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Tracking H akan Ard o February 22, 2012 H akan Ard o Tracking - - PowerPoint PPT Presentation

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Introduction Tracking Tracking H akan Ard o February 22, 2012 H akan Ard o Tracking February 22, 2012 1 / 51 Introduction Tracking Sliding Window Detection Outline Introduction 1 Sliding Window Detection Tracking 2

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SLIDE 1

Introduction Tracking

Tracking

H˚ akan Ard¨

  • February 22, 2012

H˚ akan Ard¨

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SLIDE 2

Introduction Tracking Sliding Window Detection

Outline

1

Introduction Sliding Window Detection

2

Tracking Greedy Kalman filter Particle filter

H˚ akan Ard¨

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SLIDE 3

Introduction Tracking Sliding Window Detection

Sliding window detectors

H˚ akan Ard¨

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SLIDE 4

Introduction Tracking Sliding Window Detection

Detection probability

H˚ akan Ard¨

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SLIDE 5

Introduction Tracking Greedy Kalman filter Particle filter

Greedy tracker

H˚ akan Ard¨

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SLIDE 6

Introduction Tracking Greedy Kalman filter Particle filter

UNC Chapel Hill Computer Science Slide 1

STC Lecture Series An Introduction to the Kalman Filter

Greg Welch and Gary Bishop

University of North Carolina at Chapel Hill Department of Computer Science

http://www.cs.unc.edu/~welch/kalmanLinks.html

H˚ akan Ard¨

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SLIDE 7

Introduction Tracking Greedy Kalman filter Particle filter

Sanjay Patil1 and Ryan Irwin2 Graduate research assistant1, REU undergrad2 Human and Systems Engineering

URL: www.isip.msstate.edu/publications/seminars/msstate/2005/particle/

HUMAN AND SYSTEMS ENGINEERING:

Gentle Introduction to Particle Filtering

H˚ akan Ard¨

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SLIDE 8

Introduction Tracking Greedy Kalman filter Particle filter

UNC Chapel Hill Computer Science Slide 5

Some Intuition

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SLIDE 9

Introduction Tracking Greedy Kalman filter Particle filter

UNC Chapel Hill Computer Science Slide 6

First Estimate

ˆ x

1 = z1

ˆ σ

2 1 = σ 2 z1

Conditional Density Function

14 12 10 8 6 4 2

  • 2

N(z1,σz1

2 )

z1 σ 2

z1

,

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SLIDE 10

Introduction Tracking Greedy Kalman filter Particle filter

UNC Chapel Hill Computer Science Slide 7

Second Estimate

Conditional Density Function

z2 σ 2

z2

,

ˆ x

2 =...?

ˆ σ

2 2 = ...?

14 12 10 8 6 4 2

  • 2

N(z2,σz2

2 )

H˚ akan Ard¨

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SLIDE 11

Introduction Tracking Greedy Kalman filter Particle filter

UNC Chapel Hill Computer Science Slide 8

Combine Estimates

= σ z2

2

σ z1

2 + σ z2 2

( )

[ ]z1+ σ z1

2

σ z1

2 + σ z2 2

( )

[ ]z2

ˆ

x

2

= ˆ x

1 + K2 z2 − ˆ

x

1

[ ]

where

K2 = σ z1

2

σ z1

2 + σ z2 2

( )

H˚ akan Ard¨

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SLIDE 12

Introduction Tracking Greedy Kalman filter Particle filter

UNC Chapel Hill Computer Science Slide 9

Combine Variances

1 σ 2 = 1 σ z1

2

( )+ 1 σ z2

2

( )

2

H˚ akan Ard¨

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SLIDE 13

Introduction Tracking Greedy Kalman filter Particle filter

UNC Chapel Hill Computer Science Slide 10

Combined Estimate Density

ˆ x = ˆ σ

2 = σ 2

2

ˆ

x

2

14 12 10 8 6 4 2

  • 2

Conditional Density Function N( σ 2)

ˆ

x,ˆ

H˚ akan Ard¨

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SLIDE 14

Introduction Tracking Greedy Kalman filter Particle filter

UNC Chapel Hill Computer Science Slide 11

Add Dynamics dx/dt = v + w

where v is the nominal velocity w is a noise term (uncertainty)

H˚ akan Ard¨

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SLIDE 15

Introduction Tracking Greedy Kalman filter Particle filter

Page 7 of 20 Particle Filtering – Gentle Introduction and Implementation Demo

Particle filtering algorithm step-by-step (1)

