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Prior terms can often be related to
the assumption of
approximate invariances
or approximate symmetries.
A Laplacian smoothness functional, for example,
measures the deviation from translational symmetry
under infinitesimal translations.
Consider for example a linear mapping
|
(208) |
given by the operator .
To compare with
we define a (semi-)distance defined by choosing a
positive (semi-)definite ,
and use as error measure
|
(209) |
Here
|
(210) |
is positive semi-definite
if is.
Conversely, every positive semi-definite K
can be written
=
and is thus of form (210) with
=
and
.
Including terms of the form of (210)
in the error functional forces
to be similar to .
A special case are mappings
leaving the norm invariant
|
(211) |
For real and
i.e.,
=
,
this requires
and
.
Thus, in that case
has to be an orthogonal matrix
and can be written
|
(212) |
with antisymmetric
and real parameters .
Selecting a set of
(generators)
the matrices obtained be varying the parameters
form a Lie group.
Up to first order the expansion of the exponential function reads
.
Thus, we can define an error measure with respect to an
infinitesimal transformation by
|
(213) |
Next: Approximate symmetries
Up: Covariances and invariances
Previous: Covariances and invariances
  Contents
Joerg_Lemm
2001-01-21