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Add IPCV pinhole and perspective projection
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1
src/image-processing-and-computer-vision/module1/ainotes.cls
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1
src/image-processing-and-computer-vision/module1/ainotes.cls
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../../ainotes.cls
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src/image-processing-and-computer-vision/module1/ipcv1.tex
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src/image-processing-and-computer-vision/module1/ipcv1.tex
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\documentclass[11pt]{ainotes}
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\title{Image Processing and Computer Vision\\(Module 1)}
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\date{2023 -- 2024}
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\def\lastupdate{{PLACEHOLDER-LAST-UPDATE}}
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\begin{document}
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\makenotesfront
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\input{./sections/_image_acquisition.tex}
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\end{document}
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\chapter{Image acquisition and formation}
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\section{Pinhole camera}
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\begin{description}
|
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\item[Imaging device] \marginnote{Imaging device}
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Gathers the light reflected by 3D objects in a scene and creates a 2D representation of them.
|
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|
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\item[Computer vision] \marginnote{Computer vision}
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Infer knowledge of the 3D scene from 2D digital images.
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\end{description}
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|
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\begin{description}
|
||||
\item[Pinhole camera] \marginnote{Pinhole camera}
|
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Imaging device where the light passes through a small pinhole and hits the image plane.
|
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Geometrically, the image is obtained by drawing straight rays from the scene to the image plane passing through the pinhole.
|
||||
|
||||
\begin{remark}
|
||||
Larger aperture size of the pinhole results in blurry images (circle of confusion),
|
||||
while smaller aperture results in sharper images but requires longer exposure time (as less light passes through).
|
||||
\end{remark}
|
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|
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\begin{remark}
|
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The pinhole camera is a good approximation of the geometry of the image formation mechanism of modern imaging devices.
|
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\end{remark}
|
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|
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\begin{figure}[h]
|
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\begin{subfigure}{.4\textwidth}
|
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\centering
|
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\includegraphics[width=0.8\linewidth]{./img/pinhole.png}
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\caption{Pinhole camera model}
|
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\end{subfigure}
|
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\begin{subfigure}{.45\textwidth}
|
||||
\centering
|
||||
\includegraphics[width=0.7\linewidth]{./img/pinhole_hole_size.png}
|
||||
\caption{Images with varying pinhole aperture size}
|
||||
\end{subfigure}
|
||||
\end{figure}
|
||||
\end{description}
|
||||
|
||||
|
||||
|
||||
\section{Perspective projection}
|
||||
\marginnote{Perspective projection}
|
||||
|
||||
Geometric model of a pinhole camera.\\
|
||||
|
||||
\begin{minipage}{0.65\textwidth}
|
||||
\begin{description}
|
||||
\setlength\itemsep{0.2em}
|
||||
\item[Scene point] $M$ (the object in the real world).
|
||||
\item[Image point] $m$ (the object in the image).
|
||||
\item[Image plane] $I$.
|
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\item[Optical center] $C$ (the pinhole).
|
||||
\item[Image center/piercing point] $c$ (intersection between the optical axis -- the line orthogonal to $I$ passing through $C$ -- and $I$).
|
||||
\item[Focal length] $f$.
|
||||
\item[Focal plane] $F$.
|
||||
\end{description}
|
||||
\end{minipage}
|
||||
\begin{minipage}{0.3\textwidth}
|
||||
\centering
|
||||
\includegraphics[width=\linewidth]{./img/perspective_projection1.png}
|
||||
\end{minipage}\\
|
||||
|
||||
\begin{minipage}{0.55\textwidth}
|
||||
\begin{itemize}[leftmargin=*]
|
||||
\item $u$ and $v$ are the horizontal and vertical axis of the image plane, respectively.
|
||||
\item $x$ and $y$ are the horizontal and vertical axis of the 3D reference system, respectively,
|
||||
and form the \textbf{camera reference system}. \marginnote{Camera reference system}
|
||||
\end{itemize}
|
||||
|
||||
\begin{remark}
|
||||
For the perspective model, the coordinate systems $(U, V)$ and $(X, Y)$ must be parallel.