H˚ akan Ard¨

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SLIDE 16

Introduction Tracking Greedy Kalman filter Particle filter

Page 8 of 20 Particle Filtering – Gentle Introduction and Implementation Demo

Particle filtering step-by-step (2)

H˚ akan Ard¨

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SLIDE 17

Introduction Tracking Greedy Kalman filter Particle filter

Page 9 of 20 Particle Filtering – Gentle Introduction and Implementation Demo

Particle filtering step-by-step (3)

H˚ akan Ard¨

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SLIDE 18

Introduction Tracking Greedy Kalman filter Particle filter

Page 10 of 20 Particle Filtering – Gentle Introduction and Implementation Demo

Particle filtering step-by-step (4)

H˚ akan Ard¨

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SLIDE 19

Introduction Tracking Greedy Kalman filter Particle filter

Page 11 of 20 Particle Filtering – Gentle Introduction and Implementation Demo

Particle filtering step-by-step (5)

H˚ akan Ard¨

  • Tracking

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SLIDE 20

Introduction Tracking Greedy Kalman filter Particle filter

Page 12 of 20 Particle Filtering – Gentle Introduction and Implementation Demo

Particle filtering step-by-step (6)

H˚ akan Ard¨

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SLIDE 21

Introduction Tracking Greedy Kalman filter Particle filter

UNC Chapel Hill Computer Science Slide 13

Some Details

x x A A x x w w z z H H x x . . . . . . . . = + =

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SLIDE 22

Introduction Tracking Greedy Kalman filter Particle filter

UNC Chapel Hill Computer Science Slide 14

Discrete Kalman Filter

Maintains first two statistical moments

process state (mean) error covariance z y x

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SLIDE 23

Introduction Tracking Greedy Kalman filter Particle filter

UNC Chapel Hill Computer Science Slide 16

Necessary Models

measurement model dynamic model

previous state next state state measurement

image plane

( u , v )

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SLIDE 24

Introduction Tracking Greedy Kalman filter Particle filter

UNC Chapel Hill Computer Science Slide 17

The Process Model

x k+1 = Axk + wk zk = Hxk + vk

Process Dynamics Measurement

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SLIDE 25

Introduction Tracking Greedy Kalman filter Particle filter

UNC Chapel Hill Computer Science Slide 18

Process Dynamics

x k+1 = Axk + wk

xk ∈ Rn contains the states of the process

state vector

H˚ akan Ard¨

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SLIDE 26

Introduction Tracking Greedy Kalman filter Particle filter

UNC Chapel Hill Computer Science Slide 19

Process Dynamics

nxn matrix A relates state at time step k to time step k+1

state transition matrix

x k+1 = Axk + wk

H˚ akan Ard¨

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SLIDE 27

Introduction Tracking Greedy Kalman filter Particle filter

UNC Chapel Hill Computer Science Slide 20

Process Dynamics process noise

wk ∈ Rn models the uncertainty of the process wk ~ N(0, Q)

x k+1 = Axk + wk

H˚ akan Ard¨

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SLIDE 28

Introduction Tracking Greedy Kalman filter Particle filter

UNC Chapel Hill Computer Science Slide 21

Measurement

zk = Hxk + vk

zk ∈ Rm is the process measurement

measurement vector

H˚ akan Ard¨

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SLIDE 29

Introduction Tracking Greedy Kalman filter Particle filter

UNC Chapel Hill Computer Science Slide 22

Measurement

zk = Hxk + vk

state vector

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SLIDE 30

Introduction Tracking Greedy Kalman filter Particle filter

UNC Chapel Hill Computer Science Slide 23

Measurement

mxn matrix H relates state to measurement

measurement matrix

zk = Hxk + vk

H˚ akan Ard¨

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SLIDE 31

Introduction Tracking Greedy Kalman filter Particle filter

UNC Chapel Hill Computer Science Slide 24

Measurement measurement noise

zk ∈ Rm models the noise in the measurement vk ~ N(0, R)

zk = Hxk + vk

H˚ akan Ard¨

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SLIDE 32

Introduction Tracking Greedy Kalman filter Particle filter

UNC Chapel Hill Computer Science Slide 25

State Estimates

a priori state estimate a posteriori state estimate

ˆ x –

k

ˆ xk

H˚ akan Ard¨

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SLIDE 33

Introduction Tracking Greedy Kalman filter Particle filter

UNC Chapel Hill Computer Science Slide 26

Estimate Covariances

a priori estimate error covariance a posteriori estimate error covariance

Pk

– = E[(xk- xk –)(xk - xk –)T]

