The document provides an introduction to the Kalman filter. It discusses that the Kalman filter is used to estimate the state of a system from a series of incomplete and noisy measurements. It does this optimally by using a linear feedback control. The Kalman filter works recursively on streams of noisy input data to produce a statistically optimal estimate of the underlying system state. It is effective in applications involving tracking or navigation such as object tracking. The Kalman filter assumes the true state is modeled as a linear dynamical system subject to Gaussian noise. It estimates the state of the system based on a series of measurements observed over time, with external disturbances and measurement noise.

1. Introduction to Kalman Filter

2. 2
Overview
• The Problem – Why do we need Kalman
Filter?
• What is a Kalman Filter?
• Conceptual Overview
• Kalman Filter – as linear dynamic system
• Object Tracking using Kalman Filter

3. 3
The Problem
• System state cannot be measured directly
• Need to estimate “optimally” from measurements
Measuring
Devices Estimator
Measurement
Error Sources
System State
(desired but not
known)
External Controls
Observed
Measurements
Optimal Estimate
of System State
System
Error Sources
System
Black Box

4. Tracking Example
• Noisy Measurement +
Filtering gives the final
position.
• Localization + Filtering
• why Filter ?
• The process of finding the
“best estimate” from noisy
data amounts to “filtering
out” the noise.
• Kalman filter also doesn’t
just clean up the data
measurements, but also
projects these
measurements onto the
state estimate.

5. 5
What is a Kalman Filter?
• Recursive data processing algorithm
• Generates optimal estimate of desired quantities
given the set of measurements
• Optimal?
– For linear system and white Gaussian errors, Kalman filter
is “best” estimate based on all previous measurements
– For non-linear system optimality is ‘qualified’
• Recursive?
– Doesn’t need to store all previous measurements and
reprocess all data each time step

6. Kalman Filter Assumptions
• Linearity and Gaussian White Noise
1. Because they are adequate enough to modal real time systems.
2. Can be easily manipulated because we need to calculate only two
moments.
3. Tractable mathematics due to linear property.
4. Calculation is very much simplified which is very important for online
algorithms.
5. Practical justification – central limit theorem
• Why we are adding Noise ?
1. To model disturbances in the system which are intractable.
2. To model measurement errors in the system.
3. For accurate estimation of values of a time-dependent process itself we
must include future noise.

7. 7
Kalman Filter
What if the noise is NOT Gaussian?
Given only the mean and standard deviation of noise, the
Kalman filter is the best linear estimator. Non-linear
estimators may be better.
Why is Kalman Filtering so popular?
· Good results in practice due to optimality and structure.
· Convenient form for online real time processing.
· Easy to formulate and implement given a basic
understanding.
· Measurement equations need not be inverted.

8. Optimality of Kalman Filter

9. Optimal Estimation

10. 10
Conceptual Overview
• Lost on the 1-dimensional line
• Position – y(t)
• Assume Gaussian distributed measurements
y

11. 11
Conceptual Overview
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• Sextant Measurement at t1: Mean = z1 and Variance = z1
• Optimal estimate of position is: ŷ(t1) = z1
• Variance of error in estimate: 2
x (t1) = 2
z1
• Boat in same position at time t2 - Predicted position is z1

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Conceptual Overview
• So we have the prediction ŷ-(t2)
• GPS Measurement at t2: Mean = z2 and Variance = z2
• Need to correct the prediction due to measurement to get ŷ(t2)
• Closer to more trusted measurement – linear interpolation?
prediction ŷ-(t2)
measurement z(t2)

13. Conceptual Overview

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Conceptual Overview
• Corrected mean is the new optimal estimate of position
• New variance is smaller than either of the previous two variances
measurement z(t2)
corrected optimal
estimate ŷ(t2)
prediction ŷ-(t2)
• Amount of Spread is the
measurement of
Uncertainty.
• We will try to reduce
this uncertainty in a way
trying to reduce the
variance.

15. 15
Conceptual Overview
• Lessons so far:
Make prediction based on previous data - ŷ-, -
Take measurement – zk, z
Optimal estimate (ŷ) = Prediction + (Kalman Gain) * (Measurement - Prediction)
Variance of estimate = Variance of prediction * (1 – Kalman Gain)

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Conceptual Overview
• At time t3, boat moves with velocity dy/dt=u
• Naïve approach: Shift probability to the right to predict
• This would work if we knew the velocity exactly (perfect model)
ŷ(t2)
Naïve Prediction ŷ-
(t3)

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Conceptual Overview
• Better to assume imperfect model by adding Gaussian noise
• dy/dt = u + w
• Distribution for prediction moves and spreads out
ŷ(t2)
Naïve Prediction ŷ-
(t3)
Prediction ŷ-(t3)

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Conceptual Overview
• Now we take a measurement at t3
• Need to once again correct the prediction
• Same as before
Prediction ŷ-(t3)
Measurement z(t3)
Corrected optimal estimate ŷ(t3)

