Tracking: Generating Conclusions about Motion in Image Sequences - Prof. Cornelia M. Fermu, Study notes of Computer Science

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Tracking

Definition of Tracking

• Tracking:
  • Generate some conclusions about the motion of the scene, objects, or the camera, given a sequence of images.
  • Knowing this motion, predict where things are going to project in the next image, so that we don’t have so much work looking for them.

Tracking a Silhouette by

Measuring Edge Positions

• Observations are positions of edges along normals to tracked contour

Why not Wait and Process the

Set of Images as a Batch?

• In a car system, detecting and tracking

pedestrians in real time is important.

• Recursive methods require less computing

Related Fields

• Signal Detection and Estimation
• Radar technology

The Problem: Signal Estimation

• We have a system with parameters
  • Scene structure, camera motion, automatic zoom
  • System state is unknown (“hidden”)
• We have measurements
  • Components of stable “feature points” in the images.
  • “Observations”, projections of the state.
• We want to recover the state components from

the observations

State variable a

A Simple Example of Estimation

by Least Square Method

• Goal: Find estimate of state

such that the least square error between measurements and the state is minimum

a ˆ

∑

n

i

C xi a 1

2 ( ) 2

∑ ∑ = =

∂ n

i

i

n

i

xi a x na a

C

1 1

0 ( ˆ)^ ˆ

∑

n

i

xi n

a 1

a

Measurement x

x

t

x i

a (^) x i -a

t i

Recursive Least Square Estimation

• We don’t want to wait until

all data have been collected to get an estimate of the depth

• We don’t want to reprocess

old data when we make a new measurement

• Recursive method: data at

step i are obtained from data at step i - 1

a ˆ

State variable a

Measurement x

x

t

a

x i

a^ ˆ^ i (^) − 1 i a ˆ

Recursive Least Square Estimation 3

i i i i

x a

i

a a

Estimate at step i

Predicted measure

Innovation

Gain (^) Actual measure

Gain specifies how much do we pay attention to the difference between what we expected and what we actually get

Least Square Estimation of the

State Vector of a Static System

1. Batch method

a

H 1 H i

H 2

x 1

xi

x 2

xi = H (^) i ⋅ a a

H

H

H

























=

























⇒

xn n

x

x

... ...

2

1 2

1

⇒ x = H a

measurement equation

a (H H) H X

T − 1 T ˆ =

Find estimate (^) a ˆ that minimizes

(X Ha) (X H a)

T = − − 2

C

We find

Dynamic System

i

i

i

i

A

V

X

a

A A w x i

V V A t

X X V t

i i

i i i

i i i

= +

= + ∆

= + ∆

−

− −

− −

1

1 1

1 1



























































 ∆

∆





















−

−

−

A w

V

X t

t

A

V

X

i

i

i

i

i

i 0

0

0 0 1

0 1

1 0

1

1

1

State of rocket

Measurement

⇒ a (^) i = Φ ai-1 + w

[ ] V

A

V

X x

i

i

i i + 

















 = (^1 00) ⇒ xi = Hai + V

State equation for rocket

Measurement equation

Noise

Tweak factor

a ˆ^ (^) i = Φ i a ˆ i-1 + Ki(xi − Hi Φ ia ˆ i-1 )

Recursive Least Square

Estimation for a Dynamic System

(Kalman Filter)

a (^) i = Φ i ai-1 + wi -

x (^) i = Hi ai + v i

w (^) i ~ N ( 0 , Qi )

v (^) i ~ N ( 0 , Ri )

i-1 i-1 i-1 i -

i-

T i i i-1 i

- i

T i i i

T i i i

P (I K H )P'

P' P Q

K P' H H P' H R

State equation

Measurement equation

Tweak factor for model

Measurement noise

Prediction for xi

Prediction for a i

Gain Covariance matrix for prediction error Covariance for estimation error

• Predict next state as using

previous step and dynamic model

• Predict regions

of next measurements using measurement model and uncertainties

• Make new measurements xi in

predicted regions

• Compute best estimate of next state

Tracking Steps

Measurement^ (u, v)

Prediction region

Φ ia ˆ i -

N ( H (^) i Φ ia ˆ i-1 , P'i )

a ˆ^ (^) i = Φ i a ˆ i-1 + Ki(xi − Hi Φ ia ˆ i-1 ) “Correction” of predicted state

Recursive Least Square Estimation for

a Dynamic System (Kalman Filter)

x

t

Measurement x i

State vector a i Estimation (^) a ˆ i