ces 
we introduce 
affine model, 
ich in turn is 
osition of the 
LT algorithm 
hat only four 
‘orithm, tests 
pose is to find 
n without in- 
e information 
nt on known 
) WORK 
try and com- 
areas should 
to rely much 
cient number 
esides paying 
; in computer 
em ” motion 
uncalibrated 
teratures are 
t without in- 
nstructed up 
ansformation 
et al, 1992). 
nce for object 
n ( Faugeras, 
on may date 
of Longuet- 
However, the 
basic idea behind this solution was essentially origi- 
nated from the early work of Thompson (1968), where 
he expressed the coplanarity equation via an unknown 
3 x 3 matrix which is acknowledged today as essen- 
tial matriz (c.f., Longuet-Higgins, 1981; Huang et al, 
1989; Faugeras, et al, 1990; Hartley, 1992; Hartley et 
alj1993). Photogrammetrists did not recall Thomp- 
son's idea for quite a time. Recently, Brandstatter 
(1992) employed this idea for image rectification. 
Wang’s work (1995) threw a light on this idea upon 
which a linear algorithm was designed to reconstruct 
the photogrammetric model based on the interior ori- 
entation. Most recently, the stability of this algorithm 
was studied by Barakat et al (1994), Deriche et al 
(1994), Forstner (1995) and Luong et al (1994). 
Our wotk is highly inspired by the work of Hartley et 
al, Faugeras et al and Wang. Section 3 starts from 
an affine transformation in the image plane, and then 
generalizes the Thompson and Longuet-Higgins equa- 
tion to the case of unknown interior elements. Section 
4 is focused on the affine model and the recovery of 
its components. Unlike Faugeras’s work (Faugeras, 
1992) where traditional projective geometry is uti- 
lized, we fully take advantage of the properties of the 
skew-symmetric matrix and make our development as 
parallel as possible to photogrammetry. After defin- 
ing the affine model analogous to the traditional one, 
we show some of its components can be withdrawn 
from the so-called fundamental matriz. This leads to 
a complete employment of a stereo. In section 5 we 
use a 3D affine transformation to fully recover the 
object. Unlike the well-known DLT algorithm where 
minimum six known points are required on each image 
of a stereo, our algorithm allows that one image may 
have only four of them. Tests with an aerial stereo 
in section 6 show that our algorithm is robust both 
to the configuration of known points and to the affine 
image deformation. Fully compatible (or even slightly 
better) results with the DLT algorithm are obtained 
as well. 
3. THOMPSON AND LONGUET-HIGGINS 
EQUATION 
In this section we derive the Thompson and Longuet- 
Higgins equation which plays a fundamental role in 
our problem. A short comment is thereafter made on 
the fundamental matrix. 
We write the well-known coplanarity equation as 
(Slama,1980, pp.54-56) 
x! [b * (Rx2)] =0 (1) 
In Eq.(1) 
b={8; B B; (2) 
International Archives of Photogrammetry and Remote Sensing. Vol. XXXI, Part B3. Vienna 1996 
     
  
    
   
    
  
  
  
  
    
   
    
    
    
   
   
    
    
    
   
    
     
     
  
     
    
   
     
      
   
     
     
    
    
    
     
  
   
   
   
   
  
   
    
is the base component vector. R is the orthogonal ro- 
tation matrix of the second image relative to the first 
one which is assumed to be as a reference. And 
xim of )" X2—(2z2 y» —fo y 
(3) 
are coordinates of conjugate image points pi, p» in the 
first and second image spaces respectively. In Eq.(1) 
* denotes the scalar product of two vectors. 
For any 3 x 1 vector x we have 
b*xxzBx (4) 
where B is a 3 x 3 skew-symmetric matrix whose en- 
iries are composed of the elements of b, i.e., 
0 —Bz By 
B = Bz 0 —Bx (5) 
—By Bx 0 
Applying Eq.(4) to Eq.(1) yields 
x! Ex = 0 (6) 
where 
E=BR (7) 
Eq.(6) is namely the Thompson and Longuet- Higgins 
equation which was initially derived by Thompson 
(1968) and rediscovered by Longuet-Higgins(1981). 
Matrix E, which is the product of the base component 
matrix B and the orthogonal rotation matriz R, is 
named as essential matriz by Longuet-Higgins(1981) 
and thereafter widely accepted and studied in com- 
puter vision (Huang,et al,1989; Faugeras et al ,1990). 
It is straightforward to generalize Eq.(6) to the case 
when the interior orientation is not done. Suppose 
image points are measured in an oblique image coor- 
dinate system (Z,ÿ) which is generally considered as 
a linear or an affine transformation of (z, y), namely 
we have 
x; = A1X1 X9 = AÀ»X»9 (8) 
where 
aj; 012 ais 011 312 13 
A1 = | aa 422. 23 As =| a2 az ass 
(9) 
FE noi TU "8 n T 
xin ni) =(2 5 l1) (10) 
xi and X» are essentially homogeneous coordinates of 
image points. Substituting Eq.(8) back to Eq.(6) we 
obtain 
XT Ex» =0 (11) 
where 
E = ATEA, = ATBRA, (12)