Zisserman - 12 
a b c 
Figure 5: (a) Original image containing two SORs, two canal surfaces, and two poly- 
hedra. (b) The linked edges computed from (a). Profiles are extracted and grouped 
from these edges by the class-based groupers, (c) Extracted profiles superimposed 
on original image (with SOR axes shown). All the correct instances of a class have 
been grouped, and no false instances grouped. 
object pose or camera calibration. 
For example, the profile (image outline) of a surface of revolution (SOR) can 
be separated into two ‘sides’ by the projected symmetry axis. The two sides are 
tightly constrained — they are related by a particular four degree of freedom plane 
projective transformation — a planar harmonic homology [19]. This relationship is 
exact. The transformation is represented by a non-singular 3x3 matrix T, where 
T 2 = I [26]. Two pairs of point correspondences determine T. This transformation is 
fundamental to processing profiles of SORs: the line of fixed points of T is the imaged 
symmetry axis; T provides point to point correspondence between the sides of the 
profile; this disambiguates the matching of bitangents (used to form invariants); and 
finally, T can be used to repair missing profile portions, filling in gaps by transforming 
over points from the other side of the profile. 
This relationship is then employed in grouping image curves into those that 
could have arisen from an SOR. For example, if a scene contains several SORs, the 
grouper first identifies curve pairings that could have arisen from a SOR (that satisfy 
the harmonic homology constraint), and then groups those arising from the same 
SOR. Such identification and grouping is possible because the 2D image curve which 
results from imaging the 3D class is tightly constrained. In turn these constraints 
can be used during grouping to test the validity of the SOR assumption. An example 
of using grouping constraints of this type is shown in figure 5. Further details are 
given in [29]. 
For certain classes of surface (straight homogeneous generalised cylinders) a sin 
gle cross-section of the cylinder can be used as a grouping template which is swept 
tangent to putative profile curve portions. In this manner complete profiles of GCs 
(and thence their volumetric descriptions) can be recovered from fragments of the 
outline in the presence of occlusion, clutter and extraneous edges [28]. Again, the 
grouping is viewpoint invariant.