IEEE Geoscience and Remote Sensing Magazine - June 2020 - 70

local adaptive filter [15], [40] and refined Lee filter [14]. To
overcome the window-number constraint, the directional
window is adaptively estimated by retrieving the local frequency to determine the direction of the phase fringe [15].
The refined Lee filter proposed in [14] has modifications to
improve the robustness of the directional window selection
and statistical parameter estimation.
LOCAL-FREQUENCY ESTIMATORS
Local-frequency estimators are based on polynomial-phase
models, such as linear [42]-[45] and nonlinear [46]-[49].
These methods estimate the local frequency of the interferometric phase to derive the phase fringe. Usually, the local
phase in a small window is modeled as a sine wave with one
major component, which is shown as
S win
z ^ k, l h . a k, l $ exp 6 j2r ^ k $ fx + l $ fy h@ + n v ^ k, l h,

(16)

where S win
z ^ k, l h is a local-window form of S z(k, l) centered
at site (k, l), and a k, l and ( fx, fy) are the coefficient and 2D
frequency, respectively. Then, ( fx, fy) can be approximated
using spectrum estimators, such as ML [42], vector covariance matrix [42], and multiple-signal classification [45].
Based on the approximated fringe frequencies, the filtered
phase can be estimated as
u
u
zu x (k, l) = + a / / S win
z ^ k, l h $ exp 7- j2r _ k $ fx + l $ fy iAk,
k

l

(17)
where _ ufx, ufy i is the estimated 2D frequency. In practice, the
linear-frequency model is not adequately accurate because
it neglects other frequency components. To overcome this
limitation, several innovative strategies, such as achieving
iteration by varying the window size [18] and using nonlinear-phase modeling [47]-[49], are applied to preserve the
phase details of the fringe and texture. The order of the selected model is crucial to this group of methods. The higher
the chosen order, the more accurate the phase estimation
becomes. However, there is a cost in terms of increased computational load. As a result, there must be a balance between
the order selection and computational complexity. Integration methods including phase-frequency estimation and
unwrapping have been studied [51]-[57], but they are not
discussed here.
HYBRID METHODS
The procedure for these methods is usually decomposed
into two-step estimations. The basic idea is that the interferometric phase can be separated into principal and
residual components, such as a low-resolution fringe and
high-resolution pattern [18], [41], [60], [61]. The principal
component can usually be extracted using the local-frequency estimation technique and so on. After the principal phase compensation, the interferometric phase's
residual component can be estimated more precisely. It is
70

obvious that the phase is estimated in a hybrid manner,
which is shown as
zu x = zu xLR + zu xHR,

(18)

where zu xLR and zu xHR are the estimations of the principal
and residual phases, respectively. For the local filters, the
sample-selection strategy, such as the window form, is
critical to the phase/frequency estimation. The developed
window can be roughly classified as an oriented/directional window [13]-[15], an adaptive size-varying window [18], [43], and a region-growing mask [59], [60]. The
region-growing (also known as the intensity-driven adaptive
neighborhood) method [60] employs a local window with
an adaptive shape and size to search for similar pixels that
are stationary and homogenous. Among traditional local
filters, the region-growing method can select the most relevant high-quality samples, establishing it as one of best of
the sample selectors.
COMPARISON OF LOCAL FILTERS
In this section, numerical experiments using the aforementioned interferometric data are performed to compare typical local filters, including the pivoting mean
filter [19], pivoting median filter [39], Lee filter [13], ML
method [42], and region-growing method [60]. Figure 7
presents the filtered images; the rows display the filtered
phase, phase difference (between filtered and noisy examples), residue, and coherence, respectively. It can be
seen that the pivoting mean and median filters tend to
introduce phase discontinuity in the highly sloped areas,
producing many residues. The ML method has a better
performance in the high-topography regions. However, it
produces excessive filtering, which leads to a loss of phase
details. In addition, the size of the filter window greatly
affects the ML results. The Lee filter performs better for
noise reduction and detail preservation. However, it introduces artifacts in the areas of very low coherence. Among
the filters, the region-growing method's performance is
the best, although its phase estimation in highly sloped
regions is quite limited. We can conclude that traditional
local filters are not very effective for dealing with high
topography variations, which is an inherent limitation.
REVIEW OF TRANSFORMED-DOMAIN FILTERS
Instead of filtering in the spatial domain, InSAR phase
denoising can be performed in the transformed domain.
Transformed-domain filters possess the superior quality
of enhancing the phase signatures to be more separable
from the noise. In this section, we introduce transformeddomain filters and divide them into two groups. For clarity,
their processing flowchart appears as Figure 8.
FREQUENCY-DOMAIN METHODS
Frequency domain usually refers to the Fourier transform and
discrete cosine transform. The most typical and effective of
these denoising techniques is the Goldstein filter [62] and
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