ue 
ly 
ch 
Fuse, Takashi 
  
  
overdetermined linear equation system: 
Gf--b (6) 
where 
UE Lh, hy 
: : ; : 
ens re o 
. ; V £ 
T: T T. 
has the least squares solution 
2 (tpi pr 
f-»-GG)Gb (8) 
provided that the inverse of. G' G exists. 
In the temporal local optimization method, small temporal local neighborhood is n frames, and then we can obtain n 
observation equations. Multispectral constraints method has three observation equations. These overdetermined 
linear equation systems have the least squares solution in the same way as spatial local optimization method. In the 
case of second order derivative method, three observation equations are obtained, re: 
LAT, ; 
G- In. L| 5r (9) 
1, 1, Ly, 
2.3 Imposition of the Smoothness Condition 
Another approach of solving the gradient constraint equation is imposition of condition, that is: 
(a) spatial smoothness of optical flow (spatial global optimization method) (Barron, Fleet and Beauchemin, 1994, 
Beauchemin and Barron, 1997, Horn and Schunck, 1981, Schunck, 1984); 
(b) temporal smoothness (temporal global optimization method); 
(c) their combination. 
One way to express the additional condition is to minimize the square of the magnitude of the gradient of the optical 
flow velocity: 
Qu Y (ou ? 3S P {3 : 
v v 
e. £e Qv. ov. 10 
The total error, E, to be minimized as 
: 2 2 2 
TY 3 Qu Y (du dv dv 
E = LA (uti, 1,) + a 5] 5 > + = + n : (11) 
The minimization is to be accomplished by finding suitable values for the optical flow velocity @, v). Using the 
calculus of variation, following equations are obtained. 
Hàn HI,v zo? V3 —1,, 
(12) 
Iu 1) v2a  V'v- II, 
xty 
  
International Archives of Photogrammetry and Remote Sensing. Vol. XXXIII, Part B5. Amsterdam 2000. 271