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3. Compute 3D points X, by intersecting backprojected rays using 
P = [I| 0] P' = [M'| e'] 
where I is the 3x3 identity. 
Projective structure can be utilised directly for certain visual tasks. For example, it 
is possible to distinguish points in 3-space on either side of any plane (Binary Space 
Partition) [22]. 
Much of the current research is aimed at reducing this projective ambiguity to 
affine or Euclidean (metric) so that H is an affine or Euclidean matrix. Of course, 
structure can be “upgraded” to affine by identifying parallel lines/planes or sup 
plying ratios of lengths of collinear segments or areas. Similarly, structure can be 
promoted to Euclidean by identifying perpendicular lines, for example, or supplying 
length ratios. However, in computer vision the aim is to achieve this upgrading by 
constraints available on the motion or camera without resorting to scene specific 
knowledge. 
2.3.1 Affine and Metric Structure 
In order to upgrade 3D projective geometry to metric certain geometric objects 
have to be identified, namely the plane at infinity 7(affine geometry) and the 
absolute conic Doo (Euclidean geometry) [24]. The image counterparts of these 
entities for a pair of views are the infinite homography Hqo (for affine reconstruction), 
and the dual of the image of the absolute conic in each view, K and K' (for metric 
reconstruction). The infinite homography is the 3x3 point transformation matrix 
for vanishing points, i.e. v' = HqoV, where v and v' are corresponding vanishing 
points. Having the dual (i.e. the inverse) of the image of the absolute conic K, is 
equivalent to knowing the internal parameters of the cameras, i.e. the 3x3 upper 
triangular camera calibration matrix C, since K = CC T . Once Hqo is known, an affine 
reconstruction can proceed from the projection matrices P = [l|0], P' = [Hoo|e']. 
Also, the infinite homography between two views constrains the camera calibration 
matrix [10, 16] by: 
K' = H„KH^ (3) 
Affine structure can be utilised directly for certain visual tasks. Affine constructions 
such as centroids and mid-points have been used for control of fixation by track 
ing object centroids [21], and obstacle avoidance by servoing on mid-points of free 
space [3]. 
A number of methods have been developed for obtaining the Hqo and K image 
entities: 
1. Affine structure can be recovered from two uncalibrated images with pure 
(but unknown) translation between the views and unchanging (but unknown) 
internal parameters (i.e. K' = K) [18]. In this case Hqo = I. Subsequent rotated 
views can then be used to generate linear constraints on the internal camera 
parameters via equation (3), K = HoqKH^ [1].