Motion model selection in tracking humans

ISSC2006, Dublin Institute of Technology, June 28-30

Motion Model Selection

in

Tracking

Humans

Damien Kellyt and Frank Boland* Department of Electronic and Electrical Engineering

Trinity College Dublin IRELAND

E-mail: tkellyl16tcd. ie *fbo1andQtcd ie

Abstract The performance of many human trackingalgorithms relyonaccurate motionmodels. Due tothe natureofhuman motion it is oftendifficulttodetermine thesuitabilityof achosen model. It is typically the case that over the tracking duration the characteristics of the observed motion will fit many different models. Commonly used motion models in the area oftracking include the constant po-sition (CP), constant velocity (CV) and constant acceleration (CA)

motion models. This paper applies the Kalman filtering algorithm to the problem of tracking a person's position using a finite set of different motion models. The statistical properties ofthe innovation sequence are used as a basis for motion model selection.

Keywords Kalman Filter, Motion Modelling, Tracking

I INTRODUCTION

Thesuccessful determination ofamovingperson's positionrequiresthe filtering of useful information from state observations made in the presence of noise. In order to extract this information it is necessarytomake assumptions on the characteris-ticmotion ofthe person being tracked. Although

this may be a simple assumption such as for ex-ample, the person existing in 2D space, it may also be an assumption on the nature of their

un-derlying movements. In general the more precise a priori knowledge of the person's characteristic motion that can be determined, the more useful information can be extracted from noisy position measurements.

Itisoftenthe caseinobjecttracking algorithms that a priori knowledge is incorporated into the estimation procedure through the assumption of a motion model. Many motion models have been proposedin the area oftracking such as, the con-stant position, constant acceleration andconstant velocity models [1]. Although these motion mod-els have foundapplicationsinmanyareasof track-ing, it is often difficult to assign an object's mo-tion toone particular model. It may besufficient, forexample, toassume constant acceleration when

trackingthetrajectory ofafallingobject, although

thisassumption couldnotbejustifiably appliedto

tracking motion of a changeable nature such as

that of human motion. Typically, in moving, a

human will transgress over many different

mod-els. For this reason in tracking humans it is often difficult to determine asuitable model. One com-mon model used in people tracking which reflects unpredictability is that of the Brownian motion model. This particular model assumes indepen-dent directionofmotion ateach time step. It does not however reflect the intuition that often over long durations human motion is correlatedintime. One technique in tracking which addresses the problem of changing dynamics is the use of adap-tive motion models which enable the target's mo-tion characteristics to change appropriately over time [2]. The problem with this approach is in

defining the criteria by which the motion model should be adapted. This requires continuous monitoring of the tracking filter's performance

whereby any deviation from it's expected perfor-mance would indicate a degradation in the accu-racy ofthe position estimates.

Thispaper examines motion model selection as applied to the problem of tracking human move-ments through the use ofthe Kalman filter algo-rithm. The statistical properties of theinnovation sequence of the Kalman filter areexamined in de-tecting divergence in the Kalman filter resulting from modelling inaccuracies. The remaining sec-tions of this paper are organised as follows. Sec-tion II gives an overview of kinematic state

esti-mation. Section III examines asimulatedtracking

Acommon approach intrackingistoapproximate the position Xk of a target by it's Taylor series expansion [2] i.e.,

T2

Xk+1 = Xk++Txk+ xk + (1)

2

where the time interval k to k+ 1 is of duration T. It is well known that a correct approxima-tion requires allderivatives of the position. Where such information is not possible to determine a truncated Taylor series is used. This is equiva-lent to assuming the target trajectory to follow some polynomial of finite order. In order to re-lax this assumption thehighest order derivative of the truncated Taylor series is often modelled as a

random process. Thederivatives of position define the moving target's kinematic state model Xk i.e. Xk =

[Xkk...

X k] whereXk denotes the

position

and the kinematic state model is ofdimension n.

