From bc65a7ff2b93c442dda38a7d9507fa3d2f2409aa Mon Sep 17 00:00:00 2001 From: NotXia <35894453+NotXia@users.noreply.github.com> Date: Tue, 16 Apr 2024 20:10:34 +0200 Subject: [PATCH] Add IPCV2 projective space --- .../module2/ainotes.cls | 1 + .../module2/img/_formation_system.pdf | Bin 0 -> 44388 bytes .../module2/img/_pixelization.pdf | Bin 0 -> 50213 bytes .../img/_projective_point_inifinity.pdf | Bin 0 -> 40213 bytes .../module2/img/_projective_space.pdf | Bin 0 -> 42844 bytes .../module2/img/point_infinity_example1.png | Bin 0 -> 449135 bytes .../module2/img/point_infinity_example2.png | Bin 0 -> 50142 bytes .../module2/ipcv2.tex | 13 + .../module2/sections/_image_formation.tex | 253 ++++++++++++++++++ 9 files changed, 267 insertions(+) create mode 120000 src/image-processing-and-computer-vision/module2/ainotes.cls create mode 100644 src/image-processing-and-computer-vision/module2/img/_formation_system.pdf create mode 100644 src/image-processing-and-computer-vision/module2/img/_pixelization.pdf create mode 100644 src/image-processing-and-computer-vision/module2/img/_projective_point_inifinity.pdf create mode 100644 src/image-processing-and-computer-vision/module2/img/_projective_space.pdf create mode 100644 src/image-processing-and-computer-vision/module2/img/point_infinity_example1.png create mode 100644 src/image-processing-and-computer-vision/module2/img/point_infinity_example2.png create mode 100644 src/image-processing-and-computer-vision/module2/ipcv2.tex create mode 100644 src/image-processing-and-computer-vision/module2/sections/_image_formation.tex diff --git a/src/image-processing-and-computer-vision/module2/ainotes.cls b/src/image-processing-and-computer-vision/module2/ainotes.cls new file mode 120000 index 0000000..146fd3c --- /dev/null +++ b/src/image-processing-and-computer-vision/module2/ainotes.cls @@ -0,0 +1 @@ +../../ainotes.cls \ No newline at end of file diff --git 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a/src/image-processing-and-computer-vision/module2/sections/_image_formation.tex b/src/image-processing-and-computer-vision/module2/sections/_image_formation.tex new file mode 100644 index 0000000..2915b91 --- /dev/null +++ b/src/image-processing-and-computer-vision/module2/sections/_image_formation.tex @@ -0,0 +1,253 @@ +\chapter{Camera calibration} + +\begin{description} + \item[World reference frame (WRF)] \marginnote{World reference frame (WRF)} + Coordinate system $(X_W, Y_W, Z_W)$ of the real world relative to a reference point (e.g. a corner). + + \item[Camera reference frame (CRF)] \marginnote{Camera reference frame (CRF)} + Coordinate system $(X_C, Y_C, Z_C)$ that characterizes a camera. + + \item[Image reference frame (IRF)] \marginnote{Image reference frame} + Coordinate system $(U, V)$ of the image. + They are obtained as a perspective projection of CRF coordinates as: + \[ + u = \frac{f}{z}x_C + \hspace{3em} + v = \frac{f}{z}y_C + \] + + \begin{figure}[H] + \centering + \includegraphics[width=0.8\linewidth]{./img/_formation_system.pdf} + \caption{Example of WRF, CRF and IRF} + \end{figure} +\end{description} + + + +\section{Forward imaging model} + + +\subsection{Image pixelization (CRF to IRF)} +\marginnote{Image pixelization} + +The conversion from the camera reference frame to the image reference frame +is done in two steps: +\begin{descriptionlist} + \item[Discetization] \marginnote{Discetization} + Given the sizes (in mm) $\Delta u$ and $\Delta v$ of the pixels, + it is sufficient to modify the perspective projection to map CRF coordinates into a discrete grid: + \[ + u = \frac{1}{\Delta u}\frac{f}{z_C}x_C + \hspace{3em} + v = \frac{1}{\Delta v}\frac{f}{z_C}y_C + \] + + \item[Origin translation] \marginnote{Origin translation} + To avoid negative pixels, the origin of the image has to be translated from the piercing point $c$ to the top-left corner. + This is done by adding an offset $(u_0, v_0)$ to the projection (in the new system, $c = (u_0, v_0)$): + \[ + u = \frac{1}{\Delta u}\frac{f}{z_C}x_C + u_0 + \hspace{3em} + v = \frac{1}{\Delta v}\frac{f}{z_C}y_C +v_0 + \] + + \begin{figure}[H] + \centering + \includegraphics[width=0.9\linewidth]{./img/_pixelization.pdf} + \caption{Pixelization process} + \end{figure} + + \item[Intrinsic parameters] \marginnote{Intrinsic parameters} + Parameters needed to convert from CRF to IRF. + + By fixing $f_u = \frac{f}{\Delta u}$ and $f_v = \frac{f}{\Delta v}$, the projection can be rewritten as: + \[ + u = f_u\frac{x_C}{z_C} + u_0 + \hspace{3em} + v = f_v\frac{y_C}{z_C} +v_0 + \] + Therefore, there is a total of 4 parameters: $f_u$, $f_v$, $u_0$ and $v_0$. +\end{descriptionlist} + + +\subsection{Roto-translation (WRF to CRF)} +\marginnote{Roto-translation} + +The conversion from the world reference system to the camera reference system +is done through a roto-translation wrt the optical center. + +Given: +\begin{itemize} + \item A WRF point $\vec{M}_W = (x_W, y_W, z_W)$, + \item A rotation matrix $\matr{R}$, + \item A translation vector $\vec{t}$, +\end{itemize} +the coordinates $\vec{M}_C$ in CRF corresponding to $\vec{M}_W$ are given by: +\[ + \vec{M}_C = \begin{pmatrix} x_C \\ y_C \\ z_C \end{pmatrix} = + \matr{R}\vec{M}_W + \vec{t} = + \begin{pmatrix} + r_{11} & r_{12} & r_{13} \\ + r_{21} & r_{22} & r_{23} \\ + r_{31} & r_{32} & r_{33} \\ + \end{pmatrix} + \begin{pmatrix} + x_W \\ y_W \\ z_W + \end{pmatrix} + + + \begin{pmatrix} + t_1 \\ t_2 \\ t_3 + \end{pmatrix} +\] + +\begin{remark} + The coordinates $\vec{C}_W$ of the optical center $\vec{C}$ are obtained as: + \[ + \nullvec = \matr{R}\vec{C}_W + \vec{t} + \iff (\nullvec - \vec{t}) = \matr{R}\vec{C}_W + \iff \vec{C}_W = \matr{R}^T (\nullvec - \vec{t}) + \iff \vec{C}_W = -\matr{R}^T\vec{t} + \] +\end{remark} + +\begin{description} + \item[Extrinsic parameters] \marginnote{Extrinsic parameters} + \phantom{} + \begin{itemize} + \item The rotation matrix $\matr{R}$ has 9 elements of which 3 are independent (i.e. the rotation angles around the axes). + \item The translation matrix $\vec{t}$ has 3 elements. + \end{itemize} + + Therefore, there is a total of 6 parameters. +\end{description} + + +\begin{remark} + It is not possible to combine the intrinsic camera model and the extrinsic roto-translation to + create a linear model for the forward imaging model. + \[ + u = f_u \frac{r_{11}x_W + r_{12}y_W + r_{13}z_W + t_1}{r_{31}x_W + r_{32}y_W + r_{33}z_W + t_3} + u_0 + \hspace{1.5em} + v = f_v \frac{r_{21}x_W + r_{22}y_W + r_{23}z_W + t_2}{r_{31}x_W + r_{32}y_W + r_{33}z_W + t_3} + v_0 + \] +\end{remark} + + + +\section{Projective space} + +\begin{remark} + In the 2D Euclidean plane $\mathbb{R}^2$, parallel lines never intersect and points at infinity cannot be represented. + \begin{figure}[H] + \centering + \begin{subfigure}{0.45\linewidth} + \centering + \includegraphics[width=0.45\linewidth]{./img/point_infinity_example1.png} + \end{subfigure} + \begin{subfigure}{0.45\linewidth} + \centering + \includegraphics[width=0.8\linewidth]{./img/point_infinity_example2.png} + \end{subfigure} + \caption{Example of point at infinity} + \end{figure} +\end{remark} + +\begin{remark} + Point at infinity is a point in space while the vanishing point is in the image plane. +\end{remark} + +\begin{description} + \item[Homogeneous coordinates] \marginnote{Homogeneous coordinates} + Without loss of generality, consider the 2D Euclidean space $\mathbb{R}^2$. + + Given a coordinate $(u, v)$ in Euclidean space, its homogeneous coordinates have an additional dimension + such that: + \[ (u, v) \equiv (ku, kv, k) \,\forall k \neq 0 \] + In other words, a 2D Euclidean point is represented by an equivalence class of 3D points. + + \item[Projective space] \marginnote{Projective space} + Space $\mathbb{P}^n$ associated with the homogeneous coordinates of an