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Edward M. Mikhail 
  
unknowns from 16 to 14. For any new 3D object point whose image coordinates are observed in images 1 and 2, the 
image coordinates of the new point in image 3 are computed by solving two linear equations of the type (1). 
In a new F matrix technique, all three possible F matrices are used in a simultaneous, constrained least squares 
adjustment; see Figure 1. In order to obtain initial estimates for the 24 elements (8 per F matrix), an unconstrained 
linear solution using at least eight conjugate points per image pair should be performed first. Since the coefficients of 
the 3 F matrices, i.e. the 24 parameters, are not independent, it is necessary to enforce the epipolar constraint in order to 
improve the numerical stability and robustness. According to the epipolar constraint, the three pairs of epipoles (e;», 
€13, €21, @23, €31, and e3») must lie on the same plane that contains the 3 camera stations (C;, C», and C3) [Faugeras and 
Papadopoulo, 1997]. See Figure 2. 
One form of writing the epipolar constraint equations is in (2a), where e»; is the vector of homogeneous coordinates of 
the epipole on image 2 for C;, and e is the epipole vector on image 3 for C». Both es; and es, are estimated from Fo; in 
Equation (2b): 
Fe, — Fe, =0 (2a) 
Fye;, =0,  Fj5e4 20 (2b) 
  
  
The three epipolar constraint equations (2a), 
represented in matrix form, are valid only to a 
scale factor. Thus we obtain two independent 
equations from the three by dividing the i" 
equation by the third equation, for i = 1 and 2 as 
  
  
  
  
  
  
  
follows: 
€ 23 € 32 
Fy,G)| e,» Fy)| e, 
1 1 (3) 
= =0 
€ 123 € x32 
F3) € 23 FQ) € 32 
1 1 
  
Figure 1. Epipolar Constraint 
  
  
where F(i) is the i^ 1 by 3 row vector of 3 by 3 
matrix F. The two epipolar constraint equations, G, = 0 and Gs = 0, are obtained for i — 1,2, by clearing fractions in 
Equation (3). Recall that the first three constraint equations are that the determinants of the F matrices equal zero, or 
G, =|F, =0,G, =|F,,|=0,G, =|F,, =0. 
  
  
  
The constraint equations, G4 and Gs, are written in terms of the epipoles, e»; and e», which are needed to facilitate the 
writing of the epipolar constraints. These 4 variables, e,25, €y23, €32, €y3», are called "added parameters" since they are 
not used as actual parameters in the writing of the condition equations. Since there are four added parameters, we 
require four additional constraint equations, derived by taking the first two rows of each Equation (2) as follows: 
€ 32 €132 € 323 € 23 
G, = F4, () €,3 =0, G, = F,, (2) € y32 =0, G; = Fla) € y23 20, G,= Fi (2) € 23 -0 (4) 
1 1 1 1 
In summary, for each 3D object point appearing on three images, we write three condition equations of the form of 
equation (1). A linear solution is implemented to solve for estimates of the F matrix elements that will be used as initial 
approximations in the next step. Then, a non-linear solution with constraints and added parameters, and the same three 
condition equations per point, is performed [Mikhail, 1976]. 
  
International Archives of Photogrammetry and Remote Sensing. Vol. XXXIII, Part B3. Amsterdam 2000. 585