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\title{A State Vector Augmentation Technique for Incorporating Indirect
Velocity Information into the Likelihood Function of the SIR Video Target Tracking Filter}



% author names and affiliations
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\author{\IEEEauthorblockN{
Jesyca C.~Fuenmayor Bello\thanks{This 
work was supported in part by the U.S.~Army Research Laboratory
and the U.S.~Army Research Office under grant W911NF-08-1-0293.}
%Chuong T.~Nguyen, 
and
Joseph P.~Havlicek}
\IEEEauthorblockA{School of Electrical and Computer Engineering\\ University of
Oklahoma, Norman, OK, USA}
%\\ Jesyca.C.Fuenmayor.Bello-1@ou.edu, chuong@ou.edu, joebob@ou.edu}
}

\begin{document}

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\begin{abstract}
We consider the problem of tracking moving targets in heavily cluttered video sequences 
and introduce a new state vector augmentation technique to incorporate indirect velocity
information into the likelihood function of the SIR particle filter.  While the importance 
of motion information in video tracking has been well recognized, the standard SIR filter 
typically weights particles using a likelihood function that considers the appearance model 
only.  Our goal is to prevent particles with poor velocity estimates from receiving large 
weights. The key modifications involve saving the previous values of the state variables in 
the state update equation and reformulating the measurement model to deliver both the 
current and previous observations.  This leads to a straightforward calculation of
likelihood across pairs of temporally adjacent frames.  Our preliminary experimental 
results show that the proposed method is effective for avoiding track losses due to the 
filter locking onto structured clutter.
\end{abstract}

\begin{IEEEkeywords}
video tracking, particle filter, SIR, velocity estimation
\end{IEEEkeywords}


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\section{Introduction}
\label{sec:intro}

Particle filters have become immensely popular for video object tracking in
recent years.  %Among them, 
The sampling importance resampling (SIR) 
filter~\cite{Arulampalam2002} in particular is {\em widely\/} used.  First-order
Markovian state dynamics are almost always assumed~\cite{LiuCircSys2015,PanSPL2011}.  
The SIR filter's choice
of the prior density function as the proposal density combined with its
resampling and equal weighting of the particles at the end of every time step then
imply that the particle weights are given by the likelihood function, which
is straightforward to design and implement in a wide variety of practical
scenarios.

It is common to assume that the target kinematics obey a constant velocity
model~\cite{RongTAE03,ZhouChellaTIP2004}.  In a typical formulation, the
state vector is given by, e.g.,
\begin{equation}
\label{eq:typical_x_k}
\bx_k = [x_k \; \dot x_k \; y_k \; \dot y_k \; \gamma_k \; \dot\gamma_k\; 
         \theta_k \; \dot\theta_k]^T,
\end{equation}
where $[x_k\;y_k]^T$ is the %spatial coordinate of the 
target centroid,
$[\dot x_k\;\dot y_k]^T$ is the velocity, 
$\gamma_k$ and $\theta_k$ are magnification and planar rotation
relative to a reference appearance model, 
and $\dot\gamma_k$ and $\dot\theta_k$ are the time derivatives
of $\gamma_k$ and $\theta_k$.
The state update %model 
and measurement 
models are given by~\cite{Arulampalam2002}
\begin{eqnarray}
\bx_{k+1} &=& \bff_k(\bx_k,\bv_k)  \label{eq:typical_state_eq}\\
\bz_k      &=& \bh_k(\bx_k,\bn_k), \label{eq:typical_meas_eq}
\end{eqnarray}
%where $\bv_k$ and $\bn_k$
%are i.i.d.~noises that are usually assumed mutually uncorrelated.
where $\bv_k$ and $\bn_k$ are mutually uncorrelated i.i.d.~noises.
The measurement $\bz_k$ is typically taken as the current video frame or %as 
a set of features extracted from the current frame.
%It is often the case that 
Often, $\bff_k$ and $\bh_k$ are independent of $k$ so that
$\bff_k(\cdot) = \bff(\cdot)$ and $\bh_k(\cdot) = \bh(\cdot)$.