|
||||
\end{remark}
|
||||
\end{minipage}
|
||||
\begin{minipage}{0.35\textwidth}
|
||||
\centering
|
||||
\includegraphics[width=\linewidth]{./img/perspective_projection2.png}
|
||||
\end{minipage}
|
||||
|
||||
\begin{description}
|
||||
\item[Scene--image mapping] \marginnote{Scene--image mapping}
|
||||
The equations to map scene points into image points are the following:
|
||||
\[ u = x \frac{f}{z} \hspace*{3em} v = y \frac{f}{z} \]
|
||||
|
||||
\begin{proof}
|
||||
This is the consequence of the triangle similarity theorems.
|
||||
|
||||
\begin{minipage}{0.45\textwidth}
|
||||
\[
|
||||
\begin{split}
|
||||
\frac{u}{x} = -\frac{f}{z} &\iff u = -x \frac{f}{z} \\
|
||||
\frac{v}{y} = -\frac{f}{z} &\iff v = -y \frac{f}{z}
|
||||
\end{split}
|
||||
\]
|
||||
The minus is needed as the axes are inverted
|
||||
\end{minipage}
|
||||
\begin{minipage}{0.50\textwidth}
|
||||
\begin{figure}[H]
|
||||
\centering
|
||||
\includegraphics[width=0.7\textwidth]{./img/_perspective_projection_eq_proof.pdf}
|
||||
\caption{\small Visualization of the horizontal axis. The same holds on the vertical axis.}
|
||||
\end{figure}
|
||||
\end{minipage}
|
||||
|
||||
By inverting the axis horizontally and vertically (i.e. inverting the sign),
|
||||
the image plane can be adjusted to have the same orientation of the scene:
|
||||
\[ u = x \frac{f}{z} \hspace*{3em} v = y \frac{f}{z} \]
|
||||
\end{proof}
|
||||
|
||||
\begin{remark}
|
||||
The image coordinates are a scaled version of the scene coordinates.
|
||||
The scaling is inversely proportioned with respect to the depth.
|
||||
\begin{itemize}
|
||||
\item The farther the point, the smaller the coordinates.
|
||||
\item The larger the focal length, the bigger the object is in the image.
|
||||
\end{itemize}
|
||||
|
||||
\begin{figure}[H]
|
||||
\centering
|
||||
\includegraphics[width=0.4\textwidth]{./img/perspective_projection_proportion.png}
|
||||
\caption{Coordinate space by varying focal length}
|
||||
\end{figure}
|
||||
\end{remark}
|
||||
|
||||
\begin{remark}
|
||||
The perspective projection mapping is not a bijection:
|
||||
\begin{itemize}
|
||||
\item A scene point is mapped into a unique image point.
|
||||
\item An image point is mapped onto a 3D line.
|
||||
\end{itemize}
|
||||
Therefore, reconstructing the 3D structure of a single image is an ill-posed problem (i.e. it has multiple solutions).
|
||||
|
||||
\begin{figure}[H]
|
||||
\centering
|
||||
\includegraphics[width=0.3\textwidth]{./img/perspective_projection_loss.png}
|
||||
\caption{Projection from scene and image points}
|
||||
\end{figure}
|
||||
\end{remark}
|
||||
\end{description}
|
||||
|
||||
|
||||
\subsection{Stereo geometry}
|
||||
|
||||
\begin{description}
|
||||
\item[Stereo vision] \marginnote{Stereo vision}
|
||||
Use multiple images to triangulate the 3D position of an object.
|
||||
|
||||
\item[Stereo correspondence] \marginnote{Stereo correspondence}
|
||||
Given a point $L$ in an image, find the corresponding point $R$ in another image.
|
||||
|
||||
Without any assumptions, an oracle is needed to determine the correspondences.
