adaptive particle filter based on the kurtosis of...

Masters Thesis

Adaptive Particle Filter based on the

Kurtosis of Distribution

Songlin Piao

Hanyang Universty, Graduate School

February 2011



by

Songlin Piao

A Thesis Presented to the

FACULTY OF THE GRADUATE SCHOOL

HANYANG UNIVERSITY

In Partial Fulfillment of the

Requirements for the Degree

MASTER OF SCIENCE

in the Department of Electrical and Computer Engineering

February 2011

Copyright 2010 Songlin Piao



by

Songlin Piao

Approved as to style and content by:

Sang-Won Nam

(Co-Chair of Committee)

Jong-Il Park

(Member)

Whoi-Yul Kim

(Member)

Hanyang Universty, Graduate School

February 2011

TABLE OF CONTENTS

TABLE OF CONTENTS I

ABSTRACT V

I Introduction 1

1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Related work . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

II Particle filter 6

2.1 Auxiliary particle filter . . . . . . . . . . . . . . . . . . . . . . 7

2.2 Gaussian particle filter . . . . . . . . . . . . . . . . . . . . . . 7

2.3 Unscented particle filter . . . . . . . . . . . . . . . . . . . . . 8

2.4 Rao-Blackwellized particle filter . . . . . . . . . . . . . . . . . 9

IIIProposed method 11

3.1 Basic concept . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

3.2 Concept of Kurtosis . . . . . . . . . . . . . . . . . . . . . . . 15

3.3 Proposed method in 1D . . . . . . . . . . . . . . . . . . . . . 17



3.6 Proposed method in general case . . . . . . . . . . . . . . . . 22

IV Experiment 24

4.1 Simulation in 1D . . . . . . . . . . . . . . . . . . . . . . . . . 24

4.2 Simulation in 2D . . . . . . . . . . . . . . . . . . . . . . . . . 27

I

4.3 Simulation in 3D . . . . . . . . . . . . . . . . . . . . . . . . . 28

4.4 Real particle tracking . . . . . . . . . . . . . . . . . . . . . . 31

4.5 Face tracking . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

V Conclusion 41

VI Appendix 43

BIBLIOGRAPHY 44

Acknowledgment 52

II

LIST OF FIGURES

3.1 Transition example . . . . . . . . . . . . . . . . . . . . . . . . 13

3.2 Example in 2D case . . . . . . . . . . . . . . . . . . . . . . . 14

3.3 Kurtosis of Gaussians . . . . . . . . . . . . . . . . . . . . . . 16

3.4 Proposed distribution in 1D . . . . . . . . . . . . . . . . . . . 18

3.5 Sampling from the specific probability density function . . . . 18

3.6 Proposed pdf looks similar with water wave . . . . . . . . . . 20

3.7 Motion vector in spherical system . . . . . . . . . . . . . . . . 21

3.8 Proposed pdf looks similar with shockwave . . . . . . . . . . 22

4.1 Simulation of fluctuation case . . . . . . . . . . . . . . . . . . 25

4.2 Simulation of non fluctuation case . . . . . . . . . . . . . . . 27

4.3 Simulation in 2D . . . . . . . . . . . . . . . . . . . . . . . . . 29

4.4 Simulation in 3D . . . . . . . . . . . . . . . . . . . . . . . . . 30

4.5 Particles detection result . . . . . . . . . . . . . . . . . . . . . 31

4.6 Motions in each frame . . . . . . . . . . . . . . . . . . . . . . 33

4.7 Motion angle . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

4.8 Tracking result . . . . . . . . . . . . . . . . . . . . . . . . . . 34

4.9 RMS error comparison . . . . . . . . . . . . . . . . . . . . . . 35

4.10 Face is tracked and detected . . . . . . . . . . . . . . . . . . . 37

4.11 Face is tracked but not detected . . . . . . . . . . . . . . . . 38

4.12 Motion Angle . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

4.13 Speed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

4.14 Face tracking result . . . . . . . . . . . . . . . . . . . . . . . . 39

4.15 Analysis data . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

III

LIST OF TABLES

3.1 Kurtosis of Gaussians . . . . . . . . . . . . . . . . . . . . . . 16

6.1 Random number generation test using 5000000 samples . . . 43

IV

ABSTRACT

Adaptive Particle Filter based on the Kurtosis of Distribution

Songlin Piao

Department of Electrical and Computer Engineering,

Hanyang University

Directed by Professor Whoi-Yul Kim

Kurtosis based adaptive particle filter is presented in this paper. The concept

of belief is proposed to each particle sampling and the distribution of particles

can be adaptively changed according to the belief and motion information so

that particles could track object in higher accuracy. The belief and motion

information could be defined as a distance function of observation vector. In

order to achieve this goal, we change the way of normal re-sampling technique.

We introduce a framework that particles are re-sampled based on the distance

function. We demonstrate the advantages of proposed method in two steps.

First, we did strict simulation tests in 1D, 2D and 3D spaces to show that our

method can give better result. Furthermore, we did the experiments in the

real cases. One is real particle tracking in the hydraulic engineering area and

the other is normal face tracking based on the color feature. We compared

the result in each step to the result obtained from standard particle filter.