Theprimaryaim intrackingisto estimatethe evo-lution of the state over time.

a) Kinematic State Estimation

The estimatedkinematicstateprocess with control input Uk and measurement process Zk requiredto implement the Kalman tracking filter are,

Xk+1 -

Akkk

+BkUf ±CkVk (2a)

Zk+1 =

Hk+lRk+1

+

Gk±lCk+1

(2b)

where

Hk+1

is the measurement matrix and Bk, Ck and Gk denote thegain matrices

correspond-ing to the control input Uk, process noise vk and measurement noise WVk respectively. The process noise Vk and measurement noise wk are assumed to be zero mean, independent Gaussian random processes with associated covariance matrices,

6klQk

- £

[Ckbkl;E;]

, (3a)

6klRk

- E

[Gkwkwl

;] (3b)

where 6k1 denotes the kronecker delta function. Using thestatemodelof

(2)

theequations which describe the discrete Kalman filter are

Xk+llk+l

=

Xk+1Ik

+

Kk+lik+l

(4)

V'k+l

-Zk+1 -

Hk+1Xk+llk

(5)

Xk+±Ik =AkXkIk (6)

Pk+llk

-Ak+lPkikAk+l +Qk+l (7)

-K-- ,I -l

Kk+l =Pk+llkHk+l k+l- (10)

where x isfiltered stateestimate, P is the

covari-ance of the error in the state estimate, K is the Kalman gain, io is the innovation sequence and S isthecovarianceof the innovation. The notations

klk

- 1 and kjk are used to denote the prediction

andfilteringstepsof the Kalman filterrespectively. Thestatetransitionandnoisegain matricesof typ-ical motion models and the models referred to in this paper areshown in Table 1 for reference.

Model Ak

Ck

CP [1 [T]

cv

1

T

2i;

CA

0iT

T]

TL~

Table. 1: Transition and process noise gain matrices for a constant position model CP, constant velocity model CV andconstantacceleration modelCA.

Where the statemodel of(2) represents thetrue state model, the Kalman gain is optimal and the error in the state estimate is minimum in a mean square sense. In general this is not the case and the state model of(2) represents only anestimate of some true state model,

Xk+l -AkXk +BkUk + CkVk

Zk+l -Hkxk+l +

Gk+lWk+l-(hla) (llb)

In theremainderof thisdiscussionthe estimated observation matrix Hk is taken to be Hk = Hk i.e. the estimated observation matrix is know to

be correct.

b)

Divergence

in the Kalman Filter

Divergence in the Kalman filter occurs when the error in the filtered state estimate does not

cor-respond to that determined by the Kalman filter equations (4) - (10). Divergence in the Kalman filter can be classified as either true divergence,

ISSC2006, DublinInstitute ofTechnology, June 28-30

in the filtered state estimates [3]. This paper is concerned with apparent divergence in the track-ing Kalman filter due to inaccuracies in motion modelling.

Itcanbe seen from equation (5) that the innova-tion sequence represents theerror inthe predicted measurement based on model (2). For this

rea-son it does not represent the true innovation. It is clear that theinnovation in relation to the true

state model of (11) can be obtained by replacing

Zk+1 in equation (5) by Zk+l. The actual innova-tion istherefore [4],

Vk+1 = Zk+l-

Hk+lXk+llk

(12)

Substitutingthis expression for the true innova-tion intoequation (4), thetrue error inthefiltered stateestimate,

ek+llk+l

--X1 k+lk+1,

(13)

after some manipulation isfound to be,

ek+llk+l -

qk+1Akekjk

+

Ok+lBk(Uk

-

Uk)

(14)

+

qk+1

AAkXk +

bk+lABkUk

+

Ok+1CkVk

-

Kk+lGk+lWk

where

qk+

1 [I-

Kk+±Hk+

1], AAk - Ak -Ak

and XBk -

Bk-Bk-In the case where the motion model of (2) rep-resents the true motion, the innovation sequence can be shownto be zero mean andorthogonal

[5].

In expanding equation

(12),

it can be seen that all modelling errors manifest in theinnovation

se-quence i.e.,

-k+l=

Hk+lAkeklk

+

Hk+lAAkXk

(15)

+

Hk+

Bk(Uk

Ulk)

-±Hk+lABkuk +

Hk+lCkvk

+ Gk+lWk+l1

Itisclear thatcontinually monitoring the statis-ticalproperties oftheinnovationsequenceprovides

a basis for detecting modelling errors. This is the basis ofmany existing methods ofdivergence de-tection and control in the Kalman filter

[6].

Of particular interest inthe area ofmotion model

se-lection fromafinitesetofmodels, istheproblemof

determining the most appropriate model from the observed divergence. The

following

section

exam-ines afinite set ofmotion models and innovation sequences ofthecorresponding trackingfilters in a

computer simulated tracking problem.

