IEEE Geoscience and Remote Sensing Magazine - June 2020 - 76

SPARSE REGULARIZATION
These sparse methods employ Bayesian rules to restore the
filtered image through a maximum a posteriori (MAP) estimator [97]-[100]. Usually, the MAP estimation is transferred to solve an energy-minimum problem, which can be
expressed in a generalized formulation as
min
E^z 1, z 2, x h
x

E(z 1, z 2, x) = U(z 1, z 2, x) + l $ iW(x)i1,

(35)

where U(z 1, z 2, x) is the data-fidelity term from the likelihood function, iW(x)i1 is the L 1- norm term using a sparse
prior, and l is a regularized coefficient to balance the two
terms. In (35), U(z 1, z 2, x) usually have three different formulations according to the noise reduction in different
domains. Taking the interferogram regularization as an example, U(z 1, z 2, x) can be derived from a joint likelihood
of (5) as
U(z 1, z 2, x) =

/

2 log R +

a 1 d; z 1;, a 2 d; z 2;, z z d z z

a 12 + a 22 - 2a 1a 2b
, (36)
R(1 - c 2)

where x is composed of R and z x [involved in b, shown
as (4) and (5)]. On the other hand, the format of iW(x)i1
greatly determines the regularization power. There are
many different sparse-regularization approaches, such as
total variation (TV) and wavelet-domain regularizations.
TOTAL VARIATION REGULARIZATION [23], [98]
Usually, TV regularization applies the gradient operation
on the 2D phase image for noise reduction, which is expressed as
iW(x)i1 = / u d(z ux )u,

(37)

u

where d(z ux ) denotes the 2D gradient operator on phase
z x at site u. The phase denoising benefits from the local
smooth effect using the gradient operation. To improve the
robustness of the phase reduction, the joint TV regularization of the phase and amplitude of InSAR data has been
studied, with the regularization formulated as
iW(x)i1 = / max _ d( R u ) , m $ d^z ux h i,

(38)

u

where R u denotes the pixel of amplitude image R at size
u, and m is used to balance the regularization between the
amplitude and phase. Sparse TV regularization is not well
suited to phase denoising for scenes that include high topography and high heterogeneity because it does not satisfy
the sparsity assumption on the phase gradient.
WAVELET-DOMAIN REGULARIZATION [100], [101]
The generalized formulation of sparse wavelet regularization is given by
iW(x)i1 = / uW uxu,
u

76

(39)

where W ux denotes the pixel of wavelet image W x at site u.
In this framework, the phase-noise reduction is achieved by
sparse regularization of the phase in the wavelet domain. The
amplitude and phase can be jointly regularized in the wavelet domain to improve the sparsity degree, which is helpful
to noise reduction. Accordingly, both amplitude and phasenoise reduction can be achieved through this approach.
SPARSE CODING
As opposed to sparse regularization, which uses a fixed dictionary, sparse-coding methods employ the dictionary learning of W to improve the sparse representation of the phase
image. The generalized formulation is shown as [102]-[104]
min / iy u - Ws u i 2 + l u $ is u i1,
2

s, W

(40)

u

where y u is a vector of one patch of InSAR observations (that
is, S z /I z) centered at site u, and s u is the corresponding patch
of the noise-free image. The processing flowchart for the
sparse-coding methods can be summarized in three steps:
1) for each pixel, collect the square window patch and arrange it in a vector form
2) use all of the patches for sparse coding to obtain the filtered results with a perfect sparse-signal estimation
3) aggregate the filtered patches to their original locations
through data fusion. In this manner, the filtered phase
image can be obtained.
EXPERIMENTAL ANALYSIS
Two sparse-regularization algorithms are employed to illustrate the sparse methods' performance. First, experiments
based on RADARSAT-2 data are performed using sparse TV
[23] and wavelet [100] regularizations. Figure 13 presents
the filtered results and shows that both algorithms can
effectively reduce noise in urban areas. Next, the wavelet
regularization is applied to the Mount Etna interferometric
data, and the filtered results are given as Figure 14.
For clarity, Figure 15 provides more details of the algorithms' results. As Figure 15 shows, the Lee filter introduces
some artifacts of fringes that are ambiguous and broken
in areas that have a high topography and low coherence.
Compared to the wavelet filter, the region-growing and
sparse-wavelet methods can improve the performance of
interferometric phase-noise reduction in smooth areas and
effectively suppress the phase residues of grainy noise. The
sparse-wavelet method has the best performance for dealing with blurring and the discontinuities of the phase fringes. More details can be found in [100].
COMMENTS
In addition to the sparse methods, other technologies
have been tried in InSAR phase denoising, such as the
Markov random field [105]-[108], tensor decomposition
[109], convolutional neural networks [110], and optimization integration with phase unwrapping [111]-[115].
Most of them can be roughly classified as machine- and
IEEE GEOSCIENCE AND REMOTE SENSING MAGAZINE

JUNE 2020
IEEE Geoscience and Remote Sensing Magazine - June 2020

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