V

I. Introduction

1.1 Background

The analysis and making inference about a dynamic system arise in a

wide variety of applications in many disciplines. The Bayesian framework is

the most commonly used method for the study of dynamic systems. There

are two components needed in order to describe Bayesian framework. First,

a process model describing the evolution of a hidden state of the system and

second, a measurement model on noisy observations related to the hidden

state. If the noise and the prior distribution of the state variable is Gaussian,

the predicted and posterior densities can be described by Gaussian densi-

ties. Kalman filter is one of the cases, which yields the optimized solution in

MMSE. But there are two big problems when people apply Bayesian frame-

work to the real world. One is that a realistic process and measurement

model for a dynamic system in the real world is often nonlinear, and the

other one is that the process noise and measurement noise sources could be

non-Gaussian. Simultaneous localization and mapping (SLAM) [1] problem

in robotic research area is a typical example. Kalman filter performs poor

when the linear and Gaussian conditions are not satisfied. This has motivated

intensive research for nonlinear filters for over 40 years. Nonlinear filters have

involved finding suboptimal solutions and may be classified into two major

approaches: a local approach, approximating the posterior density function

by some particular form, and a global approach, computing the posterior

density function without making any explicit assumptions about its form [2].

1

Gaussian filter is one of the typical examples in the local approach. There

are various forms of Gaussian filter proposed until now, for example, extended

Kalman filter (EKF), iterated Kalman filter (IKF), unscented Kalman filter

(UKF), and so on. Another approach approximating non-Gaussian density

is a Gaussian-sum representation. Based on the fact that non-Gaussian den-

sities can be approximated reasonably well by a finite sum of Gaussian densi-

ties, Alspach and Sorenson introduced the Gaussian-sum filter for nonlinear

system [3].

A global approach approximates the densities directly, so that the inte-

grations involved in the Bayesian estimation framework are made as tractable

as possible. Particle filter is just a special example in this case. It uses a

set of randomly chosen samples with associated weights to approximate the

true density. As one of the Sequential Monto Carlo (SMC) methods, when

the number of samples becomes larger, the posterior function becomes more

accurate. However, the large number of samples often makes the use of SMC

methods computationally expensive. But thanks to the development of com-

puter technology, it becomes possible even if the system is comprised of very

huge particles [4].

1.2 Related work

Particle filter, one of dynamic state estimation techniques, is commonly

used in many engineering applications, especially in signal processing and

object tracking. Various forms of the particle filters and their applications

are proposed until now. Chang and Lin proposed a 3D model-based tracking

algorithm called the progressive particle filter [5], Gil et el. solved Multi-

robot SLAM problem using a Rao-Blackwellized particle filter [6], Jing and

Vadakkepat proposed an interacting MCMC particle filter to track maneuver-

2

ing target [7], Lakaemper and Sobel proposed an approach to building partial

correspondences between shapes using particle filter [8], Mukherjee and Sen-

gupta proposed a more generalized likelihood function model in order to get

higer performance [9], Pistori et el. combined different auto-adjustable obser-

vation models in a particle filter framework to get a good trade-off between

accuracy and runtime [10], Sajeeb et el. proposed a semi-analytical particle

filter, requiring no Rao-Blackwell marginalization, for state and parameter

estimations of nonlinear dynamical systems with additively Gaussian process

and observation noises [11].

Shao et el. used a particle based approach to solve the constrained

Bayesian state estimation problems [12], Xu et el. proposed an ant stochastic

decision based particle filter to solve the degradation problem when it is ap-

plied to the model switching dynamic system [13], Chung and Furukawa

presented a unified framework and control algorithm using particle filter

for the coordination of multiple pursuers to search for and capture multi-

ple evaders [14], Handel proved that bootstrap-type Monte Carlo particle

filters approximate the optimal nonlinear filter in a time average sense uni-

formly with respect to the time horizon when the signal is ergodic and the

particle system satisfies a tightness property [15], Hlinomaz and Hong pre-

sented a full-rate multiple model particle filter for track-before-detect and

a multi-rate multiple model track-before-detect particle filter to track low

signal-to-noise ratio targets which perform small maneuvers [16], Hotta pre-

sented an adaptive weighting method for combining local classifiers using a

particle filter [17], Lu et el. proposed a method to track and recognize ac-

tions of multiple hockey players using the boosted particle filter [18], Maggio

and Cavallaro proposed a framework for multi target tracking with feedback

that accounts for scene contextual information [19], Moreno et el. addressed

the SLAM problem for stereo vision systems under the unified formulation

3

of particle filter methods based on the models computing incremental 6-DoF

pose differences from the image flow through a probabilistic visual odometry

method [20].

Wang et el. proposed camshift guided particle filter for tracking object

in video sequence [21], Zheng and Bhandarkar proposed an integrated face

detection and face tracking algorithm using a boosted adaptive particle fil-

ter [22], Choi and Kim proposed a robust head tracking algorithm with the

3D ellipsoidal head model integrated in particle filter [23], Li et el. presented

a temporal probabilistic combination of discriminative observers of different

lifespans into the particle filter to solve the tracking problem in low frame

rate video with abrupt motion poses [24], Lin et el. presented a particle

swarm optimization algorithm to solve the parameter estimation problem for

nonlinear dynamic rational filters [25], Olsson and Ryden did the research

on the asymptotic performance of approximate maximum likelihood estima-

tors for state space models obtained via sequential Monte Carlo method [26],

Pantrigo et el. solved the multi-dimensional visual tracking problems using

scatter search particle filter [27], Sankaranarayanan et el. proposed a new

time-saving method for implementing particle filter using the independent

Metropolis Hastings sampler [28], Simandl and Straka proposed a new func-

tional approach to the auxiliary particle filter so that it could provide closer

filtering probability density function in term of point estimates [29], Wu et

el. proposed a novel fuzzy particle filtering method for online estimation of

nonlinear dynamic systems with fuzzy uncertainties [30], Yee et el. proposed

approximate conditional mean particle filter which is a combination of the

approximate conditional mean filter and the sequential importance sampling

particle filter [31], Grisetti used Rao-Blackwelized particle fitler to solve the

