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Camillo Ressl 
coordinate-system’s axes: Op = 0, b = O3, R, = E3,3 (the so-called ‘relative orientation by successive images’). The 
matrix F,2 represents an alternative formulization of the relative orientation of two images in a linear way. In the 
computer vision community F, is termed ‘essential matrix (e.g. [Longuet-Higgins 1981]) in the case of calibrated 
images and "fundamental matrix’ in the case of uncalibrated images (e.g. [Luong, Faugeras 1996]) whereas in the 
photogrammetric community Fy, is always termed ‘correlation matrix’ (e.g. [Brandstitter 1991], [Niini 1994]). 
The latter name is derived from F;;'s property that p, resp. p» is mapped by F;; to the corresponding epipolar-line in the 
2 image (Fi pi) resp. 1* image (F^, pj) on which the homologous partner p» resp. p, is constrained to lie (called 
‘epipolar constraint’). So by means of Fy; not only the relative orientation of 2 images is described in a linear way, but 
it also allows a compact representation of the epipolar geometry inherent in two images. As a consequence, the epipoles 
and epipolar-lines can easily be computed. The epipoles v4; and v», are the left and right kernels of F5; i.e. Fj4:v,;- 0 
and F"12-Vi = 0. Fy is also used for the so-called ‘epipolar transfer’, i.e. given 3 images and the fundamental matrices 
Fi, Fi, F5 and the homologous image-points in 2 images, the homologous image-point in the 3" image can be 
computed by means of intersecting epipolar lines. But it should be noted that this epipolar transfer fails if the 
corresponding object-point and the three projection-centers lie in one common plane. All these properties may be the 
reason why Fy, is also termed ‘essential’ resp. ‘fundamental’. 
2.3 The homography induced by a plane 
A 'homography' (according to [Shashua, Werman 1995] ‘a projective transformation of planes) is a mapping 
(collineation) of points from one image y, to the points of another image V». By means of a 3x3 matrix Hj», the 
homography can be expressed in the following way: 
P.~H,,,p, (2.4) 
The points p, and p» which are related in this way by Hj», are the image-points of a point in 3d-space. The 3d-space- 
points of all pairs of image-points which satisfy (2.4) lie in one common plane - the homography-plane o. One may 
say: "Hizs is a homography from image y, to image y» induced by the plane 6". Homographies can be used e.g. for the 
detection of obstacles sited on a plane on which a robot is moving. If the XOR and IOR of the two images V; and y» are 
given in the same way as in section 2.1 and if the plane 6 is given by n, -P = d; with n, being the plane's normal- 
vector, then H;5, has the following structure: 
H,,, =C, RIE, (0, -0,)a! |, € (2.5) 
1 
d-n!-O, 
The homography H;,; due to the same plane 6 but from image > to image V, looks the same, except for exchanging 
the indices 1 and 2. Generally rank(H;,,) will be 3. Depending on rank(H;»;) the following relations between H;;, and 
the epipoles v,? and v5, hold; [Ressl 1997]: 
  
  
| rank(H,,,)=3: rank(H,,,)-2: rank(H,,, )-1: 
V4 7Hy, vy 0,0 (d-nT-0,)=0 O,ec e (d-n7-0,)-0 
0=H,,,V Vay VY Dyes 
12,6 V 12 nn a SS 
H,,:p,€ 5: Vp, Vy, : Vp,e 5 
with s, nT-R,-C,-p,-0 (alineiny,) | "I^ n; R,-Cy-p,-0 (aline iny,) 
  
  
  
  
Finally there is also a relation between H;;, (rank »1)and the fundamental-matrix Fy, of the two images: 
F, [Yo JA (2.6) 
24 A few basics of tensor calculus 
À tensor is an indexed system of numbers. There are two kinds of indices: sub-indices are called ‘co-variant’ and super- 
1 1 ‘ . , . . . + : 
Indices ‘contra-variant’. A tensor with contra-variant valence p and co-variant valence q has n"? components with n 
being the dimension of the underlying vector-space; i.e. each index runs from 1 to ri. Using these indices and Einstein's 
  
International Archives of Photogrammetry and Remote Sensing. Vol. XXXIII, Part B3. Amsterdam 2000. 771