The SIR filter maintains a set of $N_s$ weighted particles 
$\{\bx_k^i,w_k^i\}_{i=1}^{N_s}$ that approximate the posterior density 
via~\cite{Arulampalam2002}
\begin{equation}
\label{eq:particle_density}
p(\bx_k|\bz_{1:k}) \approx \sum_{i=1}^{N_s} w_k^i\delta(\bx_k - \bx_k^i).
\end{equation}
Each particle $\bx_k^i$ is an instance of the state vector that makes a
hypothesis about the true state of the target.
One of the main advantages of the SIR formulation is that the weights are
easy to calculate: they are given by the likelihood function
\begin{equation}
\label{eq:weights}
w_k^i \propto p(\bz_k|\bx_k^i),
\end{equation}
where the proportionality constant is chosen so that the $w_k^i$ sum to one.
Elements of the state vector such as the target centroid may be estimated by
taking the expected value with respect to~\Eqref{eq:particle_density} according to
\begin{equation}
\label{eq:tracked_centroid}
[\widehat x_k\;\widehat y_k]^T = \sum_{i=1}^{N_s} w_k^i [x_k^i\;y_k^i]^T
\end{equation}
or by taking (MAP) estimates from the largest-weighted (i.e., ``best")
particle according to
\begin{equation}
\label{eq:MAP_centroid}
[\widehat x_k\;\widehat y_k]^T = [x_k^{i^*}\;y_k^{i^*}]^T,
\mbox{\ \ } i^* = \argmax_i w_k^i
\end{equation}
prior to resampling.  Iterative application of~\Eqref{eq:tracked_centroid}
or~\Eqref{eq:MAP_centroid} is the standard method for tracking video targets.

Tracking failures can occur for several reasons.  One is that, in all but
the simplest cases, the appearance model must be dynamically updated. Over
time, it can become corrupted by clutter leakage and drift ultimately
leading to track loss.
The question of how best to perform robust appearance model updates has
received considerable attention recently~\cite{WangTIP12,SaltiTIP12,ZhouChellaTIP2004}.
Another problem is that ``bad" particles can sometimes receive large
weights.  This happens frequently in cases where there is strongly structured clutter
together with significant target appearance changes.
The likelihood $p(\bz_k|\bx_k^i)$ is increased for bad particles 
that have a poor state hypothesis but partially match the clutter, while 
$p(\bz_k|\bx_k^i)$ is decreased for good particles that have a good state hypothesis but
only partially match the target due to an ineffective appearance model update strategy.
The combined effect can be substantial.  Furthermore, it is exacerbated by the fact 
that the
decreases in $p(\bz_k|\bx_k^i)$ for the good particles that should be
{\em heavily\/} weighted tends to increase the proportionality constant 
in~\Eqref{eq:weights}, further amplifying the weights of the bad particles.


