From 73834f5ef1c7be73d054baf4cf5f14a39fef17dc Mon Sep 17 00:00:00 2001 From: Jon Whiteaker Date: Mon, 5 Mar 2012 11:12:04 -0800 Subject: jon's final pass part one --- algorithm.tex | 77 +++++++++++++++++++++++++++++++++++++++++++++++++++-------- 1 file changed, 67 insertions(+), 10 deletions(-) (limited to 'algorithm.tex') diff --git a/algorithm.tex b/algorithm.tex index b3e7648..9bc0fd6 100644 --- a/algorithm.tex +++ b/algorithm.tex @@ -1,29 +1,69 @@ \section{Algorithms} \label{sec:algorithms} -In Section~\ref{sec:uniqueness}, we showed that a nearest-neighbor classifier can accurately predict if a skeleton belongs to the same person if the error of the skeleton measurements is small. In this section, we suggest a probabilistic model for skeleton recognition. In this model, a skeleton is classified based on the distance from average skeleton profiles of people in the training set. +In Section~\ref{sec:uniqueness}, we showed that a nearest-neighbor classifier +can accurately predict if two sets of skeletal measurements belong to the same +person if the error of the skeleton measurements is small. In this section, we +suggest a probabilistic model for skeleton recognition. In this model, a +skeleton is classified based on the distance from average skeleton profiles of +people in the training set. \subsection{Mixture of Gaussians} \label{sec:mixture of Gaussians} -A mixture of Gaussians \cite{bishop06pattern} is a generative probabilistic model, which is typically applied to modeling problems where class densities are unimodal and the feature space is low-dimensional. The joint probability distribution of the model is given by: +A mixture of Gaussians \cite{bishop06pattern} is a generative probabilistic +model, which is typically applied to modeling problems where class densities +are unimodal and the feature space is low-dimensional. The joint probability +distribution of the model is given by: \begin{align} P(\bx, y) = \cN(\bx | \bar{\bx}_y, \Sigma) P(y), \label{eq:mixture of Gaussians} \end{align} -where $P(y)$ is the probability of class $y$ and $\cN(\bx | \bar{\bx}_y, \Sigma)$ is a multivariate normal distribution, which models the density of $\bx$ given $y$. The mean of the distribution is $\bar{\bx}_y$ and the variance of $\bx$ is captured by the covariance matrix $\Sigma$. The decision boundary between any two classes is known to be is linear when all conditionals $\cN(\bx | \bar{\bx}_y, \Sigma)$ have the same covariance matrix \cite{bishop06pattern}. In this setting, the mixture of Gaussians model can be viewed as a probabilistic variant of the nearest-neighbor (NN) classifier in Section~\ref{sec:uniqueness}. +where $P(y)$ is the probability of class $y$ and $\cN(\bx | \bar{\bx}_y, +\Sigma)$ is a multivariate normal distribution, which models the density of +$\bx$ given $y$. The mean of the distribution is $\bar{\bx}_y$ and the variance +of $\bx$ is captured by the covariance matrix $\Sigma$. The decision boundary +between any two classes is known to be is linear when all conditionals $\cN(\bx +| \bar{\bx}_y, \Sigma)$ have the same covariance matrix \cite{bishop06pattern}. +In this setting, the mixture of Gaussians model can be viewed as a +probabilistic variant of the nearest-neighbor (NN) classifier in +Section~\ref{sec:uniqueness}. + +The mixture of Gaussians model has many advantages. First, the model can be +easily learned using maximum-likelihood (ML) estimation \cite{bishop06pattern}. +In particular, $P(y)$ is the frequency of $y$ in the training set, +$\bar{\bx}_y$ is the expectation of $\bx$ given $y$, and the covariance matrix +is computed as $\Sigma = \sum_y P(y) \Sigma_y$, where $\Sigma_y$ represents the +covariance of $\bx$ given $y$. Second, the inference in the model can be +performed in a closed form. In particular, the model predicts $\hat{y} = +\arg\max_y P(y | \bx)$, where: -The mixture of Gaussians model has many advantages. First, the model can be easily learned using maximum-likelihood (ML) estimation \cite{bishop06pattern}. In particular, $P(y)$ is the frequency of $y$ in the training set, $\bar{\bx}_y$ is the expectation of $\bx$ given $y$, and the covariance matrix is computed as $\Sigma = \sum_y P(y) \Sigma_y$, where $\Sigma_y$ represents the covariance of $\bx$ given $y$. Second, the inference in the model can be performed in a closed form. In particular, the model predicts $\hat{y} = \arg\max_y P(y | \bx)$, where: \begin{align} P(y | \bx) = \frac{P(\bx | y) P(y)}{\sum_y P(\bx | y) P(y)} = \frac{\cN(\bx | \bar{\bx}_y, \Sigma) P(y)}{\sum_y \cN(\bx | \bar{\bx}_y, \Sigma) P(y)}. \label{eq:inference} \end{align} -In practice, the prediction $\hat{y}$ is accepted when the classifier is confident. In other words, $P(\hat{y} | \bx) \! > \! \delta$, where $\delta \in (0, 1)$ is a threshold that controls the precision and recall of the classifier. In general, the higher the threshold $\delta$, the lower the recall and the higher the precision. -In this work, we use the mixture of Gaussians model for skeleton recognition. Skeleton measurements are represented by a vector $\bx$ and each