III COMPUTER SIMULATED TRACKING PROBLEM

The class ofmotion models examined in this

sec-tion is defined inTable 2. This table defines a set

of motion models for the tracking scenarios of a person not moving (CP), moving at constant ve-locity (CV) and movingwithconstantacceleration

(CA). Also shown in the table is the actual simu-lated model (Act.). This model represents a

con-trollable system whereby all states are reachable through acceleration ordeceleration. The control-lable acceleration component corresponds to the

entry a in matrix Uk.

Inrelationtoequation (16), itcanbeseenthatif E[Wkj iszero mean thenthe error dueto an inac-curate motion model propagates in the expected value of the innovation sequence through AAk, ,ABk, Bk and Vk. In this simulated case, E[vk] is the change in acceleration over the time step T. Over periods of constant acceleration there-fore, E[Vk] -0. Taking Uk =0 from Table 2, the

expected value of the innovation sequence in the simulatedcase becomes,

E[vk+1] =

Hk+l,AkE

[eklk]

+

+Hk+lAAkXk+Hk+lBkUk +Hk+lABkuk+Hk+lCkE

[Vkl

(16)

- Hk+l

[AkE[eklkl

+AAkXk +BkUk

+

CkE[Vk11

The following examines E[Vk] in tracking the simulated motion using the CP, CV and CA mod-els.

* Using the CP MotionModel

AAk-[o ~T2 ] ]

/\A-k

0

O

T2]

BkU a

LO

0 1

L0

T2 T3

E

[vk+1j

=E

Leklk]

+TiXk

+-

Xk

±- E

[Vkj

(17)

2 6

* Using the CV Motion Model

O O

/Aik=

O O

L

O

T2

2

T I BkUk

[

T2 T3

E

[vk+±]

=E

[eklk]

+- Xk+

E[Vk}

* Usingthe CA Motion Model

Ak,

= °0 0 BkUk = a

T3

E

[Vk+ll

=E

[eklk]

+-6 E

[Vk]

0

a

0

I

(18)

I

Table. 2: Matrices foraconstantposition modelCP, constantvelocity modelCV andconstant acceleration modelCA.

Summarised inTable 3 istheexpected value of the innovation sequence E[vkl using the CP, CV and CA models, under the simulated scenarios of a person not moving (k O), moving with

con-stant velocity (xk 0) and movingwith constant

acceleration.

From Table 3 itcan beseen thatin usinga CP

motionmodeloverperiodsofconstantacceleration

(xk

a),

the

expected

value of the innovation sequence increases linearly. Using a CV motion

model results in an offset in theexpected value of theinnovation sequence. Asthe CAmotionmodel estimates allstatesof thesimulated motion it isthe optimal model and is therefore zero mean. Simi-larly,overperiodsofconstantvelocity both the CV and CA motion modelsare zero meanandanoffset is observed in E["ki where the CP modelis used. This is illustrated in fig. 1, where the simulated tracking problem isthat ofaperson starting from

aninitialstationary positiontoastateofconstant

velocity and thendecelerating backto astationary

position.

IV MOTION MODEL SELECTION IN A REAL

TRACKING APPLICATION

The motion models of Table 2 were applied to the problem oftracking a person in an image se-quence. The image sequence used, consisted ofa

personwalkingon agrid1 of knownwidth (4.35m)

and length (5m). The video tracking method of

background subtraction was used to locate the

person in each frame. The captured background was taken from a section of the image sequence

wherethe grid was unoccupied. The useof

back-ground subtraction and the application ofa

suit-ablethreshold to account for changes in lighting, enabled the person to be located in each frame of the image sequence. The person's position onthe

grid was determined through a mapping of frame

co-ordinates to grid co-ordinates. The mapping of frame co-ordinates to positions onthe grid was

achieved using a technique which exploitsthefact

that the intersection of the diagonals of a

trape-zoid areinvariant under perspectivedistortion [7]. Thisrecursivealgorithmfirstly determinesthe

per-son's ypositiononthe gridin a similarmannerto

the secant method for root finding. Secondly the person'sxpositionisdetermined inrelationtothe

grid boundaries. Knowing the grid dimensionsand alsothelocation of thecornersof the grid therefore enabled the geometry of the frame to be mapped

to that of the geometry ofthe grid. An example frame from the image sequence used is shown in

fig. 2. In this sequence the person walks horizon-tally acrossthe full width of the grid.