%------------------------------- double column state equation ---------------------------
\begin{figure*}[!b]
\normalsize
\vspace*{1pt}
\hrulefill
%                   save the equation counter
\setcounter{SaveEqn}{\value{equation}}
%
% RESET equation counter to ONE LESS than the double column equation
%
\setcounter{equation}{7}
\begin{equation}
\label{eq:dbl_state}
\bx_{k+1} =
\left[\begin{array}{l}
x_{k+1}\\ \dot x_{k+1}\\ y_{k+1}\\ \dot y_{k+1}\\ \gamma_{k+1}\\ \dot\gamma_{k+1}\\
   \theta_{k+1}\\ \dot\theta_{k+1}
\end{array}\right] =
%------------------ start of A matrix --------------------------
%\left[\begin{array}{*8{c}}
%1 & \Delta & 0 & 0      & 0 & 0      & 0 & 0      \\   % x
%0 & 1      & 0 & 0      & 0 & 0      & 0 & 0      \\   % x
%0 & 0      & 1 & \Delta & 0 & 0      & 0 & 0      \\   % y
%0 & 0      & 0 & 1      & 0 & 0      & 0 & 0      \\   % y
%0 & 0      & 0 & 0      & 1 & \Delta & 0 & 0      \\   % gamma
%0 & 0      & 0 & 0      & 0 & 1      & 0 & 0      \\   % gamma
%0 & 0      & 0 & 0      & 0 & 0      & 1 & \Delta \\   % theta
%0 & 0      & 0 & 0      & 0 & 0      & 0 & 1           % theta
%\end{array}\right] 
%-------------------- end of A matrix --------------------------
%------------------ start of A matrix --------------------------
\left[\begin{array}{*4{c}}
\cvmatrix \\
 & \cvmatrix & \makebox(0,0){\ \hspace{36pt}\raisebox{10pt}[0pt][0pt]{\Huge 0}} \\
 & & \cvmatrix \\
 & \makebox(0,0){\ \hspace{-36pt}\raisebox{3pt}[0pt][0pt]{\Huge 0}\ } & & \cvmatrix
\end{array}\right] 
%-------------------- end of A matrix --------------------------
\left[\begin{array}{l}
x_k\\ \dot x_k\\ y_k\\ \dot y_k\\ \gamma_k\\ \dot\gamma_k\\ \theta_k\\ \dot\theta_k
\end{array}\right] +
\left[\begin{array}{l}
0\\ v^x_k\\ 0\\ v^y_k\\ 0\\ v^\gamma_k\\ 0\\ v^\theta_k
\end{array}\right]
\equiv \bA \bx_k + \bv_k
\end{equation}
% restore equation counter
\setcounter{equation}{\value{SaveEqn}}
\end{figure*}
%--------------------------- end double column state equation ---------------------------
%
%\begin{equation}
%\left[\begin{array}{l}
%x_{k+1}\\ \dot x_{k+1}\\ y_{k+1}\\ \dot y_{k+1}\\ \gamma_{k+1}\\ \dot\gamma_{k+1}\\
%   \theta_{k+1}\\ \dot\theta_{k+1}
%\end{array}\right] = 
%\diag\left(\cvmatrix\right)
%\left[\begin{array}{l}
%x_k\\ \dot x_k\\ y_k\\ \dot y_k\\ \gamma_k\\ \dot\gamma_k\\ \theta_k\\ \dot\theta_k
%\end{array}\right] +
%\left[\begin{array}{l}
%0\\ v^x_k\\ 0\\ v^y_k\\ 0\\ v^\gamma_k\\ 0\\ v^\theta_k
%\end{array}\right]
%\end{equation} 

When a bad particle receives a large weight, it is turned into {\em many\/}
bad particles by the SIR resampling operation.
Since the total number of particles is
fixed, this reduces the number of particles that are available in the
next time step for searching the ``good" part of the state space where 
the true target state lies. This generally has a deleterious effect on
the tracking performance.  Moreover, the presence of many particles in
the ``bad" parts of the state space implies that increasing numbers of
bad particles may partially match the clutter and receive large weights
in subsequent time steps due to the stochastic nature of $\bv_k$ 
in~\Eqref{eq:typical_state_eq}.  The tracker often fails when this occurs.

The importance of motion information in video tracking has been well
recognized~\cite{SullivanICCV2001,ZhouChellaTIP2004,LiuCircSys2015,PanSPL2011}.
However, in practical video tracking applications the sensor
produces one video frame at each time step and there is no way to obtain
direct velocity measurements without incorporating additional sensors.
Thus, even if the motion model is good,
direct velocity measurements are not available in $\bz_k$ and
according to~\Eqref{eq:weights}
there is no way of using the weight calculation to
explicitly penalize particles with poor velocity 
hypotheses.  
Indeed, the likelihood calculation is often implemented in a way that
considers the appearance variables only and omits the velocity information
altogether.  This has two unfortunate consequences:
1) a particle with a state hypothesis $\bx_k^i$ that
matches the true target well will receive the same weight as 
a particle with an identical appearance hypothesis but a poor
velocity hypothesis; and 2)
even in the presence of ego motion and moving clutter, it is unlikely
for the clutter to match the target in both appearance and velocity. Nevertheless,
a particle that partially matches the clutter in appearance may receive
a large weight even though this should be preventable based on velocity
information.

Of course, one can save the previous frame and difference the positions
of best matching blocks between it and the current frame to produce estimated
velocity measurements.  
However, a na\"{i}ve implementation of this approach approximates the derivative with
a zeroth-order Taylor series which is notoriously poor and, more importantly, can
suffer from imprecision and uncertainty in the block matching algorithm.

