An algorithm for associating the features of two images
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Strang G. 1988 Linear algebra and its applications 3rd edn Appendix A. San Diego: Harcourt Brace Jovanovich.
maximizes P:Gisnow the problem of finding the 1:1 Ullman S. 1979 The interpretation of visual motion ch. 2 and
mapping from is to js that minimizes the sum of the 3. Cambridge Massachusetts: M .I.T. Press.
squared distances r^between the members of each pair. Wertheimer M. 1912 Experimentelle Studien uber das
But if the positions of the circles and the crosses are
indeed connected by an affine transformation with a
positive definite tensor component then the mapping
between them does indeed minimize the sum of the
squares of the distances. So at last we can see why first
of all the P matrix delivered by our algorithm
approximates to a permutation matrix; and secondly
why the permutation represented by the matrix
faithfully recovers the 1:1 mapping originally induced
by the affine transformation.
D IS C U S S IO N In its classical formulation the correspondence
problem has to do with human vision and the
parameters that describe motion perception. But it is
also a problem for computer vision engineers and this
is how we have chosen to address it. In the cir
cumstances it is reassuring to find that our own
solution to the problem has a certain similarity to
Ullman's minimal mapping scheme a theory that
accounts for so many observations in the area of
motion perception. There are however important
differences. Like us Ullman aims to extremize the inner product
of two matrices: one having to do with `affinities I -or
`costs ' -and the other a matrix of correspondences. He
has much to say about such measures though he does
not explicitly consider the Gaussian proximity matrix
devoting much of his discussion to a cost measure that
is linear in the separation. This leads him to minimize
where possible the sum of the first powers of the
distances between corresponding points rather than
the sum of their squares which we find to be Sehen von Bewegung. Z. Psychol. 61 161-265.