Abnormal Crowd Behavior Detection Based on the Energy Model(4)

IV.ESTIMATIONOFCROWDDENSITY

InspiredbyZhiZhong[16]andthegravitationalpotentialenergy,authorsin[18]presenttheimagepotentialenergymodel.ObsevingfromthepinholeperspectiveprojectionmodelinFig.3[19],weobtaintwocrucialperspectiveeffectsasfollows:

a)Theapparentsizeofobjectsdependsontheirdistancefromthecamera:forexample,theimages ′and ′oftheposts and havethesameheight,but isactuallyhalfthesizeof [19].

b)Fig.6showsanothercrucialeffect:thefartherobjectsappearclosertotheoriginontheimageplane,thatistosay,theY-axisvalueissmaller.Forinstance: and aretwotimesfartherthan ,soontheimageplane ′ , ′and ′areclosertotheorigin ′intheverticaldirectionthan ′

.

Intheabovede nition,weuse tore ecttheeffectofocclusion. isaconstantinthede nitionwhileitmaychangeindifferentscenes.

V.CROWDDISTRIBUTIONINDEX

Inthissection,wede neCrowdDistributionIndex( )todetectpeoplegatheringinalocalregion.Firstly,wepresenttheforegroundhistogramson and axisrespectively,andthende netheCrowdEntropy( )torepresentthedispersionofforeground. isde nedbyCrowdEntropy( )andCrowddensity( ).A.ForegroundHistogram

Inourapproach,webuildforegroundhistogramsbypro-jectingtheforegroundto and axisrespectively.Theforegroundhistogramson and axisaredenotedby ( )and ( ):

( )={ ,0< ≤ 1},(3)

( )={ ,0< ≤ 2},

(4)

Fig.3.Pinholeperspectiveprojection

Basedontheabovecrucialeffects,theimagepotential

energycanbede nedasfollows:

=

∑∑ =1 =1

( + ),(1)

where 1denotesthenumberofhistogrambinsonthe

horizontaldirection( axis)and 2denotesthenumberofhistogrambinsontheverticaldirection( axis). impliesthenumberofforegroundpixelsprojectedtothe aixsatthe thbinand denotethenumberofforegroundpixelsprojectedtotheYaxisatthe thbinforeachframe. 1and 2aredeterminedbythedistancefromthecamera.Whenthepedestrainsinthescenesarefartherfromthecamera,therelatedpixelsontheimageplanearefewer,sothebins 1and 2shouldbebigger.Inthispaper,welet 1=20,and 2

=12.

where istheimagepotentialenergy, denotesthemassofpixelwithcoordinates( , ). {0,1},with1denotingtheforegroundand0denotingthestaticbackground.Similartothegravitationalpotentialenergy,welet beaconstantinthispaper, =10;whilewecande ne bealargervalueinspeci careatodetectanomalies. isaroughmeasurementoftheclosestdistancefromcamera. isconstanttoa xedcamera. and denotethewidthandtheheightofframerespectively. istheimagecoordinateofY-axis.Inde nition(1), istheonlyparameterrequiredtobeestimated.Moredetailscouldbefoundin[18].

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