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\begin{centering}
	{\large 16-385 Computer Vision, Fall 2020\\}
	\vspace{2ex}
	{\LARGE Take-home Quiz 10\\}
	\vspace{2ex}
	{\large Due Date: Tuesday November 24, 2020 23:59\\}
\end{centering}

\bigskip

\section*{Question 1}

As we saw in class, the \emph{brightness constancy} equation
\begin{equation}
I_x(x,y)u(x,y) + I_y(x,y)v(x,y) + I_t(x,y)=0,
\end{equation}
does not provide sufficient constraints for uniquely determining the optical flow $(u(x,y), v(x,y))$ at each pixel location $(x,y)$ of an image. A standard way to deal with this ambiguity, first proposed by Lukas and Kanade in 1981, is to assume that, for all pixels $(x,y)\in W$ within a patch $W$ of size $N\times N$, the optical flow is constant:
\begin{equation}
(u(x,y), v(x,y)) = (u_W, v_W) \equiv\V_W, \, \forall (x,y)\in W.
\end{equation}
We will call this the \emph{constant flow} assumption.
\begin{enumerate}
\item Show that the constant flow assumption can be equivalently expressed in the form of a heterogeneous linear system of the form $\A \V_W = \bb$, and derive expressions for the matrix $\A$ and the vector $\bb$.
\item Show that solving this heterogeneous linear system in the least-squares sense requires inverting the \emph{covariance matrix}:
\begin{equation}
{\cal M}_W= \left[\begin{array}{cc} \sum\limits_{(x,y)\in W} I_x(x,y)I_x(x,y) & \sum\limits_{(x,y)\in W} I_x(x,y)I_y(x,y)\\ \sum\limits_{(x,y)\in W} I_y(x,y)I_x(x,y) & \sum\limits_{(x,y)\in W} I_y(x,y)I_y(x,y)
\end{array}\right].
\end{equation}
\item What are the implications of the above about the patches $W$ in an image for which we can reliably estimate the optical flow $\V_W$? \emph{Hint:} Think about where else in class we have seen the covariance matrix ${\cal M}_W$, and about the properties this matrix needs to have in order for the system $\A \V_W = \bb$ to have a stable solution.
\end{enumerate}

\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}

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