\section{Skeleton uniqueness}
\label{sec:uniqueness}

The most obvious concern raised by trying to use skeleton measurements as a
recognizable biometric is their uniqueness. Are skeletons consistently and
sufficiently distinct to use them for person recognition?

\subsection{Face recognition benchmark}

A good way to understand the uniqueness of a metric is to look at how
well an algorithm based on it performs in the \emph{pair-matching
problem}. In this problem you are given two measurements of the metric
and you want to decide whether they come from the same individual
(matched pair) or from two different individuals (unmatched pair).

This benchmark is standard for face recognition using the \emph{Labeled Faces
in the Wild} \cite{lfw} database.  Raw data of this benchmark is publicly
available and has been derived as follows: the database is split into 10
subsets. From each of these subsets, 300 matched pairs and 300 unmatched pairs
are randomly chosen. Each algorithm runs 10 separate leave-one-out
cross-validation experiments on these sets of pairs. Averaging the number of
true positives and false positives across the 10 experiments for a given
threshold then yields one point on the receiver operating characteristic (ROC)
curve, which plots the true-positive rate against the false-positive rate as
the threshold of the algorithm varies. Note that in this benchmark the identity
information of the individuals appearing in the pairs is not available, which
means that the algorithms cannot form additional image pairs from the input
data. This is referred to as the \emph{Image-restricted} setting in the LFW
benchmark.

\subsection{Experiment design}

In order to run an experiment similar to the one used in the face pair-matching
problem, we use the Goldman Osteological Dataset \cite{deadbodies}. This
dataset consists of skeletal measurements of 1538 skeletons uncovered around
the world and dating from throughout the last several thousand years. Given the
way these data were collected, only a partial view of the skeleton is
available, we keep six measurements: the lengths of four bones (radius,
humerus, femur, and tibia) and the breadth and height of the pelvis.  Because
of missing values, this reduces the size of the dataset to 1191.

From this dataset, 1191 matched pairs and 1191 unmatched pairs are generated.
In practice, the exact measurements of the bones of living subjects are not
directly accessible. Therefore, measurements are likely to have an error rate,
whose variance depends on the method of collection (\eg measuring limbs over
clothing versus on bare skin). Since there is only one sample per skeleton, we
simulate this error by adding independent random Gaussian noise to each
measurement of the pairs.

\subsection{Results}

We evaluate the performance of the pair-matching problem on the dataset by using a proximity
threshold algorithm: for a given threshold, a pair will be classified
as \emph{matched} if the Euclidean distance between the two skeletons is
lower than the threshold, and \emph{unmatched} otherwise. Formally, let
$(s_1,s_2)$ be an input pair of the algorithm
($s_i\in\mathbf{R}_+^{6}$, these are the six bone measurements), 
the output of the algorithm for the threshold $\delta$ is
defined as:
\begin{displaymath}
  A_\delta(s_1,s_2) = \begin{cases}
    1 & \text{if $d(s_1,s_2) < \delta$}\\
    0 & \text{otherwise}
  \end{cases}
\end{displaymath}

\begin{figure}[t]
  \begin{center}
    \includegraphics[width=10cm]{graphics/roc.pdf}
  \end{center}
  \vspace{-1.5\baselineskip}
  \caption{ROC curve (true positive rate
  vs. false positive rate) for several standard deviations of the
  noise and for the state-of-the-art \emph{Associate-Predict} face
  detection algorithm}
  \label{fig:roc}
\end{figure}

Figure \ref{fig:roc} shows the ROC curve of the proximity threshold
algorithm for different values of the standard deviation of the noise,
as well as the ROC of the best performing face detection algorithm in
the Image-restricted LFW benchmark: \emph{Associate-Predict}
\cite{associate}.

The results show that with a standard deviation of 3mm, skeleton
proximity thresholding performs quite similarly to face detection at
low false-positive rate. At this noise level, the error is smaller
than 1cm with 99.9\% probability. Even with a standard
deviation of 5mm, it is still possible to detect 90\% of the matched
pairs with a false positive rate of 6\%.

This experiment gives an idea of the noise variance level above which
it is not possible to consistently distinguish skeletons. 
For this problem, a classifier can be built by first learning
a \emph{skeleton profile} for each individual from all the
measurements in the training set. Then, given a new skeleton
measurement, the algorithm classifies it to the individual whose
skeleton profile is closest to the new measurement. In this case,
there are two distinct sources of noise:
\begin{itemize}
\item the absolute deviation of the estimator: how far is the estimated profile
  from the exact skeleton profile of the person due to figure position or
  motion (\ie from walking).
\item the noise of the new measurement: this comes from the device
  doing the measurement.
\end{itemize}
The combination of these two noise sources is what can be compared to the
noise represented on the ROC curves. Section \label{sec:kinect} will
give more insight on the structure of the noise.

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