Identifying low reflection amplitude and low level phase noise points for permanent scatterer (PS) interferometry

Yitzhak August, Dan G. Blumberg, Stanley R. Rotman

Ben-Gurion University of the Negev, Earth and planetary image facility lab, Beer-Sheva, Israel

Abstract — The PSI (Persistent Scatterers Interferometry) method relies on identifying a small group of scatterers that maintain high phase reliability over a relatively long period of time. This study demonstrates a new algorithm to identify natural PSC (persistent scatterer candidates) targets in non-inhabited areas. The application of our PSC selection process is conducted for a natural arid scene as opposed to the more common use of the PS technique, which is done mostly for urban areas with structures exhibiting strong reflection (manmade objects). We present a novel robust method to identify PSC in open fields and in places of low backscattering (natural areas). Our method is based on the amplitude time history signature of each point. The main difference between urban areas and open field areas is the low reflectance and less deterministic behavior of the scatterer; hence it is a challenge to detect these low reflection and stable points. Conventional methods for PSC detection require a preprocessing with fine calibration and are mainly suitable to use in urban areas, but may fail when used in the open fields. One of the advantages of our method is the use of a simple process of calibration which is based only on the flight geometry and gain factors without any auxiliary data or assumptions. Consider a vector consisting of the measurement of a PS point as a function of time. We can express this signal as an amplitude times a phase. The amplitude differs between PS points; however potential PS points should correlate spatially and temporally in terms of the phase, independent of their amplitude. Our method improves locates several candidate points with a narrow phase distribution and thus, enables the location of PSCs in natural open areas.

Index Terms — Calibration, Identifying Persistent Scatterers, Interferometry, InSAR.

I. Introduction

The PS technique provides a way to estimate the fine topographic changes and the motion of objects and structures. In contrast to the classical SAR interferometry technique, the PS technique relies only on a small group of pixels. The PS technique relies on a group of point targets whose amplitude and phase are stable as a function of time (i.e. invariant of look angle (baseline), soil moisture content and are geometrically and physically stable). Point-like target scatter is a scatter that dominates the scattering from a resolution cell, it was shown that for high reflection target the internal pixel phase noise is low [1]. The first notification of this kind of points was noted by Hanssen and Usai [2]. It was shown that some certain man made targets exhibited high coherence, even when using interferograms with long time spans and large spatial baselines (spans of look angles). Using those low level phase noise pixels provides the key for atmospheric phase estimation and fine topographic change estimation. By

searching for pixels that include a single dominant stable scatterer, a network of so called Persistent Scatterers (PS) can be established. One of the main difficulties and challenges of using and applying the PS method is the first selection of permanent scatterer candidates group (PSC) (i.e.), the process of selecting an elementary resolution cell which presents a reflection from a stable point scatter. The process of point searching needs to be applied in a very careful way. Selection of un-stable points or non-point scatter will lead to a process failure or to high errors in the topographic change estimation. In general, a persistent scatterer point can be metallic structures or a corner reflector, a corner of a building or a solid rock formation in rural areas. Different ways for PSC detection were presented in [3]-[4] and mainly were used in populated areas and urban areas. In this paper we present a new strong and reliable method to identify permanent scatter candidates in open fields and in places of low backscattering areas (natural areas). We call this method Amplitude Time Signature (ATS).

II. ATS algorithm for PSC selection

Our method is based on the amplitude time history signature (ATS) of each point (voxel). The application of the PSC selection process is present to be used in natural regions where most of the areas not cover with manmade objects. The process of selecting PSC is based on the pixel absolute reflectance signature over a time series. The stage of PSC searching contains three main parts. The first part is calibration of the SAR image data, this stage is based on the ESA presented algorithm [5] but with some simplifications. Our calibration process is done for the geometric properties of the image acquisition i.e., only calibration on the spatial domain is done. We exclude the time dependence part of the calibration from our process; the process of the PSC model deals with this part.

The second part is the estimation of the PSC amplitude time signature model. In this step we use the Principal Component Analysis (PCA) method in order to estimate the signature of PSC ATS. The PSC model in this part is the product of the SAR system changes over time, some geometric contribution, the atmosphere and also other unpredictable changes over time. The amplitude signature of the PSC is estimated using only a small and high reliable group of points in the set of images.