|
||||
\end{description}
|
||||
|
||||
\begin{description}
|
||||
\item[Standard stereo geometry] \marginnote{Standard stereo geometry}
|
||||
Given two reference images, the following assumptions must hold:
|
||||
\begin{itemize}
|
||||
\item The $X$, $Y$, $Z$ axes are parallel.
|
||||
\item The cameras that took the two images have the same focal length $f$ (coplanar image planes) and
|
||||
the images have been taken at the same time.
|
||||
\item There is a horizontal translation $b$ between the two cameras (baseline).
|
||||
\item The disparity $d$ is the difference of the $U$ coordinates of the object in the left and right image.
|
||||
\end{itemize}
|
||||
|
||||
\begin{theorem}[Fundamental relationship in stereo vision] \marginnote{Fundamental relationship in stereo vision}
|
||||
If the assumptions above hold, the following equation holds:
|
||||
\[ z = b\frac{f}{d} \]
|
||||
|
||||
\begin{proof}
|
||||
Let $P_L = \begin{pmatrix}x_L & y & z\end{pmatrix}$ and $P_R = \begin{pmatrix}x_R & y & z\end{pmatrix}$ be the
|
||||
coordinates of the object $P$ with respect to the left and right camera reference system, respectively.
|
||||
Let $p_L = \begin{pmatrix}u_L & v\end{pmatrix}$ and $p_R = \begin{pmatrix}u_R & v\end{pmatrix}$
|
||||
be the coordinates of the object $P$ in the left and right image plane, respectively.
|
||||
|
||||
By assumption, we have that $P_L - P_R = \begin{pmatrix} b & 0 & 0 \end{pmatrix}$, where $b$ is the baseline.
|
||||
|
||||
\begin{minipage}{0.6\textwidth}
|
||||
|
||||
By the perspective projection equation, we have that:
|
||||
\[ u_L = x_L\frac{f}{z} \hspace{3em} u_R = x_R\frac{f}{z} \]
|
||||
Disparity is computed as follows:
|
||||
\[ d = u_L - u_R = x_L\frac{f}{z} - x_R\frac{f}{z} = b\frac{f}{z} \]
|
||||
We can therefore obtain the $Z$ coordinate of $P$ as:
|
||||
\[ z = b\frac{f}{d} \]
|
||||
\end{minipage}
|
||||
\begin{minipage}{0.3\textwidth}
|
||||
\begin{center}
|
||||
\includegraphics[width=\textwidth]{./img/_standard_stereo_geometry.pdf}
|
||||
\end{center}
|
||||
Note: the $Y$/$V$ axes are not in figure.
|
||||
\end{minipage}\\
|
||||
\end{proof}
|
||||
|
||||
\begin{remark}
|
||||
Disparity and depth are inversely proportional:
|
||||
the disparity of two points decreases if the points are farther in depth.
|
||||
\end{remark}
|
||||
\end{theorem}
|
||||
|
||||
\begin{description}
|
||||
\item[Stereo matching] \marginnote{Stereo matching}
|
||||
If the assumptions for standard stereo geometry hold,
|
||||
to find the object corresponding to $p_L$ in another image,
|
||||
it is sufficient to search along the horizontal axis of $p_L$ looking for the same colors or patterns.
|
||||
|
||||
\begin{figure}[h]
|
||||
\centering
|
||||
\includegraphics[width=0.5\textwidth]{./img/stereo_matching.png}
|
||||
\caption{Example of stereo matching}
|
||||
\end{figure}
|
||||
\end{description}
|
||||
|
||||
\item[Epipolar geometry] \marginnote{Epipolar geometry}
|
||||
Approach applied when the two cameras are no longer aligned according to the standard stereo geometry assumption.
|
||||
Still, the focal lengths and the roto-translation between the two cameras must be known.
|
||||
|
||||
Given two images, we can project the epipolar line related to the point $p_L$ in the left plane onto the right plane
|
||||
to reduce the problem of correspondence search to a single dimension.