simultaneous localization and mapping problem(SLAM) efficiently by reusing

already computed proposal distribution [1], Hong and Wicker proposed a mul-

4

tiresolutional particle filter in the spatial domain using thresholded wavelets

to reduce significantly the number of particles without losing the full strength

of a particle filter [32], Li and Chua presented a particle filter solution for

non-stationary color tracking using a transductive local exploration algo-

rithm [33], McKenna and Nait-Charif proposed a method to track human

motion using auxiliary particle filters and iterated likelihood weighting [34],

Rathi et el. formulated particle filtering algortihm in the geometric active

contour framework that can be used for tracking moving and deforming ob-

jects [35], Shan et el. proposed a real-time hand tracking algorithm using

a mean shift embedded particle filter [4], Clark and Bell proposed a parti-

cle PHD filter which propagates the first-order moment of the multi-target

posterior instead of the posterior distribution so that the tracker could track

multi-target in real time [36], Emoto et el. proposed a cyclic motion model

whose state variable is the phase of a motion and estimated these variables

using particle filter [37], Fearnhead et el. introduced novel particle filters for

class of partially-observed continuous-time dynamic models where the signal

is given by a multivariate diffusion process [38], Tamimi et el. proposed a

novel method which could localize mobile robots with omnidirectional vi-

sion using particle filter and performance improved SIFT feature [39], Bolic

et el. proposed novel re-sampling algorithms with architectures for efficient

distributed implementation of particle filters [40], Khan et el. intergraded

Markov random field into particle filter to deal with tracking interacting tar-

gets [41], Sakka et el. proposed a new Rao-Blackwellized particle filtering

based algorithm for tracking an unknown number of targets [42], Schon et el.

implemented marginalized particle filter by associating one kalman filter to

each particle so that the proposed method could reduce time complexity [43].

5

II. Particle filter

Particle filtering is a technique used for filtering nonlinear dynamical sys-

tems driven by non-Gaussian noise processes. The purpose of particle filter is

to estimate the states {S1, , St} recursively using the sampling technique.

To estimate the states, the particle filter approximates the posterior distribu-

tion p(St|Z1:t) with a set of samples {S(1)t , , S(p)t } and a noisy observation

Z1, , Zt. In particle filtering, the probability density distribution of the

target state is represented by a set of particles. The posterior density of the

target state for a given input image is calculated and represented as a set

of particles. In other words, a particle is an hypothesis of the target state,

and each hypothesis is evaluated by assessing how well the hypothesis fits

the current input data. Depending on the scores of the hypotheses, the set

of hypotheses is updated and regenerated in the next time step. The parti-

cle filter consists of two components, state transition model and observation

model. They can be written as

TranslationModel : St = Ft(St1, Nt),

ObservationModel : Zt = Ht(St,Wt). (2.1)

The transition function Ft approximates the dynamics of the object being

tracked using previous state St1 and the system noise Nt. The measurement

Ht models a relationship among the noisy observation Zt, the hidden state

St, the observation noise Wt. We can characterize transition probability

P (St|St1) with the state transition model, and likelihood P (Zt|St) with the

observation model.

6

2.1 Auxiliary particle filter

The auxiliary particle filter is a particle filtering algorithm introduced by

Pitt and Shephard in 1999 [44] to improve some deficiencies of the sequential

importance resampling (SIR) algorithm when dealing with tailed observation

densities.

Assume that the filtered posterior is described by the following M weighted

samples:

p(xt|z1:t) Mi=1

(i)(xt x(i)t ). (2.2)

Then, each step in the algorithm consists of first drawing a sample of the

particle index k which will be propagated from t 1 into the new step t.

These indexed are auxiliary variable only used as an intermediate step, hence

the name of the algorithm. The indexes are drawn according to the likelihood

of some reference point (i)t which in some way is related to the transition

model xt|xt1 (for example, the mean, a sample, etc.):

k(i) P (i = k|zt) (i)t p(zt|(i)t ). (2.3)

This is repeated for i = 1, 2, ,M , and using these indexes we can now

draw the conditional samples:

x(i)t p(x|x

k(i)t1). (2.4)

Finally, the weights are updated to account for the mismatch between the

likelihood at the actual sample and the predicted point k(i)t :

(i)t

p(zt|x(i)t )p(zt|k

(i)

t ). (2.5)

2.2 Gaussian particle filter

The Gaussian particle filter and Gaussian sum particle filter were firstly

introduced by Jayesh H. Kotecha et el. in [45]. Gaussian particle filter ap-

7

proximates the posterior distributions by single Gaussians, similar so Gaus-

sian filters like the extended Kalman filter [46] and its variants. The un-

derlying assumption is that the predictive and filtering distributions can be

approximated as Gaussians. Unlike the EKF, which also assumes that these

distributions are Gaussians and employs linearization of the functions in the

process and observation equations, the GPF updates the Gaussian approxi-

mations by using particles.

2.3 Unscented particle filter

The procedure of the latest presented unscented particle filter could be

described as below:

1. Initialization, t = 0:

For i = 1, , N , draw particle xi0 p(x0) and set t = 1.

2. Importance sampling step:

For i = 1, use UKF with the main model to generate the im-

portance distribution N(x1t , P1t ) from particle x

1t1. Sample x

1t

N(x1t , P1t ).

For i = 2, 3, , N , use UKF with the auxiliary model to generate

the importance distribution N(xit, Pit ) from particle x

i1t . Sample

xit N(xit, P it ).

