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Overlapping images at
the scale differences are
scales, in which case the
ying principles of scale
xamining the behavior of
(corresponding to low-
icluded. When matching
stablish correspondences
ail or perform poorly in
s are also of radiometric
ong conjugate features,
sely matching conjugate
through scale space, and
occurrence is dependent
ect shape combinations.
pected to receive much
re, as it is inherently
imagery (e.g. MOMS)
available [Schneider &
loves towards the fusion
ery for geoinformation
he integration of digital
tion systems [Agouris et
ry of various scales is
f complex digital image
oblems occurring when
ures whose images differ
employs scale space
accommodation of scale
THEORY
is encoded in its values
ons occur over a wide
enna 1996
range of spatial extents, with macro-variations expressing
major signal trends, and micro-variations expressing highly
localized trends, manifesting themselves within spatially
limited areas. The visual perception and distinction of
macro- and micro-variations in images is an intricate human
cognitive process, involving perception, reasoning and
often intuition. As such, this task is fundamentally complex
to be algorithmically duplicated and functionally mimicked
by machine-supported operations.
The concept of examining the behavior of signals in
multiple scales can be traced back to the seventies with
research in hierarchical information structures [e.g.
Tanimoto & Pavlidis, 1975]. However, scale space theory
has been formally introduced and developed in the signal
processing community only during the previous decade, with
the papers of Witkin credited as introducing the concept
[Witkin, 1983; Witkin 1986]. It deals with the
identification and classification of trends encoded in the
values of signals by analyzing the behavior of those signals
in various resolutions. The scale space of an m-dimensional
signal defined in the space spanned by (Xp X5 > Xp )iS the
(m+1)-dimensional space (x, X5... s)if and only if the
Xn
additional parameter s expresses the resolution of the signal.
Digital images are two-dimensional discrete intensity
functions defined in the (x,y) space, and therefore their scale
space is the three-dimensional (x,y,s) space. A discrete
representation of the continuous in s scale space of a signal
f(x,y), comprising a set of 27 derivative signals
(f G,y,s,)) representing the original one in various
resolutions (termed scale levels), corresponding to n distinct
values (59,5;,...5, .,) of the scale parameter s, is an n-order
scale space family of the original signal. Figure ] shows a
scale space family and demonstrates how the original signal
is decomposed at coarser scale levels.
Bi M uc d AT
nr. Ne B
SSI AY gees
Sine | nA NCC pets
oe oe Pn TN rT
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som AS i
Fig. 1: A scale space family of a signal. The original
signal is at the bottom and resolution decreases upwards.
For different scale parameter values, different scale space
families of an original signal can be generated. This
generation is performed through the numerical manipulation
of the original signal. The aim of generating scale space
families of signals is to provide representations in which
the information content of a signal changes in a systematic
and therefore exploitable manner. In order for this goal to be
met, scale space generation has to follow certain rules
[Lindeberg, 1990; Lindeberg 1994]:
e The scale space is generated through the convolution of
the original signal with a single scale-generating
function (or its discrete kernel) k(x,y, s)
FG ys.) 8 KG ys) * f(x, y) Eq. 1
e The scale generating function has to be selected in such
manner that, through its application, signal resolution
will change monotonically for respective changes of
the scale parameter s.
Both rules aim at the optimization of the interpetation
potential of the generated scale space: the use of more than
one scale-generating function (e.g. different functions for
different scale parameter ranges) would make practically
impossible the comparison of different scale space versions
of a signal. The non-monotonic change of resolution would
have similar implications.
Scale generating functions have to possess certain
properties, in order to satisfy the above rules [Burt, 1981;
Babaud et al., 1986; Meer et al., 1987], among which the
most important are:
e symmetry, in order for direction independance to be
satisfied,
e normalization, for ensuring the (essential in terms of
data handling and processing) compatibility in value
range of the multiresolution versions of a signal,
e unimodality, to avoid semantic distortions due to the
disproportionate participation of distant information
during scale space generation, and
e separability, for the alleviation of the computational
requirements associated with scale space generation and
manipulation.
Considering two-dimensionality, as is the case for digital
imagery, the separability property of a scale generating
kernel k(x,y) allows its decomposition into two one-
dimensional signals
k(x,y) = [ky (01 ky (9) Eq. 2
and thus permits the use of different scale values in x and y,
effectively allowing us to consider the scale space of images
as a four-dimensional one. Actually, even for m-dimensional
signals we could, in the same manner, consider the scale
space as a 2m-dimensional space. Scale space generation
applied on digital imagery leads to the generation of digital
image pyramids [Burt, 1984; Meer et al., 1987].
Arguably, the most important operation associated with
scale space is to link the information of all scale space
members together. This is achieved through feature tracing,
which can be defined as the problem of identifying global
features out of local signal properties, and of tracing the
position and behavior of these features through various
levels of the signal’s scale space. Features are typically
identified at the coarser signal levels, where overlaying high
International Archives of Photogrammetry and Remote Sensing. Vol. XXXI, Part B3. Vienna 1996
