ion is not 
hand side 
rpretation 
(28) 
(29) 
ition that 
rm an or- 
(30) 
Se 
~ 13 and 
(32) 
he optical 
te system 
xu) - 0, 
he optical 
Ns in turn 
(33) 
ogonal to 
| basis, g 
es that g 
as fa is 
; must be 
jivalent to 
suffices to 
d by fs, ia 
and (13) 
e 
(34) 
ilar result 
A. implies 
RA is); 
(35) 
Since b and BR; can be expressed as linear combinations of 
f3,is and us and these three vectors are linearly dependent, 
it follows that b, BRis and is are also linearly dependent. 
But by construction BRi; = b x z where z = Riz is the 
optical axis for the first image. Thus BRis is the vector 
perpendicular to both the baseline and the optical axis of 
the first image. Since is is the direction of the optical axis 
in the second image, we have that the optical axis in the 
second image, the baseline and the vector perpendicular to 
the optical axis in the first image and to the baseline are all 
coplanar. This completes the proof. 0 
In conclusion, we have shown that there are two classes of 
degenerate imaging configurations in which two focal lengths 
cannot be uniquely recovered from the fundamental matrix, 
but that in all other cases unique recovery is possible. As 
noted in the introduction, the fact that the first class (in 
which the optical axes and the baseline are coplanar) is de- 
generate is of practical significance, since many existing ar- 
tificial vision systems are restricted to such configurations. 
The form of the second class has a pleasing symmetry with 
respect to the form of the first, but it is of little practical 
importance: imaging systems are usually constructed so that 
the optical axes tend to be roughly parallel with each other 
and roughly orthogonal to the baseline. Note, however, that 
it is still possible for there to be significant overlap of the 
camera fields of view for configurations in this class. 
5 DERIVING A SOLUTION FOR A SINGLE 
FOCAL LENGTH 
We now turn to the special case where the two focal lengths 
are known a priori to be equal: this will occur if a single 
camera with an unknown focal length is used to take both 
images. For general configurations the problem can obvi- 
ously be solved by the algorithm presented above: we are 
interested in deriving a solution that will still recover the fo- 
cal length in some of the degenerate configurations identified 
under Proposition 2. 
To achieve this we return to (14). In the case where the 
focal lengths are equal a priori, we have A = A’ = A”, so 
multiplying each side of the equation on the left by A^! and 
on the right by A gives 
—2 T 
FA?FT A? — ) i LT QUE | (36) 
uf A-?us 
where A is an arbitrary scale factor. We now show how to 
derive a quadratic from this system that will uniquely identify 
the focal length f in almost all configurations. To do this we 
note that forming the inner products of the matrices in (36) 
w.r.t the vectors u; and u» gives: 
WEAF Ay, = A. = ul FATT Alnus. (3T) 
Recalling (16), setting u — (f? —1) and noting that uZ F = 
oxvi shows that (37) can be written as 
civi (Ed pisi?) F7 (I -- pioi? )u; — 
02V3 (1+ pisid)FT(I--puisif)uo. (38) 
Expanding out both sides of the equations and collecting 
terms gives the following quadratic in u 
579 
International Archives of Photogrammetry and Remote Sensing. Vol. XXXI, Part B3. Vienna 1996 
0=0—-0 + 
[(QuTis)? 4 (vfis)?) o1 — ((uZis)? - (vZis)^) o2] n 
[(uris)(viis)ei — (uzis)(v2ia)o2] Fas p^. (39) 
This quadratic has at most two solutions unless all the co- 
efficients vanish identically. If the highest order coefficient 
does not vanish we have observed that the roots pj, uo of the 
quadratic always appear to be real and satisfy uj « —1 « 42. 
but have not been able to prove this. If this does indeed hold 
true, (18) shows that this will give one real and positive focal 
length and one imaginary focal length: the latter can be dis- 
carded immediately leaving the former as the unique solution. 
Thus more than two solutions will only occur if all coeffi- 
cients vanish. A necessary condition for this is that 01 = 02, 
i.e. that the matrix be an essential matrix. This by itself is not 
sufficient, however a complete analysis of exactly what con- 
figurations will achieve this is not attempted here: we simply 
note that they are likely to be rather special configurations 
that are already known to be degenerate, such as when there 
is no rotation (see [1, 2] for a more detailed analysis of these 
situations). 
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