3. Importance weight step:

For i = 1, , N , evaluate the importance weights using likelihood

function.

For i = 1, , N , normalize importance weight of all the particles.

8

4. Resampling step:

Resample N particles xi1:t from the xi1:t according to the normal-

ized importance weights.

set it = 1N .

2.4 Rao-Blackwellized particle filter

The advantage of the Rao-Blackwellized particle filter is that it allows

the state variables to be splitted into two sets, being of them analytically

calculated from the posterior probability of the remaining ones. It has been

applied to SLAM, non-linear regression, multi-target tracking, appearance

and position estimation. In the particle filter framework, if the dimension

of the state space becomes higher, it would be inefficient sampling in high-

dimensional spaces [47]. However, the state can be separated into tractable

subspaces in some cases. If some of these subspaces can be analytically

calculated, the size of the space over which particle filter samples will be

drastically reduced. This kind of concept was first proposed in [48].

If we denote a state space model as st and observation model as zt and

observations are assumed to be conditionally independent given the process

st of marginal distribution p(st|zt). The aim is to estimate the joint posterior

distribution p(s0:t|z1:t). The pdf can be written in the recursive way

p(s0:t|z1:t) =p(zt|st)p(st|st1)p(s0:t1|z1:t1)

p(zt|z1:t1), (2.6)

where p(zt|z1:t1) is a proportionality constant.

In multi-dimensional spaces, obtained integrals are not always tractable.

However, if the hidden variables had a structure, we could devide state st

into two groups, rt and kt such that p(st|st1) = p(kt|rt1:t, kt1)p(rt|rt1).

9

In such case, we can marginalize out k0:t from the posterior, reducing the di-

mensionality problem. Following the chain rule, the posterior is decomposed

as follows

p(r0:t, k0:t|z1:t) = p(k0:t|z1:t, r0:t)p(r0:t|z1:t), (2.7)

where the marginal posterior distribution p(r0:t|z1:t) satisfies the alternative

recursion like

p(r0:t|z1:t) p(zt|rt)p(rt|rt1)p(r0:t1|z1:t1). (2.8)

10

III. Proposed method

It is acknowledged that a successful implementation of particle filter re-

sorts to two aspects:

How to select appropriately samples, i.e., how to avoid degeneration in

which a number of samples are removed from the sample set due to the

lower importance weights.

How to design an appropriate proposal distribution to facilitate easy

sampling and further to achieve a large overlap with the true state

density function.

This paper is focused on the second aspect. The choice of proposal im-

portance distribution is one of the critical issues in particle filter, while the

performance of particle filter heavily depends on the proposal importance

function. The optimal proposal importance distribution is q(xt|x0:t1, y1:t) =

p(xt|xt1, yt), because it fully exploits the information in both xt1 and yt.

In practice, it is impossible to sample from this distribution due to its com-

plication. The second choice of proposal function is the transmission prior

function q(xt|x0:t1, y1:t) = p(xt|xt1) for its easiness to sample. This is the

most popular choice. But since this function does not use the latest infor-

mation yt, the performance depends heavily on the variance of observation

noise. When the observation noise variance is small, the performance is poor.

The third choice is to use the method of local linearization to generate the

proposal importance distribution [49].

11

We proposed a kurtosis based sampling technique to improve the accu-

racy of the current particle filter framework. The word kurtosis means a

measure of the peakedness of the probability distribution of a real-valued

random variable. In particle filter framework, each time state has its own

distribution for particles and it could be used to estimate the exact state

associated with observed data. The distribution is changing at every time

step. In the previous work, people assume the distribution of particles equals

to the distribution of state transition and assume the same Gaussian noise

while doing prediction. In this article, kurtosis based framework is proposed,

which means the distribution of the particles is changing according to some

meaningful features, motion vector in this case.

3.1 Basic concept

We denote state vector at time t as Vt = {v1, v2, , vn} and the obser-

vation state vector at time t as Zt = {z1, z2, , zk}, here n is the number

of state dimension and k is the number of observation dimension, usually k

is smaller than n. Then the derivative of the each observation state can be

represented z1t ,z2t , ,

zkt . For a vector Zt = {z1, z2, , zk}, we define

the distance function of Zt as

D(Zt) =

((z1t

)2 + (z2t

)2, ,+(zkt

)2) 1

2

. (3.1)

Even though the dimension of the observation vector is k, it is possible to

use only part of this. That is to say, the value of D(Zt) could be replaced by

the value of D(Z t), where Zt = {z1, z2, , zq} and {z1, z2, , zq} is subset

of {z1, z2, , zk}. Here {z1, z2, , zq} are the interesting features we want

to use.

Fig. 3.1 shows an example of the state transition in the normal particle

12

Figure 3.1 Transition example

filter framework. The relationship between St and St1 could be represented

using St = f(St1) + nt, here nt is assumed to const Gaussian distribution

most of cases. In the case we predict current state St with the velocity

information vt1 from the previous state, the relationship could be written

as St = St1 + vt1 + nt. We change the distribution of particles according

to the value of vt1, more specifically, if the absolute value of vt1 become

larger, then the kurtosis of the particles distribution would become higher,

too. In order to introduce this concept more easily, we take a simple example.

Consider the case that if the radius and angle values are known and they

comply the Gaussian distribution as described in (3.2) and (3.3)

P =12

exp

((motion)

2

22

), (3.2)

P =12

exp

(( motionAngle)

2

22

), (3.3)

x = cos ,

y = sin . (3.4)

Here the value of radius means the length of the motion vector and the

value of angle means the orientation of the motion vector. From the equation

(3.2) and (3.3), it is known that they comply a specific Gaussian distribution.