The third and final stage is the stage of PSC detection. This stage is based on using the ATS that we found in the second step. Pixels that have similar signatures as the PSC ATS model are marked as PSC. The amplitude of the pixel signature vectors are normalized to one, hence strong and weak reflections target both have the same opportunity to be selected as PSC (in the case of similar signatures compare to the presented model). The similarity between the models to all pixels in the image is based on the Spectral Angle Mapper (SAM), it uses only the vector "direction" of the spectra (in our case, time), and not the "vector length". Using this method is insensitive to the unknown gain factor, and all size of sub pixel targets. The SAM algorithm generalizes the geometric interpretation of the scalar product to a high dimensional space. SAM determines the similarity of an unknown spectrum to a reference model. Using some thresholds, the product of this step is a binary map of the PSC positions in the spatial domain of the interferogram or on the image.

A. PSC amplitude signature

For the simplicity of the description, we start from a schematic 2D scatter-plot of different types of scatter natures. Each of the axes of the 2D scatter-plot represents the absolute reflection coefficient at a different time of SAR acquisition. From this schematic description of the problem we describe the ATS algorithm for PSC selection.

The PSC amplitude signature is based on a small high quality group of PSC targets (in the words high quality we mean that the targets represent point scatters and that the group is homogeneous). From the group statistics, we can calculate the PSC ATS model. This small group is selected under high restrictions, in order for these restrictions to ensure the high probability for PSC detection. The high restrictions for the detection, result in a very low rate of false alarms, but also in high rates of missed targets. It is more important in this step to identify small high quality groups of pixels then (in the cost of high miss target rate) to identify a bigger group with a reduced quality of targets (low quality targets are targets that the deterministic part is smaller). The SAR image contains different types of areas and targets i.e., open fields, sand area rocks, manmade objects, roads and areas covered with vegetation and also agriculture fields. Using the SAR detected and calibrated data amplitude, we select pixels on the based on high reflection values and the number of times that it reaches high values. The motivation for this kind of selection is the high reflection coefficient which is likely to be a point scatter than a distributed scatter. The other criterion is the number of times these points remain above a certain threshold value. Using this criterion ensures that the scatter does not move (as a stone in a river path or as manmade changes over time) and does not change. Using this algorithm for PSC selecting is not effective for detecting large groups in open fields. Using this algorithm in the open fields will results in a high rate of missing targets (this algorithm provides very small numbers of PSC points which cannot be dense enough for the used with

the PS method). For the building of the signature model, it provides enough points. The detection of PSC points for the model is based on amplitude thresholds and its frequency of appearance. The SAR calibrated amplitude matrix )(tD is presented in equation 1. This matrix is for the time which is the time of the image acquisition.

(1)

)( τ=tA is a binary matrix which presents the product of

first restriction. In equation 2 the elements of )( τ=tA are present as )(, τa ji . The pixel in the position )(, τa ji in on the image time τ is marked as hot in the case where its amplitude value is above or equal to some fixed threshold τTH which is slightly change for each τ in the set.

⎭⎬⎫

⎩⎨⎧ ≥

01)(

)(elseTHτd

=τa τijji, (2)

The τTH is a derivation from a global set threshold th. Next we will show the set of τTH values from the th value. The th threshold value is chosen for the set of images and its value is 0 < th<1 (corresponds to 0% to 100% percent’s of pixel that their amplitude is above some threshold). th value is converting for an absolute value τTH for each image. In order to find τTH we do an histogram for the data )(τD , as written in equation 3.

{ })(τDHistogram=S Fτ (3)

τS is the histogram vector, its length is F samples. The histogram contains F elements where each element size equals { } { }( ) Fdd ijij /)(min)(max ττ − . In order to find the term position f (in with above it certain percent of the pixels values appears), we use equation 4.

