cr et 94 U CN 
= ET 
bid 
  
ISPRS Commission III, Vol.34, Part 3A ,Photogrammetric Computer Vision“, Graz, 2002 
  
  
  
  
  
  
  
  
  
  
  
  
  
Property Remark 
| (a) | Vai c Ja: 3s | provided rank(J,) = 3 a 
general eigenvalue problem (Jy — pu - J,)-X=0, 1,y,z € {1,2,3}, pairwise diff. 
el + 
(b) gen. eigenvalue pu; = rr gen. eigenvector Xi ~ Vis 
el. = 
(c) | gen. eigenvalue u2 = us = Em gen. eigenvector X23 y A -V12 4- 8. À * o; 
  
  
  
Table 1: Properties of the homographic slices J. 
  
  
  
  
  
  
  
  
  
Property Remark 
(a) rank(1.) «2 
(b). | rünk(Y 3. Ge las 
(c) R'.9$94 =0 | using R = [P1, D, Ps] with I, - A = O 
(d) L'-Ÿ% =0 | using L = [A1, Az, As] with I] X.=0 
  
  
  
Table 2: Properties of the correlation slices I,; c.f. [Papadopoulo, Faugeras 1998] 
image-coordinates in six homologous triples, convenient 18 
coordinates are kept fixed. In this way a minimal pa- 
rameterization of the tensor is achieved. The unknowns 
themselves are obtained as (up to 3) solutions of a cu- 
bic equation. Due to this fixing of erroneous observations 
in the images one might be suspicious that errors in the 
calculated tensor may be induced, furthermore no correct 
minimization of the measurement-errors in all observations 
is possible. And as it is shown by the results in [Torr, Zis- 
serman 1997] the standard deviation depends on the choice 
of the 6 points resp. the fixed 18 coordinates, which is not 
obvious in the beginning. However, this method of keep- 
ing the proper number of image-coordinates fixed, could 
be helpful also for other tasks, where a minimal parame- 
terization is needed, but cannot be formulated easily. 
[Papadopoulo, Faugeras 1998] introduce a minimal param- 
eterization together with a set of 12 sufficient constraints 
- not minimal, since any number of constraints greater 
than eight must contain dependencies. Their set of con- 
straints are entirely based on the correlation slices I; and 
are made of the properties (b), (c), (d) shown in Table 2. 
Their minimal parameterization looks like the following: 
'The left kernels of the correlation slices are parameter- 
ized using 2 parameters for their common epipole v3; and 
1 parameter (a direction angle) for each kernel - thus 5 
parameters in total. With other 5 parameters the right 
kernels and epipole V21 are parameterized. With the left 
and right kernels the correlation slices I; can be param- 
eterized by 8 coefficients. This way of parameterization 
results in a very large number of maps (9-3*-3°) and it is 
not clear how this parameterization is applicable in case 
of rank(I,) < 2 - because the kernels need to be lines. 
In [Canterakis 2000] the first set of minimal constraints is 
presented, which are entirely based on the homographic 
slices J, and are derived from the properties shown in 
Table 1, i.e. each general eigenvalue problem set up with 
two homographic slices has one general eigenvalue with 
multiplicity 2 (u2 = ua) (— 1 constr.), the correspond- 
ing general eigenvector is 2-dimensional (— 2 constr.), the 
general eigenvector X1 corresponding to the single general 
eigenvalue ji is the same (up to scale) for all three pairs of 
J. (— 2 constr.). This general eigenvalue problem can be 
independently set up twice yielding the required number of 
8 constraints. Open questions with this set of constraints 
are, how are they applicable in case of rank(Jz) < 3 and 
how to implement them efficiently in a computer program 
(e.g. constraint (Xi(of pair (x,y)) ^ Xi(of pair (x,z)) re- 
quires this general eigenvector to be expressed in terms of 
the 27 tensor elements). 
In the following sections a new set of minimal constraints 
together with a minimal parameterization will be pre- 
sented. Both are derived very easily, having very sim- 
ple geometric properties. Their implementation is rather 
simple (actually the minimal parameterization is easier to 
realize than the constrained version). 
5 A NEW MINIMAL SET OF 
CONSTRAINTS 
The basic input for this set of constraints are the correla- 
tion slices I,, therefore we will take a closer look at these 
matrices. 
5.1 The correlation slices I, - Revisited 
The correlation slices in equation (12) describe a mapping 
of lines /; in image V» to points P3 in image vs via the 
principal ray ri;, meaning that pa is the projection of the 
intersection point of ri; with the projection plane of la. 
In general, rank(I,) = 2, since the columns of I, are 
linear combinations of two vectors (B-e, and Ÿ31) - or 
the rows are linear combination of two vectors (A-e, and 
Ÿ21). For the same reason, any linear combination of the 
correlation slices S az + 1, will also have rank = 2 in 
general; c.f. [Papadopoulo, Faugeras 1998]. 
Using equ. ((5) - (9), (12)), we can find the cases where 
rank(I.) «2: 
rank(I,) — 1 will result if B. e; — $31 (—^ Os € ris 
and I, — Vai: Vj3) or if (Ae; — $21) (— Os € ri, and 
I. — 932-921). 
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