%However, this approach approximates the derivative with
%a zeroth-order Taylor series which is notoriously poor and, more importantly, would
%suffer from imprecision and uncertainty in the block matching algorithm.

% 
% this argument is kind of weak because the state model is not actually a true
%  derivative... so a better way would be to seek a finite difference based model
%  for the const vel model... which would probably be pretty good.  The bigger
%  concern is the error that would result from the block matching implementation

In this paper, we address these problems by considering the question of how to 
prevent particles with a poor velocity hypothesis from receiving large weights.  
In Section~\ref{sec:method}, we introduce a new state vector augmentation method
to incorporate indirect velocity information into the likelihood function.
Preliminary experimental evaluation of the new method is presented in 
Section~\ref{sec:experiments}, while discussion and conclusions are reserved 
for Section~\ref{sec:conclusion}.



\section{Incorporating Velocity Information into the Likelihood Function}
\label{sec:method}

In this section, we introduce our proposed method in the context of template 
tracking~\cite{SullivanICCV2001,BrunoTSP04,ZhouChellaTIP2004,HagerPAMI98,KwonPAMI14}.
The reason is that we are most interested in tracking targets in monochromatic infrared
video signals that are characterized by low contrast, poor SNR, and strong clutter.
Template tracking often works better than other approaches for such signals.
However, the proposed method extends easily to appearance models based on, e.g.,
histograms, HOG, SIFT, LBP or other features.

\addtocounter{equation}{1}
Suppose that the sensor delivers a sequence of video frames $\bz_k$ and let
$\bT$ be a globally defined appearance model (template) for the target.
We begin by considering a very standard conventional SIR formulation 
where~\Eqref{eq:typical_state_eq} takes the form shown 
in~\Eqref{eq:dbl_state} below. In~\Eqref{eq:dbl_state}, $\Delta$ is the
frame time and $v^\cdot_k$ are mutually uncorrelated i.i.d.~noises that model the 
temporal second derivatives of position, magnification, and rotation.  
Note that~\Eqref{eq:dbl_state} {\em defines\/} the matrix $\bA$ and
the vector $\bv_k$.
The function $\bh$ in~\Eqref{eq:typical_meas_eq} creates a frame of zeros,
inserts the target model $\bT$ at position $[x_k\;y_k]^T$ with magnification
$\gamma_k$ and rotation $\theta_k$, and adds measurement noise $\bn_k$
(there is no explicit background model).

For a particle $\bx_k^i$, the hypothesized target appearance is
$\bz_k^i = \bh(\bx_k^i,\bzero)$. Let $\bOmega_k^i$ be the spatial support of the
magnified and rotated template in $\bz_k^i$.  The SIR filter assigns the particle
weight~\Eqref{eq:weights} using likelihood
\begin{equation}
\label{eq:likelihood}
p(\bz_k|\bx_k^i) = e^{-\kappa(1-\rho_k^i)},
\end{equation}
where $\kappa$ is a gain that must be tuned and
$\rho_k^i$ is normalized cross correlation given by
\begin{equation}
\label{eq:NCC}
\rho_k^i = \frac{\sum_{\bOmega_k^i} \bz_k \bz_k^i}
{\sqrt{\sum_{\bOmega_k^i}\bz_k^2 \sum_{\bOmega_k^i}(\bz_k^i)^2}}.
\end{equation}
Eqs.~\Eqref{eq:typical_meas_eq}-\Eqref{eq:weights}, \Eqref{eq:dbl_state}-\Eqref{eq:NCC},
and~\Eqref{eq:tracked_centroid} or~\Eqref{eq:MAP_centroid} define the conventional
SIR track filter that will be used as a baseline for comparison with the proposed method.

