Tracking with Linear Dynamic Models

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Tracking with Linear Dynamic Models
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Introduction

• Tracking is the problem of generating an
  inference about the motion of an object given a
  sequence of observations.
• Applications
• Motion capture
• Recognition from motion
• Surveillance
• Targeting

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Tracking

• Very general model
• We assume there are moving objects, which have an
  underlying state X
• There are measurements Y, some of which are
  functions of this state
• There is a clock
• at each tick, the state changes
• at each tick, we get a new observation
• Examples
• object is ball, state is 3D positionvelocity,
  measurements are stereo pairs
• object is person, state is body configuration,
  measurements are frames, clock is in camera (30
  fps)

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Linear Dynamics

• Tracking can be thought of as an inference
  problem
• The moving object has some form of internal
  state, which is measured at each frame.
• We need to combine our measurements as
  efficiently as possible to estimate the objects
  state.
• Assume both dynamics and measurements are linear.
  (Non-linearity introduces difficulties.)

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Three main issues
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Simplifying Assumptions
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Tracking as induction

• Assume data association is done
• well talk about this later a dangerous
  assumption
• Do correction for the 0th frame
• Assume we have corrected estimate for ith frame
• show we can do prediction for i1, correction for
  i1

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Base case
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Induction step
Given
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Induction step
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Linear dynamic models

• Use notation to mean has the pdf of, N(a, b)
  is a normal distribution with mean a and
  covariance b.
• Then a linear dynamic model has the form
• This is much, much more general than it looks,
  and extremely powerful

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Examples

• Drifting points
• we assume that the new position of the point is
  the old one, plus noise.
• For the measurement model, we may not need to
  observe the whole state of the object
• e.g. a point moving in 3D, at the 3kth tick we
  see x, 3k1th tick we see y, 3k2th tick we see
  z
• in this case, we can still make decent estimates
  of all three coordinates at each tick.
• This property, which does not apply to every
  model, is called Observability

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Examples

• Points moving with constant velocity
• Periodic motion
• Etc.
• Points moving with constant acceleration

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Points moving with constant velocity

• We have
• (the Greek letters denote noise terms)
• Stack (u, v) into a single state vector
• which is the form we had above

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Points moving with constant acceleration

• We have
• (the Greek letters denote noise terms)
• Stack (u, v) into a single state vector
• which is the form we had above

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Velocity
Position
Position
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x-axis position y-axis velocity
Position
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The Kalman Filter

• Key ideas
• Linear models interact uniquely well with
  Gaussian noise - make the prior Gaussian,
  everything else Gaussian and the calculations are
  easy
• Gaussians are really easy to represent --- once
  you know the mean and covariance, youre done

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The Kalman Filter in 1D

• Dynamic Model
• Notation

Predicted mean
Corrected mean
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Notation and A Few Identities
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Prediction
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Prediction for 1D Kalman filter

• The new state is obtained by
• multiplying old state by known constant
• adding zero-mean noise
• Therefore, predicted mean for new state is
• constant times mean for old state
• Predicted variance is
• sum of constant2 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
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Correction for 1D Kalman filter

• Pattern match to identities given in book
• basically, guess the integrals, get
• Notice
• 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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This is figure 17.3. The notation is a bit
involved, but is logical. We plot state as open
circles, measurements as xs, predicted means as
s with three standard deviation bars, corrected
means as s with three standard deviation bars.
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Smoothing

• Idea
• We dont have the best estimate of state - what
  about the future?
• Run two filters, one moving forward, the other
  backward in time.
• Now combine state estimates
• The crucial point here is that we can obtain a
  smoothed estimate by viewing the backward
  filters prediction as yet another measurement
  for the forward filter
• so weve already done the equations

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Backward Filter
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Forward-Backward Algorithm
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Forward filter, using notation as above figure
is top left of 17.5. The error bars are 1
standard deviation, not 3 (sorry!)
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Backward filter, top right of 17.5, The error
bars are 1 standard deviation, not 3 (sorry!)
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Smoothed estimate, bottom of 17.5. Notice how
good the state estimates are, despite the
very lively noise. The error bars are 1 standard
deviation, not 3 (sorry!)
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Data Association

• Nearest neighbors
• choose the measurement with highest probability
  given predicted state
• popular, but can lead to catastrophe
• Probabilistic Data Association
• combine measurements, weighting by probability
  given predicted state
• gate using predicted state

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Applications

• Vehicle tracking
• Surveillance
• Human-computer interaction