Abnormal Crowd Behavior Detection Based on the Energy Model(6)

B.ForegroundProbabilityDistribution

Theforegroundprobabilitydistributionon and axiscanbedirectlyestimatedfrom ( )and ( )bydividingtheentriesbythetotalnumberofforegroundpixels. ( )=

( )

,0< ≤ 1, ∈ ,(5)

( )= ( )

,0< ≤ 2, ∈ ,

(6)

Where isthetotalforegroundpixels.

C.CrowdEntropy

Inspiredbythede nitionofentropyfromtheinformationtheory[20],wede neforegroundentropybasedonfore-groundprobability.Foregroundentropyisde nedasfollows:

( ∑ 1)=

( )log(

1

( )

), ( )=0.(7) =1 ( )=∑

2 ( )log(1

), ( )=0(8)

=1

( ).

Foregroundentropydenotesthedispersionofforeground

onthehorizontalandverticaldirections.Forexample,iftheprobabilityis1atbin ,then ( )=1,sotheentropyofprobabilitydistributionis =1 log(1)=0.Iftheprobabilityisequallydistributedonallbins,weget = 1/ log(1/(1/ ))=log( ).Therefore,adistributionwithasinglesharppeakyieldstoalowentropyvalue,whereasadisperseddistributioncorrespondstoahighentropyvalue.D.CrowdDispersion

( ), ( )isthedispersionofforegroundonthehor-izontalandverticaldirectionrespectively.CrowdDispersionisusedtore ectthedispersionoftheframeglobally.CrowdDispersion( )isde nedasfollows:

= ( ) ( ).

(9)

E.CrowdDistributionIndex

Basedontheabovede nition,wecande neCrowdDistributionIndexasfollows:

= 2

3,

(10)Where isthecrowddensityand isCrowdDispersion.Equation(10)meansthatwhenpeoplegatherinalocalregion, willbelargewhile besmall,whichyieldstoalarge value.Basedonathreshold,wecandetectpedestraingathering.However,whenthepedestrainsmovinginthescenesarrangeinalineofhorizontalorverticaldirectionjustasFig.5(a), willbeverysmallwhile ismedium,whichmakes bealargevaluejustasthesituationwhenpeoplegatheringinalocalregion(Fig.5

(b)).

498

(a) =1.1, =4,

2 3

=

16(b) =2.22, =8,

2 3

=9

Fig.5.ThereasonwhyweusePiecewiseFunctiontode neCDI.

InFig.5(a),4peoplewalkinalineintheverticalnoabnormalactivitieshappen,however, 2

direction,

region 3=16.InFig.5(b),thepeoplegatheringinalocalindicatesome

accidentstakingplace,but 2

3=9.So(10)willleadtoafalsealarm.Toavoidthis

kindoffalsealarm,weuseaPiecewiseFunctiontomodifythede nition:

{

=

1.1 , ≤2, 2

3,

>2.(11)

Themodi ed caneffectivelysingleoutpeoplegath-eringandishelpfultodetectpedestrainrunning.

VI.KINETICENERGY

OF

CROWD

ThispaperadoptstheHarriscornerasfeatures.MotionvectorsareobtainedbytrackingfeaturesofaseriesofimagesthroughtheLucas-Kanadeoptical owapproach[21].Fig.6showtheresultoftheoptical ow

computation.

(a)Original

frame

(b)Motionvector

Fig.6.

Optical owcomputation.

Thekineticenergyofeachframeisde nedasfollows:

=

∑ 2

,

(12)

=1

Where isthekineticenergyofthe thframe, is

CrowdDistributionIndexand isthevelocity. providenotonlytheinformationofcrowddensitybutalsothatofcrowddistribution.Whenthepeoplegatherinalocalregion,occlusionswilloccur,whichmeansthatMotionFeaturestendtobesmallerthanthatinascenewherepersonsaresparselyscattered.However,intheabovesituation, willbelargerwhichcompensatesthein uenceofocclusions.

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