IEEE Geoscience and Remote Sensing Magazine - June 2021 - 21

PU OF ML
The ML method belongs to probabilistic machine learning,
which is based on the ML estimators proposed in [51]-[53].
ML directly estimates the unknown terrain height h(s)
maximizing the likelihood function,
fs hsi^h{ ();( ) (( )si
{
is the wrapped phase of the sth pixel in interferogram
(, , ..., ),
ii = 12 M and M denotes the number of the interferograms
based on the statistical distribution of the interferometric
phase noise [54] using the phase-height relation
of (1). The likelihood function of the sth pixel in the ith
interferogram is defined in (11), shown at the bottom of the
page, where ()si
c is the coherence value of the sth pixel in
the ith interferogram, and Bi is the normal baseline length
of the ith interferogram. Because the likelihood function of
(11) is periodic, it yields an infinite number of PU results.
In this case, multiple interferograms with different normal
baseline lengths are used to obtain the unique PU solution
based on the ML criteria, which is given by
M
t ()
hs ML = argmax ()%fs hs();( ).
h
{i
i =1
It is worth noting that the maximization in (12) has to
be performed pixelwise, i.e., the height estimation of each
pixel is independent of every other. Under this condition, a
reliable and accurate PU result can only be obtained when
the number of interferograms M is large, especially in the
presence of low-coherence areas. To solve this problem, the
authors in [55] explore the ML method using local planes
(MLLP), which assumes that the heights of a pixel and its
neighboring ones belong to the same local plane; therefore,
the height estimation of the sth pixel is from a local cluster
of data using the ML method.
PU OF MAP
The MAP method, taking into consideration the priori statistical
distributions for DEM reconstruction together with
MB InSAR data using the Bayes' approach, has been introduced
in [56]. In this framework, a local Gaussian MRF was
proposed to describe the priori distribution of the height
profile, given by
gh t (, ) ()Z t
v =v
1
exp)
/ <
S
s =1
(( )( 1))2
2 t ss,
hs hs
2
v
where S is the number of pixels; vt
vt
--
1
F3,
(13)
is
a hyperparameter,
which indicates the spatial correlation between the neighboring
pixels; and (( ))Z1
fs hs h
i
^{ ();( ) = 2r
1
# 1 +
]]
Z
[
\
]
]
1 c{
() cosc
ii()
() cos
ss
2
2 c
ii
ss
--
1
c{()
c
() 2
i s
mi
r
4 Bi
rs
---
-
mi
r
rs
() ()sin
;1 ss
2
() cos
JUNE 2021 IEEE GEOSCIENCE AND REMOTE SENSING MAGAZINE
··
/
c{()ii () ()sin
mi
r
rs ·
hs ss
2 c
()mcoscosc
-1 ; ii
()
c{()
4 Bi
()
rs
hs mE
12
44
·
() ()sin
Bi
hs m
()
mi
r
Bi
() ()sin
hs mE
()
_
`
a
b
b
.
bb
(11)
is the partition function.
(12)
t
MAP
=
=- ln
h
hf sh sg h
fs hs
argmax ln
argmin
h
(
(<
S
+
% %
/ /
/
s =1
<
s =1 i =1
M
s =1 i =1
S
(( ); ())( ,)
(( ); ())
{v
{
i
i
(( )( 1))2
-hs
hs
2
t
vss, -1
2
F3.
(14)
To obtain the final height result, the iterated conditional
modes algorithm is employed to optimize the posteriori distribution
in (14). Following a similar method, Ferraiuolo et
al. developed a graph-cut-based MB PU method [58] that can
decrease execution time and improve the robustness of the
MAP method based on the total-variation model using the
graph-cut optimization algorithm. Moreover, Shabou et al.
applied an evolution of the graph-cut-based MB PU method
into the reconstruction of urban areas using very-high-resolution
images with the help of the amplitude of SAR data
[59]. Baselice et al. [60] gave a review paper of ML- and MAPbased
MB PU methods, and a comparison study of the DEM
reconstruction accuracy between these two MB PU methods
was provided in [61], which implies that, compared with the
ML method, the MAP-based method is more robust when
the interferogram contains the serve phase noise because
the contextual information is exploited. However, the computational
burden of MAP is much heavier than that of ML,
especially for large-size images. In addition, the parameters
of MAP do not directly relate to PU meanings, which need
to be well-chosen to achieve a satisfactory PU result. Besides
this, the ML and MAP methods cannot maintain the fringe
congruency of the input interferogram.
PU OF CA
The CA method translates the MB PU problem into a CA
problem. Yu et al. first proposed a fast CA MB PU method
to unwrap multiple interferograms [62]. In the following,
F
S
M
t 2
Specifically, if the value of vt is high, the similarity between
h(s) and () is large; otherwise, for a low value of v ,t
the similarity between h(s) and () is small. The hyhs
1hs
1perparameters
vt can be estimated from the available noisy
wrapped phase using the expectation-maximization algorithm
[57]. For more details on hyperparameter determination,
the reader is referred to [56].
Under the Bayes' rule, the MAP solution for DEM reconstruction
can be calculated by the product of the likelihood
function of (12) and the priori distribution of (13), which
is defined by
21
IEEE Geoscience and Remote Sensing Magazine - June 2021

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