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COMP9517ComputerVision
Tracking
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Mo@onTracking• Trackingistheproblemofgenera@nganinferenceaboutthe
mo@onofanobjectgivenasequenceofimages– Whatdoweinferaboutanobject’sfutureposi@onfromasequenceof
measurements?
• Applica@ons– Mo@oncapture
• Controlacartoon• Modifythemo@onrecordtoobtainslightlydifferentmo@ons
– Recogni@onfrommo@on• Determinetheiden@tyoftheobject• Tellwhatitisdoing
– Surveillance• Monitorac@vi@esandgiveawarningifitdetectsaproblemcase
– Targe@ng• DecidewhattoshootandhiRngit
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Mo@onTracking• Whenmovingpointsarenottaggedwithuniquetextureorcolourinforma@on,thecharacteris@csofthemo@onitselfmustbeusedtocollectpointsintotrajectories
• Assump@on:– Theloca@onofanobjectchangessmoothlyover@me– Thevelocity(speedanddirec@on)ofanobjectchangessmoothlyover@me
– Anobjectcanbeatonlyoneloca@oninspaceatagive@me
– Twoobjectscannotoccupythesameloca@onatthesame@me
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TrackingwithDynamicModels• Trackingcanbeconsideredastheproblemofgenera@ngan(probabilis@c)inferenceaboutthemo@onofanobjectgivenasequenceofimages
• Trackingisproperlythoughtofasanprobabilis@cinferenceproblem– Themovingobjecthasinternalstate,whichismeasuredateachframe
– Measurementsarecombinedtoes@matethestateoftheobject
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TrackingwithDynamicModels• State:therepresenta@onofanobjectat@me(frame)t– Posi@on– Transforma@onparameters– Class– Etc
• Measurement:theobserva@on– Colour– Texture– Etc
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TrackingwithDynamicModels• Given– Xi:thestateoftheobjectattheithframe– Yi:themeasurementobtainedintheithframe
• Therearethreemainproblems:– Predic@on:predictthestatefortheithframebyhavingseenasetofmeasurementsy0,y1,…,yi-1
– Dataassocia@on:selectthemeasurementsthatarerelatedtotheobject’sstate
– Correc@on:correctthepredictedstatebyobtainedrelevantmeasurementsyi
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),...,,|( 111100 −− === iii yYyYyYXP
),...,,|( 1100 iii yYyYyYXP ===
TrackingwithDynamicModels• IndependenceAssump@ons(MarkovAssump@on)– Onlytheimmediatepastma`ers
– Measurementsdependonlyonthecurrentstate
– Theseassump@onsmeanthatatrackingproblemhasthestructureofinferenceonahiddenMarkovmodel
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)|,...,()|()|,...,,( ikjiiikji XYYPXYPXYYYP =
)|(),...,,|( 1121 −− = iiii XXPXXXXP
Xi-1
Yi-1
Xi
Yi
Xi+1
Yi+1
LinearDynamicModels• LinearDynamicModels– Arandomprobabilitydistribu@onwithmeanandcovariance
– Thestateisadvancedbymul@plyingitbysomeknownmatrixandthenaddinganormalrandomvariable
– Themeasurementisobtainedbymul@plyingthestatebysomematrixandthenaddinganormalrandomvariableofzeromeanandknowncovariance
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).,(~ ∑µNx
).,(~ 1 idiii xDNx ∑−
).,(~imiii xMNy ∑
ProbabilityDensityPropaga@on
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ProbabilityDensityPropaga@on
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TrackingwithDynamicModels• TrackingasInference– Predic@on
– Correc@on
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111011
111011101
11101110
),...,,|()|(
),...,,|(),...,,,|(
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−−−−
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∫∫
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iiiii
iiiiii
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dXyyyXPXXP
dXyyyXPyyyXXP
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iiiii
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i
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iiii
dXyyyXPXyPyyyXPXyP
yyyPyyyPyyyXPXyP
yyyPyyyPyyyXPyyyXyP
yyyPyyyXPyyyXP
∫ −
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=
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),...,,(),...,,,(),...,,|(
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TrackingwithDynamicModels• Someconvenienttransforma@ons
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),0;(),;( vxgvxg µµ −=
),;(),;( vmngvnmg =
)/,/;(),;( 2avaxgvaxg µµ =
),;(),0;(),;( baba vvxgduvugvuxg +=−∫∞
∞−µµ
