[AMR] More fixes

electives/amr/parts/02_Sensors-Actuators/03_cameras.tex

@@ -12,9 +12,12 @@ with $f$ the distance to the lens and $z$ the full distance
 \newpage
 $u = c_u + f \cdot x'$ and $v = c_v + f \cdot y'$ where $x' = t_x \div t_z$ and $y' = t_y \div t_z$
 where $u, v$ are the pixel $x, y$ coords, $\vec{c} = [c_u, c_v]^\top$ is optical centre of cam in pixel coords, $f$ scale factor,
-and $\vec{{_C}\vec{t}_P} = [t_x, t_y, t_z]^\top$
+% and $\vec{{_C}\vec{t}_P} = [t_x, t_y, t_z]^\top$
 
-The full proj: $\vec{u} = \begin{bmatrix}
+The full proj:
+$\vec{u} =
+    {\scriptsize
+    \begin{bmatrix}
         \lambda u \\
         \lambda v \\
         \lambda
@@ -28,23 +31,27 @@ The full proj: $\vec{u} = \begin{bmatrix}
     \begin{bmatrix}
         t_x \\t_y\\t_z
     \end{bmatrix}
+    }
     = \mat{K}\; {_C}\vec{t}_P$
 
-If p. in diff frame ${_W} \vec{t}_P$, then $\vec{u} = \mat{K}[\mat{R}_{CW}\; {_C}\vec{t}_{CW}] = {_W}\vec{t}_P$
+If p. in diff frame ${_W} \vec{t}_P$, then $\vec{u} = \mat{K}[\mat{R}_{CW}\; {_C}\vec{t}_{CW}] {_W}\vec{t}_P$
 
 \subsubsection{Pinhole Camera Projection with distortion}
-\shortdefinition Model: $\vec{u} = \vec{k}(\vec{d}(\vec{p}({_C}\vec{t}_P)))$, with:
+\shortdefinition Model: $\vec{u} = \vec{k}(\vec{d}(\vec{p}({_C}\vec{t}_P)))$, with ($\vec{c}$ as above):
 
 (Projection) $\vec{x'} = \vec{p}({_C}\vec{t}_P) = t_z^{-1} \cdot [t_x, t_y]^\top$
 
-(Distortion model, $r^2 = x'^2 + y'^2$, $\vec{x''} = \vec{d}(\vec{x'})$\\
+(Distortion model) $r^2 = x'^2 + y'^2$, $\vec{x''} = \vec{d}(\vec{x'})$\\
 $\vec{x''} = \! \frac{1 + k_1 r^2 + k_2 r^4 + k_3 r^6}{1 + k_4 r^2 + k_5 r^4 + k_6 r^6} \vec{x'}
-    + \begin{bmatrix}
+    +
+    {\scriptsize
+    \begin{bmatrix}
         2p_1 x' y' + p_2(r^2 + 2x'^2) \\
         p_1 (r^2 + 2y'^2) + 2p_2 x'y'
-    \end{bmatrix}$
+    \end{bmatrix}
+    }$
 
-(Scale and Centre) $\vec{u} = \vec{k}(\vec{x''}) = \text{diag}([f_u, f_v]) \cdot \vec{x''} + \vec{x}$
+(Scale and Centre) $\vec{u} = \vec{k}(\vec{x''}) = \text{diag}([f_u, f_v]) \cdot \vec{x''} + \vec{c}$
 
 All with $k_i$ radial distortion params, optional for $i > 2$, $p_i$ tang. dist. param, $f_u, f_v$ focal length in pixels
 

electives/amr/parts/02_Sensors-Actuators/04_depth-range.tex

@@ -13,17 +13,23 @@ $z = \frac{b \cdot f}{u_r - u_l}$ with $u_l = f\cdot \frac{x}{z} + c_u$ and $u_r
 \subsubsection{Time of Flight, Projection}
 {\color{ForestGreen}No occlusions/shadows}, {\color{red}Interference with other dev, multipath leading to larger distances sensed}
 
