2_ullo_presentation.pdf

2011 IEEE International Geoscience and Remote Sensing Symposium

A NEW ALGORITHM FOR NOISE REDUCTION AND
QUALITY IMPROVEMENT IN SAR INTERFEROGRAMS
USING INPAINTING AND DIFFUSION

Silvia Liberata Ullo, Maurizio di
Bisceglie, Carmela Galdi1

Universit` degli Studi del Sannio
a
Benevento, ITALY

July 28, 2011

1 This work is supported by Centro Euro-Mediterraneo per i Cambiamenti Climatici

within a framework project by Italian Ministry of Environment.

Introduction

Main objective of SAR interferometry is the generation of high–quality and
high–resolution digital elevation maps (DEM’s).

Accuracy of DEM’s is strongly related to the quality of the generated
interferograms.

2 of 23 2011 IEEE International Geoscience and Remote Sensing Symposium

In this context
• We first present an inpainting–diffusion algorithm recently introduced 2

to improve interferograms quality and therefore final quality of DEM’s

• Secondly we go through the Complex Ginzburg–Landau (CGL) equation
successfully applied to image restoration 3 , and in our work applied to the
inpainting scheme for restoration of SAR interferograms

• At the end we propose a modified version of starting algorithm and
evaluate the efficiency of two algorithms also in the presence of noise

2 A. Borz` M. di Bisceglie, C. Galdi, L. Pallotta, and S. L. Ullo, Phase retrieval in
ı,
SAR interferograms using diffusion and inpainting, proceedings of IEEE Transactions
on Geoscience and Remote Sensing Symposium, Honolulu, Hawaii, July 2010
3 A. Borz` H. Grossauer, and O. Scherzer, Analysis of iterative methods for solving
ı,
a Ginzburg–Landau equation, International Journal of Computer Vision, vol.64,
pp.203–219, September 2005


Preliminary Considerations
• We start from an interferogram, unwrapped via the Minimum Cost Flow
(MCF) approach, centered on Ariano Irpino (AV), Italy, produced
through a couple of SAR images acquired with ERS–1 and ERS–2 on the
13th and the 14th of July 1995.

• A selected area of the interferogram and its corresponding coherence map
are shown as follows


• Based on the coherence map and fixing an appropriate threshold, a new
interferogram is produced by discarding phase values whose correlation
coefficients are lower than the threshold.
• Dark pixels in the figure represent the discarded phase values.


Threshold Setting
The choice for the threshold is subject to some considerations:

• a correlation coeﬃcient of 0.25 is suﬃcient to guarantee height errors of
7.5 meters or less (thus approaching a suitable error for topographic
mapping) 4

• for coherence values under 0.05 the corresponding unwrapped phases are
very noisy

• coherence values are regarded as low if they vary between 0.05 and 0.20

Therefore we make the threshold vary from 0.05 to 0.25 in our experiments.

4 H. A. Zebker, C. L. Werner, P.A. Rosen, and S. Hensley, Accuracy of topographic

maps derived from ERS–1 interferomtric radar, IEEE Transactions on Geoscience and
Remote Sensing, vol.32, no. 4, pp.823–836, July 1994


Inpainting is a well known method for restoration of missing or damaged
portions of images or paintings.
Complex Ginzburg–Landau inpainting is a technique that can be
conveniently considered to ﬁll fragmentary areas.