ˆ ˆ

Pk = E[(xk- xk)(xk - xk)T]

ˆ ˆ

H˚ akan Ard¨

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SLIDE 34

Introduction Tracking Greedy Kalman filter Particle filter

UNC Chapel Hill Computer Science Slide 27

Filter Operation

Time update (a priori estimates) Measurement update (a posteriori estimates)

Project state and covariance forward to next time step, i.e. predict Update with a (noisy) measurement

  • f the process, i.e. correct

H˚ akan Ard¨

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SLIDE 35

Introduction Tracking Greedy Kalman filter Particle filter

UNC Chapel Hill Computer Science Slide 28

Time Update (Predict)

state error covariance

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SLIDE 36

Introduction Tracking Greedy Kalman filter Particle filter

UNC Chapel Hill Computer Science Slide 29

Measurement Update (Correct)

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SLIDE 37

Introduction Tracking Greedy Kalman filter Particle filter

UNC Chapel Hill Computer Science Slide 30

Time Update (Predict)

a priori state and error covariance

ˆ

xk+1 = Axk

–

Pk+1 = APk A + Q

–

H˚ akan Ard¨

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SLIDE 38

Introduction Tracking Greedy Kalman filter Particle filter

UNC Chapel Hill Computer Science Slide 31

Measurement Update (Correct)

a posteriori state and error covariance Kalman gain

ˆ

xk = xk + Kk (zk - Hxk )

ˆ ˆ

– –

Pk = (I - Kk H)Pk

–

Kk = Pk HT(HPk HT + R)-1

– –

H˚ akan Ard¨

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February 22, 2012 38 / 51

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SLIDE 39

Introduction Tracking Greedy Kalman filter Particle filter

UNC Chapel Hill Computer Science Slide 32

Filter Operation

Time Update (Predict) Measurement (Correct) Update

H˚ akan Ard¨

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SLIDE 40

Introduction Tracking Greedy Kalman filter Particle filter

Kalman filter tracker

H˚ akan Ard¨

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SLIDE 41

Introduction Tracking Greedy Kalman filter Particle filter

Kalman filter tracker

H˚ akan Ard¨

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SLIDE 42

Introduction Tracking Greedy Kalman filter Particle filter

Kalman filter tracker

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  • Tracking

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SLIDE 43

Introduction Tracking Greedy Kalman filter Particle filter

Kalman filter tracker

H˚ akan Ard¨

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SLIDE 44

Introduction Tracking Greedy Kalman filter Particle filter

Particle filter

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SLIDE 45

Introduction Tracking Greedy Kalman filter Particle filter

Patricle filter states

Current state

  • x(1)

k , x(2) k , x(3) k , · · · , x(n) k

  • Predicted state (a priori state extimate)
  • x−(1)

k+1 , x−(2) k+1 , x−(3) k+1 , · · · , x−(n) k+1

  • Corrected state (a posteriori state extimate)
  • x(1)

k+1, x(2) k+1, x(3) k+1, · · · , x(n) k+1

  • H˚

akan Ard¨

  • Tracking

February 22, 2012 45 / 51

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SLIDE 46

Introduction Tracking Greedy Kalman filter Particle filter

Patricle filter states

Current state

  • x(1)

k , x(2) k , x(3) k , · · · , x(n) k

  • Predicted state (a priori state extimate)
  • x−(1)

k+1 , x−(2) k+1 , x−(3) k+1 , · · · , x−(n) k+1

  • Corrected state (a posteriori state extimate)
  • x(1)

k+1, x(2) k+1, x(3) k+1, · · · , x(n) k+1

  • H˚

akan Ard¨

  • Tracking

February 22, 2012 45 / 51

slide-47
SLIDE 47

Introduction Tracking Greedy Kalman filter Particle filter

Patricle filter states

Current state

  • x(1)

k , x(2) k , x(3) k , · · · , x(n) k

  • Predicted state (a priori state extimate)
  • x−(1)

k+1 , x−(2) k+1 , x−(3) k+1 , · · · , x−(n) k+1

  • Corrected state (a posteriori state extimate)
  • x(1)

k+1, x(2) k+1, x(3) k+1, · · · , x(n) k+1

  • H˚

akan Ard¨

  • Tracking

February 22, 2012 45 / 51

slide-48
SLIDE 48

Introduction Tracking Greedy Kalman filter Particle filter

Dynamic model

Any sthocastic model from which we can sample f (xk+1 |xk ) Example: The dynamic model from the kalman filter xk+1 = Axk + wk f (xk+1 |xk ) = N(Axk, Q)

