IEEE Geoscience and Remote Sensing Magazine - September 2021 - 46

hyperspectral) became an intensive research topic. The
main idea in this regard has been to reduce the various
and often correlated image bands into fewer components
through some dimension-reduction tool and extract APs
independently from each of them, followed by their subsequent
merging.
The initial extension of APs to hyperspectral images was
proposed in [19], namely, extended AP (EAP), which consists
of employing principal component analysis (PCA) to
this end-an unsupervised yet often suboptimal tool due
to its linear nature. Alternatives to PCA that have been additionally
studied include independent component analysis
(ICA) [20], [21], kernel PCA (KPCA) [22], feature space
discriminant analysis (FSDA) [23], and other supervised
methods, such as discriminant analysis feature extraction
(DAFE) [24], nonparametric weighted feature extraction
(NWFE) [25], and the sparse Hilbert-Schmidt independence
criterion and surrogate kernel [26], among others.
Given that an independent AP calculation from each
band (or each image component, assuming dimension reduction
has been applied to the image bands) ignores any
and all correlational information among them, [27] proposed
a vector alternative calculating APs collectively and
simultaneously from all available bands/components. The
core idea relies on representing all bands through a single
hierarchical representation, leading to vector APs.
It would be also possible to compute APs on derived features
of optical remote sensing data, such as edge/contour
information obtained by Sobel gradient filtering of panchromatic
images [10] or the normalized difference vegetation
index from multispectral images [28] for urban and
crop field classification. Then, to deal with VHR optical
images, where highly textural information becomes significant,
another adaptation of APs was proposed in [29]. The
raw input data are replaced by the textural features, thus
considerably improving the classification performance on
the tested textured images.
Furthermore, although APs are usually explored for the
supervised classification of optical images (either panchromatic,
multispectral, or hyperspectral), some studies have
investigated them within unsupervised scenarios. For example,
in [30], the authors exploited the differential APs
(i.e., which compute the difference between successive APs
to form differential profiles) for unsupervised anomaly detection
in hyperspectral images, where they observed that
the anomalies and background of an image are enhanced
in the thinning and thickening profiles, respectively. Some
other examples are the retrieval of building height using
panchromatic angular images [31] and change detection in
temporal panchromatic images [32].
While there has been strong focus on the application
of APs to optical remote sensing data alternative remote
sensing image types have received far less attention. One
may witness some tentative works on synthetic aperture
radar (SAR) and polarimetric SAR images for segmentation
[33], building detection [34], crop field and land cover
46
classification [35], [36], and change detection [37]-[39] using
the original APs and differential APs; on passive microwave
remote sensing image analysis [40]; on lidar data for
building detection [41] and land cover classification [16],
[28], [42]-[44]; on satellite image time series classification
using Sentinel-2 data [45], [46]; and on the fusion of APs and
extinction profiles (EPs) (a variant of APs discussed in the
" Extensions and Generalization of APs " section) of hyperspectral
and lidar data using composite kernel support vector
machine (SVM)[47], [48], and deep learning approaches
[49], [50] for land cover classification. This is still an open
topic for ongoing and future research.
TREE CONSTRUCTION
Although AFs were introduced more than 20 years ago [8],
the relatively late popularity of APs is mainly due to computational
issues revolving around the efficient calculation
of CCs, which were resolved, to a significant degree,
through the tree representation of images [11]. They are of
paramount importance for the computation of APs since
the trees need to be computed only once and then multiple
filtering outputs can be derived easily from them.
Even though the seminal work of Dalla Mura et al. [7]
relies on component trees for the implementation of APs,
the type of tree is independent from the rest of the procedure;
thus, component trees can be replaced by alternative
tree types. Consequently and not surprisingly, a number
of reported works explore such options; e.g., tree of shapes
(ToS) as well as alpha and omega trees, each with its own
set of properties. Despite a plethora of tree representations
for modeling CCs in mathematical morphology
(not all of which have been implemented with the purpose
of AP construction), they can all be categorized into
inclusion and partitioning hierarchies. We invite interested
readers to refer to the recent comprehensive survey of
partition and inclusion hierarchies of images conducted
by Bosilj et al. [51].
Inclusion hierarchies constitute partial partitions of a
given image with nested supports, and their components
are formed by creating,
inflating, and merging image
blocks [52]. Moreover, inclusion trees require the presence
of a total ordering relation imposed on the set of image
pixel values, which evidently renders their extension to
multivariate images challenging since ordering vectors is
not straightforward.
Partitioning hierarchies, on the other hand, constitute
full partitions of a given image, where the leaves of the hierarchy
form the finest partition and are iteratively merged
until a single root node is formed. In addition, partitioning
hierarchies, or simply partitioning trees, most often require
only a similarity metric for determining the merging or not
between neighboring components, hence making them
particularly suitable for processing multivariate images.
The vast majority of the reported work on APs relies on
max- and min-trees, which belong both to the category
of inclusion trees. More formally, given a grayscale image
IEEE GEOSCIENCE AND REMOTE SENSING MAGAZINE SEPTEMBER 2021
IEEE Geoscience and Remote Sensing Magazine - September 2021

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