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Edward M. Mikhail 
  
for one of the nonlinear solutions that contain constraints among the parameters. These refined solutions are discussed 
in the remaining subsections. 
2.2.2 Constrained Solution with Four Dependent Equations (Model 1). There are four constraints among the 27 T 
coefficients in Equations (7) that are commonly used. The first is that the 27" (i.e., the last) element of the vector T is 
set equal to one. The next three are that the determinants of three 3 by 3 matrices, constructed from the T's, are equal to 
zero; see [Hartley, 1997]. 
Although this solution with three constraints tends to alleviate some of the instability problems, the solution still is not 
rigorous because it uses four algebraicly dependent equations. Since it is not possible to linearize with respect to the 
observations (the corresponding coefficient matrix becomes rank deficient), the objective function to be minimized is 
the sum of the squared errors of each equation (7) instead of the sum of the squared residuals to the observations. 
2.2.3 Constrained Solutions with Three Independent Equations (Models 2,3). There are two models to be 
discussed that use three independent equations per point. Both of these two models use the first three of the four 
trilinearity equations; i.e., Equations (7). Since the three condition equations are algebraicly independent, we linearize 
them with respect to the observations in order to obtain a rigorous solution. 
Both models carry 24 parameters with six constraints since there should be only 18 independent parameters. In Model 
2, the parameters are 24 of the T coefficients (T; through 75,). In Model 3, however, the values for the elements of Pa 
and P,3; (aj; and bj) are computed from the initial estimates of the 27 T coefficients, and are used as the actual 
parameters in the adjustment. 
The six constraint equations in terms of T coefficients are found in [Faugeras and Papadopoulo, 1997]. The six 
constraint equations in terms of the elements of P, and P,; are described in [Hartley, 1996]; however, since the 
reference does not give them explicitly, they are listed below: 
G =a,a,+a,a,, T 4,0, 70 G, 2450, t 450, t d5d,, 20 
G, =Jau + a, +ay, —1=0 (8) 
G, 54-120 
G, 2a4,, t 4505, t d4,, 20 
G, 24b, + bi, + bi, —1=0 
  
2.3 Collinearity Model (Model 4) 
All published derivations to trilinearity use the two scale factors, 
allow more than one value for 7, when in theory there should be 
only one. The following derivation, called "relative collinearity" 
uses a single scale factor, 7, and four algebraically independent 
equations. As in the trilinearity derivation, we compute the 3D 
projective object coordinates from image 1, and then project the 
object point into images 2 and 3; see Figure 2. This leads to the 
following four condition equations: 
   
(A. VA, I) 
Figure 2. Relative Collinearity Model 
  
  
  
  
F, 2x (ax as yas, t a,t)- (a,xaj,y aj, * a,,1)20 
Fz y (as, X a4, y - a, +ayut)— (ay X+a,y+ay, +a,t)=0 
” 
y =X (by x + by, y+ by, + byt) (bx +b, y +b, +b,1)=0 
F, = y (bx t b, y + ba Tbat)- (bx t by + by, *tb4t)z20 
(9) 
These four condition equations are independent; however they contain the additional unknown parameter 7, which is a 
unique scale factor for each point observed. The other parameters involved are the 24 total elements of P,» and P, (aj 
and b;), which are computed from the initial estimates of the 27 T coefficients. Note that the T coefficients are 
estimated from the linear solution of the trilinearity equations, (7). 
  
International Archives of Photogrammetry and Remote Sensing. Vol. XXXIII, Part B3. Amsterdam 2000. 587