Tracking Computer Vision I -

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Computer Vision I - Tracking

Carsten Rother

Computer Vision I: Tracking 17/02/2015

17/02/2015

[slide credits: Alex Krull]

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What is Tracking?

• „Tracking an object in an image sequence means continuously identifying its location when either the object or the camera are moving“ [Lepetit and Fua 2005]

• This can mean estimating in each frame:

• 2D location or window

• 6D rigid body transformation

• More complex parametric models

• Active Appearance Models

• Skeleton for human pose etc.

17/02/2015 Computer Vision I: Tracking 2

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What is Tracking?

• „Tracking an object in an image sequence means continuously identifying its location when either the object or the camera are moving“ [Lepetit and Fua 2005]

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What is Tracking?

• „Tracking an object in an image sequence means continuosly

identifying its location when either the object or the camera are moving“ [Lepetit and Fua 2005]

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Tracking vs Localization

• Tracking of objects is closely related to:

• camera pose estimation in a known environment

• localization of agents (eg. Robots) in a known environment

• Reminder: SLAM has unknown location (agent, camera) and unknown environment

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Outline

• This lecture

• The Bayes Filter

• Explained for localization

• Next lecture

• The Particle Filter

• The Kalman Filter

• Pros and Cons

• Beyond tracking and localization

• Case study:

• 6-DOF Model Based Tracking via Object Coordinate Regression

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The Bayes Filter / Convolute and Multiply

• We have:

• Probabilistic model for movement

• Probabilistic model for measurement

• Based on map of the environment

• Where is the ship?

• Using all previous and current observations

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1

z

t

z

t

1

x

t

x

t

The Hidden Markov Model

Observations:

Depth measurement

States:

Positions

time

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The Hidden Markov Model

Observation Model:

What is the likelihood of an observation, given a state?

𝑝 𝑧

_𝑡

𝑥

_𝑡

= 1

𝑐 𝑒

^{− 𝑧}^𝑡^{−𝑑 𝑥}^𝑡 ²^/(2𝜎²⁾

1

z

t

z

t

1

x

t

x

t

time

𝑑 𝑥_𝑡 is the known true depth

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The Hidden Markov Model

Motion Model:

Probability for state transition

𝑝 𝑥

_𝑡+1

𝑥

_𝑡

1

z

t

z

t

1

x

t

x

t

time

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Probability distribution for the state given all previous and current observations:

• This is what we are interested in

• Eg. use maximum as current estimate

The Posterior distribution

z

t

x

t

1

z

t

1

x

t

z

0

x

0

...

𝑝 𝑥

_𝑡

𝑧

_0:𝑡

𝑥_𝑡^∗ = 𝑎𝑟𝑚𝑎𝑥_𝑥_𝑡 𝑝 𝑥_𝑡 𝑧_0:𝑡

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The Prior distribution

1

x

t

x

t

1

z

t

1

x

t

z

0

x

0

...

𝑝 𝑥

_𝑡+1

𝑧

_0:𝑡

Probability distribution for the next state given only previous observations:

• Intermediate step for calculating the next posterior

z

t

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Important Distributions

𝑝 𝑥_𝑡+1 𝑧_0:𝑡

1

xt

1

zt

1

xt

z0

x0 ^...

zt

xt

1

zt

1

xt

z0

x0 ^...

𝑝 𝑥_𝑡 𝑧_0:𝑡

1

xt

zt

xt

𝑝 𝑧_𝑡 𝑥_𝑡 𝑝 𝑥_𝑡+1 𝑥_𝑡

Observation Model: Motion Model:

Posterior: Prior:

• Likelihood of

observation given state

• Continuous Gaussian around real depth

• Probability of new state given old one

• Discrete Gaussian

• Probability of state given previous and current observations

• Probability of state given only previous observations

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Probabilities - Reminder

• A random variable is denoted with 𝑥 ∈ {0, … , 𝐾}

• Discrete probability distribution: 𝑝(𝑥) satisfies _𝑥 𝑝(𝑥) = 1

• Joint distribution of two random variables: 𝑝(𝑥, 𝑧)

• Conditional distribution: 𝑝 𝑥 𝑧

• Sum rule (marginal distribution): 𝑝 𝑧 = _𝑥 𝑝(𝑥, 𝑧)