13

(a) Distribution (b) Contour (c) Normal case

Figure 3.2 Example in 2D case

If their distributions are known, then the joint distribution of (x, y) looks like

in Fig. 3.2(a). In this example, the mean of is set to 4.3633 radian (about

250 degree) and the mean of is set to 6. And their covariance matrix is set

to

1 00 2

as in Fig. 3.2(b). It is time to explain our concept now. If theposition Pt1 of object at previous state is located in (0, 0), the velocity of

the object is 6 and the orientation of this velocity is 4.3633 radian, then the

distribution of position Pt of object at current state could be predicted as in

Fig. 3.2(a). The distribution of the position Pt is associated with the motion

information from the previous state. In the normal particle filter framework,

people always use the fixed Gaussian (Fig. 3.2(c)) or uniform distribution

to predict the objects position in the future state. Actually, this is Nt in

translation model in equation (2.1). Instead, in the proposed method, we

use the noise model Nt associated with motion information. In order to go

further, it is necessary to introduce the concept of kurtosis a little.

14

3.2 Concept of Kurtosis

Kurtosis is defined as the fourth cumulant divided by the square of the

second cumulant, which is equal to the fourth moment around the mean

divided by the square of the variance of the probability distribution minus 3

as

kurtosis =44 3. (3.5)

For a sample of n values the sample kurtosis is

kurtosis =m4m22 3 =

1n

ni=1(xi x)4

( 1nn

i=1(xi x)2)2 3, (3.6)

where m4 is the fourth sample moment of about the mean, m2 is the second

sample moment about the mean, xi is the ith value, and x is the sample

mean. In the case of particle filter, the kurtosis of a distribution could defined

similarly as

kurtosis =

ni=1 i(xi x)4

(n

i=1 i(xi x)2)2 3, (3.7)

where i is the weight of the ith particle. As mentioned before, kurtosis is

measure of the peakedness of some probability distribution. In the case

of particle filter, this concept could be used to measure the new predicted

position of the particles. The kurtosis changes according to the belief of

particles and the length of the motion information. When the belief of particle

is much more reliable and the strength of the motion vector is large, the value

of kurtosis become smaller. In the case of Gaussian distribution, the larger

the kurtosis is, then the larger the standard deviation is. This is actually the

key concept of the proposed method in this article.

Fig. 3.3 and Table 3.1 show several Gaussian distributions in 1D and

their corresponding kurtosis values. It can seen that the larger the sigma

value is, the larger the kurtosis is.

15

Figure 3.3 Kurtosis of Gaussians

Table 3.1 Kurtosis of Gaussians

kurtosis

red 0 1.0 11.17963

blue 0 1.2 14.01556

green 0 1.4 16.85140

yellow 0 1.6 19.68429

magenta 0 1.8 22.48823

black 0 2.0 25.16695

16

3.3 Proposed method in 1D

In the case of 1D, there are only two options for the angle. One is to

translate along the positive direction, the other is to translate along the

negative direction. If assume that the motion information is known, the

position of the current state is at 0 and the state transition satisfies St+1 =

St + vt + nt+1, then the probability of St+1 could represented as

Pst+1 =

=Pp,Pn

12

exp

((positionX motion)

2

22

), (3.8)

=

{Pp positive direction ,

Pn negative direction , (3.9)

Pp + Pn = 1. (3.10)

The state transition distribution could be seen in Fig. 3.4. But it is shown

that there is a gap at position 0, which is the position of the current state. In

order to solve this problem, we used cubic spline [50] to smooth the selected

five points so that the gap is gone. The five points selected are two peak

points and two half middle points from left and right Gaussian distribution

and the point at 0. The smoothed curve is drawn with green color in Fig.

3.4. The left column has mean 1.5 and -1.5, the middle column has mean

2.0 and -2.0, the right column has mean 2.5 and -2.5. The above row has

standard deviation 1.2 and the nether row has standard deviation 1.8. The

positive direction has weight 0.7 and the negative direction has weight 0.3.

For N particles at time step t 1, {p1t1, p2t1, , pNt1}, they would

be filtered firstly by resampling step and then produce new location of the

particle using the proposed probability density function like in Fig. 3.5(a).

For example, pit = pit1 + qt1, here qt1 is the proposed pdf at time step

17

(a) = 1.5 = 1.2 (b) = 2.0 = 1.2 (c) = 2.5 = 1.2

(d) = 1.5 = 1.8 (e) = 2.0 = 1.8 (f) = 2.5 = 1.8

Figure 3.4 Proposed distribution in 1D

(a) Required pdf (b) Sampling result

Figure 3.5 Sampling from the specific probability density function

18

t1. All the motion information has already been considered inside the pdf,

and the pdf would change according to the motion information and the belief

of the particles. The concept is very different from the normal particle filter

where pit = pit1 + vt1 +nt, vt1 is the speed of object at time step t 1 and

nt is usually a gaussian noise.


In the case of 2D, motion vector could be represented as a 2D vector with

length and phase. We denote these two variables using and as described

in equation (3.2) and (3.3). Similar with the case of 1D, motion based dis-

tribution would be used to predict the next state of each particle instead of

gaussian noise. The proposed distribution has a peak at the location (,),

here, denotes the length of the motion vector and denotes the phase of

motion vector. If we assume and are independent with each other and

the kurtosis of their each distribution is changing adaptively. One of the

examples is shown in Fig. 3.2(a). The shape of the sampling probability

density function is much more like the a ripple on the water when a object

is moving inside the water Fig. 3.6(b) [51].