|
||||
|
||||
\begin{figure}[H]
|
||||
\centering
|
||||
\includegraphics[width=0.3\textwidth]{./img/_epipolar_geometry.pdf}
|
||||
\caption{Example of epipolar geometry}
|
||||
\end{figure}
|
||||
|
||||
\begin{remark}
|
||||
It is nearly impossible to project horizontal epipolar lines and
|
||||
searching through oblique lines is awkward and computationally less efficient than straight lines.
|
||||
\end{remark}
|
||||
|
||||
\begin{description}
|
||||
\item[Rectification] \marginnote{Rectification}
|
||||
Transformation applied to convert epipolar geometry to a standard stereo geometry.
|
||||
\begin{figure}[H]
|
||||
\centering
|
||||
\begin{subfigure}{0.35\linewidth}
|
||||
\centering
|
||||
\includegraphics[width=\linewidth]{./img/rectification_no.png}
|
||||
\caption{Images before rectification}
|
||||
\end{subfigure}
|
||||
\begin{subfigure}{0.35\linewidth}
|
||||
\centering
|
||||
\includegraphics[width=\linewidth]{./img/rectification_yes.png}
|
||||
\caption{Images after rectification}
|
||||
\end{subfigure}
|
||||
\end{figure}
|
||||
\end{description}
|
||||
\end{description}
|
||||
|
||||
|
||||
\subsection{Ratios and parallelism}
|
||||
|
||||
Given a 3D line of length $L$ lying in a plane parallel to the image plane at distance $z$,
|
||||
then its length $l$ in the image plane is:
|
||||
\[ l = L\frac{f}{z} \]
|
||||
|
||||
In all the other cases (i.e. when the line is not parallel to the image plane),
|
||||
the ratios of lengths and the parallelism of lines are not preserved.
|
||||
|
||||
\begin{figure}[h]
|
||||
\centering
|
||||
\includegraphics[width=0.3\textwidth]{./img/_perspective_projection_ratio.pdf}
|
||||
\caption{Example of not preserved ratios. It holds that $\frac{\overline{AB}}{\overline{BC}} \neq \frac{\overline{ab}}{\overline{bc}}$.}
|
||||
\end{figure}
|
||||
|
||||
\begin{description}
|
||||
\item[Vanishing point] \marginnote{Vanishing point}
|
||||
Intersection point of lines that are parallel in the scene but not in the image plane.
|
||||
|
||||
\begin{figure}[h]
|
||||
\centering
|
||||
\includegraphics[width=0.7\textwidth]{./img/_vanishing_point.pdf}
|
||||
\caption{Example of vanishing point}
|
||||
\end{figure}
|
||||
\end{description}
|
||||
|
||||
|
||||
|
||||
\section{Lens}
|
||||
|
||||
\begin{description}
|
||||
\item[Depth of field (DOF)] \marginnote{Depth of field (DOF)}
|
||||
Distance at which a scene point is on focus (i.e. when all its light rays gathered by the imaging device hit the image plane at the same point).
|
||||
|
||||
\begin{remark}
|
||||
Because of the small size of the aperture, a pinhole camera has infinite depth of field
|
||||
but requires a long exposure time making it only suitable for static scenes.
|
||||
\end{remark}
|
||||
|
||||
\item[Lens] \marginnote{Lens}
|
||||
A lens gathers more light from the scene point and focuses it on a single image point.
|
||||
|
||||
This allows for a smaller exposure time but limits the depth of field (i.e. only a limited range of distances in the image can be on focus at the same time).
|
||||
|
||||
\begin{description}
|
||||
\item[Thin lens equation] \marginnote{Thin lens equation}
|
||||
$\frac{1}{u} + \frac{1}{v} = \frac{1}{f}$
|
||||
\end{description}
|
||||
\end{description}
|
||||
Reference in New Issue
Block a user