Let $\bx_{k-1}^j$ be the particle
from which $\bx_k^i$ was resampled in time step $k-1$ and define
$\bx_{k-1}^{(i)} \equiv \bx_{k-1}^j$.
The likelihood function~\Eqref{eq:likelihood} makes no explicit
consideration of velocity information.  Furthermore, no direct measurements of
velocity are available.  However, if particle $\bx_k^i$ has a good appearance
hypothesis then $\rho_k^i$ should be large.  If the velocity hypothesis is also
good, then $\rho_{k-1}^j$ should also have been large.  We want the particle
$\bx_k^i$ to receive a large weight only if both of these conditions are met.
This suggests a way to
{\em indirectly\/} incorporate velocity information into the likelihood function 
$p(\bz_k|\bx_k^i)$ if the previous measurement $\bz_{k-1}$ can be retained in $\bz_k$
and the previous hypothesis $\bx_{k-1}^{(i)}$ can be retained in $\bx_k^i$. State vector
augmentation provides a means to do this.
%
We denote quantities in the augmented system model with a tilde.
With respect to~\Eqref{eq:typical_x_k}, the augmented state vector is given by
$\widetilde\bx_k = [\bx_k \; \bx_{k-1}]^T$.  The augmented state update model is then
\begin{equation}
\label{eq:state}
\widetilde\bx_{k+1} =
\left[\begin{array}{*2{c}} \bA & \bzero \\ \bI & \bzero\end{array}\right] \widetilde\bx_k 
+ \left[\begin{array}{c} \bv_k \\ \bzero\end{array}\right],
\end{equation}
where $\bI$ is the $8\times 8$ identity matrix.  Now let 
$\breve\bz_k = \bh(\bx_k,\bn_{k+1})$ be a second realization of
$\bz_k$ that is generated at time step $k+1$ instead of time step $k$.  
The difference between 
this second realization and the original measurement $\bz_k$ is 
$\breve\bz_k - \bz_k = \bn_{k+1} - \bn_k$.  We define the augmented measurement
$\widetilde \bz_k = [\bz_k \; \breve\bz_{k-1}]^T$.  The augmented measurement model
is then
\begin{equation}
\label{eq:meas}
\widetilde\bz_k = \left[\begin{array}{c}\bz_k \\ \breve\bz_{k-1}\end{array}\right]=
\widetilde\bh(\widetilde\bx_k,\bn_k)
\equiv
\left[\begin{array}{c}\bh(\bx_k,\bn_k) \\ \bh(\bx_{k-1},\bn_k)\end{array}\right].
\end{equation}

For an augmented particle $\widetilde\bx_k^i=[\bx_k^i \; \bx_{k-1}^{(i)}]^T$ 
with measurement hypothesis
$\widetilde\bz_k^i = [\bz_k^i \; \breve\bz_{k-1}^i]^T$, 
let $\breve\bOmega_{k-1}^i$ be the spatial support of the
magnified and rotated template in $\breve\bz_k^i$.  The likelihood may now be
defined according to
\begin{equation}
\label{eq:likelihood2}
p(\widetilde\bz_k|\widetilde\bx_k^i) = e^{-\kappa(1-\widetilde\rho_k^{\,i})},
\end{equation}
where
%\begin{eqnarray}
%\lefteqn{\widetilde\rho_k^{\,i} =
%  \left(\sum_{\bOmega_k^i} \bz_k \bz_k^i + 
%   \sum_{\breve\bOmega_{k-1}^i} \bz_{k-1} \breve\bz_{k-1}^i\right)} \hspace{48pt}
%                                                                            \nonumber \\ 
%& &    \times \left(\sum_{\bOmega_k^i}\bz_k^2 +
%             \sum_{\breve\bOmega_{k-1}^i}\bz_{k-1}^2\right)^{-\frac{1}{2}} \nonumber \\ 
%& &    \times \left(\sum_{\bOmega_k^i}(\bz_k^i)^2 +
%             \sum_{\breve\bOmega_{k-1}^i}(\breve\bz_{k-1}^i)^2\right)^{-\frac{1}{2}}.
%\label{eq:NCC2}
%\end{eqnarray}
\begin{equation}
\label{eq:NCC2}
\widetilde\rho_k^{\,i} =
\frac{{\displaystyle \sum_{\bOmega_k^i}} \bz_k \bz_k^i +
   {\displaystyle \sum_{\breve\bOmega_{k-1}^i}} \bz_{k-1} \breve\bz_{k-1}^i
}
{
       \sqrt{{\displaystyle\sum_{\bOmega_k^i}}\bz_k^2 +
            {\displaystyle \sum_{\breve\bOmega_{k-1}^i}}\bz_{k-1}^2}
       \sqrt{{\displaystyle \sum_{\bOmega_k^i}}(\bz_k^i)^2 +
             {\displaystyle \sum_{\breve\bOmega_{k-1}^i}}(\breve\bz_{k-1}^i)^2}}
\end{equation}
is the normalized cross correlation between the hypothesis of augmented particle 
$\widetilde\bx_k^i$ and two consecutive video frames acquired from the camera.
Intuitively, $\breve\bz_{k-1}$ is nothing more than a {\em theoretical\/} construct that
provides us with a rigorous way to consider a realization of $\bz_{k-1}$ in the
likelihood $p(\widetilde\bz_k|\widetilde\bx_k)$. Note that $\breve\bz_{k-1}$ does
not appear in~\Eqref{eq:NCC2}; rather, it is the actual saved
video frame $\bz_{k-1}$ that is correlated with the lagged particle hypothesis 
$\breve\bz_{k-1}^i$.
Eqs.~\Eqref{eq:state}-\Eqref{eq:NCC2}, \Eqref{eq:particle_density}, \Eqref{eq:weights},
and~\Eqref{eq:tracked_centroid} or~\Eqref{eq:MAP_centroid} define our new method
for incorporating indirect velocity information into the SIR filter likelihood function. 










