Zisserman - 11 
i 
Figure 4: (a) Acquisition image for bracket. Algebraic plane projective invariants 
are measured from this image, (b) The recognised bracket is highlighted in white. 
model and features in the configuration). If the projected model edges overlap 
image edges to a sufficient extent then the hypothesis is verified. More details are 
given in [23]. Camera calibration is not required at any stage. An example is shown 
in figure 4 for an object (the bracket) which can be modelled by algebraic curves 
(namely 5 lines and a conic). It is recognised despite being partially occluded. 
3.3 3D Object Recognition 
For planar objects the original and image spaces are the same dimension and the 
projection matrix P reduces to a projective transformation represented by a 3 x 3 
matrix. For three-dimensional objects, the original and image spaces are no longer 
of the same dimension. Nevertheless, it is still possible to measure invariants of 
3D structure (i.e. invariants under a projective transformation of 3-space, for exam 
ple) from single image projections for certain classes of object. These classes are 
defined geometrically and include: polyhedra, surfaces of revolution, general cones, 
tube surfaces, repeated structures and algebraic surfaces. Further details are given 
in [29]. The invariants are used to identify models in a library, and thereby generate 
recognition hypotheses for particular objects in a similar manner to the planar case. 
For the remainder of this section we concentrate on grouping. For such model classes 
there are exact invariant relationships (transformations) on the object outline in any 
image. These relationships can be harnessed to group outline curves. 
In any object recognition system a major and primary task is to associate those 
image features, within an image of a complex scene, that arise from an individual 
object. The key idea here is that a geometric class defined in 3D induces relationships 
in the image which must hold between points on the image outline (the perspective 
projection of the object). The resulting image constraints enable both identification 
and grouping of image features belonging to objects of that class. The constraints 
and grouping methods are viewpoint invariant, and proceed with no information on