Part I
Images as Multidimensional Signals
Part I provides the theoretical background for the rest of the book.
It introduces the concept of still images interpreted as twodimen
sional signals, as well as the generalization to multidimensional
interpretation of moving images and threedimensional (spatial)
image information. Once this general notion is introduced, the sig
nal theoretical concepts, after generalization to the twodimensional
or multidimensional case, can be utilized for image processing and
analysis. This concept proved very successful in enabling the for
malization (and consequently optimization) of many approaches to
image acquisition, processing, and analysis that were originally
designed as heuristic or even not feasible.
A characteristic example comes from the area of medical tomo
graphic imaging: the intuitively suggested heuristic algorithm of
image reconstruction from projections by backprojection turned out
to be very unsatisfactory, giving only a crude approximation of the
proper image, with very disturbing artifacts. Later, relatively com
plex theory (see Chapter 9) was developed that led to a formally
derived algorithm of
filtered
backprojection, widely used nowadays,
that is theoretically correct and provides very good images, even
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under practical limitations. Both algorithms are quite similar, with
the only difference being the filtering of each individual projection
added to the original procedure in the later method — seemingly an
elementary step, but probably impossible to discover without the
involved theory. The alternative methods of image reconstruction
from projections rely heavily on other aspects of the multidimen
sional signal theory as well.
Part I introduces the basic image processing concepts and ter
minology needed to understand further sections of the book. Broader
and deeper treatment of the theory can be found in the numerous
literature that is partly listed in the references to this section,
e.g., in [4], [5], [6], [18], [22], [23], [25], [26]. Other sources used but
not cited elsewhere are [1], [2], [8], [12], [14]–[17], [19], [21], [24].
In context of the theoretical principles, we shall introduce the
concepts of twodimensional systems and operators, twodimensional
transforms, and twodimensional stochastic fields. The text is con
ceived to be selfcontained: the necessary concepts of the onedimen
sional signal theory will be brieﬂy included, however, without detailed
derivations. A prior knowledge of the signal theory elements, though
definitely advantageous, is thus not necessary. With respect to the
purpose of the book, we shall mostly limit ourselves to the twodimen
sional case; the generalization to three and fourdimensional cases is
rather straightforward and will be mentioned where necessary.
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 Fall '07
 Singh
 Francis Group, Taylor & Francis Group

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