Analysis of the coherence map generation techniques for SAR image pair processing is presented. The areas of map coherence acceptable values for the averaging window are calculated. The map coherence enhancement method based on the results is submitted. The SAR interferogram calculation using submitted method is performed. The result accuracy improvement in comparison with the classical technique is shown.
Coherence map is an image of the SAR pair correlation coe cients eld. It
gives information about deviation degree of the absolute phase from its true
value. Such deviation may be caused by the phase noise, a surface variability,
and the stability of the radio signal re ected from various parts of the Earth
surface [
The coherence map development is performed at the stage of the SAR
interferogram creating. According to the traditional technique, value of each element
is calculated as the correlation coe cient between the values of the rst and
second SAR images in the pair. The classical method of coherence map
development is based on multiplying the rst (reference) image of the interferometric
pair by the second (auxiliary) one that is complex-conjugate to itself [
j PxM=01 PyN=01 Z1(x; y) Z2 (x; y)j qPM 1 PyN=01 jZ1(x; y)j2 x=0
PM 1 PyN=01 jZ2(x; y)j2 x=0 ; (1) where Z1, Z2 are the radar images of the reference and auxiliary signals respectively, M , N are the dimensions of the averaging region size, is the complex conjugation operator.
Despite the fact that the correlation function has been thoroughly studied in
details [
Firstly, it is necessary to obtain the minimum averaging region size N with is used during the coherence map development. The plot of dependence of the bias of estimate of coherence magnitude d^ at zero correlation on the averaging region size is constructed. The size N in terms of expression (1) is similar to the sample size used in calculating the correlation coe cient. Due to the niteness of the sample, a bias of the correlation estimate may occur in the coherence map development. The displacement value will increase with decreasing coherence and reach the maximum values at zero. For calculations, the test image corresponding to a at terrain without any relief changes is used. Test images are combined with the Gaussian noise by complex multiplication operation. The range of the averaging window is from 3 to 65 elements. The results are shown in Fig.1.
As it follows from the gure, the minimum averaging region size is the size
of 11 elements for the classical method of coherence map development. Using
the smaller one, the bias in the coherence estimate exceeds 0.1. This e ect may
deeply distort the interpretation of the output product. The dependence has the
form of a hyperbolic curve slowly converging to zero value, which corresponds to
the classical notions of the signi cance of estimating the bias from the window
size [
To analyze the behavior of the estimation using the classical method, the dependence STD of estimate of coherence magnitude on the varied coherence ^ = ^( ) is performed. The STD estimate changes in the coherence interval from 0 to 1 with step of 0.005. The calculations are made for the averaging region size 11 11; 19 19; 27 27; 35 35; 43 43; 51 51; 67 67. Averaging at 600 points is performed. The obtained dependences are presented in Fig. 2. The abscissa axis shows a range from 0 to 0.3 for clarity.
As it follows from the gure, behavior of the dependence ^ = ^( ) is
similar to the lower Rao-Kramer boundary [10, 12{14]. The discrepancy with the
theoretical dependency is due to the fact that the Rao-Cramer formula shows
reliable values for su ciently large sampling values. Also, the dropping peaks
are distinctly distinguished at the calculated values, after which the value o
thef STD estimate drops sharply to the minimum values. The reason for this
is that there is a bias in the STD estimate (Fig. 1). Position of the dropping
peak depends on the window size: the smaller the window size, the farther from
the zero is the peak. This fact is caused by a decrease in the STD estimate
of coherence magnitude with increasing sample size. It follows that the range
of brightness values on the output coherence map corresponding to the set of
coherence values (which are located between the zero and the dropping peak) is
incorrect and have to be removed from the output coherence map. A summary
of the obtained results is presented in Table 1.
dp
0.055
0.055
0.055
0.050
0.050
0.050
0.050
0.045
0.045
0.045
To illustrate the data of Table 1, the coherence map for two images of SAR pair
was modeled using the classical expression (1) (the data are taken from [
As it follows from the Fig. 4, during increasing averaging region size, the e ective area of the coherence map increases too. It becomes possible to build a mask of pixels, which are incorrect. At the same time, it should be noted that during increasing averaging region size, the nal product details decrease. The present result will be used to improve the interferogram accuracy for the SAR image pair in the tasks of DEM creation. For interferogram accuracy of obtained products, the STD of absolute phase deviation from a reference DEM was calculated as a quality indicator for the interferometric coherence estimate. The better estimates should give a less- uctuation decreasing dependence for at least for low- and medium-valued coherences with a possible wider range of both coherence and STD values. The method based on coherence map masking has shown the more accurate result: the range of deviation is wider by 11 percent in comparison with traditional techniques.
The reference DEM covered a territory of 8 5 km, which contained average hills and river valleys. The averaging window size was 15 15, the value of responding dropping peak dp was 0.13. Two coherence maps were used for interferogram creation: one was obtained with the traditional method, and another was generated according to the proposed method. The results are shown in Fig. 4. The traditional method of the coherence map generation was investigated. Some statistics results related to a classical expression were obtained. The new method of interferogram creation based on dropping peak position was proposed. The method has shown the more accurate result: the range of deviation is wider by 3.8 percent in comparison with traditional techniques.
This work was supported by the Ural Federal University's Center of Excellence in \Geoinformation technologies and geophysical data complex interpretation methods" (according to the Act 211 Government of the Russian Federation, contract 02.A03.21.0006)