• Independent probability distribution: 𝑝 𝑥, 𝑧 = 𝑝 𝑧 𝑝 𝑥

• Product rule: 𝑝 𝑥, 𝑧 = 𝑝 𝑧 𝑥 𝑝(𝑥)

• Bayes’ rule: 𝑝 𝑥|𝑧 = ^{𝑝(𝑧|𝑥)𝑝 𝑥}

𝑝(𝑧)

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Probabilities - Reminder

• Sum rule (marginal distribution): 𝑝 𝑧 = _𝑥 𝑝(𝑥, 𝑧)

• Product rule: 𝑝 𝑥, 𝑧 = 𝑝 𝑧 𝑥 𝑝(𝑥)

• Bayes’ rule: 𝑝 𝑥|𝑧 = ^{𝑝(𝑧|𝑥)𝑝 𝑥}

𝑝(𝑧)

• Sum rule (marginal distribution): 𝑝 𝑧|𝐴 =

_𝑥

𝑝(𝑥, 𝑧|𝐴)

• Product rule: 𝑝 𝑥, 𝑧|𝐴 = 𝑝 𝑧 𝑥, 𝐴 𝑝(𝑥|𝐴)

• Bayes’ rule: 𝑝 𝑥|𝑧, 𝐴 =

𝑝(𝑧|𝑥,𝐴)𝑝 𝑥|𝐴 𝑝(𝑧|𝐴)

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Probabilities - Reminder

𝐴 𝐵

𝐴

𝐶

𝐵

Independence

𝐴 and 𝐵 are not connected in graph

• 𝐴 does not contain information about 𝐵

• 𝑝 𝐴, 𝐵 = 𝑝 𝐴 𝑝 𝐵

• 𝑝 𝐴|𝐵 = 𝑝 𝐴

• 𝑝 𝐵|𝐴 = 𝑝(𝐵)

Conditional Independence

𝐴 and 𝐵 are connected only via 𝐶

• 𝐴 does not contain information about 𝐵 when 𝐶 is known

• 𝑝 𝐴, 𝐵|𝐶 = 𝑝 𝐴|𝐶 𝑝 𝐵|𝐶

• 𝑝 𝐴|𝐵, 𝐶 = 𝑝 𝐴|𝐶

• 𝑝 𝐵|𝐴, 𝐶 = 𝑝(𝐵|𝐶)

Two dice rolls

It rains

People have

umbrellas Tram is

crowded

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Step by Step

𝑡 = 0

• Assume prior for first frame:

• Make first measurement: 𝑧₀

𝑝(𝑥

₀

)

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Step by Step

𝑡 = 0

• Calculate likelihood 𝑝(𝑧₀|𝑥₀) for every possible state 𝑥₀:

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Step by Step

𝑡 = 0

• Calculate the posterior by

• multiplying with prior

• Normalizing

• Reducing uncertainty

𝑝 𝑥

₀

𝑧

₀ ⁼ ^𝑝(𝑥⁰^)𝑝(𝑧⁰^|𝑥⁰⁾

𝑥𝑝(𝑥₀ = 𝑥)𝑝(𝑧₀|𝑥₀ = 𝑥)

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The Bayes Filter / Convolute and Multiply

𝑡 = 0

𝑝 𝑥₁ 𝑧₀ =

𝑥

𝑝 𝑥₀ = 𝑥 𝑧₀ 𝑝 𝑥₁ 𝑥₀ = 𝑥

• Calculate the prior by Convolution with motion model

• Adding uncertainty

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The Bayes Filter / Convolute and Multiply

𝑡 = 1

• Make new measurement: 𝑧₁

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The Bayes Filter / Convolute and Multiply

𝑡 = 1

• Calculate likelihood 𝑝(𝑧₁|𝑥₁) for every possible state 𝑥₁:

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The Bayes Filter / Convolute and Multiply

𝑡 = 1

• Normalizing

𝑝 𝑥

₁

𝑧

_0:1 ⁼ ^𝑝(𝑥¹^|𝑧⁰^)𝑝(𝑧¹^|𝑥¹⁾

𝑥𝑝(𝑥₁ = 𝑥|𝑧₀)𝑝(𝑧₁|𝑥₁ = 𝑥)

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The Bayes Filter / Convolute and Multiply

