IEEE Geoscience and Remote Sensing Magazine - June 2017 - 15

being the ith connected component of the upper- or lowerlevel sets of an image]. If P holds true, then the region is
kept unaltered; otherwise, the region can be set to the grayscale value of the adjacent region with the closest gray level,
thereby merging the connected components. When the region is merged to the adjacent region of a lower (or greater)
gray level, the operation performed is a thinning (or thickening). A criterion is said to be increasing if it is verified
for a connected component, in which case the criterion's
components will also be increasing. This property leads to,
e.g., P (C j) = true when P (C i) = true for any C j 3 C i .
When the criteria (e.g., the area and volume associated
with increasing attributes) are increasing, the attribute thinning and thickening transformations are also increasing,
leading to attribute opening and attribute closing [58], respectively. For nonincreasing criteria (e.g., gray-level homogeneity, shape descriptors, and region orientation), we recall
that different outputs of the filter are obtained according to
the filtering rules selected [48]. If P holds true for a connected component, it will be merged to a darker or brighter
surrounding region according to the transformation. Given
a sequence of ordered threshold values m = {m 1, m 2, f, m n}
(i.e., m i 1= m j, with i 1 j), an AP is obtained by applying a
sequence of attribute thinning and thickening operations
to the gray-scale image f,
AP ( f ) = {{ n ( f ), f, { 1 ( f ), f, t 1 ( f ), f, t n ( f )},

(5)

where { i and t i denote the attribute thinning and thickening transformations with reference values m i, respectively.
AFs process the image without distorting or inserting new
edges but only by merging existing flat regions [7], as shown
in Figures 8(a) and 9(a). Compared to MPs, the APs permit
the modeling of other characteristics (e.g., geometrical, textural, and spectral) rather than the size of the objects. We
refer the reader to [11] for further details.
However, as connected operators, both geodesic reconstruction [49] and AFs [11] suffer the same leakage effects
(again, called over-reconstruction in [10]). Some objects that
should disappear at a certain threshold, however, remain

present after filtering, as shown in Figures 5(b), 6(b), 8(a),
and 9(a). In LiDAR data [see Figure 6(b)], the residential
roof and trees are connected and are similar in elevation
characteristics; consequently, they are treated as a single object after reconstruction. As a result, different objects (even
though they are in different categories) are considered as
a single object if, in the original image, they are connected
through a narrow line in the image. In typical remotesensing scenes, many objects are arranged in a complex
manner (such as the image in Figure 1), i.e., roads are connected to many other objects such as parking lots (with the
same gray level) and buildings (with a different gray level).
For example, objects P11, P21 in Figure 3 are expected to disappear in the image when the sizes of the area attributes are set
to 350 and 450, respectively [see Figure 8(a)]. However, they
remain present even when the area size is set to 800 in the
original attribute thinning and thickening [11]. While these
two objects are assumed to remain the same with the MI [11]
attributes of 0.3 and 0.8, they, however, disappear at MI values of 0.2 and 0.3, respectively [in Figure 9(a)]. This means
that, in MPs/APs, the pixels from the road are not characterized by the attributes (e.g., size, area, and MI) of the road but
by the attributes of the whole connected object (and these
connected objects belong to different classes). Clearly, this
leads to a poor performance on post applications.
morPhologicAl feATures wiTh
PArTiAl reconsTrucTion
To overcome the leakage effects, partial reconstruction is
used for morphological operators and AFs. A MATLAB application that implements partial reconstruction for both
morphological operators and AFs is available at http://telin
.ugent.be/~wliao/Partial_Reconstruction/.
MORPHOLOGICAL OPERATORS BY
PARTIAL RECONSTRUCTION
In the geodesic reconstruction process, a pixel is reconstructed if it is connected to another pixel that was not deleted after the opening or closing. In other words, a pixel
is reconstructed if the geodesic distance d in the mask g of

(a)

(a)

(b)

(b)

figure 8. The attribute thinning with an area attribute on a syn-

figure 9. The attribute thinning with an MI attribute. From

thetic image. From left: the size of the area was set to 200, 350, 450,
600, and 800 pixels. (a) Original attribute thinning and (b) attribute
thinning with partial reconstruction.

left: the value of the MI was set to 0.15, 0.17, 0.2, 0.3, and 0.8.
(a) Original attribute thinning and (b) attribute thinning with
partial reconstruction.

june 2017

ieee Geoscience and remote sensing magazine

15
http://http:// http://www.ugent.be/~wliao/Partial_Reconstruction/
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