Tracking H akan Ard o March 4, 2013 H akan Ard o Tracking - - PowerPoint PPT Presentation

Introduction Tracking with static camera Tracking with moving camera

Tracking

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• March 4, 2013

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Introduction Tracking with static camera Tracking with moving camera State space Sliding Window Detection

Outline

1

Introduction State space Sliding Window Detection

2

Tracking with static camera Greedy Kalman filter Particle filter

3

Tracking with moving camera Self-motion

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Introduction Tracking with static camera Tracking with moving camera State space Sliding Window Detection

State space

State: position X = (X, Y , Z, 1), velocity, v = (vx, vy, vz, 0), ... Observation: detection in image x = (x, y, 1) Observation model: λx = PX Dynamic model: Xt+1 = Xt + vt and vt+1 = vt Sthocastic dynamic model: Introduce noise, random numbers q and w Xt+1 = Xt + vt + q vt+1 = vt + w Sthocastic observation model: Introduce noise, a random number r λx = PX + r

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Introduction Tracking with static camera Tracking with moving camera State space Sliding Window Detection

State space

State: position X = (X, Y , Z, 1), velocity, v = (vx, vy, vz, 0), ... Observation: detection in image x = (x, y, 1) Observation model: λx = PX Dynamic model: Xt+1 = Xt + vt and vt+1 = vt Sthocastic dynamic model: Introduce noise, random numbers q and w Xt+1 = Xt + vt + q vt+1 = vt + w Sthocastic observation model: Introduce noise, a random number r λx = PX + r

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Introduction Tracking with static camera Tracking with moving camera State space Sliding Window Detection

State space

State: position X = (X, Y , Z, 1), velocity, v = (vx, vy, vz, 0), ... Observation: detection in image x = (x, y, 1) Observation model: λx = PX Dynamic model: Xt+1 = Xt + vt and vt+1 = vt Sthocastic dynamic model: Introduce noise, random numbers q and w Xt+1 = Xt + vt + q vt+1 = vt + w Sthocastic observation model: Introduce noise, a random number r λx = PX + r

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Introduction Tracking with static camera Tracking with moving camera State space Sliding Window Detection

State space

State: position X = (X, Y , Z, 1), velocity, v = (vx, vy, vz, 0), ... Observation: detection in image x = (x, y, 1) Observation model: λx = PX Dynamic model: Xt+1 = Xt + vt and vt+1 = vt Sthocastic dynamic model: Introduce noise, random numbers q and w Xt+1 = Xt + vt + q vt+1 = vt + w Sthocastic observation model: Introduce noise, a random number r λx = PX + r

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Introduction Tracking with static camera Tracking with moving camera State space Sliding Window Detection

State space

State: position X = (X, Y , Z, 1), velocity, v = (vx, vy, vz, 0), ... Observation: detection in image x = (x, y, 1) Observation model: λx = PX Dynamic model: Xt+1 = Xt + vt and vt+1 = vt Sthocastic dynamic model: Introduce noise, random numbers q and w Xt+1 = Xt + vt + q vt+1 = vt + w Sthocastic observation model: Introduce noise, a random number r λx = PX + r

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Introduction Tracking with static camera Tracking with moving camera State space Sliding Window Detection

State space

State: position X = (X, Y , Z, 1), velocity, v = (vx, vy, vz, 0), ... Observation: detection in image x = (x, y, 1) Observation model: λx = PX Dynamic model: Xt+1 = Xt + vt and vt+1 = vt Sthocastic dynamic model: Introduce noise, random numbers q and w Xt+1 = Xt + vt + q vt+1 = vt + w Sthocastic observation model: Introduce noise, a random number r λx = PX + r

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Introduction Tracking with static camera Tracking with moving camera State space Sliding Window Detection

Sliding window detectors

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Introduction Tracking with static camera Tracking with moving camera State space Sliding Window Detection

Detection probability

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Introduction Tracking with static camera Tracking with moving camera Greedy Kalman filter Particle filter

Greedy tracker

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Introduction Tracking with static camera Tracking with moving camera 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

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Introduction Tracking with static camera Tracking with moving camera 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

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Introduction Tracking with static camera Tracking with moving camera Greedy Kalman filter Particle filter

UNC Chapel Hill Computer Science Slide 5

Some Intuition

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Introduction Tracking with static camera Tracking with moving camera 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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Introduction Tracking with static camera Tracking with moving camera 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 )

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Introduction Tracking with static camera Tracking with moving camera 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

( )

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Introduction Tracking with static camera Tracking with moving camera 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

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Introduction Tracking with static camera Tracking with moving camera 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

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Introduction Tracking with static camera Tracking with moving camera 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)

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Introduction Tracking with static camera Tracking with moving camera Greedy Kalman filter Particle filter

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

Particle filtering algorithm step-by-step (1)

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Introduction Tracking with static camera Tracking with moving camera Greedy Kalman filter Particle filter

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

Particle filtering step-by-step (2)

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Introduction Tracking with static camera Tracking with moving camera Greedy Kalman filter Particle filter

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

Particle filtering step-by-step (3)

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Introduction Tracking with static camera Tracking with moving camera Greedy Kalman filter Particle filter

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

Particle filtering step-by-step (4)

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Introduction Tracking with static camera Tracking with moving camera Greedy Kalman filter Particle filter

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

Particle filtering step-by-step (5)

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Introduction Tracking with static camera Tracking with moving camera Greedy Kalman filter Particle filter

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

Particle filtering step-by-step (6)

