tracking by sajid ali(nitttr,chandigarh)

8/6/2019 Tracking By Sajid Ali(Nitttr,Chandigarh)

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Presented By:SajidAliCSE (Regular)-2010

NITTTR, Chandigarh

Tracking with Linear Dynamic

Models

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Contents:

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y Tracking

y Three main issues

y Assumptions

y Tracking as Inductiony Linear Dynamic Models

y Kalman Filter

y References


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Tracking

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y Tracking is the problem of generating an inference about themotion of an object given a sequence of images.

y The key technical difficulty is maintaining an accurate

representation of the posterior on object position givenmeasurements, and doing so efficiently.

y Targeting: a significant fraction of the tracking literature isoriented towards (a) deciding what to shoot and (b) hitting it.Typically, this literature describes tracking using radar or infra-

red signals (rather than vision).


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Tracking(Cont..)

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y Very general model:

y We assume there are moving objects, which have anunderlying state X

y There are measurements Y, some of which are functions ofthis state

y There is a clock

y at each tick, the state changes

y at each tick, we get a new observation

y Examples

y object is ball, state is 3D position+velocity, measurements arestereo pairs

y object is person, state is body configuration, measurementsare frames, clock is in camera (30 fps)


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Three main Issues in Tracking


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Simplifying Assumptions

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Tracking as induction

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y Assume data association is done

y well talk about this later; a dangerous assumption

y Do correction for the 0th frame

yAssume we have corrected estimate for ith framey show we can do prediction for i+1, correction for i+1

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Base case

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Induction step

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Given

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Induction step (Cont.)

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Linear dynamic models

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yUse notation ~ to mean hasthe pdf of , N(a, b) is anormal distribution withmean a and covariance b.

y Then a linear dynamic modelhas the form

y This is much, much moregeneral than it looks, andextremely powerful

y We are on a boat at night andlost our position

y We know: star position

yi

! N Mix

i;7

m i

xi

! N Di1xi1;7di

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Examples

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y Drifting points

y we assume that the new position of the point is the old one, plusnoise.

y For the measurement model, we may not need to observe thewhole state of the objecty e.g. a point moving in 3D, at the 3kth tick we see x, 3k+1th tick we see

y, 3k+2th tick we see z

y in this case, we can still make decent estimates ofall three coordinates ateach tick.

y This property, which does not apply to every model, is calledObservability

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Examples

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y Points moving with constant velocity

y Periodic motion

y Points moving with constant acceleration

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Points moving with constant velocity

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y We have

y (the Greek letters denote noise terms)

y Stack (u, v) into a single state vector

y which is the form we had above

ui

ui 1

(tvi1 Ii

vi! v

i1 :i

v

i !

1 (t

0 1

u

v

i1noise

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Points moving with constant

acceleration

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y We have

y (the Greek letters denote noise terms)

y Stack (u, v) into a single state vector

y which is the form we had above

ui

ui 1

(tvi1

Ii

vi! v

i1 (ta

i1 :

i

ai

! ai

1 \i

v

i

!

1 (t 0

0 1 (t

0 0 1

u

v

i1

noise

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The Kalman Filter

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y Key ideas:

y Linear models interact uniquely well with Gaussian noise -make the prior Gaussian, everything else Gaussian and thecalculations are easy

y Gaussians are really easy to represent --- once you know themean and covariance, youre done

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What is it used for ?

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y Tracking missiles

y Tracking heads/hands/drumsticks

y Extracting lip motion from video

y Fitting Bezier patches to point datay Lots of computer vision applications

y Economics

y Navigation


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The Kalman Filter in 1D

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y Dynamic Model

y Notation

Predicted mean

Corr

ected mean

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Prediction for 1D Kalman filter

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y The new state is obtained by

y multiplying old state by known constant

y adding zero-mean noise

y

Therefore, predicted mean for new state isy constant times mean for old state

y Predicted variance is

y sum of constant^2 times old state variance and noise variance

Because:

old state is normal random variable, multiplying normal rv by constant

implies mean is multiplied by a constant variance by square of constant, adding

zero mean noise adds zero to the mean, adding rvs adds variance

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Correction for 1D Kalman filter

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y Pattern match to identities given in book

ybasically, guess the integrals, get:

y

Notice:y if measurement noise is small,

we rely mainly on the measurement,

if its large, mainly on the

prediction

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In higher dimensions,

derivation follows the

same lines, but isnt as

easy. Expressions here.

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Smoothing

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y Idea

y We dont have the best estimate of state - what about thefuture?

y Run two filters, one moving forward, the other backward intime.

y Now combine state estimates

y The crucial point here is that we can obtain a smoothed estimate byviewing the backward filters prediction as yet another measurement for

the forward filtery so weve already done the equations

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Data Association

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y Nearest neighbours

y choose the measurement with highest probability givenpredicted state

y popular, but can lead to catastrophe

y Probabilistic Data Association

y combine measurements, weighting by probability givenpredicted state

y gate using predicted state

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References:

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y http://www.cs.utexas.edu/~grauman/courses/fall2008/slides/lecture23_tracking.ppt

y http://luthuli.cs.uiuc.edu/~daf/book/bookpages/Slides/Tracking.ppt


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Thanks

tracking by sajid ali(nitttr,chandigarh)

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