ISPRS Commission III, Vol.34, Part 3A „Photogrammetric Computer Vision‘, Graz, 2002 
  
rank(1,) — 0 will result if B-e, - 91, = Ya -e7-A' 
(— Os; = Os € ri, and 1, = 0). 
Since the correlation slices are always singular, there al- 
ways exists a nontrivial null space. The right null space 
of I, is a line 9, in image v» in general. It represents 
the line for which the correlation matrix I, yields no valid 
point (i.e. 0) in image vs. Geometric reason: The projec- 
tion plane due to pg, contains the principal ray ri;. Analo- 
gously, the left null space is a line À, in image 3 in general. 
It represents the line on which all mapped points I, -@5 lie. 
The lines 9; and À, are the epipolar lines of the principal 
ray Ti, in image V» and vs, respectively. Epipolar lines 
always pass through the respective epipole (V21 resp. Vai). 
Thus the matrices L = [A1, A2, As] and R = [P1, P, fa] are 
also of rank — 2 in general; c.f. [Papadopoulo, Faugeras 
1998]. 
If one considers the pencil of lines with a carrier 69 € f. 
then all of its lines (7^ 9;) map to the same point ps on 
Às; he. ps — p3(€2). Thus, I, represents a 1-dimensional 
collineation of the points C2 ON f, in image v» to the points 
ps on X, in image Vs.! If we consider particularly the 
pencil of lines with carrier V21, then we can prove that all 
of its lines 45, (X 5.) are mapped to Vai. 
f = [V21 le po with po lb V21 and p2 g f 
Eu 1. : Cos mew A ^M 3 AT 3 [Va1 ]-: P2 (13) 
ne 
scalar s 
(5) = 
s el AT. [-C d. R5. 0]. p2 
equ. (3 °F - * 
gol 09 e; Ci [02], - R2. C2: 5220 — pa d fio 
— Á—M— 
-0!—O»5€ri,  Z0—psoX92i 
So we see, that as long as Oz £ ri; (— rank(I,) = 2) any 
line Cos, "^ ps through the epipole V21 is mapped by any 
I; to Vai. Analogously, any line £j,, % Äx through the 
epipole v3; is mapped by any IT to V21 - provided O3 @ 
Tis. Table 3 summarizes these mapping properties (the 
others can be derived similarly to (13)) for I, depending 
on its rank. 
5.2 The minimal set of constraints 
The underlying geometric properties become clearer, when 
we consider the columns of these matrices: I, — [à, b, e], 
Is = [d, ë, f], Is = [&, b, i]. Since I. describes a mapping 
of lines £j in image V» to points pa in image v3 these 
columns can be interpreted as being points in image Vs - - 
the mappings of the v»-lines (1,0,0) ' , (0,1,0)", (0,0, 1)" 
- (again ^ emphasizes that qu B CENE are projective 
points in a specific scale). It always holds that rank(I,) < 
2. Thus we already have 3 constraints: 
Det(l,)=0 ze {1,23 (14) 
The geometric interpretation of (14) is, that the columns 
of I, interpreted as image points are collinear: {a, b,c) c 
Ar; {d, é,f} € À and {g,h, i} € As. These three lines 
{A1 Aa, da} are the left kernels of the correlation slices. 
Following section (5.1) we saw that these left kernels are 
epipolar lines, which always pass through the respective 
epipole (V31 in this case). This yields the next constraint. 
  
!If the carrier Ca @ p,,then I. describes a regular 1- dimensional 
correlation between the lines of this pencil and the points on À, 
The matrix L made of the three left kernels must be singu- 
lar; provided all three correlation slices I; have rank — 2 
- otherwise the kernel of I, will not be a line X 
Det(L) = 0 (15) 
These 4 constraints have already been presented in e.g. 
[Papadopoulo, Faugeras 1998]. 
The 4 remaining constraints are new and will be explained 
in the following. In section (5.1) we saw, that any line 
{si Ÿ Aa through V31 is mapped by IT to the epipole 
V21 in image v»; provided, rank(I,) — 2. So, we can 
formulate the following constraints: 
T TAG 
(1. zin: v3, — 0 P,a,r € {1,2,3}, 
16) 
(I, —pa-1})- 2e ; 
=0 pairwise different 
Obviously, the relations (16) produce 4 independent equa- 
tions (6 equations - 2 additional unknown scales (ji u2)). 
However, it needs to be proven that theses relations are 
also independent of the determinant constraints (14) and 
(15). 
If the determinant constraints are satisfied, then 
the correlation slices can be parameterized in 
the following way (without loss of generality): 
I, = [à,b, 
I; = [d,6,f] = [$2, v2: $2 -- k- Va, w2- $2 t t: Vaif17) 
e» 
= [8, $1 j- Vai, wi 81 s Y3i] 
1: = [3.5.1 = [$s , V3 - $3 +1 - V31 , W3 - $3 + U - Va1] 
This parameterization just means, that the columns of the 
three matrices I, are represented as linear combinations 
of à vector V31 common to all three matrices and indi- 
vidual vectors $,. The vector V31 (~ V31) is the common 
perpendicular of the three left kernels (Ai, Az, Az} - and 
its scale is chosen appropriately (hence * instead of^). If 
we choose any line /;,, through Y31, ^ to any left kernel, 
we get: 
jg M e 
I, Lig, = v1 8; og, e vi 
wi 8 «bys w1 
P St. $a 1 
I, ‘vy = V2 - S9 Cig, ^ V2 (18) 
waz - 82 V31 wa 
> 853 "Lin 1 
II “hy, = US $i “ls; ew v3 
W3 + S3 ZH w3 
Since all the right sides in (18) should be similar to the 
same vector (i.e. epipole V21), this only can be achieved, if 
for the coefficients holds: v4 — v? — v = v and w, = we = 
ws — w - and thus the constraints (16) are independent of 
the determinant constraints (14) and (15). 
Actually the constraints (16) correspond to the already 
known property (b) in Table 2 with a — (1,0, pi). resp. 
a = (0,1, 42). The constraints (16) hold for any line T 
S. through V31, therefore the components pa to e 
of the column vectors (Ig — ji - I5) resp. (I, — ua - I, ) 
are of no concern and only the components AA UNE to 
Vai need to be considered. Consequently we get a more 
preferable form for the constraints (16) by: 
Male Inv A [Var] Te + XF -[Va:], Is (19) 
A - 280