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\title{Recovery of 3-D coordinates of Environmental Points}
\author{R. Dutta}
\begin{document}
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\section{Introduction}
This is a supplementary write-up for the paper entitled
{\em A Data Set for Quantitative Motion Analysis}. It contains
the following
\begin{enumerate}
\item Further details on the data collection procedure for
determining the 3-D position of selected environmental points.
\item Procedure for conversion of raw data to 3-D coordinates.
\item The 3-D position of selected environmental points at the various
sites at which the image sequences were collected.
\end{enumerate}

The primary focus of this report is to give estimates of the reliability
of the data.

\section{Data Collection Procedure}
The data was collected in two stages. In both stages two theodolites 
were used. One of the theodolites had a laser range finder with which
to check the results. The description of the two stages follow.
\begin{description}
\item [Stage 1 ] Researchers from UMass measured the horizontal and vertical
angles for various environmental locations while collecting the image
sequences. The number of environmental points for which such readings were
taken were very few (The only exception being the rocket field site).
Stakes were put at the positions at which theodolites had been
placed.
\item [Stage 2 ] After a few weeks researchers from Martin Marietta visited
the sites and placed the theodolites at the positions at which the stakes
had been put and took additional measurements for a larger number of points
at the various sites.
\end{description}

{\bf Coordinate System -} Two theodolites were used. They are referred to as
Theodolite-M and Theodolite-D. Theodolite-M was equipped with a laser range
finder. As shown in Fig 1 the coordinate of Theodolite M is (0,0,0). The
position of enviromental objects are given in the XYZ coordinate system.
The positive Y-axis is directed into the paper in Fig. 1.

{\bf Drawbacks} - We could not take all the readings while
taking the image sequences because of shortage of personnel and time.
Also, before taking readings 
it was not possible to determine if the index error in the vertical
circle of the theodolites was zero or not. However, later on it was
concluded from other measurements that for Theodolite-M the index error was
not very different from
zero. Hence the Z-coordinates of the environmental points have been
determined with the vertical readings of Theodolite-M. The heights of the
theodolites, even when they are placed on the same position on different
days,
would vary somewhat. This would cause the Z-coordinates of the
same environmental point to differ on different days. While it is
theoretically possible to make a correction for this it could not be
done in our case becuase we did not always have reliable
common points in Stage 1
and Stage 2 \footnote{Sometimes the one or two
common points we had gave different correction
factors. Therefore, no correction was done.}



\section{Data Conversion Procedure}
Let $d$ be the distance between the theodolites as measured by
the laser range finder. Let $\gamma$ be the vertical angle measured
by Theodolite-M as shown in Fig 1. Let P be an environmental point
with environmental location $(x,y,z)$.
Let $\theta_m$ and $\phi_m$ be the horizontal and vertical angles
of P measured with Theodolite-M. Let $\theta_d$ be the horizontal
angle of P measured with Theodolite-D.
Then
\begin{equation}
x =\frac
{d\; cos(\gamma)\; sin(\theta_d)\; cos(\theta_m)}
{sin(\pi-\theta_m-\theta_d)}
\end{equation}
\begin{equation}
y =\frac
{d\; cos(\gamma)\; sin(\theta_d)\; sin(\theta_m)}
{sin(\pi-\theta_m-\theta_d)}
\end{equation}
\begin{equation}
z = tan(\phi_m) \; \sqrt{(x^2+y^2)}
\end{equation}

\section{Data}
The data for the various sites is given in the attached sheets. The
environmental points are labelled in the corresponding images.
For environmental points which were physically reachable by the
researchers the distance given by the laser range finder and the
the theodolites agreed very closely (e.g. in the rocket field
sequence the average error was considerably less than 1\% at a
distance of about 30 meters).

\section{Conclusion}
The overall conclusion is that
the X and Y coordinates of environmental points
are quite reliable. The Z-coordinate is somewhat
less reliable because of the following reasons:
\begin{itemize}
\item The difference in heights between the same theodolite on different days.
\item The small vertical angles for most environmental points.
\item The approximation of the index error in the vertical circle.
\end{itemize}

However for depth determination the somewhat inaccurate Z-coordinates do not
matter very much because the magnitude of the Z-coordinate is usually
very small when compared to the other coordinates. Physically this
means that most the objects were rather far and spread widely on the ground
but were not very large in height (exceptions being mountain tops, pole tops
etc.).

\end{document}