-\bi{Proj.} $\vec{z} = \begin{bmatrix}
+\bi{Proj.} $\vec{z} =
+    {\scriptsize
+    \begin{bmatrix}
         \vec{u} \\ d
     \end{bmatrix} =
     \begin{bmatrix}
         \vec{k}(\vec{d}(\vec{p}({_C}\vec{t}_P))) \\
         [0, 0, 1] {_c}\vec{t}_P
-    \end{bmatrix}$
+    \end{bmatrix}
+    }$
 \bi{Back}: ${_C}\vec{t}_P
-    = \begin{bmatrix}
+        =
+        {\scriptsize
+            \begin{bmatrix}
                 d \vec{x'} \\d
-    \end{bmatrix}$
+            \end{bmatrix}
+        }$
 
 
 \subsubsection{Range Sensors}

electives/amr/parts/03_multi-sensor-estimation/00_linearization.tex

@@ -3,5 +3,5 @@ $\vec{f}(\vec{x}) \approx \vec{f}(\vec{\overline{x}}) + \mat{J}_{\vec{f}} \big|_
 
 \shortdefinition[Jac.] $\mat{J}_{\vec{f}}$ rows for eq of $\vec{f}$ cols for vars of each eq.
 
-Approximation using finite differences $\frac{f(\overline{x} + h) - f(\overline{x})}{h}$,\\
+Part. diff; Approx. using finite differences $\frac{f(\overline{x} + h) - f(\overline{x})}{h}$,\\
 or central differences (vector of $\frac{\vec{f}(\vec{\overline{x}}) + h_i \vec{e_i}}{h_i}$, with $\vec{e_i}$ unit vec)

electives/amr/parts/03_multi-sensor-estimation/01_least-squares.tex

@@ -3,8 +3,8 @@
 
 \bi{Man. sol.}: comp. $M = A^\top A$, $b' = A^\top b$, then $Mx = b'$.
 
-\bi{Prob. sol.}: $\text{argmax} p(\vec{x} \divider \vec{z})$ with
+\bi{Prob. sol.}: $\text{argmax} \; \P(\vec{x} \divider \vec{z})$ with Maximum \dots
 \begin{itemize}
-    \item \bi{Max. Like} $p(\vec{x} | \vec{z}) \propto p(\vec{z} | \vec{x}) = \prod_{i = 1}^N p(\vec{z}_i | \vec{x})$
+    \item \bi{Likelihood} $\P(\vec{x} | \vec{z}) \propto \P(\vec{z} | \vec{x}) = \prod_{i = 1}^N \P(\vec{z}_i | \vec{x})$
-    \item \bi{M a Post} $p(\vec{x} | \vec{z}) \propto p(\vec{z} | \vec{x}) p(\vec{x}) = p(\vec{x}) \prod_{i = 1}^N p(\vec{z}_i | \vec{x})$
+    \item \bi{a Post} $\P(\vec{x} | \vec{z}) \propto \P(\vec{z} | \vec{x}) \P(\vec{x}) = \P(\vec{x}) \prod_{i = 1}^N \P(\vec{z}_i | \vec{x})$
 \end{itemize}

electives/amr/parts/03_multi-sensor-estimation/02_nonlinear-least-squares.tex

@@ -1,5 +1,5 @@
 \subsection{Non-Linear Least Squares}
-Find $\vec{x}^* = \text{argmax} p(\vec{x} | \vec{z}) = \argmin{}(-\log(p(\vec{x}|\vec{z})))$
+Find $\vec{x}^* = \text{argmax}\; \P(\vec{x} | \vec{z}) = \argmin{}(-\log(\P(\vec{x}|\vec{z})))$
 
 \bi{Gauss-Newton} % TODO: Do we really need these? If so, use from NumCS (much simpler notation)
 

electives/amr/parts/03_multi-sensor-estimation/03_bayes-filters.tex

@@ -1,4 +1,4 @@
-\subsection{Bayes Filter}
+\subsection{Bayes Filter (in DAG)}
 $\vec{x}_k^R$ state at time k, $\vec{z}_k^p$ dist. meas., $\vec{u}^p_k$ wheel odometry (= meas.).
 Typ. care ab. curr. state: altern. pred. \& update. 
 % TODO: Do we really need the below?