The Complex Ginzburg–Landau inpainting scheme
The basic element of the algorithm is the CGL differential diffusion equation
stated in the space–time coordinates in the form
∂u 1
= ∆u + 2 1 − |u|2 u = 0 (1)
∂t ε
where
• an inpainting domain Ω is defined where there are the phase values to be
restored
• the function u : D → C is the complex valued solution of the equation
such that Ω ⊂ D
• ε is a suitably chosen parameter


• It has been proved [3] that the solution u of the equation lies on the unit
radius sphere
• The real part of u, u, includes the interferogram values scaled and
normalized, such that u may assume any value in the interval [−1, +1]
• The imaginary part (u) is computed as

2
(u) = 1 − (u) (2)

such that |u| = 1
• a sequence of images are generated where the information along the
borders is smoothly propagated inside the region of missing data
• this is achieved by moving mostly along the level line directions to
preserve edges
• the steady state is achieved when the smoothness of the image is nearly
constant along the level lines ( ∂u = 0)
∂t


Application of the algorithm


Evaluation of the algorithm
To get a measure of performance for the proposed algorithm we used the
Signal–to–Noise Ratio (SNR) as expressed by the equation:

Nt
SN R = 20 log (3)
Nr

where Nt is the number of total pixels in the interferogram and Nr is the
number of residuals, where the residuals are those points around which a
close integral of the phase diﬀerences gives a non–zero result 5 .

5 U. Wegmuller, C. Werner, T. Strozzi, and A. Wiesmann, Phase unwrapping with

GAMMA ISP, Technical Report, Gamma Report Sensing, May 2002


• SNR=27.3483 dB with the MCF algorithm

Table: SNR for the CGL inpainting algorithm.

Threshold SNR Number of Number of
[dB] iterations residuals [Nr ]
0.05 27.3549 21 10636
0.10 27.3860 47 10598
0.15 27.4287 185 10546
0.20 27.6009 639 10339
0.25 27.9436 3509 9939

Results show that the proposed algorithm works better than the MCF based
algorithm.


Modified CGL based algorithm

• The CGL inpainting equation works by filling the regions where there are
missing values with zeros at the first iteration and later on through the
reaction diffusion and inpainting procedure explained before.

• In a new version of the algorithm we modify the inpainting equation (1)
and the low–coherence phase values are not discarded but used as initial
conditions. The CGL equation is forced to use these values at the first
iteration to drive the inpainting scheme.
The idea is that even if these phase values have low–coherence they may
contain a part of useful information in any case.


Results for the modified and the previous algorithm are shown in the table

Table: SNR for the new and the old CGL inpainting algorithm.

Threshold SN R1 SN R0 [Nr1 ] [Nr0 ] ∆ [Nr ]
[dB] [dB]
0.05 27.3565 27.3549 10634 10636 -2
0.10 27.3863 27.3860 10596 10598 -2
0.15 27.4584 27.4287 10510 10546 -36
0.20 27.6480 27.6009 10283 10339 -56
0.25 28.0199 27.9436 9852 9939 -13
0.30 28.6393 28.6336 9174 9180 -6

Modified algorithm performs even better and reaches its best performance
for a threshold of 0.20
This appears to be reasonable. With the modified version of the algorithm
we re–used as initial conditions the discarded pixels.
Obviously as the threshold increases besides a certain value all this work
sounds to be worthless.


Inpainting in the presence of noise
In another set of experiments the algorithm is tested in the presence of noise.
The idea is to introduce noise only in some parts of the interferogram, in a
controlled manner.
We could add noise to the interferogram in a random way, but we have
preferred to use the following method.
• first a region of the interferogram, with high–coherence phase values
only, is selected
• then a mask is created from a different part of the interferogram, where
also low–coherence phase values are present: the mask is composed by all
these low-coherence pixels and keeps their shape
• the high–coherence region is marked using this mask to make a footprint
• at the end noise is added only over the footprint: practically through the
mask, some high–coherence phase values are identified first and
corrupted with noise later on.
Since we know exactly the true values before adding the noise, after
implementing our algorithm, we can use them for comparison.


In ﬁgure the red rectangle includes a part of interferogram where
high–coherence phase values only are present; the black circles on the right
and on the left of red rectangle are regions where also low–coherence phase
values are present. These last two regions will be used as masks to establish
where, in the good interferogram, noise will be added.