Showninfig. 3(a) are the ID position

measure-mentsZk, ofthe person for each of the 188 frames ofthe image sequence. Due to the frame resolu-tionof 576x720and the areaoccupied bythegrid within theframe,

!~~X I' ° 0 1 0 0 T 0oI[o]

CV

Xk

[]

I T ° ° ° ° °2-O 0 0

[fl1

0 0

[O]

CA [±k] [0 1°][010] 1 0 0] [1 0 0] [0]

Xk Ak Bk Ck Gk Uk

ISSC 2006,DublinInstituteofTechnology, June 28-30

the position measurements weredetermined to be

accurate to O.Olmoverthe mid sectionof the grid.

For this reason the variance of the measurement

noise Wk was chosento be o2 0.0001m2. In

ex-amining the position measurements, suitable val-uesfor theprocess noise 2 weredetermined tobe

02

2 5~

2= 0.04m2

c,

=

0.25m

and i = 0.25s4 for

the CP, CV and CA Motion Models respectively. Similarly in the case of the computer simulated

tracking problem the control input was assumed

to be uik = 0. The innovation sequences of the Kalman filtersapplied tothe tracking problem us-ingthesethree motion models areshownin fig. 3. As can be seen in fig. 3 (b) the offset in the meanof the innovation sequence correspondingto the CP model indicates thepersonis movingwith

constant velocity. Thesuitability ofa CVmotion

model is seen in fig. 3 (c) where after the initial

divergence ofthefilter, the innovation sequenceis zero mean over the period ofconstant velocity. It isalso seen infig. 3 (d) that the higher dimension CAmodel also results in a zero mean innovation

sequence but time required for the filter to con-verge afterthe initial divergence is greater.

(d)

Fig. 1: Innovation sequences for the CP (a), CV (b) and CA (c) motion models in a computer simulated tracking

problem. Thesimulated datais thatofapersonstartingat

aninitialstationarypositionatk=0 andacceleratingover

k= 30 to k 60, maintaining aconstant velocity (5m/s) from k = 60 to k = 90 and then decelerating back to a

stationarypositionat k= 120. (T=0.25s).

Table. 3: Expected value of the true innovation k±+l for

different estimating models under motionscenarios of

con-stant acceleration (CA) andconstantvelocity (CV).

Fig.2: Framefrom theimagesequenceofapersonwalking

across agrid. The videotrackingtechniqueofbackground

subtraction wasusedtodetermineabounding box for the

personineachframe.

V DISCUSSION AND CONCLUSIONS

The successful tracking of humans using the Kalman filter requires accurate motion models. Due to the nature of human motion it is difficult

to determine the suitabilityofa particular model.

Inaccurate motion modelling causes bounded

di-vergence in the Kalman filter resulting in biased position estimates. In order to correct the esti-mates obtained from the filter it is necessary to control thedivergence byadaptingorchangingthe

model.

300E

300~ ~ ~ ~ ~ ~~~. _0 , . ._

200~ ~ ~ ~ ~ ~~ M _ /

100 _ /~~~~~~~~~0

(c)

Motion Scenario

CV

CAI

r < c] +~T,k E

[ekik]

+Tk + T2 Xk

I E

[eklk]T

E

[ekik]+

2 Xk

CA E

[eklk]

E

[eklk]

I

300 250 200 100_

O0

O So 100 ISO 201

k (188frames/7.52s)

(a)

(b)

(c)

CA Motion Model

o 50 100 k 7T=0.04s)

(d)

Fig. 3: Innovation sequences of the Kalman filter for the CP (a), CV (b) and CA (c) motion models as applied to

theproblemoftrackingthe person intheimagesequence.

(T=0.04s)

The performance of a finite set of motion mod-els was examined under varioussimulated motion scenarios. It was seenthat the deviationinthe ex-pected value of theinnovation sequence provideda

basisfor not only detecting divergenceinthefilter but also for classifying the motion being tracked.

The particular set of motion models used en-abledthe observedmotion tobeclassifiedaseither fitting a CP, CVor CA model. In thisway moni-toringtheinnovation sequence was seen toprovide a basis for motion model selection.

VI ACKNOWLEDGEMENTS

DamienKelly gratefully acknowledges thesupport of the Irish Research Council for Science, Engineer-ing and Technology (IRCSET), through an Em-bark Initiative research scholarship.

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1Audio and video data ofpeoplewalkingandtalkingon

a grid was collected for investigations in tracking people

using audio and video. An online databaseofthecollected datais currently beingcompiled.

0.1,II

-0.1|,

0.08 0.06 0.04 0.02

.1 0 -0.02 -0.04 -0.06 -0.08