e to be 
uted by 
e in the 
d to be 
images 
os with 
and the 
ms of a 
ctor of 
(1) 
(2) 
/ectors 
(3) 
|o]. T 
andom 
error 
995 
223 
  
  
MATHEMATICAL MODEL KALMAN FILTER 
Fig. 6. System model and Kalman filter. 
The coordinates of the projected vertex in the image plane make up the measurement vector 
z,- (xy) (4) 
with S denoting the sensor coordinate system of the camera. The nonlinear measurement equation is given by 
Zu zh UA (5) 
The function h, with 
C C 
T S Xk S Yk 
hy (x,) 2 (hy, hy) , h, 2X, 2f.—c, h,z ykf c (6) 
represents the perspective projection assuming an ideal pin hole camera with focal length f. The relation between video 
memory coordinates (pixels) and coordinates at the CCD sensor (sels) as well as the internal parameters of the real 
camera (focal length, lens distortion, etc.) are known from camera calibration. The measurement noise v, is modeled as 
zero mean gaussian white noise with covariance matrix R,. System and measurement noise are assumed to be 
uncorrelated. 
The measurement equation is linearized by Taylor series expansion, which yields the following equations of an 
Extended Kalman filter in discrete time (Gelb, 1974). The propagation of the state estimate and the error covariance 
matrix is expressed by : 
=F 2 and FP, =F PF +Q,. (7) 
The update equations are given by 
X, 2 X, *K, Iz, -hy A) and Pr 1-Kr Hr (5). (8) 
with the Kalman gain matrix 
IAPRS, Vol. 30, Part 5W1, ISPRS Intercommission Workshop "From Pixels to Sequences", Zurich, March 22-24 1995