  We introduce a uniform representation of general objects that captures the
regularities with respect to their structure. It allows a representation of a
general class of objects including geometric patterns and images in a sparse,
modular, hierarchical, and recursive manner. The representation can exploit any
computable regularity in objects to compactly describe them, while also being
capable of representing random objects as raw data. A set of rules uniformly
dictates the interpretation of the representation into raw signal, which makes
it possible to ask what pattern a given raw signal contains. Also, it allows
simple separation of the information that we wish to ignore from that which we
measure, by using a set of maps to delineate the a priori parts of the objects,
leaving only the information in the structure.
  Using the representation, we introduce a measure of information in general
objects relative to structures defined by the set of maps. We point out that
the common prescription of encoding objects by strings to use Kolmogorov
complexity is meaningless when, as often is the case, the encoding is not
specified in any way other than that it exists. Noting this, we define the
measure directly in terms of the structures of the spaces in which the objects
reside. As a result, the measure is defined relative to a set of maps that
characterize the structures. It turns out that the measure is equivalent to
Kolmogorov complexity when it is defined relative to the maps characterizing
the structure of natural numbers. Thus, the formulation gives the larger class
of objects a meaningful measure of information that generalizes Kolmogorov
complexity.
