\documentstyle[12pt,fullpage,psfig,subfigure]{article}

\newcommand{\lecturetitle}[4]{
  \noindent
  CS 280, Computer Vision (Prof. Jitendra Malik) \hfill #3\\
  Lecture \##1 (#2) \hfill {\bf DRAFT} Notes by #4\\[-.05in]
  \mbox{}\hrulefill \mbox{}\\}

\begin{document}

% MODIFY THE FOLLOWING TO INCLUDE THE LECTURE #, TOPIC, DATE, and YOUR NAME
\lecturetitle{9}{Rotation translation ambiguity, Stereopsis}{Feb 22, 1999}{Ren\'e Vidal}

% INCLUDE AN OUTLINE OF THE CLASS TOPICS

\begin{itemize} 
  \item Rotation v/s translation ambiguity 

\item Stereopsis: description, vergence and version movements, disparity
\end{itemize}

% THE NOTES WILL GO HERE

\section{Rotation v/s translation ambiguity}
Figure \ref{rtambig} shows the optical flow produced by a rotation
around Y axis and a translation along X axis. It can be observed that
near the origin both flows are pretty similar, which difficults the
task of estimating the motion, specially in the presence of
noise. Errors in motion estimation would also affect depth
estimation. For example, if we overestimate the rotational part, depth
estimation will be flat and if we overestimate the translational part,
depth will be cute. This is because depth affects only the
translational part of the optical flow. To avoid this problem (called
{\bf Bas-Relief Ambiguity}) it is necessary to have a large field of
view (FOV).


\begin{figure}[h]
\centering
\mbox {
\subfigure[Rotation around Y axis]{\psfig{file=hw1/ry.eps,height=2.5in}} \quad
\subfigure[Translation along X axis]{\psfig{file=hw1/tx.eps,height=2.5in}}
}
\caption{Bas-Relief Ambiguity}
\label{rtambig}
\end{figure}


\section{Stereopsis}

\subsection{Description}

Consider two cameras LE (left eye) and RE (right eye), with fixed
optical centers OL and OR, located at a distance $b$ (See Figure
\ref{stereo}). Each camera has 3 degrees of freedom (d.o.f.)
corresponding to rotation around the 3 axes. In the case of human
beings, these 6 d.o.f. are restricted by the following:

\begin{itemize}
\item {\bf Listing's law}: the axis of rotation of the eye lies on the
front parallel plane, i.e., rotation around the optical axis (Z axis)
is restricted.


\item {\bf Fixating system}: the eyes move in such a way that both
point to the same arbitrary point in the space, namely fixation point.
\end{itemize}
These two restrictions reduce the d.o.f. of the stereo vision system
to 3.

\begin{figure}[h]
\centerline {\psfig{file=stereo.eps,height=3in}}
\caption{Stereopsis}
\label{stereo}
\end{figure}

\subsection{Vergence and version movements}

For the time being, restrict ourselves to fixation points located
in a horizontal plane. In this case we can classify the motion of the
eyes in two different movements (See Figure \ref{vergence}):

\begin{itemize}
\item {\bf Vergence movement}: it is a movement in which both eyes
rotate in the same angle, but in opposite directions. In this case the
{\bf angle of vergence} changes while the fixation point moves along
the {\bf gaze direction}. Note that if the angle of vergence is not
too small, it can be used to measure depth.

\item {\bf Version movement}: it is a movement in which gaze direction
changes, but angle of vergence remains constant. From Euclidean
geometry we now that this happens when the fixation point lies on a
circle. This circle is called {\bf Viet-Muller Circle}.

\end{itemize}



\subsection{Disparity}

Consider that the stereo system fixated at a point $P$, whose
projections in LE and RE are $P_L$ and $P_R$, respectively (See Figure
\ref{disparity}). Now look at a point $P^{'}$ with projections
$P_L^{'}$ and $P_R^{'}$. Define $\theta_L = \angle P_L O_L P_L^{'}$
and $\theta_R = \angle P_R O_R P_R^{'}$. Then the {\bf disparity} is
defined as $\theta_R - \theta_L$.  Note that the disparity remains
constant if the angle of vergence is fixed.


\begin{figure}
\centering
\mbox { 
\subfigure[Vergence movement]{\psfig{file=vergence.eps,height=3.5in}} 
\qquad \qquad
\subfigure[Version movement]{\psfig{file=version.eps,height=2.6in}}}
\caption{\label{vergence}Vergence and version movements}
\end{figure}

%\begin{figure}[b]

%\end{figure}


\subsubsection{Example}

Consider the configuration of Figure \ref{example}. For $\theta$
small we have

\begin{eqnarray*}
Disparity &=& 2 \delta \theta \\
\theta &\approx& \tan \theta = {\frac b {2z}}  \\
\theta -\delta\theta&\approx& {\frac b {2(z+\delta z)}}  \\
\delta\theta &\approx& {\frac b {2z}}-{\frac b {2(z+\delta z)}} \\
Disparity &\approx& {\frac {b\delta  z}{z^2}}
\end{eqnarray*}

In the case of human beings, best stereo accuracy is obtained for a
disparity of approximately 5''= $2.42\times 10^{-5}$ radians. 

For $b=6$ cm, $z=100$ cm,  $\delta z = 2.42\times 10^{-5}\times
100^2/6=0.04$ cm (good precision).

For $b=6$ cm, $z=1000$ mt,  $\delta z = 2.42\times 10^{-5}\times
1000^2/6=404$ mt (bad precision).

Therefore a stereo system works well for short distances.

\begin{figure}[h]
\centerline {\psfig{file=disparity.eps,height=2.6in}}
\caption{Disparity}
\label{disparity}
\vspace{1cm}
\centerline {\psfig{file=example.eps,height=2.6in}}
\caption{Example}
\label{example}
\end{figure}

\end{document}