𝑡 = 1

• Calculate the prior by Convolution with motion model

• Adding uncertainty 𝑝 𝑥₂ 𝑧_0:1 =

𝑥

𝑝 𝑥₁ = 𝑥 𝑧_0:1 𝑝(𝑥₂|𝑥₁ = 𝑥)

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The Bayes Filter / Convolute and Multiply

𝑡 = 2

• Make new measurement: 𝑧₂

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The Bayes Filter / Convolute and Multiply

𝑡 = 2

• Calculate likelihood 𝑝(𝑧₂|𝑥₂) for every possible state 𝑥₂:

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The Bayes Filter / Convolute and Multiply

𝑡 = 2

• Normalizing

𝑝 𝑥

₂

𝑧

_0:2 ⁼ ^𝑝(𝑥²^|𝑧^0:1^)𝑝(𝑧²^|𝑥²⁾

𝑥𝑝(𝑥₂ = 𝑥|𝑧_0:1)𝑝(𝑧₂|𝑥₂ = 𝑥)

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The Bayes Filter / Convolute and Multiply

𝑡 = 2

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The Bayes Filter / Convolute and Multiply

Algorithm:

1. Make observation

2. Calculate likelihood for every position 3. Multiply with last prior and normalize

• Calculate posterior

4. Convolution with motion model

• Calculate new prior 5. Go to 1.

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𝑝 𝑥

₀

𝑧

₀

Calculating the Posterior

𝑝(𝑥

₀

) z

0

x

1

x

0

z

1

...

= 𝑝(𝑥₀)𝑝(𝑧₀|𝑥₀)

𝑥𝑝(𝑥₀ = 𝑥)𝑝(𝑧₀|𝑥₀ = 𝑥)

(Multiply and normalize)

Observation model

(see proof 1)

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Proof 1

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Calculating the next Prior

𝑝(𝑥

₀

) 𝑝 𝑥

₀

𝑧

₀

𝑝 𝑥

₁

𝑧

₀

...

=

𝑥

𝑝 𝑥₀ = 𝑥 𝑧₀ 𝑝 𝑥₁ 𝑥₀ = 𝑥

(Convolution)

Observation model

z

0

x

1

x

0

z

1

(see proof 2)

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Proof 2

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Calculating the Posterior

𝑝(𝑥

₀

) 𝑝 𝑥

₀

𝑧

₀

𝑝 𝑥

₁

𝑧

₀

...

Observation model

x

1

x

0

Observation model

𝑝 𝑥

₁

𝑧

_0:1 ⁼ ^𝑝(𝑥¹^|𝑧⁰^)𝑝(𝑧¹^|𝑥¹⁾

𝑥𝑝(𝑥₁ = 𝑥)𝑝(𝑧₁|𝑥₁ = 𝑥)

z

1

z

0

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𝑝 𝑥

_𝑡

𝑧

_0:𝑡

Calculating the Posterior (General Case)

𝑝(𝑥

_𝑡

|𝑧

_0:𝑡−1

) z

t

1

x

t

x

t

1

z

t

...

= 𝑝(𝑥_𝑡|𝑧_0:𝑡−1)𝑝(𝑧_𝑡|𝑥_𝑡)

𝑥 𝑝(𝑥_𝑡 = 𝑥|𝑧_0:𝑡−1)𝑝(𝑧_𝑡|𝑥_𝑡 = 𝑥)

(Multiply and normalize)

Observation model

...

(see proof 3)

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Proof 3

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Calculating the next Prior (General Case)

𝑝(𝑥

_𝑡

|𝑧

_0:𝑡−1

) 𝑝 𝑥

_𝑡

𝑧

_0:𝑡

𝑝 𝑥

_𝑡+1

𝑧

_0:𝑡

...

=

𝑥

𝑝 𝑥_𝑡 = 𝑥 𝑧_0:𝑡 𝑝(𝑥_𝑡+1|𝑥_𝑡 = 𝑥)

(Convolution)

Observation model

...

z

t

1

x

t

x

t

1

z

t

(see proof 4)

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Proof 4

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Overview Continuous Approaches

• How to apply it in continuous space?