\section{Experiments}
\label{sec:experiments}

We compared the proposed method against the standard SIR filter on 
two synthetic video sequences and two of the longwave IR sequences
described in~\cite{NguyenAsilomar14}.  
Appearance model updates were not performed in order to isolate performance
differences between the SIR filters with standard likelihood~\Eqref{eq:likelihood}
and with velocity information incorporated via state
augmentation and likelihood~\Eqref{eq:likelihood2}.
The synthetic sequences were simulated by inserting the
target of Fig.~\ref{fig:Examples}(a) into benign and complex
backgrounds as shown in Figs.~\ref{fig:Examples}(c) and (b)
with trajectory determined by~\Eqref{eq:dbl_state}.
The horizontal and vertical velocity drift noises were Gaussian with variances
0.63 and 0.75, while the magnification and rotation drift noises were uniform
with variances 3.6 $\times$ $10^{-5}$ and 6.4 $\times$ $10^{-3}$.
For the IR sequences shown in Figs.~\ref{fig:Examples}(g)-(o), ground truth was
compiled manually and the noise variances were estimated by approximating derivatives
with finite differences.  
The initial particle sets were distributed normally (position) and uniformly 
(magnification/rotation) with means and variances given by ground truth from the first frame.

The tracking results are given in Fig.~\ref{fig:Examples} and Table~\ref{tab:Examples}.
As we expected, incorporating velocity information had negligible impact against
the benign background of Figs.~\ref{fig:Examples}(c), (d) but provided a substantial
performance gain against the complex background of Figs.~\ref{fig:Examples}(e), (f)
where the standard SIR filter lost the target and locked onto clutter.
Against IR sequence 1, the proposed method again provided a significant advantage when
the standard SIR filter became distracted by clutter as shown in 
Figs.~\ref{fig:Examples}(i), (j).  As in the benign synthetic sequence, both filters
performed comparably against the second IR sequence shown in
Figs.~\ref{fig:Examples}(l)-(o).




