[AMR] Vision intro

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

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+\subsection{Keypoints}
+\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 $M$;
+$\mat{R} = \det(M) - \kappa \cdot \text{trace}(M)^2 = \lambda_1\lambda_2 - \kappa(\lambda_1 + \lambda_2)^2$;
+$\displaystyle M = \sum_{x, y \in P} \begin{bmatrix}
+        I_x^2   & I_x I_y \\
+        I_x I_y & I_y^2
+    \end{bmatrix}$
+
+
+\shade{gray}{Blob Detection} ($I$ is the image)
+
+\bi{Laplacian of Gaussian} (LoG): $L = g(x, y, t) \cdot I(x, y)$.
+Then apply Laplacian Operator $\nabla_\text{norm}^2 L = t\left( \frac{\partial^2 L}{\partial x^2} + \frac{\partial L}{\partial y^2} \right)$
+
+\bi{Diff. of Gaussians} (DoG): $\Delta L = L(x, y, t) - L(x, y, kt)$
+
+\bi{SIFT Detector} \bi{(1)} Subsample + Blur \bi{(2)} DoG on each res. image \bi{(3)} Keypoints extrema in DoG pyramid

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

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+\subsection{Bootstrapping}
+
+\bi{PnP Problem} {\scriptsize Persp. n-P.}
+Find sol. for camera pose \textit{directly}
+
+\bi{RANSAC} {\scriptsize RANdom SAmpling Consensus} for find. outliers \& correct
+
+\bi{Stereo Triang.} Given two rays (known poses for points in 2D).
+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]
+\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]
+\[
+    \mat{A} = \begin{bmatrix}
+        1       & \vec{q} \\
+        \vec{q} & 1
+    \end{bmatrix}
+    \quad
+    \vec{b} = \begin{bmatrix}
+        \vec{e}_1^\top \cdot ({_W}\vec{t}_{C_2} - {_W}\vec{t}_{C_1}) \\
+        -\vec{e}_2^\top \cdot ({_W}\vec{t}_{C_2} - {_W}\vec{t}_{C_1})
+    \end{bmatrix}
+\]
+
+\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/02_place-recognition.tex

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+\subsection{Place Recognition}
+Idea: Build vocab from ``visual words'' (in Training).
+\bi{Runtime} Detect keypoints; Extract descriptors; Build histogram; Query DB for similarity; If no match, insert into DB, else use to compute with e.g. RANSAC