The coefficients in (26) can all be calculated directly from the 
data once the SVD of F has been computed. Moreover once 
(26) has been solved, the focal lengths f and f' can be read 
off directly from the values of wı,w2 and ws through (18) 
and (19). Finally there is a unique correspondence between 
focal lengths and coefficients once the arbitrary scale factor 
A has been eliminated. 
To conclude this section, note that if F is rescaled to «F, 
then the coefficients in the vector on the left hand side and 
in the first column on the right hand side scale as «?, while 
the coefficients in the remaining two columns are invariant. 
Hence any numerical estimate of F should be rescaled to en- 
sure that (26) is a well-conditioned system. A simple sensible 
choice for this is to attempt to ensure that elements in the 
first column have roughly the same magnitude as elements 
in the other columns (i.e. are of order unity) by scaling F so 
that ||£]| 2 1 as proposed in the discussion leading up to (9). 
4 CHARACTERISING DEGENERATE 
CONFIGURATIONS 
We now explore degenerate configurations for which multi- 
ple factorisations of the form in (8) may exist with A and 
A" having the form in (2). Assuming the underlying camera 
model is correct, by Proposition 1 such a factorisation is pos- 
sible if and only if A, A' and F satisfy (14). As each step in 
the remainder of the derivation is reversible, it follows that, 
given the special form of A and A’, a unique factorisation 
exists for (8) if and only if (26) has a unique solution. 
For convenience, let us express (26) formally as s = Cw, 
where s = (o2,0,02)" and w = (Wa; Wa, wa)”. Then it is 
a fundamental result of linear algebra that (26) has a unique 
solution if and only if C is invertible. If C is not invertible, 
then (26) will have either multiple solutions or no solutions 
depending on whether or not s is in the range of C. The 
case where (26) has no solutions is of little interest. Since 
we are assuming we have data from a real world system there 
must be at least one solution: assuming that the underlying 
camera model is essentially correct, then the fact that s is 
outside the range of C can be put down to measurement 
errors and appropriate allowances made (e.g. (26) could be 
solved in a total least squares sense as described in [5]). 
The condition that C is not invertible is equivalent to 
det(C) = 0. From (26) and a little algebra we have that 
det(C) = [(ui f3)(ui is) + (uj fs) (uj is)] 
(uf f3)(u7 is) — (ui fs) (uiis)]. (27) 
Thus det(C) vanishes if and only if either of the factors on the 
right hand side of (27) is identically zero. The next proposi- 
tion now interprets these conditions in terms of the geometry 
of the imaging system. 
Proposition 2 Two focal lengths cannot be uniquely identi- 
fied from the fundamental matrix if and only if the geometry 
of the imaging system is in either of the following classes of 
configurations: 
(i) The optical axes of the two cameras and the baseline 
between them are coplanar. 
(ii) One optical axis, the baseline and the vector perpen- 
dicular to the baseline and the other optical axis are 
coplanar. 
International Archives of Photogrammetry and Remote Sensing. Vol. XXXI, Part B3. Vienna 1996 
Proof. We have established that a unique solution is not 
possible if and only if one of the factors on the right hand side 
of (27) is zero. It remains to give the geometric interpretation 
of each of the conditions 
[(uf fs) (uf is) -- (u2f3)(u2i3)] = 
| 
[(uZ£)(ufis) - (uZ£)(ufi;)] = 
We first show that (28) is equivalent to the condition that 
F33 — 0. To see this, note that since the u; form an or- 
thonormal basis 
3 
f, 2 3 (uifs)us. (30) 
k=] 
Since F7 us — 0, it follows that ff us — 0 and that 
(ult) (alia) + (ulfs)(ulia) = fli; — Fas. (31) 
Thus (28) is equivalent to Fas = 0. Since Ais — is and 
A’ 'i3 ~ is, we have that 
(emus a9 TESI Z ITA CTTBEATH. 
^ i3 BRis 
But is is also the unit vector in the direction of the optical 
axis in each image expressed in the local coordinate system 
of each image. Thus (32) is equivalent to z' - (b x z) — 0, 
where z and z' are unit vectors in the direction of the optical 
axes expressed in the global coordinate system. This in turn 
implies that the vectors z, z' and b are coplanar. 
We next examine (29). If we define the vector g by 
g — (ul fi)u; - (uf fi)u;, (33) 
then it is straightforward to check that g is orthogonal to 
fs. Moreover since the uj form an orthonormal basis, g 
is orthogonal to us. Finally (29) effectively states that g 
is orthogonal to i3. Since g is non-zero as long as fs is 
non-zero, this implies that the vectors f5,i; and u3 must be 
linearly dependent (i.e. they are coplanar). 
We Sow show that this coplanarity condition is equivalent to 
condition (ii) of the proposition. To see this, it suffices to 
show that b and BRiz are in the subspace spanned by f3, i3 
and us. To establish this first for b, note that (2) and (13) 
imply 
br A us (1—(f' + DES 
= bz À, U3 + A213, (34) 
for some constants A; and A». To establish a similar result 
for BRis, we first note that the particular form of A implies 
that A^lis ^ is. Thus 
BRi; ^» BRA i; 
— A A'TBRA-^!is 
e(T-(/7 + DLAI BRA, 
= ATTBRAT Hs 4 (FT + 1)0T A TT BRAT iu) 
— Asfs 4 A4ds, (35) 
for some constants À3 and A4. 
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