H˚ akan Ard¨

  • Tracking

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slide-49
SLIDE 49

Introduction Tracking Greedy Kalman filter Particle filter

Dynamic model

Any sthocastic model from which we can sample f (xk+1 |xk ) Example: The dynamic model from the kalman filter xk+1 = Axk + wk f (xk+1 |xk ) = N(Axk, Q)

H˚ akan Ard¨

  • Tracking

February 22, 2012 46 / 51

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SLIDE 50

Introduction Tracking Greedy Kalman filter Particle filter

Particle filter prediction step

Propagate each particle i, separately The prediction x−(i)

k+1 is choosen as a sample from

f

  • xk+1
  • x(i)

k

  • H˚

akan Ard¨

  • Tracking

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slide-51
SLIDE 51

Introduction Tracking Greedy Kalman filter Particle filter

Measurement model

Any sthocastic model from which we can calculate likelihoods f (zk |xk ) Example: The measurement model from the kalman filter zk = Hxk + vk f (zk |xk ) = N(Hxk, R)

H˚ akan Ard¨

  • Tracking

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slide-52
SLIDE 52

Introduction Tracking Greedy Kalman filter Particle filter

Measurement model

Any sthocastic model from which we can calculate likelihoods f (zk |xk ) Example: The measurement model from the kalman filter zk = Hxk + vk f (zk |xk ) = N(Hxk, R)

H˚ akan Ard¨

  • Tracking

February 22, 2012 48 / 51

slide-53
SLIDE 53

Introduction Tracking Greedy Kalman filter Particle filter

Particle filter correction step

The measurement model gives a weight for each particle, i w(i)

k

= f

  • zk
  • x−(i)

k

  • Resample the set of particles using the weights

Each corrected particle, x(i)

k , is choosen randomly

x(i)

k

= x−(j)

k

for some random j The probability of choosing sample j is w(j)

k

  • l w(l)

k

The same particle may be choosen several times

H˚ akan Ard¨

  • Tracking

February 22, 2012 49 / 51

slide-54
SLIDE 54

Introduction Tracking Greedy Kalman filter Particle filter

Particle filter correction step

The measurement model gives a weight for each particle, i w(i)

k

= f

  • zk
  • x−(i)

k

  • Resample the set of particles using the weights

Each corrected particle, x(i)

k , is choosen randomly

x(i)

k

= x−(j)

k

for some random j The probability of choosing sample j is w(j)

k

  • l w(l)

k

The same particle may be choosen several times

H˚ akan Ard¨

  • Tracking

February 22, 2012 49 / 51

slide-55
SLIDE 55

Introduction Tracking Greedy Kalman filter Particle filter

Particle filter correction step

The measurement model gives a weight for each particle, i w(i)

k

= f

  • zk
  • x−(i)

k

  • Resample the set of particles using the weights

Each corrected particle, x(i)

k , is choosen randomly

x(i)

k

= x−(j)

k

for some random j The probability of choosing sample j is w(j)

k

  • l w(l)

k

The same particle may be choosen several times

H˚ akan Ard¨

  • Tracking

February 22, 2012 49 / 51

slide-56
SLIDE 56

Introduction Tracking Greedy Kalman filter Particle filter

Particle filter correction step

The measurement model gives a weight for each particle, i w(i)

k

= f

  • zk
  • x−(i)

k

  • Resample the set of particles using the weights

Each corrected particle, x(i)

k , is choosen randomly

x(i)

k

= x−(j)

k

for some random j The probability of choosing sample j is w(j)

k

  • l w(l)

k

The same particle may be choosen several times

H˚ akan Ard¨

  • Tracking

February 22, 2012 49 / 51

slide-57
SLIDE 57

Introduction Tracking Greedy Kalman filter Particle filter

Particle filter correction step

The measurement model gives a weight for each particle, i w(i)

k

= f

  • zk
  • x−(i)

k

  • Resample the set of particles using the weights

Each corrected particle, x(i)

k , is choosen randomly

x(i)

k

= x−(j)

k

for some random j The probability of choosing sample j is w(j)

k

  • l w(l)

k

The same particle may be choosen several times

H˚ akan Ard¨

  • Tracking

February 22, 2012 49 / 51

slide-58
SLIDE 58

Introduction Tracking Greedy Kalman filter Particle filter

Particle filter

H˚ akan Ard¨

  • Tracking

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slide-59
SLIDE 59

Introduction Tracking Greedy Kalman filter Particle filter

Particle filter tracker

H˚ akan Ard¨

  • Tracking

February 22, 2012 51 / 51