Find good point in 3D. Fast sol: \bi{Midpoint Method}:
     \end{bmatrix}
 \]
 
-\rmvspace[0.7]
 \bi{3} Pick midp. ${_W}\vec{t}_P \! = \! 0.5(\tau_1 \! + \! \tau_2)$; $\tau_n \! = \! {_W}\vec{t}_{C_n} + \lambda_n{_W}\vec{e}_n)$

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

@@ -0,0 +1,34 @@
+\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

electives/others/amr/parts/04_vision/04_dense-tracking_loop-closure.tex

@@ -0,0 +1,2 @@
+\subsection{Dense Tracking and Loop Closure}
+Idea: Min. sum of sq. errors over all pixels w/ Gauss-Newton.