We denote state vector as St = {xt, yt} ande velocity vector Vt = {vtx, vty}

at the time step t, then we generate the new state of each sample using

Sit+1 = Sit + F (Vt), here F is the proposed sampling function based on the

motion information at each time step. In the case of normal particle filter,

the equation is like Sit+1 = Sit + vt+Nt. So the proposed sampling method is

different original sammpling method in the prediction state. The proposed

sampling method has a big advantage that it can change pdf or integrate

other factors into the sampling state.

19

(a) Proposed pdf (b) Water wave

Figure 3.6 Proposed pdf looks similar with water wave


In the case of 3D, when the state vector is St = {xt, yt, zt}, which is the

location of the object at the current time step, the motion vector could be

represented as spherical coordinate system as in Fig. (3.7). The relation-

ship between the spherical coordinate (r, , ) of a point and its cartesian

coordinates (x,y,z) is like

x = r sin cos,

y = r sin sin,

z = r cos . (3.11)

If we denote current state as St and current velocity as Vt, then the tran-

sition probability becomes P (St+1|St) = f(Vt), where f(Vt) is the proposed

pdf which depends on the current motion of the object. The relationship

20

Figure 3.7 Motion vector in spherical system

between proposed pdf and motion could be expressed as

Vt = Xti+ Ytj + Ztk,

=X2t + Y

2t + Z

2t ,

= acos(Zt

),

= g(Xt, Yt),

f(Vt) = P(,,)(, , ). (3.12)

It is shown that the proposed pdf is the joint probability of (, , ). If

we assume the probability of , , is independent with each other and they

comply the Gaussian distribution, then the joint probability of these three

variables looks similar to the shape of shockwave. Fig. 3.8(a) is the proposed

pdf when Xt = 2, Yt = 3, Zt = 4, it can be seen that the shape of the pdf

is similar to the shockwave in Fig. 3.8(b). It means the probability of the

position which locates along the motion direction has higher value. Of course,

the distribution of , , is not necessarily the Gaussian distribution, they

21

(a) Proposed pdf (b) Shockwave

Figure 3.8 Proposed pdf looks similar with shockwave

could be any distribution you want to define. But whatever the distribution

is, the kurtosis of the distribution would be changed according to the motion

information.

3.6 Proposed method in general case

We have proposed new sampling methodology in 1D, 2D, 3D space. The

proposed method could be extended to higher dimensional space. We denote

state vector at time t as St = {s1, s2, , sm} and observation state vector

as Zt = {z1, z2, , zk}, here m is the number of state dimension and k is the

number observation dimension and k is smaller than m or equal to m. Then

the problem can classify into two groups, one is the case when k is equal to

m, and the other is the case when k is smaller than m. The observation state

vector could be obtained using the corresponding state vector, we denote

this relationship as: Zt = H(St). We need to first calculate the gradient of

the current observation vector and then calculate the distance D(Zt) using

equation (3.1). Then we calculate the gradient of St from the gradient of Zt,

22

the inference procedure is like

Zt = H(St),

Ztt

=H(St)

t=H(St)

(St) Stt

,

gradient(Zt) =Ztt

,

gradient(St) =Stt

,

gradient(St) = gradient(Zt) (H(St)

(St))1. (3.13)

As long as the gradient of state St is estimated, then we could calculate

the angle between gradient vector of St and unit base vector in the state

space as

i = acos

[gradient(St), Si|gradient(St)| |Si|

], (3.14)

where i is the ith unit base vector in the state space. In the case m di-

mensional space, we only need m 1 parameters like . We make a joint

probability like

P = P (D(St), 1, 2, , i1), (3.15)

each random variable in this joint probability changes adaptively according

the value of D(St), it mean the kurtosis of the sampling probability changes

adaptively. Finally the joint probability would be used to predict state of

object in the next time step.

23

IV. Experiment

In order to verify proposed method, we designed test simulations in 1D,

2D, 3D spaces, respectively. Then we also applied the proposed method to

the real cases to verify the accuracy.

4.1 Simulation in 1D

We did two kind of simulations in 1D space, one is the fluctuation case

and the other is the non-fluctuation case. In order to take the test in the

fluctuation case, we chose the state transition model and observation model

as

Xt = Xt1 + 8 cos(1.2 (t 1)) +Nt,

Zt = Xt +Wt, (4.1)

where Nt and Wt are both defined as a normal Gaussian distribution with

the standard deviation 1.0. In order to change transition density probability

adaptively according to the motion information, in equation (3.8) and

Pp, Pn in equation (3.9) are defined as

= 8.8 exp(|motion| 0.17), (4.2)

24

(a) Pdf estimation (b) Result compared with particle filter

(c) Result with EKF (d) RMSE for 50 Monte Carlo runs

Figure 4.1 Simulation of fluctuation case

Pn =1

2.14 + |motion| 0.64,

Pp = 1 Pn, (4.3)

Pp =1

2.14 + |motion| 0.64,

Pn = 1 Pp, (4.4)

Pp = 0.5,

Pn = 0.5. (4.5)

The above three equations define Pp and Pn when motion is positive,

negative, zero, respectively. The proposed filter is used to estimate the state

and the results are compared with that of a usual particle filter and extended

25

kalman filter(EKF). From the definition of , the larger the motion is, the

smaller the is, so the kurtosis of the distribution becomes smaller, too.