• Two popular alternatives:

• Particle filter

• Represent prior and posterior with samples

• Kalman Filter

• Represent prior and posterior distributions as Gaussians

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Important Distributions (Particle Filter)

𝑝 𝑥_𝑡+1 𝑧_0:𝑡

1

xt

1

xt

1

zt

1

Zt

xt ^...

xt

1

zt

1

Zt

z0

x0

zt

zt ^...

xt

1

zt

1

xt

z0

Posterior: Prior:

• Likelihood of

• Continuous Gaussian around real depth

• Continuous Gaussian

• Continuous, represented as set of Samples (Particles)

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Particle Filter

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Discrete Bayes Filter vs. Particle Filter

• Discrete Bayes Filter:

1. Make observation

• Particle Filter:

2. Calculate likelihood for every position

3. Multiply with last prior and normalize

4. Convolution with motion model

5. Go to 1.

1. Make observation

2. Calculate likelihood for every sample -> weights 3. Resampling according to weights

4. Randomly move samples according to motion model (Sampling)

5. Go to 1.

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The Bayes Filter / Convolute and Multiply

𝑡 = 0

• Represent continuous prior with particles 𝑥_𝑡¹ , … , 𝑥_𝑡^𝑛

• Make measurement: 𝑧_𝑡

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Particle Filter / Resampling

• Calculate weight 𝑤_𝑡^𝑖= 𝑝(𝑧_𝑡|𝑥_𝑡 = 𝑥_𝑡^𝑖) for every particle 𝑥_𝑡^𝑖:

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Particle Filter / Resampling

𝑡 = 0

• Draw samples 𝑥_𝑡¹ , … , 𝑥_𝑡^𝑛 from posterior by resampling from 𝑥_𝑡¹ , … , 𝑥_𝑡^𝑛 using the weights 𝑤_𝑡^𝑖

0 

i i

wt

(46)

Particle Filter / Resampling

𝑡 = 0

random

0 

i i

wt

random random

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Particle Filter / Resampling

𝑡 = 0

0 

i i

wt

(48)

Particle Filter / Resampling

Why is this allowed?

Resampling is like multiplication and normalization:

• Sample density in posterior depends linearly on

• Density of prior samples 𝑥_𝑡¹ , … , 𝑥_𝑡^𝑛

• Likelihood 𝑤_𝑡^𝑖

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Discrete Bayes Filter vs. Particle Filter

1. Make observation

• Particle Filter:

5. Go to 1.

1. Make observation

2. Calculate likelihood for every sample -> weights 3. Resampling according to weights

4. Randomly move samples according to motion model (Sampling)

5. Go to 1.

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Particle Filter / Sampling

𝑡 = 0

• Obtain samples 𝑥_𝑡+1¹ , … , 𝑥_𝑡+1^𝑛 from the new prior by moving each

particle according to motion model

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Particle Filter / Sampling

𝑝 𝐵 𝐴 = 1 𝑝 𝐵 𝐴 = 2 𝑝 𝐴 = 1 = 𝑝 𝐴 = 2 = 0.5

Why is this allowed?

• We want to sample: Bⁱ~𝑝 𝐵

• Don‘t know 𝑝 𝐵

• We sample first: 𝐴^𝑖~𝑝 𝐴

• And then 𝐵^𝑖~𝑝 𝐵|𝐴 = 𝐴^𝑖

1

1 2 3 4 5 6 1 2 3 4 5 6

𝐴 𝐵

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A B Particle Filter / Sampling

1

x

t

x

t

1

z

t

1

x

t

z

0

x

0

...

z

t

• We want to sample 𝑥_𝑡+1^𝑖 ~𝑝 𝑥_𝑡+1 𝑧_0:𝑡

• Don‘t know 𝑝 𝑥_𝑡+1 𝑧_0:𝑡

Why is this allowed?

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Particle Filter / Sampling

17/02/2015 53

1

x

t

x

t

1

x

t

x

0

...

1

z

t

z

0

z

_t

• We use samples 𝑥_𝑡^𝑖~𝑝 𝑥_𝑡 𝑧_0:𝑡

Why is this allowed?

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Particle Filter / Sampling

• We use samples 𝑥_𝑡^𝑖~𝑝 𝑥_𝑡 𝑧_0:𝑡

• We sample 𝑥_𝑡+1^𝑖 ~𝑝 𝑥_𝑡+1 𝑥_𝑡 = 𝑥_𝑡^𝑖 = 𝑝 𝑥_𝑡+1 𝑥_𝑡 = 𝑥_𝑡^𝑖, 𝑧_0:𝑡

1

x

t

x

t

1

x

t

x

0

...

