2D image mosaic building 2D3 - Ifremer

Time update equations (“predict”)
x = A xˆ
+ Bu
− ˆk + 1
−
k +1
k
P = A P A + Q
k
k
k
T
k
k
k
Measurement update equations (“correct”)
K
x ˆ
k
k
− T − T
= P H ( H P H − R )
k k k k k k
−
−
= xˆ
+ K ( z − H xˆ
)
k k k k k
P I K H ) P
−
= ( −
k
k k k
−1
In these formulae, the variables stand for:
Project Exocet/D page 13/16
−
• xˆ : A priori state vector at time k (given the process before step k)
k
• xˆ : A posteriori state vector at time k (given the measurement at time k)
k
• z : Measurement vector at time k
k
• u : Vector of the control entry
k
• A : Matrix linking the states at time k and k+1
k
• B : Matrix linking the control entry to the state vector
• K : Kalman gain matrix
k
−
• P : Matrix of covariance of the prediction error
k
• P : Matrix of covariance of the a posteriori error
k
• H : Matrix linking the state to the measurement
k
• w : Process noise, assumed to be white and gaussian
k
• v : Measurement noise, assumed to be white and gaussian
k
• R : Covariance matrix of the measurement noise
k
• Q : Covariance matrix of the process noise
k
In the case of video mosaicing, the state variables are the positioning (X_utm and Y_utm)
and a term to correct the pixel size. The measurement vector consists of X_utm and Y_utm
given by navigation system.
The experiment we have led is to perform a route with the underwater vehicle and to perform
the same route in the other direction. We can notice in Figure 5 that there is a shift of the
mosaic if we don’t use navigation in the algorithm whereas when using dead-reckoning
navigation in the Kalman filter, the mosaic drift is well corrected.
Deliverable N° 2D3
Report on image mosaic building
DOP/CM/SM/PRAO/06.224
Grade : 1.0 27/09/2006