Fig. 4.1 shows the simulation result. Fig. 4.1(a) shows the probability

density at the time step 20, Fig. 4.1(b) show the comparison with the usual

particle filter, Fig. 4.1(c) shows the estimation result form EKF and Fig.

4.1(d) shows the comparison of the average RMSE among the filters over

the 50 time samples. It is shown that the estimation result compared with

normal particle filter is very close and RMSE error is also similar with the

normal particle filter but much smaller than the EKF. And the bandwidth of

the pdf of the proposed filter is narrower than the other two filters. It is also

shown in Fig. 4.1(b) that when the speed of the object changes, there exists

estimation errors. This is because we didt consider the acceleration in this

case. Actually, we only considered the first derivative of the state but not

the second derivative of the state. And the usual particle filter estimation

is a little more accurate than the proposed method. It is because transition

model is known while generating new particles in the normal particle filter

method, but we dont even know the transition model, we only predict the

state using the velocity(derivative of the state) information. In real cases,

the transition model is unknown, we can only estimate state using previous

experience like first and second derivative of the state and so on. Besides, the

parameter functions used in this case could be changed to any other forms.

Fig. 4.2 shows the test result of the non fluctuation case. The speed is always

positive in this case, so the acceleration is not needed. It is shown that the

estimation result is almost the same as the true state.

26

Figure 4.2 Simulation of non fluctuation case


In order to make a test in 2D space, we have manually defined a specific

trajectory which could be expressed as

vx = 5 cos(1

4),

vy = 5 sin(1

4),

t = 0 : 1 : 300,

X = vx t+ 1100

t1.5,

Y = vy t 0.4t1.4. (4.6)

27

Then the speed, , at time step t can be expressed as

Vtx = vx+ 0.015t0.5,

Vty = vy 0.56t0.4,

=V 2tx + V

2ty,

= g(Vtx, Vty). (4.7)

And the distributions of and are defined as

= ||,

= 8.8 exp(|| 0.17),

= ||,

= 4.8 exp(|| 0.25). (4.8)

Fig. 4.3 shows the simulation result. Fig. 4.3(a) is original trajectory,

Fig. 4.3(b) is the tracking result using particle filter, Fig. 4.3(c) is the

tracking result using proposed method, Fig. 4.3(d) shows the RMS error

comparison between proposed method and particle filter. The result shows

that the RMS error of proposed method is smaller than the general particle

filter through the whole iteration time.


In order to make a test in 3D space, we have manually defined a specific

trajectory which could be expressed as

t = 0 :pi

100: 10 pi,

x = 30 sin(t),

y = 10 cos(t),

z = sin(0.6 t), (4.9)

28

(a) Original trajectory (b) Particle filter result

(c) Proposed sampling result (d) RMS error comparison

Figure 4.3 Simulation in 2D

29

(a) Proposed method result (b) Particle filter result

(c) True state (d) RMS error

Figure 4.4 Simulation in 3D

the distributions of , , are defined as

= ,

= 8.8 exp( 0.17),

= ,

= 4.8 exp( 0.25),

= ,

= 4.8 exp( 0.25). (4.10)

Fig. 4.4 shows the simulation result in 3D, where Fig. 4.4(a) is the

result from proposed method, the Fig. 4.4(b) is the result from particle filter

30

(a) 10th frame (b) 20th frame (c) 30th frame

Figure 4.5 Particles detection result

method and the Fig. 4.4(c) is the true state. It is difficult to see the trajectory

clearly, so we calculated root mean square error of proposed method and

particle filter, respectively. Fig. 4.4(d) shows RMS error comparison between

proposed method and particle filter. The result shows that the RMS error of

particle filter is 1.9766 and the RMS error of the proposed method is 1.9042,

which is smaller than the general particle filter when iteration was finished.

4.4 Real particle tracking

Test set from halcon software [52] in the hydraulic engineering area is

used to do experiment. The set consists of fifty sequential images. In or-

der to make ground truth data, the positions of particles and their motion

information in each frame are recorded by detecting each frame. Subpixel

algorithm is used to improve the accuracy. And then we implemented the

tracking algorithm using particle filter associated with the proposed method

to compare the result with the ground truth data and the other results ob-

tained using general particle filter method. And only one particle would be

tracked in this experiment.

Fig. 4.5 shows the detection result from 10th, 20th, 30th frame in the

31

sequence and the particle surrounded by red circle would be tracked in this

experiment. The detection procedure is like below:

Calculate the optical flow [53] between two images.

Estimate the region with moving particles using the length of the vec-

tors of the calculated vector field.

Find the position of particles in the original image using an estimated

ROI.

Estimate the speed of the particles using the result from optical flow.

Fig. 4.6 and Fig. 4.7 show the detected motion length and angle in each

frame, respectively. Fig. 4.8 shows the tracking result. The parameters used

in this test are the same as in equation (4.8) and the likelihood used here is

Gaussian distance. The green line is the estimated position of the particle

using proposed method, the red line is the estimated position of the particle

using general particle filter and the the blue dotted line is the estimated

position by detecting each frame. Fig. 4.9 shows the RMS error comparison

between the proposed method and the original particle filter. During the

experiments, we found that the performance of the proposed method depends

on the two factors. One is the measurement noise and the other is the noise

nt which is added to the original particle filter. Because original particle

filter depends on the measurement noise, too, we can ignore it here. We only

discuss the impact of the noise nt. Actually, there is some threshold in nt.

When the value of nt is larger than this threshold then the performance of

the proposed method becomes better, if the value of nt is smaller than this

threshold then the performance of the original method becomes better.