),,,(),;(),;(),;( dcbafdb
bddbcdadxgdcxgbaxg
++
+=
TrackingwithDynamicModels• KalmanFiltering– Dynamicmodel:
– Define:
– Predic@on:
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);,(~)|( 211 idiiii xdNXXP σ−−
).,(~)|( 2imiiii xmNXYP σ
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)(,
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))(,;(),;(
),...,,|()|(),...,,|(
+−
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iidiiii
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i
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σσσ
σσ
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).,...,,|(__var),,...,,|(__ 1010 iiiiii yyyXPofianceyyyXPofmeanX −−++
σ
).,...,,|(__var),,...,,|(__ 110110 −
−
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−−− iiiiii yyyXPofianceyyyXPofmeanX σ
TrackingwithDynamicModels• KalmanFiltering– Correc@on
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⎟⎟⎠
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+=⎟
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)()(
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),...,,|()|(),...,,|()|(),...,,|(
iim
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mmymX
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XXgXmyg
yyyXPXyP
dXyyyXPXyPyyyXPXyPyyyXP
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σσ
σσσ
σσ
σσ
σσ
TrackingwithDynamicModels• KalmanFiltering–thealgorithm(1-D)– StartAssump@on: areknown– UpdateEqua@on:Predic@on
– UpdateEqua@on:Correc@on
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−−
00 ,σx
21
2
1
)( +−
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+=
=
iidi
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d
xdx
iσσσ
⎟⎟⎠
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+=
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)()(
)()(
iim
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mymx
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σσ
σσσ
σσ
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TrackingwithDynamicModels• KalmanFiltering–thealgorithm(2-D)– StartAssump@on: areknown– UpdateEqua@on:Predic@on
– UpdateEqua@on:Correc@on
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−−Σ00 ,x
iiidi
iii
DDxDx
i
+−
−
+−
−
+Σ=
=
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σσ
[ ][ ]
[ ] −+
−−+
−−−
Σ−=Σ
−+=
Σ+ΣΣ=
iiii
iiiiii
mTiii
Tii
MKIdxMyKxx
MMMKi
1
TrackingwithDynamicModels• Par@cleFiltering
– Anon-lineardynamicmodel– Alsoknownvariouslyas:
• Sequen@alMonteCarlomethod,• bootstrapfiltering,• thecondensa@onalgorithm,• interac@ngpar@cleapproxima@onsand• survivalofthefi`est
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TrackingwithDynamicModels• Par@cleFiltering
– Thecondi@onalstatedensityat@metisrepresentedbyasetofsamplingpar@cleswithweight(samplingprobability).
– Theweightdefinetheimportanceofasample(observa@onfrequency)
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TrackingwithDynamicModels• Par@cleFiltering– Thecommonsamplingschemeisimportancesampling
• Selec@on:selectNrandomsamples• Predic@on:generatenewsampleswithzeromeanGaussianErrorandnon-nega@vefunc@on
• Correc@on:computeweightscorrespondingtothenewsamplesusingthemeasurementequa@onwhichcanbemodelledasaGaussiandensity
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TrackingwithDynamicModels• Par@cleFiltering(A.VlakeandM.Isard,Ac@veContour)
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TrackingwithDynamicModels• Par@cleFiltering–thealgorithm(SIS),A.VlakeandM.Isard
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TrackingwithDynamicModels• Par@cleFiltering(A.VlakeandM.Isard,Ac@veContour)
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Trackingcanbecomplex• Lossofinforma+oncausedbyprojec@onofthe3Dworldona2Dimage,
• Noiseinimages,• Complexobjectmo+on,• Non-rigidorar@culatednatureofobjects,• Par@alandfullobjectocclusions,• Complexobjectshapes,• Sceneillumina+onchanges,and• Real-+meprocessingrequirements.
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ReferencesandAcknowledgements• ShapiroandStockman2001• Chapter19ForsythandPonce2003• Chapter5Szeliski2010• Imagesdrawnfromtheabovereferencesunlessotherwisemen@oned
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