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{\bf CS 280 FINAL EXAM,  1230-330pm, May 10, 1996} 
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\noindent
Make sure you have both pages of this exam. Each question is worth 20 points. 
Good luck.
\begin{enumerate}
\item Prove that under perspective projection, the image of a plane
has a vanishing line, the {\em horizon}, that depends only on the
surface
normal ${\rm{\bf n}}$ but not on the distance of the plane from the origin.


\item Prove that for any two mutually visible points $x$ and $y$ in space, the
radiance leaving point $x$ in the direction of point $y$ is the same as the
radiance impinging  on point $y$ from the direction of point $x$.

\item Junctions can be localized by finding the point that minimizes
the sum of perpendicular distances to all lines in a given image
neighborhood.
Formulate this as a least squares problem and derive its solution in
terms
of inverting a windowed image second moment matrix.

\item Show how the Bayesian formulation of teh problem of 

\item Suppose that a color--sensing system has three types of sensors
and that the spectral sensitivity of each type is a sum of scaled versions of
the human cone sensitivities. Show that two metameric colors (i.e. colors
which appear identical to human observers) will produce identical signals
in the sensors.

\item Consider the line drawing of a smooth curved surface which  occludes itself
along a limb curve. Prove that, under orthographic projection, the surface normal
is perpendicular to the projection of the limb curve in the image
plane. Give an example to show that the limb curve need not be
perpendicular to the line of sight.

\item Consider a binocular viewing geometry where both the eyes are
fixating
at a point straight ahead (gaze angle $=0$). Show that the disparity
is
$I \Delta z / z^2$ where $I$ is the interocular distance and $z$ is
the fixation distance.



\item This exercise is aimed at testing your understanding of epipolar
geometry. If the displacement between the two camera positions is
given
by $({\bf R}, {\bf t})$, show that the corresponding points ${\bf x}$,
${\bf x^\prime}$ in the two image planes satisfy a relation of the
form
${\bf x^T M x^\prime} = 0$. Here ${\bf x}$, ${\bf x^\prime}$ are in
homogeneous
coordinates.


\item  Derive a differential
technique for estimating the 1--D affine transform between two 1--D signals
$f$ and $g$. It is known that $f$ is a scaled copy of $g$
i.e. $f(x)=g(ax)$ with $a\approx 1$. You are given the values
$f(x_i)$, $g(x_i)$ at a series of grid points $x_i$. 



\item For the following object recognition techniques:
\begin{enumerate}
\item The alignment method due to Huttenlocher and Ullman.
\item The appearance-based approach based on principal component analysis.
\end{enumerate}
indicate a class of objects for which 
\begin{itemize}
\item technique a. works but b. does not.
\item technique b. works but a. does not.
\item neither is likely to work.
\end{itemize}

\item In this exercise we will study the optimum Bayes classifier for
a simple two class problem. The two classes have the same prior
probabilities. We are asked to classify on the basis of the
measurement
of the scalar feature $x$. It is known that the probability density
distribution
for the value of the feature $x$
\begin{itemize}
\item  Given that the object is from class
1,  follows a Gaussian distribution with mean $\mu_1$
and variance $\sigma_1^2$.  
\item  Given that the object is from class
2,  follows a Gaussian distribution with mean $\mu_2$
and variance $\sigma_2^2$.  
\end{itemize}
What is the optimum Bayes decision rule for this problem? The Bayes risk?
\end{enumerate}
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