person is assigned to one class $y$. In particlar, our dataset $\cD = \set{(\bx_1, y_1), \dots, (\bx_n, y_n)}$ consists of $n$ pairs $(\bx_i, y_i)$, where $y_i$ is the label of the skeleton $\bx_i$. To verify that our method is suitable for skeleton recognition, we plot for each skeleton feature $x_k$ (Section~\ref{sec:experiment}) the histogram of differences between all measurements and the expectation given the class $(\bx_i)_k - \E{}{x_k | y_i}$ (Figure~\ref{fig:error marginals}). All histograms look approximately normal. This indicates that all class conditionals $P(\bx | y)$ are multivariate normal and our generative model, although very simple, may be nearly optimal \cite{bishop06pattern}. +In practice, the prediction $\hat{y}$ is accepted when the classifier is +confident. In other words, $P(\hat{y} | \bx) \! > \! \delta$, where $\delta \in +(0, 1)$ is a threshold that controls the precision and recall of the +classifier. In general, the higher the threshold $\delta$, the lower the recall +and the higher the precision. + +In this work, we use the mixture of Gaussians model for skeleton recognition. +Skeleton measurements are represented by a vector $\bx$ and each person is +assigned to one class $y$. In particlar, our dataset $\cD = \set{(\bx_1, y_1), +\dots, (\bx_n, y_n)}$ consists of $n$ pairs $(\bx_i, y_i)$, where $y_i$ is the +label of the skeleton $\bx_i$. To verify that our method is suitable for +skeleton recognition, we plot for each skeleton feature $x_k$ +(Section~\ref{sec:experiment}) the histogram of differences between all +measurements and the expectation given the class $(\bx_i)_k - \E{}{x_k | y_i}$ +(Figure~\ref{fig:error marginals}). All histograms look approximately normal. +This suggests that all class conditionals $P(\bx | y)$ are multivariate normal +and our generative model, although very simple, may be nearly optimal +\cite{bishop06pattern}. \begin{figure}[t] \centering @@ -39,15 +79,32 @@ In this work, we use the mixture of Gaussians model for skeleton recognition. Sk \subsection{Sequential hypothesis testing} \label{sec:SHT} -The mixture of Gaussians model can be extended to temporal inference through sequential hypothesis testing. Sequential hypothesis testing \cite{wald47sequential} is an established statistical framework, where a subject is sequentially tested for belonging to one of several classes. The probability that the sequence of data $\bx^{(1)}, \dots, \bx^{(t)}$ belongs to the class $y$ at time $t$ is given by: +The mixture of Gaussians model can be extended to temporal inference through +sequential hypothesis testing. Sequential hypothesis testing +\cite{wald47sequential} is an established statistical framework, where a +subject is sequentially tested for belonging to one of several classes. The +probability that the sequence of data $\bx^{(1)}, \dots, \bx^{(t)}$ belongs to +the class $y$ at time $t$ is given by: + \begin{align} P(y | \bx^{(1)}, \dots, \bx^{(t)}) = \frac{\prod_{i = 1}^t \cN(\bx^{(i)} | \bar{\bx}_y, \Sigma) P(y)} {\sum_y \prod_{i = 1}^t \cN(\bx^{(i)} | \bar{\bx}_y, \Sigma) P(y)}. \label{eq:SHT} \end{align} -In practice, the prediction $\hat{y} = \arg\max_y P(y | \bx^{(1)}, \dots, \bx^{(t)})$ is accepted when the classifier is confident. In other words, $P(\hat{y} | \bx^{(1)}, \dots, \bx^{(t)}) > \delta$, where the threshold $\delta \in (0, 1)$ controls the precision and recall of the predictor. In general, the higher the threshold $\delta$, the higher the precision and the lower the recall. +In practice, the prediction $\hat{y} = \arg\max_y P(y | \bx^{(1)}, \dots, +\bx^{(t)})$ is accepted when the classifier is confident. In other words, +$P(\hat{y} | \bx^{(1)}, \dots, \bx^{(t)}) > \delta$, where the threshold +$\delta \in (0, 1)$ controls the precision and recall of the predictor. In +general, the higher the threshold $\delta$, the higher the precision and the +lower the recall. -Sequential hypothesis testing is a common technique for smoothing temporal predictions. In particular, note that the prediction at time $t$ depends on all data up to time $t$. This reduces the variance of predictions, especially when input data are noisy, such as in the domain of skeleton recognition. +Sequential hypothesis testing is a common technique for smoothing temporal +predictions. In particular, note that the prediction at time $t$ depends on all +data up to time $t$. This reduces the variance of predictions, especially when +input data are noisy, such as in the domain of skeleton recognition. -In skeleton recognition, the sequence $\bx^{(1)}, \dots, \bx^{(t)}$ are skeleton measurements of a person walking towards the camera, for instance. If the camera detects more people, we use tracking to distinguish individual skeleton sequences. +In skeleton recognition, the sequence $\bx^{(1)}, \dots, \bx^{(t)}$ is skeleton +measurements of a person walking towards the camera, for instance. If the +Kinect detects multiple people, we use the figure tracking of the tools in +\xref{sec:experiment:dataset} to distinguish individual skeleton sequences. -- cgit v1.2.3-70-g09d2