 # An Analysis Of Pin Hole Camera Model

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\chapter{Multiview Geometry}
This chapter deals with the camera models, projection of scene into image plane and two view geometry.
\section{Pin Hole Camera Model}
A camera is a mapping between the 3D world (object space) and a 2D image. The principal camera of interest in this thesis is central projection. This describes about Pin Hole camera model which are matrices with particular properties that represent the camera mapping.\\
\begin{figure}[h]
\includegraphics[scale = .5]{Pine_hole_camera_model.jpg}\centering \caption{Pin Hole Camera Model.} \label{fig:Pin Hole Camera Model} \small Figure shows the projection of World Points $X1$,$X2$ on image plane $x1$ and $x2$ are respective projections. $f$ is the focal length of the camera,
For ${\bf n}$ calibration points we have ${\bf 2n}$ equations:\\
\begin{equation}
{\bf AP=0}
\end{equation}

Camera calibration can be done by known patterns. Chess pattern is one of the standard pattern used for calibration. We know the number of rectangles and size of rectangles in the chess board pattern and pattern is a plane. The units that we use in calibration will be same for all other steps in SFM, using these information and $zang's$ method we can calibrate the camera. After calibration we can get following information about the camera.\\
${\bf 1)}$ Focal lenth of camera\\
${\bf 2)}$ Distortion parameters\\
${\bf 3)}$ Image centre or optic centre\\
${\bf 4)}$ Size of the pixel\\
These parameters will be constant for a camera unless it's focal length has not been changed. From this we can form camera matrix ${\bf K}$:
${\bf K}=\begin{pmatrix} f_x&0&c_x\\ 0&f_y&c_y\\ 0&0&1 \end{pmatrix}$
We will be using this camera matrix to calculate $Essential$ $matrix$ from fundamental
What is the relation betwen the corresponding image points ${\bf x}$ and ${\bf x'}$ ? As shown in figure 2.1 the image point ${\bf X}$, and camera centres are coplanar. This plane is named as ${\bf \pi}$. The rays back-projected from ${\bf x}$ and ${\bf x'}$ interset at ${\bf X}$, and the rays are coplanar, lying in ${\bf \pi}$. It is this latter property that is of most significance in searching from correspondence.\\

Supposing now that we know only ${\bf x}$, we may ask how the corresponding point ${\bf x'}$ is constrained. The ${\bf \pi}$ is determined by the baseline and the ray defined by ${\bf x}$. From above we know that the ray corresponding to the (unknown) point ${\bf x'}$ lies in ${\bf \pi}$, hence the point ${\bf x'}$ lies on the line of intersection ${\bf l'}$ of ${\bf \pi}$ with the second image plane. This line ${\bf l'}$ is the image in the second view of the ray back-projected from ${\bf x}$. In terms of stereo correspondence algorithm the benefit is that the search for the point corresponding to ${\bf x}$ need not cover the entire image plane but can be restricted to the line \${\bf

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