\begin{table*}[htp]
\centering
\caption{Performance comparison between proposed method and standard SIR filter}
\begin{tabular}{|l|c|c|c|c|c|c|c|c|}
\cline{6-9}
\multicolumn{5}{c|}{\rule[-2pt]{0pt}{9pt}}
& \multicolumn{4}{c|}{Mean Absolute Tracking Error (pixels)} \\
%
\cline{2-9}
\multicolumn{1}{c|}{\rule[-2pt]{0pt}{10pt}} & Num & $N_s$, Num & Gain $\kappa$ & Num
& \multicolumn{2}{|c|}{Standard SIR}
& \multicolumn{2}{|c|}{Proposed SIR} \\
\cline{1-1}
%
\multicolumn{1}{|c|}{\rule[-2pt]{0pt}{9pt} Case}
& Frames & Particles & \Eqref{eq:likelihood},\Eqref{eq:likelihood2}
& Runs
& E[$\cdot$] \Eqref{eq:tracked_centroid}& MAP \Eqref{eq:MAP_centroid} 
& E[$\cdot$] \Eqref{eq:tracked_centroid}& MAP \Eqref{eq:MAP_centroid} \\
\hline\hline
\rule[-2pt]{0pt}{10pt}
Synthetic/Benign Background   
&150& 700 & 10  & 10 & \bf{1.7505} & 2.0090 & 1.9985 & 2.4028 \\
\rule[-2pt]{0pt}{9pt}
Synthetic/Complex Background  
&150& 700 & 10  & 20 & 44.2557 & 43.7655 & \bf{2.4348}  & 2.5860 \\
\rule[-2pt]{0pt}{9pt}
Real IR Sequence 1            
&100& 700 & 75  &  7 & 11.7222 & 12.1286 & \bf{4.0832}   & 4.2776 \\
\rule[-2pt]{0pt}{9pt}
Real IR Sequence 2            
&80& 700 & 75  &  5 & \bf{2.4934} & 2.5651 & 2.5130 & 2.6138 \\
\rule[-2pt]{0pt}{9pt}
Real IR Sequence 2            
&80& 700 & 100 &  5 & 2.7948 & 2.8300 & \bf{2.5979}  & 2.6337 \\
\hline
\end{tabular}
\label{tab:Examples}
\end{table*}


\begin{figure*}[htp]
\centering
\begin{minipage}[b]{0.19\linewidth}
  \centering
  \centerline{\includegraphics[width=1.03in]{targ4gray}}
%  \vspace{1.5cm}
  \centerline{(a)}\medskip
\end{minipage}
%
\begin{minipage}[b]{0.19\linewidth}
  \centering
  \centerline{\includegraphics[width=1.03in]{targ4gray}}
%  \vspace{1.5cm}
  \centerline{(b)}\medskip
\end{minipage}
\begin{minipage}[b]{.19\linewidth}
\centering
  \centerline{\includegraphics[width=1.03in]{targ4gray}}
%  \vspace{2.0cm}
  \centerline{(c)}\medskip
\end{minipage}
%
\begin{minipage}[b]{.19\linewidth}
  \centering
  \centerline{\includegraphics[width=1.03in]{targ4gray}}
%  \vspace{1.5cm}
  \centerline{(d)}\medskip
\end{minipage}
%
\begin{minipage}[b]{0.19\linewidth}
  \centering
  \centerline{\includegraphics[width=1.03in]{targ4gray}}
%  \vspace{1.5cm}
  \centerline{(e)}\medskip
\end{minipage}
%
\\
%
\begin{minipage}[b]{0.19\linewidth}
  \centering
  \centerline{\includegraphics[width=1.03in]{targ4gray}}
%  \vspace{1.5cm}
  \centerline{(f)}\medskip
\end{minipage}
\begin{minipage}[b]{.19\linewidth}
  \centering
  \centerline{\includegraphics[width=1.03in]{targ4gray}}
%  \vspace{2.0cm}
  \centerline{(g)}\medskip
\end{minipage}
%
\begin{minipage}[b]{.19\linewidth}
  \centering
  \centerline{\includegraphics[width=1.03in]{targ4gray}}
%  \vspace{1.5cm}
  \centerline{(h)}\medskip
\end{minipage}
%
\begin{minipage}[b]{0.19\linewidth}
  \centering
  \centerline{\includegraphics[width=1.03in]{targ4gray}}
%  \vspace{1.5cm}
  \centerline{(i)}\medskip
\end{minipage}
%
\begin{minipage}[b]{0.19\linewidth}
  \centering
  \centerline{\includegraphics[width=1.03in]{targ4gray}}
%  \vspace{1.5cm}
  \centerline{(j)}\medskip
\end{minipage}
%
\\
\begin{minipage}[b]{.19\linewidth}
  \centering
  \centerline{\includegraphics[width=1.3in]{targ4gray}}
%  \vspace{2.0cm}
  \centerline{(k)}\medskip
\end{minipage}
%
\begin{minipage}[b]{.19\linewidth}
  \centering
  \centerline{\includegraphics[width=1.3in]{targ4gray}}
%  \vspace{1.5cm}
  \centerline{(l)}\medskip
\end{minipage}
%
\begin{minipage}[b]{0.19\linewidth}
  \centering
  \centerline{\includegraphics[width=1.3in]{targ4gray}}
%  \vspace{1.5cm}
  \centerline{(m)}\medskip
\end{minipage}
%
\begin{minipage}[b]{0.19\linewidth}
  \centering
  \centerline{\includegraphics[width=1.3in]{targ4gray}}
%  \vspace{1.5cm}
  \centerline{(n)}\medskip
\end{minipage}
%
\begin{minipage}[b]{0.19\linewidth}
  \centering
  \centerline{\includegraphics[width=1.3in]{targ4gray}}
%  \vspace{1.5cm}
  \centerline{(o)}\medskip
\end{minipage}
%
\caption{Examples. Where particles are shown, red/blue are the distribution before/after
resampling. (a) Target for synthetic sequences. (b) Target inserted in complex background
with magnification and rotation. 
(c),(d) Standard/proposed SIR result in benign background. 
(e),(f) Standard/proposed SIR result in complex background; standard SIR fails.  
(g) Target for real sequence 1. 
(h),(i),(j) Ground truth, standard SIR result, and proposed SIR result for frame 87.
Standard SIR fails.
(k) Target for real sequence 2.  (l)-(o) ground truth for frames 32, 91, 116, and 174.}
\label{fig:Examples}
%
\end{figure*}