Euclidean space $\mathbb{R}^n$. + + \begin{figure}[H] + \centering + \includegraphics[width=0.6\linewidth]{./img/_projective_space.pdf} + \caption{Example of projective space $\mathbb{P}^2$} + \end{figure} + + \begin{remark} + $\nullvec$ is not a valid point in $\mathbb{P}^n$. + \end{remark} + + \begin{remark} + A projective space allows to homogeneously handle both ordinary (image) and ideal (scene) points without introducing additional complexity. + \end{remark} + + \item[Point at infinity] \marginnote{Point at infinity} + Given the parametric equation of a 2D line defined as: + \[ + \vec{m} = \vec{m}_0 + \lambda \vec{d} = + \begin{pmatrix} u_0 \\ v_0 \end{pmatrix} + \lambda \begin{pmatrix} a \\ b \end{pmatrix} = + \begin{pmatrix} u_0 + \lambda a \\ v_0 + \lambda b \end{pmatrix} + \] + It is possible to define a generic point in the projective space along the line $m$ as: + \[ + \tilde{\vec{m}} \equiv + \begin{pmatrix} \vec{m} \\ 1 \end{pmatrix} \equiv + \begin{pmatrix} u_0 + \lambda a \\ v_0 + \lambda b \\ 1 \end{pmatrix} \equiv + \begin{pmatrix} \frac{u_0}{\lambda} + a \\ \frac{v_0}{\lambda} + b \\ \frac{1}{\lambda} \end{pmatrix} + \] + + The projective coordinates $\tilde{\vec{m}}_\infty$ of the point at infinity of a line $m$ is given by: + \[ \tilde{\vec{m}}_\infty = \lim_{\lambda \rightarrow \infty} \tilde{\vec{m}} \equiv \begin{pmatrix} a \\ b \\ 0 \end{pmatrix} \] + + \begin{figure}[H] + \centering + \includegraphics[width=0.6\linewidth]{./img/_projective_point_inifinity.pdf} + \caption{Example of infinity point in $\mathbb{P}^2$} + \end{figure} + + In 3D, the definition is trivially extended as: + \[ + \tilde{\vec{M}}_\infty = + \lim_{\lambda \rightarrow \infty} \begin{pmatrix} \frac{x_0}{\lambda} + a \\ \frac{y_0}{\lambda} + b \\ \frac{z_0}{\lambda} + c \\ \frac{1}{\lambda} \end{pmatrix} \equiv + \begin{pmatrix} a \\ b \\ c \\ 0 \end{pmatrix} + \] + + \item[Perspective projection] \marginnote{Perspective projection in projective space} + Given a point $\vec{M}_C = (x_C, y_C, z_C)$ in the CRF and its corresponding point $\vec{m} = (u, v)$ in the image, + the non-linear perspective projection in Euclidean space can be done linearly in the projective space as: + \[ + \begin{split} + \tilde{\vec{m}} &\equiv + \begin{pmatrix} u \\ v \\ 1 \end{pmatrix} \equiv + \begin{pmatrix} f_u\frac{x_C}{z_C} + u_0 \\ f_v\frac{y_C}{z_C} +v_0 \\ 1 \end{pmatrix} \equiv + z_C \begin{pmatrix} f_u\frac{x_C}{z_C} + u_0 \\ f_v\frac{y_C}{z_C} +v_0 \\ 1 \end{pmatrix} \\ + &\equiv \begin{pmatrix} f_u x_C + z_C u_0 \\ f_v y_C + z_C v_0 \\ z_C \end{pmatrix} \equiv + \begin{pmatrix} f_u & 0 & u_0 & 0 \\ 0 & f_v & v_0 & 0 \\ 0 & 0 & 1 & 0 \end{pmatrix} \begin{pmatrix} x_C \\ y_C \\ z_C \\ 1 \end{pmatrix} \equiv + \matr{P}_\text{int} \tilde{\vec{M}}_C + \end{split} + \] + + \begin{remark} + The equation can be written to take account of the arbitrary scale factor $k$ as: + \[ k\tilde{\vec{m}} = \matr{P}_\text{int} \tilde{\vec{M}}_C \] + or, if $k$ is omitted, as: + \[ \tilde{\vec{m}} \approx \matr{P}_\text{int} \tilde{\vec{M}}_C \] + \end{remark} + + \begin{remark} + In projective space, we can also project in Euclidean space the point at infinity of parallel 3D lines in CRF with direction $(a, b, c)$: + \[ + \tilde{\vec{m}}_\infty \equiv + \matr{P}_\text{int} \begin{pmatrix} a \\ b \\ c \\ 0 \end{pmatrix} \equiv + \begin{pmatrix} f_u & 0 & u_0 & 0 \\ 0 & f_v & v_0 & 0 \\ 0 & 0 & 1 & 0 \end{pmatrix} \begin{pmatrix} a \\ b \\ c \\ 0 \end{pmatrix} \equiv + \begin{pmatrix} f_u a + c u_0 \\ f_v b + c v_0 \\ c \end{pmatrix} \equiv + c\begin{pmatrix} f_u \frac{a}{c} + u_0 \\ f_v \frac{b}{c} + v_0 \\ 1 \end{pmatrix} + \] + Therefore, the Euclidean coordinates are: + \[ \vec{m}_\infty = \begin{pmatrix} f_u \frac{a}{c} + u_0 \\ f_v \frac{b}{c} + v_0 \end{pmatrix} \] + + Note that this is not possible when $c = 0$. + \end{remark} +\end{description} \ No newline at end of file