1

z

t

z

0

z

_t

Why is this allowed?

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Particle Filter (Tracking application)

• Tracking an object in a video sequence [Perez at al. 2002]

• States:

• 2D windows (x,y and size)

• Gaussian motion

• Observations:

• Color histograms

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Particle Filter (Tracking application)

• Pose tracking of an object in a kinect sequence [Krull at al. 2014]

• States:

• 6D Pose

• 3D position

• 3D rotation

• Observations:

• Depth images

• Predicted Object coordinates

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Particle Filter (Summary)

• The Particle Filter implements the Bayes Filter

• Prior and posterior are represented as sample sets (particle)

• Likelihood is only evaluated at particles

• Multiplication -> weighted resampling

• Convolution -> random movement according to motion model

(58)

Overview Continuous Approaches

• How to apply it in continuous space?

• Two popular alternatives:

• Particle filter

• Represent prior and posterior with samples

• Kalman Filter

• Represent prior and posterior distributions as Gaussians

(59)

Kalman Filter

(60)

Important Distributions (Kalman Filter)

𝑝 𝑥_𝑡+1 𝑧_0:𝑡

1

xt

1

xt

1

zt

1

Zt

xt ^...

xt

1

zt

1

zt

z0

x0

zt

zt ^...

xt

1

zt

1

xt

z0

Posterior: Prior:

• Likelihood of

• Continuous Gaussian around real position

• Continuous Gaussian

• Continuous, represented as Gaussiasn

(61)

Discrete Bayes Filter vs. Kalman Filter

1. Make observation

• Kalman Filter:

5. Go to 1.

1. Make observation

2. Calculate likelihood for

every position in closed form 3. Multiply with last prior

and normalize in closed form 4. Convolution with motion model in closed form

5. Go to 1.

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Kalman Filter

• Calculate likelihood 𝑝(𝑧_𝑡|𝑥_𝑡) as Gaussian:

(63)

Kalman Filter

• Multiplying with prior

• Normalizing

𝑝 𝑥

_𝑡

𝑧

_0:𝑡 ⁼ _𝑝(𝑥 ^𝑝(𝑥^𝑡^|𝑧^0:𝑡−1^)𝑝(𝑧^𝑡^|𝑥^𝑡⁾

𝑡 = 𝑥|𝑧_0:𝑡−1)𝑝(𝑧_𝑡|𝑥_𝑡 = 𝑥)𝑑 𝑥

• Closed form solution:

• Another Gaussian

(64)

Discrete Bayes Filter vs. Kalman Filter

1. Make observation

• Kalman Filter:

5. Go to 1.

1. Make observation

2. Calculate likelihood for

every position in closed form 3. Multiply with last prior

and normalize in closed form 4. Convolution with motion model in closed form

5. Go to 1.

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Kalman Filter

• Calculate the prior by

Convolution with motion model

𝑝 𝑥_𝑡+1 𝑧_0:𝑡 = 𝑝 𝑥_𝑡 = 𝑥 𝑧_0:𝑡 𝑝 𝑥_𝑡+1 𝑥_𝑡 = 𝑥 𝑑 𝑥

• Closed form solution:

• Another Gaussian

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Pros and Cons

Particle Filter:

Observation model can be anything

Kalman Filter:

Multimodal

Likelihood calculation per particle can be expensive

Easy to implement and parallelize

Problematic in high dimensional state space (many particles required)

Motion model can be anything

Observation model: linear transformation of state plus Gaussian noise

Unimodal

Motion model: linear transformation of last state plus Gaussian noise

Closed form solution in every step

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Pose Estimation vs. Pose Tracking

One Shot Pose Estimation [1] Pose Tracking

[1] Brachmann, E., Krull, A., Michel, F., Shotton, J., Gumhold, S., Rother, C.: Learning 6d object pose estimation using 3d object coordinates, ECCV (2014)

• Estimate 6D Pose from single RGB-D image

• Use Object Coordinate Regression

• Stream of RDB-D images

• Use information from previous frames:

• Realtime

• Increase robustness, accuracy

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• Augmented Reality

• Alteration

• Annotation

• Substitution

 Robotics

 Recognition/Tracking

 Automatic Grasping

Application Scenarios

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Object Coordinate Regression