[ ] [ ]∑∑ ⋅≥F

=iτ

f

=iτ isthis

11

(4)

][isτ represent the elements of )(τS . Using f we can translate the th in to τTH as follow

⎟⎟⎠

⎞⎜⎜⎝

⎛ −⋅

Fτdτd

f=TH ijijτ

))(min())(max( (5)

Where as usual ))(max( τdij and ))(min( τdij are the maximum and minimum values in the matrix )( τ=tD . In order to apply the frequency (temporal) restriction to our set of binary matrixes { } Tt

tA ==1)(τ we define the G matrix.

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

)(...)()(:::

)(...)()()(...)()(

)(

21

22221

11211

τdτdτd

τdτdτdτdτdτd

=τ=tD

MNMM

N

N

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

∑∑∑

∑∑∑∑∑∑

)(...)()(:::

)(...)()()(...)()(

21

22221

11211

tatata

tatatatatata

=G

MNMM

N

N

(6)

The G matrix represents the number of times for each pixel ji, to be above the th. We use equation 7 in order to define

the binary terms of matrix binG .

⎭⎬⎫

⎩⎨⎧ ≥

01)(

)(elseTHτg

=τg timeijijbin, (7)

In this way there is no importance for the order that point is present and mark as high point. The timeTH threshold value is the number of times where points were marked as hot. In general, this case can be derivatived from timeth in percent representation. The result of these two restrictions is a binary matrix binG which presents the position of the points which survived the process (equation 8).

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

MNbin,Mbin,Mbin,

Nbin,bin,bin,

Nbin,bin,bin,

bin

ggg

gggggg

=G

...:::

...

...

21

22221

11211

(8)

The next step in the creation of the PSC model is to reload all the images, and keep only the pixel vectors that are marked as ones in binG . We order these set of vectors in the form of a matrix in equation 9.

⎟⎟⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜⎜⎜

⎝

⎛

=

⎟⎟⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜⎜⎜

⎝

⎛

===

===

===

===

P

p

d

d

dd

ddd

ddd

dddddd

=D

~:

~:

~~

~...~~:::

~...~~:::

~...~~~...~~

~2

1

TtP,2tP,1tP,

Ttp,2tp,1tp,

Tt2,2t2,1t2,

Tt1,2t1,1t1,

(9)

In equation 9, each row represents the selected point measurements over time t. The index p is the point index, ( 10 −≤≤ Pp where P is the total number of hot points in binG ). Matrix 9 presents no data about the vector's spatial domain and hence we also make an index vector for the translation from the D~ to D coordinates. The d~ represents a pixel of high amplitudes and high number of appearances over time, but an agriculture field can sometimes appear in a similar way (which is not a permanent scatter). In order to release these points which do not behave like the main group, we use the PCA method follow by SEM. Equation 10 presents

the process of PCA, in this process eigenvalues and moralized eigenvectors are found for the covariance matrix of D~

⎪⎪⎪⎪

⎭

⎪⎪⎪⎪

⎬

⎫

⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

⎨

⎧

⎟⎟⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜⎜⎜

⎝

⎛

⎟⎟⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜⎜⎜

⎝

⎛

⎯→⎯

⎪⎪⎪⎪

⎭

⎪⎪⎪⎪

⎬

⎫

⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

⎨

⎧

⎟⎟⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜⎜⎜

⎝

⎛

N

n

N

n

pca

pca

pca

pca

pca

pca

pca

pca

pca

P

p

v

v

vv

d

d

dd

λ

λ

λλ

:

:

:

:

~:

~:

~~

2

1

2

1

2

1

(10)

pcav are the PCA eigenvectors and pcaλ are the eigenvalues. The eigenvectors are arranged in a way that

1+≥ ii λλ for each eigenvalue iλ there is correspond eigenvector iv and 1=iv .

In order to make the group of vectors D~ more homorganic and reliable we reject some of the vectors d~ that do not point in the direction of the main vector 1v . In order to do this, two more steps of PCA calculation are done. In the first step we calculated the d~ projection over the second PCA vectors (equation 11).

2

2

~

,~

vd

vd

i

i

i⋅

=β (11)

Vectors, which their absolute normalized projection

ββ thi ≥ is above some threshold βth are marked as non-part of the group, and rejected from the group of id

~.

