2. Kalman Filter For Object Tracking
• Kalman filter – as a linear discrete-time variant system
• Kalman Filtering Problem
• Optimal Estimation of Parameters
• Extended Kalman filter – Modeling Non-Linear Systems
• Object Tracking using Kalman filter
a) Using Linear KF model
b) Extended KF
• Mean Shift Optimal Prediction and Kalman Filter
for Object Tracking
• Object Tracking using an adaptive Kalman Filter combined
with Mean Shift
3. 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 behavior of the system.
•We keep on updating this system i.e. state of the
system or state vector based on observable data.
4. 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.
5. • 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
7. KF- Filtering Problem
Use the entire observed data, consisting of the vectors y1; y2…
yk, to find for each k ¸ 1 the minimum mean-square error
estimate of the state xk .
The problem is called filtering if i = k , prediction if i > k and
smoothing if 1<= i < k.
8. Kalman Filter – Optimal Estimates
• yk = xk + vk ;
• where xk is an unknown signal and vk is an additive noise
component. Where xk denote the a posteriori estimate of
the signal xk, given the observations y1; y2; … yk. In
general, the estimate ^xk is different from the unknown
signal xk.
• Cost function for incorrect measurements is mean square
error which is Jk = E[(xk - ^xk)2] ;
• Optimal Error Estimate :-
• ^xk = E[xk | y1, y2, …. yk] ;
9. KF – using observed data to improve
estimate of System Description
• Using the information contained in the new measurement yk
to update estimate of the unknown state xk.
• ^xk_ denote a prior estimate of the state which is already
available at time k.
• Posterior estimate after using the information in the observed
data
• ^xk = G(1,k) * ^xk_ + G(2,k) * yk ;
• Where G(1,k) and G(2,k) is to determined such that it performs
best linear estimation.
• Here we need Cost function estimate error and correct
parameters after that.
10. KF – Cost Function and Error
Calculation
• State Error Vector xk~ = xk - ^xk ;
• Using principle of Orthogonality
• E [ xk~ yiT] = 0 for i = 1 to k-1 ;
• Using error vector definition, system definition and this
orthogonality we can reduce above equations into the following
form :-
• E[(xk - G(1,k) ^xk_ - G(2,k)*Hk*xk – G(2,k)*vk)yiT] = 0 for i = 1
to k-1 ;
• Since vk is independent process of yi hence we can drop the
last term .
11. KF – Cost Function and Error
Calculation; Continued …
•Remaining expression can be written as by adding this element
“ G(1,k)*xk – G(1,k)*xk ” …
•E[(I - G(2,k) *Hk - G(1,k))*xk*yiT + G(1,k)(xk - ^xk_)yiT] = 0
for i = 1 to k-1 ;
•Second term again can be dropped using principle of
orthoganility.
•Expression reduced to (I - G(2,k) *Hk - G(1,k))*E[xk*yiT ] = 0
•Which should be true for arbitrary value xk and yi hence
following equation must hold
I – G(2,k)Hk – G(1,k) = 0 ;
• ^xk = ^xk_ + G(1,k)(yk – Hk * ^xk_) … G(1,k) is kalman gain we
will be using Gk for G(1,k) here onwards.
12. KF – Kalman Gain Calculation;
Continued …
• From the Principle of orthogonality
• E[ (xk - ^xk) yiT] = 0 and it follows that …
• E[ (xk - ^xk) ^yiT] = 0 where ^yiT is an estimate of yk given the
previous measurement y1,y2……..yk-1
• Define the innovations process yk~ = yk – ^yk ;
• Innovation process represents a measure of the “new”
information contained in yk;
• It may also be expressed as yk~ = yk – Hk* ^xk_ ;
which can be reduced to yk~ = vk + Hk* xk~ ;
• From above two equations we can deduce into the following
form E[ ( xk - ^xk)*yk~T] = 0
13. • State Error vector can be written as
• xk - ^xk = ~xk_ - Gk *(Hk*~xk_ + vk) further can be resolved
and putting back into the earlier equation :-
• E[{(I – Gk*Hk)*~xk_ - Gk*vk}*(Hk*~xk_ + vk)] = 0 ;
• (I-Gk*Hk)*E[~xk_*~xk_T]HkT - Gk*E[vkvkT]= 0
• Kalman Gain can be calculated once we know E[~xk_*~xk_T] ;
• Defining a priori covariance Pk_ =E[~xk_*~xk_T];
• And posteriori covariance Pk =E[~xk*~xkT];
• Given the “old” a posteriori covariance matrix Pk-1 , compute
the “updated” a priori covariance matrix Pk_ .