The phase noise distribution
For the generation of the random additive term necessary to degrade the
interferometric phase, an interesting theoretical model has been used 6-7 .
√
The phase noise ∆φ is the diﬀerential phase of the product S(g1 , g2 ),
often referred in the literature as the compound–Gaussian model, where g1
and g2 are complex Gaussian variables, independent of S, with zero mean
and assigned covariance matrix K.

Interestingly, since S is a real–values random variable, the pdf of ∆φ
depends on the covariance matrix of the Gaussian components only.

6 M. di Bisceglie, C. Galdi, and R. Lanari, Statistical characterization of the phase

process in interferometric SAR images, proceedings of IEEE Transactions on
Geoscience and Remote Sensing Symposium, Lincoln, NE, USA, May 1996
7 D. Just, R. Bambler, Phase Statistics of Interferograms with Applications to

Synthetic Aperture Radar, Applied Optics, Vol.33, No.20; July 1994, pp.4361-4368


Generation of the additive phase noise
Two complex vectors c1 = (X1 + jY1 ) and c2 =(X2 + jY2 ) of independent
Gaussian variables with zero mean and unitary variance are generated whose
length must cover the total number of pixels we want to corrupt with noise.
Since each scatterer is observed at two slightly different viewpoints, some
degree of correlation is expected between the two vectors. To take into
account such a correlation, the vectors c1 and c2 ,are returned into vectors g1
and g2 through the covariance matrix K given by
 
1 0 ρc ρs
K = 0 1 −ρs ρc 

 ρc −ρs
 (4)
1 0
ρs ρc 0 1

where ρc is the correlation coefficient of the components (X1n , X2n ) and
(Y1n , Y2n ); ρs is the cross–correlation coefficient of Xin , Yjn , for i, j = 1, 2
∗
and i = j. We get the noise to be added as the phase of the product g1 g2 .


Validation in the presence of noise
In sequence (left-right, top-bottom) : 1) high–coherence interferogram
represented before in the red rectangle; 2) the same region marked with the
mask in the circle on the right; 3) the same region after noise is added; 4)
the interferogram restored with the CGL algorithm.


Results and some considerations

In this case with a threshold of 0.20 the SNR is equal to 59.92 dB, very
high, both in the case of MCF and in the case of CGL algorithm.

This can be justified because first we start from a high–coherence part of
the interferogram, and moreover the mask used to mark the ” good ” region
is small. Consequently the quantity of phase values corrupted with noise has
been little.

The final amount of residuals is actually only 13.

If we use also the Mean Square Error (MSE) to appreciate the difference
between the two interferograms we get a MSE = 5 × 10−7 if the CGL
algorithm is applied without noise and a MSE = 1.8 × 10−6 if noise is added.

Results appear to be pretty good and show the algorithm works well in
reconstructing the missing values even in the presence of noise.


If instead we use as mask the one on the left of the red rectangle
SNR = 27.09 dB when noise is added

This can be explained because if the noise is added over several parts of the
interferogram, we should consider a covariance matrix K not equal for the
whole region, but varying.
Moreover, the more pixels are marked to add the noise, the more corrupted
regions are considered and the greater the number of these regions that take
a part into the algorithm, resulting in a lower performance.
We remark also that with respect to the modiﬁed algorithm that took in
consideration the true values, even if with low–coherence coeﬃcients, in this
case the algorithm starts from noisy values that are generated in accordance
with a theoretical model but are not real values.

Conclusions and future work
• An inpainting scheme based on Complex Ginzburg–Landau equation has
been successfully applied to image restoration of SAR interferograms
• A new algorithm is used to restore interferograms by using the low
coherence–phase values as initial conditions
• Results appear to be very good especially in the medium–coherence area
• The algorithm shows to work well in reconstructing the interferogram
even in presence of noise if the region is restrained
• It is under analysis the possibility to adapt the algorithm even when the
noise is added over a region spatially distributed
• Future verifications can be done also through comparison of final DEM’s
to test algorithm efficiency


THANK YOU FOR YOUR ATTENTION!


2_ullo_presentation.pdf

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