19. 19
Conceptual Overview
• Lessons learnt from conceptual overview:
– Initial conditions (ŷk-1 and k-1)
– Prediction (ŷ-
k , -
k)
• Use initial conditions and model (eg. constant velocity) to make
prediction
– Measurement (zk)
• Take measurement
– Correction (ŷk , k)
• Use measurement to correct prediction by ‘blending’ prediction
and residual – always a case of merging only two Gaussians
• Optimal estimate with smaller variance

20. Kalman Filter - as linear discrete-time
dynamical system
•Dynamic System – State of the system is time-variant.
•System is described by “state vector” – which is
minimal set of data that is sufficient to uniquely
describe the dynamical behaviour of the system.
•We keep on updating this system i.e. state of the
system or state vector based on observable data.

21. KF – as linear discrete -time system
System Description
• Process Equation :-
• xk+1 = F(k+1,k) * xk + wk ;
• Where F(k+1,k) is the transition matrix taking
the state xk from time k to time k+1.
• Process noise wk is assumed to be AWG with
zero mean and with covariance matrix defined
by :-
E [wn wkT] = Qk ; for n=k and zero otherwise.

22. • Measurement Equation :-
• yk = Hk * xk + vk ;
• Where yk is observable data at time k and Hk is
the measurment matrix.
• Process noise vk is assumed to be AWG with
zero mean and with covariance matrix defined
by :-
E [vn vkT] = Rk ; for n=k and zero otherwise.
KF – as linear discrete -time system
Measurement

23. 23
Theoretical Basis
• Process to be estimated:
yk = Ayk-1 + wk-1
zk = Hyk + vk
Process Noise (w) with covariance Q
Measurement Noise (v) with covariance R
• Kalman Filter
Predicted: ŷ-
k is estimate based on measurements at previous time-steps
ŷk = ŷ-
k + K(zk - H ŷ-
k )
Corrected: ŷk has additional information – the measurement at time k
K = P-
kHT(HP-
kHT + R)-1
ŷ-
k = Ayk-1 + Buk
P-
k = APk-1AT + Q
Pk = (I - KH)P-
k

24. KF- linear dynamical time variant
system

25. 25
Theoretical Basis
ŷ-
k = Ayk-1 + Buk
P-
k = APk-1AT + Q
Prediction (Time Update)
(1) Project the state ahead
(2) Project the error covariance ahead
Correction (Measurement Update)
(1) Compute the Kalman Gain
(2) Update estimate with measurement zk
(3) Update Error Covariance
ŷk = ŷ-
k + K(zk - H ŷ-
k )
K = P-
kHT(HP-
kHT + R)-1
Pk = (I - KH)P-
k

26. KF – Review the complete process

27. 27
Blending Factor
• If we are sure about measurements:
– Measurement error covariance (R) decreases to zero
– K decreases and weights residual more heavily than prediction
• If we are sure about prediction
– Prediction error covariance P-
k decreases to zero
– K increases and weights prediction more heavily than residual

28. Kalman Filter – as Gaussian density
propagation process
• Random component wk
leads to spreading of the
density function.
• F(k+1,k)*XK causes drift
bodily.
• Effect of external
observation y is to
superimpose a reactive
effect on the diffusion in
which the density tends to
peak in the vicinity of
observations.

29. • Object is segmented using image processing techniques.
• Kalman filter is used to make more efficient the localization
method of the object.
• Steps Involved in vision tracking are :-
• Step 1:- Initialization ( k = 0 ) Find out the object position and
initially a big error tolerance(P0 = 1).
• Step 2:- Prediction( k > 0 ) predicting the relative position of
the object ^xk_ which is considered using as search center to
find the object.
• Step 3:- Correction( k > 0) its measurement to carry out the
state correction using Kalman Filter finding this way ^xk .
Object Tracking using Kalman Filter

30. • Advantage is to tolerate
small occlusions.
• Whenever object is
occluded we will skip the
measurement correction
and keeps on predicting
till we get object again
into localization.
Object Tracking using Kalman Filter

31. Object Tracking using Kalman Filter
for Non Linear Trajectory
• Extended
Kalman Filter
- modelling
more
dynamical
system using
unconstraine
d Brownian
Motion

32. Object Tracking using Kalman Filter

33. Mean Shift Optimal Prediction and Kalman Filter
for Object Tracking

34. Mean Shift Optimal Prediction and Kalman Filter
for Object Tracking
•Colour Based
Similarity
measurement –
Bhattacharyya
distance
• Target
Localization
• Distance
Maximization
• Kalman
Prediction

35. Object Tracking using an adaptive Kalman Filter
combined with Mean Shift
• Occlusion detection
based on value of
Bhattacharyya
coefficient.
• Based on trade-
offs between weight
to measurement and
residual error matrix
Kalman Parameters
are estimated.

36. Thank You