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%\usepackage[dvips,bookmarks=false,colorlinks,urlcolor=blue,pdftitle={
\usepackage{caption}
\usepackage{subcaption}
\usepackage{amsmath}
\usepackage{amssymb}
\usepackage{amsfonts}
\usepackage{amsthm}

\def\vO{{\bf O}}
\def\vP{{\bf P}}
\def\vp{{\bf p}}
\def\vx{{\bf x}}
\def\vt{{\bf t}}
\def\vl{{\bf l}}
\def\vm{{\bf m}}
\def\vX{{\bf X}}
\def\mS{{\bf S}}
\def\mT{{\bf T}}
\def\mH{{\bf H}}
\def\mA{{\bf A}}
\def\mM{{\bf M}}
\def\mP{{\bf P}}
\def\tx{{\tilde{\bf x}}}
\def\ta{{\tilde{\bf a}}}
\def\tb{{\tilde{\bf b}}}
\def\tc{{\tilde{\bf c}}}
\def\hn{{\bf \hat{n}}}
\def\hv{{\bf \hat{v}}}
\def\hh{{\bf \hat{h}}}
\def\vh{{\bf h}}
\def\vs{{\bf s}}
\def\hs{{\bf \hat{s}}}
\newcommand{\Ev}{\mathbf{E}}
\newcommand{\Fv}{\mathbf{F}}

\usepackage[bookmarks=false,colorlinks,urlcolor=blue,pdftitle={Take-home Quiz 5},pdfauthor={Matthew O'Toole}]{hyperref}

\begin{document}

\begin{centering}
	{\large 16-385 Computer Vision, Fall 2020\\}
	\vspace{2ex}
	{\LARGE Take-home Quiz 5\\}
	\vspace{2ex}
	{\large Due Date: Monday October 19, 2020 23:59\\}
\end{centering}

\bigskip

\section*{Question 1 \emph{\small(5 points)}}

As we discussed in class, two cameras are said to form a \emph{rectified pair} if their camera coordinate systems differ only by a translation of their origins (the camera centers) along a direction that is parallel to either the $x$ or $y$ axis of the coordinate systems.
\begin{enumerate}
	\item Derive an expression for the essential matrix $\Ev$ of a rectified pair.
	\item Prove that the epipolar lines of a rectified pair are parallel to the axis of translation.
\end{enumerate}

\section*{Question 2 \emph{\small(5 points)}}

Consider three images $I_1$, $I_2$ and $I_3$ that have been captured
by a system of three cameras, and suppose the fundamental matrices
${\bf F}_{13}$ and ${\bf F}_{23}$ are known. (Notation: the matrix
${\bf F}_{ij}$ satisfies the equation $\vx_j^\top{\bf F}_{ij}\vx_i=0$
for any correspondence $\vx_i\leftrightarrow\vx_j$ between images
$I_i$ and $I_j$.) In general, given a point $\vx_1$ in $I_1$ and a
corresponding point $\vx_2$ in $I_2$, the corresponding point in
$\vx_3$ in $I_3$ is uniquely determined by the fundamental matrices
${\bf F}_{13}$ and ${\bf F}_{23}$.

\begin{enumerate}
\item Write an expression for $\vx_3$ in terms of $\vx_1$, $\vx_2$,
  ${\bf F}_{13}$ and ${\bf F}_{23}$.

\item Describe a degenerate configuration of three cameras for which
the point $\vx_3$ cannot be uniquely determined by this expression.
\end{enumerate}

\noindent \emph{Hint}: Consider the epipolar geometry of the
situation.  Draw a picture!

\section*{Instructions}

\begin{enumerate}
    \item \textbf{Integrity and collaboration:} Students are encouraged to work
    in groups but each student must submit their own work. If you work as a
    group, include the names of your collaborators in your write up. Plagiarism
    is strongly prohibited and may lead to failure of this course.

    \item \textbf{Questions:} If you have any questions, please look at Piazza
    first. Other students may have encountered the same problem, and it may be solved
    already. If not, post your question on the discussion board. Teaching staff will
    respond as soon as possible.

    \item \textbf{Write-up:} Your write-up should be typeset in \LaTeX and should consist of your answers to the theory questions. Please note that we
    \textbf{\color{red}do not} accept handwritten scans for your write-up in quizzes.

    \item \textbf{Submission:} Your submission for this take-home quiz should be a
    PDF file, \texttt{<andrew-id.pdf>}, with your write-up. \textbf{\color{red}Please do not submit ZIP files.}
\end{enumerate}

\end{document}