electives/amr/parts/03_multi-sensor-estimation/06_extended-kalman-filter.tex

@@ -10,3 +10,5 @@ Non-l. state trans. model $\vec{x}_k = \vec{f}(\vec{x}_{k - 1}, \vec{u}_k, \vec{
     \item \bi{Meas. residual}: $\vec{y}_k = \tilde{\vec{z}}_k - \vec{h}(\hat{\vec{x}}_{k | k - 1})$
 \end{itemize}
 Difference to above: $\mat{H}$ becomes $\mat{H}_k$, and $\mat{H}^\top$ is $\mat{H}_k^\top$
+
+% TODO: Consider adding examples

electives/amr/parts/04_vision/00_keypoints.tex

@@ -1,11 +1,14 @@
 \subsection{Keypoints}
-\bi{Corner det.} $SSD(\Delta_x, \Delta_y) \approx [\Delta_x \; \Delta_y] \mat{M} [\Delta_x \; \Delta_y]^\top$
+\bi{Corner det.} $SSD(\Delta x, \Delta y) \approx [\Delta_x \; \Delta_y] \mat{M} [\Delta_x \; \Delta_y]^\top$
-with $\mat{M} = \mat{R}^\top \text{diag}(\lambda_1, \lambda_2) \mat{R}$; $\lambda_i$ E.V. of $M$;
+with $\mat{M} = \mat{R}^\top \text{diag}(\lambda_1, \lambda_2) \mat{R}$; $\lambda_i$ E.V. of $M$; $\kappa$ const 0.04-0.15;
-$\mat{R} = \det(M) - \kappa \cdot \text{trace}(M)^2 = \lambda_1\lambda_2 - \kappa(\lambda_1 + \lambda_2)^2$;
+$\mat{R} = \det(M) - \kappa \cdot \text{TR}(M)^2 = \lambda_1\lambda_2 - \kappa(\lambda_1 + \lambda_2)^2$;\\
-$\displaystyle M = \sum_{x, y \in P} \begin{bmatrix}
+$M = \sum_{x, y \in P}
+    {\scriptsize
+        \begin{bmatrix}
             I_x^2   & I_x I_y \\
             I_x I_y & I_y^2
-    \end{bmatrix}$
+        \end{bmatrix}
+    }$
 
 
 \shade{gray}{Blob Detection} ($I$ is the image)

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

@@ -1,9 +1,11 @@
 \subsection{Bootstrapping}
 
-\bi{PnP Problem} {\scriptsize Persp. n-P.}
+\bi{PnP Problem} {\scriptsize Persp. $n$-P.}
 Find sol. for camera pose \textit{directly}
 
 \bi{RANSAC} {\scriptsize RANdom SAmpling Consensus} for find. outliers \& correct
+% TODO: Do we need to know this algorithm / be able to implement it?
+% -> likely not
 
 \bi{Stereo Triang.} Given two rays (known poses for points in 2D).
 Find good point in 3D. Fast sol: \bi{Midpoint Method}:
@@ -16,15 +18,22 @@ Find good point in 3D. Fast sol: \bi{Midpoint Method}:
 \rmvspace
 \bi{2} Solve normal equation $\mat{A} \vec{\lambda} = \vec{b}$ with $\vec{q} = -{_W}\vec{e}^\top_1 {_W}\vec{e}_2$:
 \[
-    \mat{A} = \begin{bmatrix}
+    \mat{A} =
+    {\scriptsize
+    \begin{bmatrix}
         1       & \vec{q} \\
         \vec{q} & 1
     \end{bmatrix}
+    }
     \quad
-    \vec{b} = \begin{bmatrix}
+    \vec{b} =
+    {\scriptsize
+    \begin{bmatrix}
         \vec{e}_1^\top \cdot ({_W}\vec{t}_{C_2} - {_W}\vec{t}_{C_1}) \\
         -\vec{e}_2^\top \cdot ({_W}\vec{t}_{C_2} - {_W}\vec{t}_{C_1})
     \end{bmatrix}
+    }
+    \quad C_i \text{ cam}
 \]
 
 \bi{3} Pick midp. ${_W}\vec{t}_P \! = \! 0.5(\tau_1 \! + \! \tau_2)$; $\tau_n \! = \! {_W}\vec{t}_{C_n} + \lambda_n{_W}\vec{e}_n)$