Energy Optimization

Compare observed and rendered images

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Energy Formulation

17/02/2015 Computer Vision I: Tracking (Part I) 70

𝐸_𝑐 H_c = λ^depthE_c^depth H_c + 𝜆^{𝑐𝑜𝑜𝑟𝑑}𝐸_𝑐^{𝑐𝑜𝑜𝑟𝑑} 𝐻_𝑐 + 𝜆^𝑜𝑏𝑗𝐸_𝑐^𝑜𝑏𝑗 𝐻_𝑐

E_c^depth H_c 𝐸_𝑐^{𝑐𝑜𝑜𝑟𝑑} 𝐻_𝑐 𝐸_𝑐^𝑜𝑏𝑗 𝐻_𝑐 𝐸_𝑐 H_c

• Arbitrary 6DoF pose hypotheses 𝐻_𝑐 is scored according to:

Channel Energy

Comparison of Depth Comparison to Forest Prediction

(71)

Efficient Search

Forest Prediction

Object Space Input

Camera Space Pose Hypothesis

(72)

How to Adapt it for Pose Tracking?

Ingredients:

Particle Filter

Object Coordinate Regression

Efficient Search

Observation Model:

Proposal Distribution:

• Find rough estimate using

• Concentrate samples around rough estimate

Energy Motion Model:

• Assume continous motion

(73)

Filtering with Object Coordinates

Sampling from Proposal Distribution Sampling from Motion Model

(74)

Filtering with Object Coordinates

• Find a rough estimate for

current pose Efficient Search

Sampling from Motion Model Sampling from Proposal Distribution

(75)

Filtering with Object Coordinates

𝑝 𝑥_𝑡+1 𝑥_𝑡 𝑞 𝑥_𝑡+1 𝑥_𝑡

(76)

Filtering with Object Coordinates

• Most samples have a very low weight

• Few samples have very low weight

𝑤_𝑡^𝑖= 𝑝(𝑧_𝑡|𝑥_𝑡 = 𝑥_𝑡^𝑖) 𝑤_𝑡^𝑖= 𝑝(𝑧_𝑡|𝑥_𝑡 = 𝑥_𝑡^𝑖)𝑝 𝑥_𝑡 = 𝑥_𝑡^𝑖 𝑥_𝑡−1 = 𝑥_𝑡−1^𝑖 𝑞 𝑥_𝑡 = 𝑥_𝑡^𝑖 𝑥_𝑡−1 = 𝑥_𝑡−1^𝑖

(77)

Filtering with Object Coordinates

• Most samples have a very low weight

• Few samples have very low weight

𝑤_𝑡^𝑖= 𝑝(𝑧_𝑡|𝑥_𝑡 = 𝑥_𝑡^𝑖) 𝑤_𝑡^𝑖= 𝑝(𝑧_𝑡|𝑥_𝑡 = 𝑥_𝑡^𝑖)𝑝 𝑥_𝑡+1 𝑥_𝑡 = 𝑥_𝑡^𝑖 𝑞 𝑥_𝑡+1 𝑥_𝑡 = 𝑥_𝑡^𝑖

(78)

Filtering with Object Coordinates

• More efficient

• Number of Particles can be reduced

• Frame rate can be increased

(79)

Evaluation: Choi and Christensen’s Dataset [2]

• A total of 4 synthetic sequences with 4 objects

• Objects placed in static environment

• Camera moving around object

• Partial occlusion

• Very exact ground truth

[2] Changhyun Choi, Henrik I. Christensen, RGB-D Object Tracking: A Particle Filter Approach on GPU, IROS, 2013

(80)

Evaluation: Our Dataset

• A total of 6 captured

sequences of with 3 objects

• Manually annotated ground truth

• Moving objects in front of dynamic background

• Fast erratic movement

• Strong occlusions

(81)

Evaluation: Our Dataset

• Compared to [1] applied to each frame separately

• Lower average error

• Almost no outliers

(82)

Conclusion

• We have adapted the system from [1] for real time pose tracking

• We have designed a proposal distribution making efficient use of object coordinates

• Our method is robust against:

• Quick motion

• Strong occlusion

• Shadows and changing light conditions

• Deformation

[1] Brachmann, E., Krull, A., Michel, F., Shotton, J., Gumhold, S., Rother, C.: Learning 6d Object Pose Estimation Using 3d Object Coordinates, ECCV '14 (2014)