[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