IEEE Geoscience and Remote Sensing Magazine - September 2021 - 47

:, 3 its upper-level sets are defined as Xt$ "
with t Z! (respectively, lower-level sets as " Xt ),# , i.e., the
XE Z "
E Z2
,
,
set of images obtained by thresholding an image at all possible
values of its pixels. The CCs (
CC )E3 composing the
upper- or lower-level sets are referred to as peak components.
These two tree types (which are dual with respect to complementation)
model the inclusion relations between these
peak components; thus, max-trees are excellent for modeling
regions that are brighter than their surroundings and
min-trees, for regions that are darker. That is why, in addition,
we employ both attribute thinnings (i.e., max-tree filtering)
and thickenings (i.e., min-tree filtering) during the
construction of APs [see (1)].
As the construction of two trees per image is both memorywise
and computationally inefficient and results in longer
feature vectors per image, thus also affecting classification
performance and complexity, self-dual APs (SDAPs)
were introduced to target these issues [53]. More specifically,
they rely on the use of the ToS [54], which has been
designed to provide a unified representation for both bright
and dark image structures.
The ToS is constructed by filling the holes of the aforementioned
peak components, and the shapes represented by
the nodes do not intersect and are either disjoint or nested.
Consequently, the ToS is an inclusion hierarchy that unlike
the component tree, is also contrast invariant and self-dual.
SDAPs have been empirically shown to outperform APs consistently
in terms of classification performance while also
producing shorter feature vectors per pixel [25], [53], [55].
Motivated by the success of the ToS and by the useful
qualities of partitioning trees in the aforementioned context,
a - and
~ -trees were recently applied to AP implementation
[56]. In particular, the -a tree is constructed based on the local
range, where every tree node corresponds to an -a CC [57]. For
instance, for
a20 , an -a CC is defined as the CC of the maximal
size such that only the neighboring pixels with gray-level
difference less than or equal to a are considered connected.
Although -a trees can lead efficiently to a complete and
self-dual image representation, due to the locality of the
metric used, gray-level variations within
a -CCs can be much
higher than a (i.e., the chaining effect [57]). This undesirable
effect can be mitigated through the use of
~ -trees [58],
which constitute a subset of a -trees, constraining every
a -CC with their global range (i.e., the maximal dissimilarity
between any two pixels belonging to that component).
The hierarchy remains self-dual, complete, and capable of
capturing regions of low, intermediate, and high gray levels,
but global range provides better grouping per level than
just a local measure.
Lefèvre et al. [59] have focused on the similarity metric
requirement of partitioning trees for extension to multiband
images and used metric learning to adapt them to
hyperspectral images. Bosilj et al. [56] and Koç et al. [60],
on the other hand, have empirically tested APs, SDAPs, and
a - and
tree-based profiles outperformed their inclusion tree counterparts
in terms of noise robustness.
ATTRIBUTE AND THRESHOLD SELECTION
Undoubtedly, APs possess many desirable practical and
theoretical properties that render them particularly suitable
for the task of spatial-spectral description in our context.
However, they are not without flaws, and the main
source of criticism so far has been in terms of their sensitivity
to parameter selection [61]-[63]; by parameters, we
mean particularly the attributes employed to characterize
every tree node and, most importantly, the set of associated
threshold values. The attribute- and threshold-selection parameters
are discussed in the " Attributes " and " Thresholds "
sections, respectively.
ATTRIBUTES
From a theoretical point of view, any function : () ,
aEP R "
where ()P E denotes the power set of E, computable on an
arbitrary collection of pixels, can, in fact, be employed as
an attribute for AP construction. In practice, it is used during
filtering by comparing a given CC's
)
(CC E3 attribute
value against a predetermined threshold in the form of a binary
predicate (e.g., in case of area, " is the CC's area greater
than 300 pixels? " ). It thus provides a great degree of freedom
as far as the object-based analysis of an image is concerned.
The pioneering article of APs [7] introduced four
such attributes: area, moment of inertia, diagonal length of
the bounding box, and standard deviation; the first three
describe a geometric property related to the shape of the
tree node under study, and the last provides its statistical
pixel intensity distribution.
Although these four attributes are, by far, the most widely
encountered in the state of the art, APs can accommodate
(from a theoretical point of view) a vast pool of attributes.
Examples include entropy and homogeneity [7] as well as
the diameter of the equivalent circle and area of the convex
hull for automatic threshold selection [64]; complexity (perimeter
over area) [65]; perimeter and area of the bounding
box used to evaluate threshold-free (TF) APs [66]; solidity
(area over area of the convex hull) and orientation (between
the major axis of the convex hull and x-axis) [67]; coefficient
of variation (CoV) and normalized radar cross section
(NRCS) tailored for SAR images [33], [39], where CoV
is the ratio of the standard deviation divided by the mean
value of pixel intensities and NRCS, expressed in decibels,
is the RCS per unit area of the surface. Furthermore, in [68],
it has been observed that, when dealing with multiband
input, one can extend the pool of attribute measures to include
multidimensional functions exploiting all available
bands simultaneously, and two new attributes have been
proposed: higher-dimensional spread and dispersion.
As far as the selection of attributes is concerned, there is
~ -APs against each other and established the superiority
of SDAPs in noise-free conditions, while partitioning
SEPTEMBER 2021 IEEE GEOSCIENCE AND REMOTE SENSING MAGAZINE
no straightforward rule or limitation. Formally, one should
use the attributes that are " meaningful " for the data under
consideration or, in other words, the attributes with respect
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IEEE Geoscience and Remote Sensing Magazine - September 2021

Table of Contents for the Digital Edition of IEEE Geoscience and Remote Sensing Magazine - September 2021
