1. Image processing and computer vision
Unscented Kalman filter
UKF v7a

2. Overview
Unscented transform
UKF
UKF v7a

3. Unscented transform
A method to use Sigma Points to approximate the distribution of data after non-linear transform.
Assume original mean (m) and covariance (M) are given
m=[12.3, 7.6]’, M=[ ; ]
Assume the non-linear transform is Cartesian t polar transformation
We want to find the new mean and covariance in the polar coordinates (transformed space)
UKF v7a

4. Example in matlab
%test unscented transform ut1.m, code for the example in
m=[12.3, 7.6]'
M=[1.44 0; ]
M_root=sqrt(M)
s(:,1)=(1/sqrt(2))*[0,2]'
s(:,2)=-(1/sqrt(2))*[sqrt(3),1]'
s(:,3)=-(1/sqrt(2))*[-sqrt(3),1]'
mplus(1,:)=M_root*s(:,1)+m
mplus(2,:)=M_root*s(:,2)+m
mplus(3,:)=M_root*s(:,3)+m
[theta(1),rho(1)] = cart2pol(mplus(1,1),mplus(1,2))
[theta(2),rho(2)] = cart2pol(mplus(2,1),mplus(2,2))
[theta(3),rho(3)] = cart2pol(mplus(3,1),mplus(3,2))
m_UT=[mean(rho),mean(theta)]' %note:order is rho first,same as that in wiki
' '
v1=([rho(1),theta(1)]'- m_UT)
v2=([rho(2),theta(2)]'- m_UT)
v3=([rho(3),theta(3)]'- m_UT)
v1*v1'
(1/3)*(v1*v1'+v2*v2'+v3*v3')
UKF v7a