• Now task is to calculate posteriori from priori estimated
covariance matrix.
KF – Kalman Gain Calculation;
Continued …
14. • From the definition of covariance matrix we can get
the following expression :-
• Pk = (I – Gk*Hk)*E[~xk_ ~xk_T ]* (I – Gk*Hk)T
+Gk*E[vk vkT]*GkT ;
• Pk = (I – Gk*Hk)*Pk_* (I – Gk*Hk)T +Gk*E[vk
vkT]*GkT we got relation between prior and posteriori
covariance matrix ;
• On simplification of above expression the relation
between posteriori and prior condition can be boils
down to following expression :-
Pk = (I – Gk*Hk)*Pk_ ;
KF – Kalman Gain Calculation;
Continued …
15. KF – Calculation of Error covariance Matrix
Continued …
• For the second stage of error covariance we must
make sure that priori estimate of the stage is defined in
terms of the “old” a posteriori estimate as follows :-
• ~xk_ = F(k,k-1)*^xk-1 ;
• ~xk_ = xk - ^xk ;
• ~xk_ = (F(k,k-1)*xk-1 + wk-1) – (F(k,k-1)*^xk-1) which
is “ F(k,k-1)*~xk-1 + wk-1 ” ;
• Pk_ = F(k,k-1)*Pk-1*F(k,k-1)T + Qk-1 ;
• Which defines priori covariance Pk_ in terms of old
covariance matrix Pk-1 ;
16. KF – Calculation of Error covariance
Matrix ; Continued …
• Now we know how to estimate posteriori
using priori covariance matrix .
• And to calculate priori matrix using “old”
posteriori covariance matrix.
• Once we have error covariance matrix we can
get the kalman gain and hence new estimate of
the position.
• Only thing left is “Initialization of Error
Covariance Matrix”..
17. KF – Calculation of Error covariance
Matrix ; Continued …
•Initialization
•X0 = E[x0] ;
•P0 = E[(x0 - E[x0] )*(x0 - E[x0] )T] ;
•Initial condition has an advantage of yielding an
unbiased estimate of the state xk.
19. 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.
20. Extended Kalman Filter – modelling
Non-Linear Systems
• Process Equation :-
Xk+1 = f(k,Xk) + wk ;
• Measurement Equation :-
Yk = h(k,Xk) + vk ;
Where as before wk and vk are independent zero-mean white
gaussian noise process with covariance matrices Rk and Qk
respectively.
• f(k,Xk) denotes a nonlinear transition matrix function that is
possibly time-variant.
• H(k,Xk) denotes a nonlinear measurement matrix that may be
time-variant too.
21. • Linearize the state-space model for very short
duration i.e. at each time instant around the most
recent state estimate which is taken to be either ^Xk or
^Xk_ expending on which particular function is being
considered.
• F (k+1,k) = df (k,xk)/dx at x = ^xk .
• H(k) = dh(k,xk)/dx at x = ^xk .
• Taylor Approximation of the nonlinear functions F(k,x)
and H(k,x) around ^Xk and ^Xk_ respectively.
Extended Kalman Filter – modelling
Non-Linear Systems
22. • F(k,Xk) ≈ F( x ;^xk) + F (xk,^xk) k+1 to K ;
• H(k,Xk) ≈ H( x ;^xk) + H(xk,^xk) k+1 to K ;
• With these quantities in hand we can approximate as
following :-
Xk+1 ≈ F k+1,kxk + wk + dk and _yk ≈ Hkxk + vk ;
• Two new introduced quantities are as follows:
_yk = yk – {h(x,^xk_) Hk*^xk_} &&
dk = f(x,^xk) – F k+1,k * ^xk ;
Extended Kalman Filter – modelling
Non-Linear Systems
23. • 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
24. • 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
25. Object Tracking using Kalman Filter
for Non Linear Trajectory
• Extended
Kalman Filter
- modelling
more
dynamical
system using
unconstraine
d Brownian
Motion
28. Mean Shift Optimal Prediction and Kalman Filter
for Object Tracking
•Colour Based
Similarity
measurement –
Bhattacharyya
distance
• Target
Localization
• Distance
Maximization
• Kalman
Prediction
29. 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.