32

Figure 4.6 Motions in each frame

Figure 4.7 Motion angle

33

Figure 4.8 Tracking result

34

Figure 4.9 RMS error comparison

35

4.5 Face tracking

In order to prove that our method could also be used in other real cases,

we did experiment on face tracking. We used open computer vision librarys

trained haar like feature to detect face every frame in order to get ground

truth data. We tracked face every frame using 250 particles. HSV color

space and back projection technique are used as features and measurement

method during tracking. As it is shown in Fig. 4.11, the program could not

detect face because the classifier is focused on the frontal face. In this case,

we use interpolation to produce the location of the face at that frame. Fig.

4.10 shows the case that both tracking and detection are successful. Fig.

4.12 and Fig. 4.13 show the detected motion length and angle in each frame,

respectively.

Fig. 4.14 shows the face tracking result. The first row shows the result

from the first experiment with Gaussian noise matrix

2.5 00 2.5

and thesecond row shows the result from the second experiment with Gaussian noise

matrix

4.0 00 4.0

. From the first experiment, it could be seen that theperformance of the proposed method and normal particle filter are similar in

the anterior 90 frames. But the performance of the proposed method becomes

better when the noise value becomes higher in the second experiment. There

is some threshold value in this noise value. If the noise of the original particle

filter is less then this threshold, then the original particle filter performs

better than the proposed method. But if the noise of the original particle

filter is greater than the threshold value, then the proposed method performs

better. It is meaningful when we would apply the proposed method into the

real world cases. Because, with the too small noise value, we cannot tracking

object robustly.

36

Figure 4.10 Face is tracked and detected

37

Figure 4.11 Face is tracked but not detected

Figure 4.12 Motion Angle

38

Figure 4.13 Speed

(a) First test trajectory (b) First test RMS

(c) Second test trajectory (d) Second test RMS

Figure 4.14 Face tracking result

39

Figure 4.15 Analysis data

Fig. 4.15 shows the analysis data sheet in order to explain why the

RMS error in Fig. 4.14(b) and in Fig. 4.14(d) changed abruptly at about

frame 92. You can see the speed at frame 92 is 11.40175425, which changed

abruptly from the previous speed. This is because the distance function in

the proposed method considers only first derivative of the observation vector,

so the proposed model seems a little weak to the abrupt change of the speed.

40

V. Conclusion

Kurtosis based adaptive particle filter is proposed in this article. This new

method uses a new sampling method, which changes particles probability

density function adaptively according the motion information, which is the

special case of the distance function defined previously. We show the accuracy

of the new method by doing simulations in 1D, 2D and 3D, respectively. We

also applied the new method in real cases in order to show the potential

applicable possibility in real world problems. The result has shown that the

new method has a good performance as we expected. This method is very

useful when the object state transition function is unknown. But there is

one thing that I want to emphasize here is the performance of the proposed

method depends on the measurement noise and Gaussian distribution noise

in the original particle filter. Because proposed method and original particle

filter are both depends on the measurement noise, we could ignore it here. So

the proposed method only depends on the noise of the original particle filter

relatively. There is some threshold value in this noise value. If the noise

of the original particle filter is less then this threshold, then the original

particle filter performs better than the proposed method. But if the noise

of the original particle filter is greater than the threshold value, then the

proposed method performs better. It is meaningful when we would apply

the proposed method into the real world cases. Because, with the too small

noise value, we cannot tracking object robustly. The future work includes

expanding this framework to the higher dimensional space. We are doing

41

research on the convergence problem when dimension becomes higher. Of

course, the proposed method has also some limitations, for example, it is a

little weak to deal with the abrupt changes as it was shown in face tracking

case. This is because the distance function considers only the first derivative

of the observation vector. So we need to add some other factors to handle the

abrupt changes. We could consider second derivative, third order derivative

or curvature in the future. The parameters we used in the experiments was

obtained by test, so how to get the optimized parameters is also the problem

we need to solve in the future. We did experiments using only Gaussian

distributions currently, but it would be extended to any random distribution

in the future.

42

VI. Appendix

Whatever you are doing simulation or implementing the proposed method

in the real cases, random number generation is very important. There are

many libraries we can use. For example, when I did my simulation test, I

used Matlab to generate the desired distribution directly. But in the case of

the real world applications, people have to implement the proposed method

using C language in order to maximize the speed. I did a small test which

compares the random number generation performance among my own imple-

mented functions, random number generation functions from boost library,

random number generation functions from GNU scientific library and random

number generation functions from open computer vision library. As uniform

distribution and gaussian distribution are the most important distributions

in particle filter, I compared the performance focused on these two distribu-

tions using the mentioned four libraries. Table 6.1 shows the performance

results using each library. The performance of open computer vision library

is the best among these methods.

Table 6.1 Random number generation test using 5000000 samples

Own function Boost function GSL function opencv opencvForloop

150ms 220ms 90ms 60ms 892ms

290ms 230ms 682ms 70ms 934ms

43

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Acknowledgment

I would like to express my sincere gratitude to my advisor Prof. Whoi

Yul Kim for the continuous support during my master study and research.

I am also thankful to my mother Xing Ai Li, my young brother Xiang Yu

Piao and my girl friend Mei Xian Fang who supported me all through with

their love.

I would like to thank to all my fellow friends who came to Hanyang

University together from Shanghai Jiao Tong University for their support

during my study. I would like to thank to my friend Sheng Zhu Kim who

gave me support from the side of the mathematics for my master thesis.

All my colleagues in Image Engineering Lab have supported me in my

research work during my master study period more or less. I want to thank

them for their help, support and valuable hints.

I would like to thank to BK21 for their financial support during my master

study period.

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