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used to compute the pose of the model in 3-space, and the projection of the entire 
model onto the image is generated. The support for this recognition hypothesis is 
assessed by measuring the overlap of the projected features with image features 
other than those of the original grouping. This approach is typical of many existing 
systems [2, 8, 14, 15]. 
As the size of the model library increases, this approach becomes computation 
ally too expensive, since if M models are in the library then recognition complexity 
is linear in M, each model being tried in turn. It is then more effective to choose 
potential models from the library based on the observed image features alone. That 
is, image feature measurements are used to index into the model base. In construct 
ing such index functions , invariance plays a major role, since a model should be 
identified irrespective of object pose. 
3.1 Geometric invariants for recognition 
Invariants are properties of geometric configurations which remain unchanged under 
an appropriate class of transformations. For example, properties such as intersection, 
collinearity, and tangency are unaffected by a projective transformation; however, 
invariant values can also be computed and these are of particular importance in 
forming index functions. Five coplanar lines, for example, have 2 invariants under 
planar projective transformations given by: 
h = | M431 || N521 | and h = | N<21 11” 532 ' (4) 
|N 4 21 1 |N 5 31 1 IN432I |N 5 21 1 
where N,-^ = (1,-,lj,U), |N,j^| is the determinant, and 1 = (/ 1 , / 2 , ^ 3 ) is the homoge 
neous representation of a line: l\X + l%y + /3 = 0. Numerous other invariants for 
planar and 3D structures are given in [20]. 
3.2 Planar Object Recognition 
The use of planar projective invariants for planar object recognition is particularly 
appropriate and straightforward because a projective transformation between object 
and image planes covers all the major imaging transformations: the plane to plane 
projectivity models the composed effects of 3D rigid rotation and translation of the 
world plane (exterior orientation), perspective projection to the image plane, and an 
affine transformation of the final image which covers the effects of camera internal 
parameters. 
The key idea is that projective invariants of the object have the same value 
when computed in any perspective image of the object. Recognition proceeds by 
grouping image features into configurations, and computing the invariants of the 
configurations. These invariants are used to index the model library. If an invariant 
value corresponds to a value in the library then a recognition hypothesis is generated 
for that object. The recognition hypothesis is verified by projecting the model 
outline onto the image (the projection is based on correspondences between the