To introduce the generic form of our embedding
metric, and establish its intuitive validity, let us first
consider the following process. Assume that the reference
is an image on a transparent rubber sheet. We
move this sheet over the sensed image and, at each possible
placement, we pull or push on the rubber sheet to
get the best possible alignment between the reference
image on the sheet and the underlying sensed image.
We evaluate each such embedding both by how good a
correspondence we were able to obtain and by how much
pushing and pulling we had to exert to obtain it.

Let us now consider a discrete version of the above
process which is both more precise and more reasonable
from an implementation standpoint. In a specific application,
we might have some information on the range
of permissible distortions that can occur between the
reference and sensed images. For instance, some subset
of the items appearing in the sensed image might always
retain their internal shape even though their relative
positions might be subject to change with respect to
their locations in the reference scene. Further, where
change of relative position is possible, we might be able
to bound the extent of such change; and, finally, we
might like to assign variable "costs" to the different
types of change of relative position or relative change in
some nongeometric attribute.

To achieve these capabilities, we replace the rubber
sheet by a reference image which is composed of a number
of rigid pieces (components) held together by
"springs." A rigid piece of the reference image can be as
small as a single resolution cell, or as large as the entire
reference image, and corresponds to a single coherent
entity in the reference image. The springs joining the
rigid pieces serve both to constrain relative movement
and to measure the "cost" of the movement by how
much they are "stretched." (Typically, the springs will
be highly nonlinear in their behavior.) In determining
the cost of an embedding, we measure the "tension" on
each spring (the tension can be a function of direction
as well as stretch or even a relative change in some
locally defined attribute), and also make a local evaluation
of how well each coherent piece is embedded as an
independent entity.