The motivation for this is based on the assumption that these rejected vectors point in the direction of the second strongest group and hence we first reject this group. The second step is a re-calculation of equation 10 and the projection of the id

~group over the new PCA's vectors. The

vector projection operator is shown in equation 12

2

2

~

,~

vd

vd

i

i

i⋅

=α (12)

Vectors with absolute normalized projection

αα thi ≥ remain in the group of id~

, all others are rejected. The motivation for this method is for narrowing the group to reach a high homogeneity. The final step is to re-calculate

pcav from the reduced group of id~

. We use the first PCA as the PSC model for the next stage.

B. Detect PSC by the amplitude signature

The detection of PSC points is based on the comparison between the PSC model to a normalized vector in the

point ji, . Our model for PSC is presented as 1v for thed~ . The comparison between the PSC and the model is based on SEM. The comparison is based only on a geometric direction in the N dimensional space. The operation is similar to the presented operation in equation 12; where we replace the d~ group with the group of all points in image d without any restriction or with a weak restriction. We choose only the points that their absolute projection (equation 13) was above a certain threshold γγ thi ≥ .

1

1

~

,~

vd

vd

i

i

i⋅

=γ (13)

In our process we used about 1/3 of the full image pixels

(we released points that were of very low reflection). Using this was in order to reduce the computational complexity and running time; we assumed that in this way we do not miss PSC points from the final step. After the detection of the PSC points we used the indexing converting to convert the D~ to D coordinates (D~ represents all the image pixels and their positions in contrast to the definition of D~ in the previous subsection). The motivation for pixel selecting that is based only on direction is presented in the theoretical part. Hence by looking in some direction and in a small angle span, we assume we will find a strong and low reflection target with high phase stability.

III. ATS algorithm for PSC selection

The final step of our study was the PSC signature detection

process validation. The validation step is the process used to confirm that the PSC signature detection process procedure can be employed for accurate results in the PSInSAR and also to compare its performance with the popular method for PSC selections. The PSC detection process is based only on the absolute amplitude value over the time, the validation results showed the behavior of the phase values for the PSC points. Phase values for interferogram contain few phase contributions. In case of PSC validation we eliminate the earth phase and topographic reference phase contribution. This is done by using the differential interferogram, the main phase contribution in this case is the atmospheric and deformation phase contribution and also some phase noise and small phase errors. In order to reduce the probability of area subsidence we used the differential interferogram with a small temporal baseline. Our validation process is based on two types of groups based on different PSC selections from the total image points. Those groups are made from the standard temporal and amplitude detection algorithm and PSC detection based on ATS. In order to evaluate the different process performance two tests were done. The tests were based on the phase

histogram the full width at half maximum (FWHM) and the full width at two third of its maximum. The results for both tests were about a twenty percent narrow phase values distribution.

IV. Conclusion

In this thesis work, a particular type of concepts is described; we adopted signature pixel selection for PS identification. One of the conclusions from this work is that we have a strong tool to look at the target statistics. In addition, two pixels with the same amplitude fluctuations and mean value can be distinguished by their signature. It allows us high performance and high reliability.

ACKNOWLEDGEMENT

The authors wish to acknowledge the assistance of the EPIF lab members at Ben-Gurion University of the Negev. We are grateful to Ruth Lubashevsky (Ben-Gurion University) for the text editing work of this document.

References

[1] F. R. Hanssen, RADAR interferometry: data interpretation and error analysis. Netherlands: Springer, 1st edition. , 2001.

[2] R. Hanssen and S. Usai, “Interferometric phase analysis for monitoring slow deformation processes”, Third ERS Symposium-Space at the Service of our Environment. pp 487–491, March 1997

[3] Adam. N, and Kampes. B, and Eineder. M, “Development of a Scientific Permanent Scatterer System: Modifications for Mixed ERS/Envisat Time Series,” Envisat ERS Symposium. ESA Special Publication, vol. 572, April 2005.

[4] Kampes. B, RADAR Interferometry: Persistent Scatterer Technique, Netherlands: Springer, 2006.

[5] P. J. Meadows, H. Laur, and B. Schttler, “The Calibration of ERS SAR Imagery for Land Applications”, Earth Observation, vol 62, pp 5-8, 1999.