IEEE Geoscience and Remote Sensing Magazine - June 2017 - 19

The figures show that many small objects that should disappear at a certain scale of area attribute remain even at a
very high scale when using attribute thinning [11], [12]. This
is much worse when the selected features were rescaled to a
lower range (e.g., [0, 10] in Figure 11). This is because more
objects are connected as the ranges of the rescaled features set
decrease. If the attributes of all connected objects are mixed
together, these connected objects remain or disappear together. In these cases, the attribute thinning and thickening
cannot well model the spatial information of the objects.

pared the performances of stacking all APs or APPRs
together, which are defined as EAP and extended APPR
(EAPPR) [EMP and EMP with reconstruction (EMPPR) for
EMPs, respectively].
The performances of each scheme are quantitatively
evaluated by measuring the following metrics:
1) the normalized mutual information ^NMI h that tests the
independence between two variables and measures the
information that they share
NMI ( f, g) =

exPerimenTAl resulTs

MI ( f, g)
,
MI ( f, f ) MI (g, g)

(9)

where the mutual information
EXPERIMENTAL SETUP
To generate MPs, we apply a circular SE with ten openings
and ten closings (ranging from one to ten with a step-size
increment of one). For the construction of the APs, we consider three different attributes: 1) a, area of the regions; 2) s,
Std of the gray-level values of the pixels in the regions; and
3) i, first moment invariant of Hu, MI. The area extracts information on the scale of the objects. The Std and MI are
not dependent on the size dimension, but they are related
to the geometry of the objects and the homogeneity of the
intensity values of the pixels, respectively. All of the images
were rescaled to the range [0, 255] and converted to integer form to be processed by the AFs. The values of different
attributes are m a = [100, 500, 1, 000, 2, 000, 3, 000, 4, 000,
5, 000, 6, 000, 7, 000, 8, 000],m s = [0.1, 0.5, 1, 2, 3, 4, 5, 6, 7, 8],
and m i = [0.1, 0.15, 0.2, 0.25, 0.3, 0.35, 0.4, 0.45, 0.5, 0.55].
Prior to applying morphological openings and closings (or the attribute thinning and thickening) to the
hyperspectral image, PCA was first applied to the original hyperspectral data set, and the first few PCs (the first three PCs
for the University of Pavia) were selected (representing 99%
of the cumulative variance) to construct the EMPs. To compare MPs and APs by reconstruction with those by partial
reconstruction, we consider both information redundancy
and their post applications to classification. We use a support vector machine (SVM) [59] classifier, as it performs
well on the classification of high-dimensional and/or multiple features [26], even with a limited number of training
samples, limiting the Hughes phenomenon [60]. The SVM
classifier with radial-basis function (RBF) kernels in the
MATLAB SVM Toolbox, a library for SVMs [61], is applied
in our experiments. An SVM with RBF kernels has two
parameters, i.e., the penalty factor C and the RBF kernel
widths c. We apply a grid search on C and c using fivefold cross-validation to find the best C within the given set
{10 -1, 10 0, 10 1, 10 2, 10 3} and the best c within the given set
{10 -3, 10 -2, 10 -1, 10 0, 10 1}.
We compared the following schemes: original image
(Raw); morphological profiles with no reconstruction
(MPNs), morphological profiles with geodesic reconstruction
(MPRs), morphological profiles with partial reconstruction (MPPRs); each single existing AP ^ APa, APs, and APih;
and single APPR ^ APPR a, APPR s, and APPR ih . We also comjune 2017

ieee Geoscience and remote sensing magazine

MI ( f, g) =

/ / p (x, y) log d p p(x()xp, y()y) n, p^x, yh

x!f y!g

is the joint probability distribution function of f and g,
and p ^ x h and p ^ y h are the marginal probability distribution functions of f and g, respectively.
2) the overall accuracy (OA) calculating the number of correctly classified samples divided by the number of all
test samples
3) the average accuracy (AA) denoting the average of class
classification accuracy
4) the kappa coefficient of agreement ^K h measuring
the percentage of agreement corrected by the amount
of agreement that could be expected due to chance
alone [64]
5) the specific class accuracy representing the percentage of
accurately classified samples for a given class.
Note that for 1), an NMI close to zero indicates independence, while a high NMI indicates dependence and feature
redundancy [63].
INFORMATION REDUNDANCY: RECONSTRUCTION
VERSUS PARTIAL RECONSTRUCTION
The most popular MPs/APs generated by using morphological reconstruction (including geodesic reconstruction
and AFs) [11], [12] contain redundant information, because
the connected objects survive in many scales. To test this
assumption, we take a Ghent Watersportbaan panchromatic image as an example to compare the NMI among each profile
(see Figures 13 and 14). These figures show that the MPNs
contain the least redundancy with the lowest NMI among
profiles. Figures 5(a), 6(a), and 7(a) show that objects smaller
than the SE will disappear.
Morphological reconstruction [11], [12] cannot model the
spatial information of the connected objects well in VHR images. Objects that are expected to disappear in the image at a
low scale are still present at the highest scales, as is shown in
Figures 5-9, 11, and 12. This is why additional geometrical
features generated with morphological reconstruction [11],
[12] have the highest NMI, i.e., contain much more redundant information. To reduce the redundancy, some algorithms were developed to automatically select a threshold for
morphological APs [65]. Recently, some artificial intelligence
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