Ilkka Niini 
  
There are currently four types of additional constraints that are needed in the projective version of the adjustment methoj 
in order to obtain the correct number of independent parameters and equations in the system. The Kruppa equation 
and the trifocal constraints are already known from earlier studies, e.g. (Niini, 1994) and (Niini, 1998). A new interig 
orientation constraint has been derived. Also a new constraint which, indeed, connects five singular correlation matrios 
between any four images together is introduced. 
2.1 A new trifocal interior orientation constraint 
In a trifocal system, the Kruppa equations (Niini, 1994), derived from the three singular correlation matrices of three image 
pairs do not take into account the fact that the rotations between these three images ¢, j, k are dependent through the cong; 
tion R;; Ri RT ip = I (identity matrix). If the interior orientation parameters are computed by using the Kruppa equation 
only, the relative rotations, when computed, do not necessarily exactly fulfill the consistency condition R;; Rj RT Sl 
This suggests that a new interior orientation constraint is hidden in this condition or in some of its variant. À candidate fo 
this constraint, though a quite complicated one, has been derived from the above condition. It relates the interior orient 
tion parameters of the three images and the singular correlation parameters between these three images, and it guarantees 
the consistency of the relative rotations of the three images. There is one such constraint per each trifocal plane or triple 
of images in a projective block: 
(e; TCI Cj jeje; Ck Chew) ey CF Cj jej)* 
-pi(el C? Ce el OT Creri)(e el CT E t d Es CT Cie RE (l 
- se, CT ieu; (6, E5 C7! C77 Ejej det(C; ) et(C Yet (Cr) =) 
Above, the interior orientation matrices C are of type 
13 —(Tp + Byp) 
C= 10 à —ayp ; (2) 
0-0 —Cp 
where xp, yp are the principal point co-ordinates, a is affinity (z/y-scale ratio), Zis the lack of orthogonality between i: 
and y-axis, and c, is the focal length. A scale Ti (ratio of the scales of singular correlation matrix and the epipole) is 
tij 
needed because the singular correlation matrices and epipoles are homogeneous quantities. The epipolar matrices hav 
the form 
> / ee 
0 ZZ 
Ei; = —z! 0 z, ; (3) 
Ye c1. 0 
and the epipole co-ordinate vectors are of form e;; = [Te, Ye, ze |f. 
e J Mix Mijejk)(e3; MI, Miei; (e ME, ICA LUTON 
ifi — t A 4 v == ————— HÀ ee > 1 B i the 
Additionally, py TT Mae FRAT MT ci) (eT ME Mes and p» i == Here, the sign of p» is 
  
same as the sign of e. scale ratio — „and it can be u... from two corresponding image co-ordinate vectors 7j, 
z4,, by using the equality AE;,z; — - ; T M ary. The scalars a, b, c, and d are the four elements of N;, (equation 4). Tk 
| Th : 
projection ray scale A is positive by definition because the object is always in front of the cameras, so the correct sign of 
Zi*. and, hence, p, can be checked. The sign of p, is automatically correct from the above formulae. Note that the terns 
m and p» depend only on the singular correlation parameters. 
The epipolar decomposition of the singular correlation matrix is (Thompson, 1968), (Niini, 1998): 
loni cie 0.0 ^0 
Mijzig 42.0 0 a b = 5 l'O]. (4 
0 0 1 "ed ib 0 
rl 
or, shortly, M;; = FIN; ; Fj;. Matrices F;; and F;; represent shifts to scaled epipoles (co-ordinates x, y,, x" y 
of which one can be NT in each image), and N;; is a transformation matrix between the centred 2 2 -D mr pencils 
extended to a singular 3 x 3 format. There are seven parameters in decomposition 4, since one of the elements in tli 
central matrix N;; (the largest one) can be fixed. The decomposition 4 can have other forms, too, depending on the large 
epipole co-ordinates. The above form was obtained by scaling the epipoles with the x co-ordinates. This decompositiol 
is used to parameterize the singular correlation matrices in the projective version of the adjustment. 
  
644 International Archives of Photogrammetry and Remote Sensing. Vol. XXXIII, Part B3. Amsterdam 2000. 
  
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