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Introduction Tracking with static camera Tracking with moving camera 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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Introduction Tracking with static camera Tracking with moving camera 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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Introduction Tracking with static camera Tracking with moving camera 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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Introduction Tracking with static camera Tracking with moving camera 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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Introduction Tracking with static camera Tracking with moving camera 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

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Introduction Tracking with static camera Tracking with moving camera 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

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Introduction Tracking with static camera Tracking with moving camera 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

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Introduction Tracking with static camera Tracking with moving camera Greedy Kalman filter Particle filter

UNC Chapel Hill Computer Science Slide 21

Measurement

zk = Hxk + vk

zk ∈ Rm is the process measurement

measurement vector

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Introduction Tracking with static camera Tracking with moving camera Greedy Kalman filter Particle filter

UNC Chapel Hill Computer Science Slide 22

Measurement

zk = Hxk + vk

state vector

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Introduction Tracking with static camera Tracking with moving camera 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

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Introduction Tracking with static camera Tracking with moving camera 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

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Introduction Tracking with static camera Tracking with moving camera 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

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Introduction Tracking with static camera Tracking with moving camera 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

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Introduction Tracking with static camera Tracking with moving camera 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

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Introduction Tracking with static camera Tracking with moving camera Greedy Kalman filter Particle filter

UNC Chapel Hill Computer Science Slide 28

Time Update (Predict)

state error covariance

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Introduction Tracking with static camera Tracking with moving camera Greedy Kalman filter Particle filter

UNC Chapel Hill Computer Science Slide 29

Measurement Update (Correct)

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Introduction Tracking with static camera Tracking with moving camera 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

–

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Introduction Tracking with static camera Tracking with moving camera 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

– –

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Introduction Tracking with static camera Tracking with moving camera Greedy Kalman filter Particle filter

UNC Chapel Hill Computer Science Slide 32

Filter Operation

Time Update (Predict) Measurement (Correct) Update

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Introduction Tracking with static camera Tracking with moving camera Greedy Kalman filter Particle filter

Kalman filter tracker

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Introduction Tracking with static camera Tracking with moving camera Greedy Kalman filter Particle filter

Kalman filter tracker

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Introduction Tracking with static camera Tracking with moving camera Greedy Kalman filter Particle filter

Kalman filter tracker

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Introduction Tracking with static camera Tracking with moving camera Greedy Kalman filter Particle filter

Kalman filter tracker

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Introduction Tracking with static camera Tracking with moving camera Greedy Kalman filter Particle filter

Particle filter

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Introduction Tracking with static camera Tracking with moving camera 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¨

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Introduction Tracking with static camera Tracking with moving camera 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

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Introduction Tracking with static camera Tracking with moving camera 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¨

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Introduction Tracking with static camera Tracking with moving camera 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)

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Introduction Tracking with static camera Tracking with moving camera 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)

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Introduction Tracking with static camera Tracking with moving camera 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˚

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Introduction Tracking with static camera Tracking with moving camera 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)

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Introduction Tracking with static camera Tracking with moving camera 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)

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Introduction Tracking with static camera Tracking with moving camera 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

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Introduction Tracking with static camera Tracking with moving camera 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

March 4, 2013 50 / 57

Introduction Tracking with static camera Tracking with moving camera 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

March 4, 2013 50 / 57

Introduction Tracking with static camera Tracking with moving camera 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

March 4, 2013 50 / 57

Introduction Tracking with static camera Tracking with moving camera 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

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Introduction Tracking with static camera Tracking with moving camera Greedy Kalman filter Particle filter

Particle filter

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Introduction Tracking with static camera Tracking with moving camera Greedy Kalman filter Particle filter

Particle filter tracker

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Introduction Tracking with static camera Tracking with moving camera Self-motion

Recording sport events

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Introduction Tracking with static camera Tracking with moving camera Self-motion

Self-motion

Detect keypoints Track each keypoint separately Use RANSAC to find camera motion Compensate for camera motion (stiching)

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Introduction Tracking with static camera Tracking with moving camera Self-motion

Self-motion

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Introduction Tracking with static camera Tracking with moving camera Self-motion

Self-motion

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Introduction Tracking with static camera Tracking with moving camera Self-motion

Master Thesis: Automatisk inspelning av sport

Detta projekt syftar till att utveckla mjukvara som automatiskt kan filma, f¨

• lja och m¨

ata tider f¨

• r ett idrottsevenemang, till exempel ett 100m-lopp i
• friidrott. F¨
• r detta kr¨

avs dels kunskaper i matematisk bildanalys, men en del kunskaper i programmering kommer ¨ aven att beh¨

• vas. Vi f˚

ar hj¨ alp av IFK Lund att spela in film och utv¨ ardera resultatet. Ett examensarbete syftar till att utveckla algoritmer f¨

• r automatisk

f¨

• ljning av lopp. Fr˚

an b¨

• rjan ¨

ar kameran stilla och efter starten skall den hela tiden f¨

• lja l¨
• parna genom att rotera och zooma. Algortimen

m˚ aste kunna kompensera och den egna kamerar¨

• relsen.

Ytterligare ett examensarbete behandlar automatisk detektion av l¨

• parbanan. Genom kamerakalibrering och banans k¨

anda m˚ att kan l¨

• parnas hastighet ber¨

aknas och diagram ¨

• ver hur hastigheten

f¨

• r¨

andrats genom loppet presenteras f¨

• r publik och tr¨
• anare. Denna

information efterfr˚ agas av aktiva l¨

• pare. Automatisk detektion av

m˚ allinje ¨ ar ¨ aven viktigt f¨

• r tidtagningen ovan.

Contact: Petter Strandmark <petter@maths.lth.se>

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