\section{Conclusions}
\label{sec:conclusion}

We introduced a new state vector augmentation method to incorporate indirect velocity
information into the likelihood function of the SIR filter for video target tracking
in clutter.  Our main objective was to prevent particles with a poor velocity
hypothesis from receiving large weights.  Our preliminary experimental results suggest
that the new method is effective, preventing the track filter from locking onto
structured clutter in cases where the standard SIR filter fails and delivering equivalent
tracking performance in cases where the standard SIR filter succeeds.

%auxiliary particle filters~\cite{PittShephard99,BrunoTSP04,Arulampalam2002}
%Auxiliary

%
% Old ZhouChella comparison
%Our method is distinct from the one in~\cite{ZhouChellaTIP2004}
%which uses velocity estimates computed from the measurements 
%to adaptively change the number of particles and the state update and
%appearance models, but does not use them explicitly in
%the likelihood.


Our method is distinct from the ones in~\cite{ZhouChellaTIP2004}
and~\cite{smalltracking08} which use velocity estimates computed from the measurements 
to adaptively change the state update and
appearance models, but do not use them explicitly in the likelihood.
It is also distinct from higher-order particle filters that
predict the current state from multiple previous states like the
one in~\cite{PanSPL2011} where the likelihood depends on multiple past states but not 
on past measurements and the one in~\cite{LiuCircSys2015} which uses
agent based crowd models.  Our likelihood function~\Eqref{eq:likelihood2}
is similar to the one in~\cite{OdobezGaticaBa-tip06} in the sense that both
depend explicitly on the current and past state and the current and past observation;
however, the method in~\cite{OdobezGaticaBa-tip06} obtains explicit velocity
estimates by block matching.  
An important component of our future work involves comprehensive performance
evaluation of our proposed method relative to all of these very interesting related
techniques.









%
%HPF - see Pan and Liu in Lit notes.
%Compared to the HPF: why not predict $z_k$ from $x_{k-m}$ ?  Well, that considers 
%all possible transitions.  It works great in the limiting theoretical case where you 
%have an infinite set of particles that actually converges to the true posterior.  
%But we are in the practical case where there are a finite number of particles.  
%We are interested in the particles we've got, and particularly in encouraging the good 
%ones and discouraging the bad ones.  
%For a bad particle, I don't care about all of the theoretically possible transitions from
%previous state to current observation - only the ones that were actually realized
%by particles in my Monte Carlo simulation.  This kind of multi-step prediction would
%weaken the good particle.  Note that I am disagreeing with (2) in~\cite{PanSPL2011}
%in a sense; I am saying that it's fine theoretically but with finite particles,
%modeling errors, etc, not practical (think about this harder).



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