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\begin{document} 
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\psetheader{Fall Semester, 1996}{Practice Problems -- Dec 3, 1996}

\leftbold{Problem: Object recognition by image alignment}

(a) In this problem, we will examine rigid objects with sharp edges
undergoing 3D rotation about the vertical axis.  Let $P_1$ and $P_2$ be
two images of an object O rotating in depth around the vertical
y-axis.  $P_2$ is obtained from $P_1$ by a rotation of angle $\theta$.
Let $P_u$ be a third image of the same object obtained from $P_1$ by a
rotation of angle $\phi$ around y-axis.  Let the projections of a
point $p=(x,y,z) \in O$ in the three images are given by:
\begin{eqnarray*}
p_1 & = & (x_1, y_1) = (x,y) \\
p_2 & = & (x_2, y_2) = (x \cos \theta + z \sin \theta, y)\\
p_u & = & (x_u, y_u) = (x \cos \phi + z \sin \phi, y)\\
\end{eqnarray*}

Show that two constants $\alpha$ and $\beta$ exist such that for every
point $p \in P_u$:
\[ x_u = \alpha x_1 + \beta x_2 \]

Answer: \\ \onebox{6in}{2in}

\clearpage

(b) Recognizing a triangle on a plane

Consider the following two views of a triangle which differ only by a
rotation. 

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\centerline{\psfig{figure=/home/sc/yip/6034/1995/alignment.ps,height=2in,width=4in}}

Which of the following 1D projections are consistent with a rotation
of the above triangle?

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\centerline{\psfig{figure=/home/sc/yip/6034/1995/alignment-options.ps,height=2in,width=4in}}


(c) Suppose we allow translation on the plane in addition to rotation.
Is it still possible to recognize the unknown view of a triangle using
only two stored views?  Why or why not?

Answer: \\ \onebox{6in}{2in}

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