Zisserman - 7 
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A drawback with the two view approach is that the fundamental matrix defines 
only a one dimensional constraint on the correspondence of an image point. For each 
point it defines an epipolar line for which its correspondence must be coincident, if 
it is to be consistent with that motion. As a consequence any outlier (either due to 
mismatching or due to independent movement) that lies by chance near the epipo 
lar line will not be clustered correctly. This drawback is ameliorated considerably 
if further views are combined to determine the clustering. Clustering can then be 
based on the “tri-focal tensor” for three views, say, which applies to both points 
and lines. This tensor is discussed further in section 4. It is also important to prop 
erly incorporate and model degenerate cases where multiple fundamental matrices 
explain the data equally well [27]. 
2.3 3D Structure Recovery 
From point matches in two uncalibrated views and no other information it is only 
possible to recover 3D structure up to a projective transformation of 3-space. This 
can be seen relatively simply. Suppose x and x' are corresponding image points in 
the first and second (left and right) cameras respectively. The projection matrices, 
P and P', and 3D structure X, are consistent with these points if x = PX, x' = P'X. 
However, the matrices (PH -1 ) and (P'H -1 ) and structure (HX), with H a 4 x 4 non- 
singular matrix, are also a solution consistent with the point matches since 
x = PH _1 HX = PX 
x' = P , H” 1 HX = P'X 
Consequently, structure is only determined from the two views up to a projective 
transformation H of 3-space. It can be shown that this is the only ambiguity. 
To summarise: If X# = (X, Y,Z, 1) T is a homogeneous vector representing the 
actual Euclidean position (X, Y, Z) of a point (or the optical centre) then from a set 
of n > 7 point matches over two images (and no other information) the recovered 
position is X = HX# where H is a general non-singular 4x4 matrix. The same 
matrix applies to all points. 
Computational algorithm 
1. Compute the fundamental matrix F from x, <-> xj. 
2. Compute e' and M' from F, where M' = [e'JxF l . 
x The vector product v x x is represented as a matrix multiplication v x x = [v] x x, where [v] x 
denotes the matrix 
Mx 
0 —v z v y 
v z 0 — v x 
—v y v x 0 
[v] x is a 3 x 3 skew-symmetric matrix of rank 2.