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

The most obvious concern raised by trying to use skeletons 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).

The \emph{Labeled Faces in the Wild} \cite{lfw} database is
specifically suited to study the face pair matching problem and has
been used to benchmark several face recognition algorithms. 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 curve (ROC curve: this is the curve of the
true-positive rate vs. 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 osteometric measurements of 1538 skeletons dating from
throughout the Holocene. 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.
With exact measurements, all skeletons are distinct and therefore every pair is
correctly classified.  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). 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}
  \caption{Receiver operating characteristic (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\%.

\todo{We should unify the language here with that in the related work (and intro)}
This experiment gives an idea of the noise variance level above which
it is not possible to consistently distinguish skeletons. This noise
level can be interpreted as follows in the person recognition
problem. For this problem, a classifier can be built be 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.
\item the noise of the new measurement: this comes from the device
  doing the measurement.
\end{itemize}
the combination of these two noises 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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