Local, Fine Co-Registration of SAR Interferometry Using the Number of Singular Points for the Evaluation

This book provides a current overview of the theoretical and experimental aspects of some interferometry techniques applied to Topography and Astronomy. The first two chapters comprise interferometry techniques used for precise measurement of surface topography in engineering applications; while chapters three through eight are dedicated to interferometry applications related to Earth's topography. The last chapter is an application of interferometry in Astronomy, directed specifically to detection of planets outside our solar system. Each chapter offers an opportunity to expand the knowledge about interferometry techniques and encourage researchers in development of new interferometry applications.


Introduction
Synthetic aperture radar (SAR) has great advantage of being able to observe large area accurately in any weather. It can measure various properties of the earth Boerner (2003), e.g., the topography, the vegetation Hajinsek et al. (2009), and the landscape changes caused by earthquakes or volcanoes Gabriel et al. (1989). One of the common usages of SAR is Interferometric synthetic aperture radar (InSAR), which can measure the landscape with an interferogram of SAR images. The SAR interferogram is made from two complex-valued maps obtained by observing identical place, named "master" and "slave." The phase information of the interferogram corresponds to the ground topography. To generate digital elevation model (DEM) from interferogram, phase unwrapping process is required. However, an unwrapping process is disturbed seriously by singular points (SPs), the rotational points existing in the phase map.
There are three origins in the SP generation. One is the low SNR caused by low scattering reflectance. Another one is the sharp cliff and layover. The landscape can be so rough that the aliasing occurs. The last one is the local distortion in the co-registration of the master and the slave. That is, the reflection, scattering and fore-shortening can be different in the two observations with slightly different sight angle, resulting in local phase distortion in the interferogram. Filtering and unwrapping methods can solve first two origins. On the other hand, the last origin, the local distortion, has been generally ignored.
Without the distortion, no SP is expected through an appropriate co-registration of non-aliasing master and slave maps. Usually the co-registration is realized by maximizing the amplitude cross-correlation of the maps in macro scale, while by maximizing the complex-amplitude correlation in micro scale. The correlations require an averaging process over a certain area for sufficient reduction of included noise. However, a wide-area averaging degrades the locality in the matching required to eliminate the distortion. This trade-off brings a limitation in the co-registration performance. In short, the difference in the reflection, scattering and fore-shortening yields local distortion, and SPs are generated inevitably by the cross-correlation process.
In this chapter, we firstly introduce the basis of InSAR and its SP problem. Secondly, we introduce a local and fine co-registration method which employs the number of SPs as evaluation criterion (SPEC method Natsuaki & Hirose (2011)). Finally, we demonstrate the effectiveness of the improvement by comparing the DEMs generated from interferograms which co-registered with and without the SPEC method. For experiment, we use the data of Mt. Fuji observed by JERS-1 which was launched by JAXA (Japan Aerospace Exploration Agency). Mt. Fuji has an ordinary single volcanic cone shape.   Figure 1 shows the observation system of InSAR. We define wave length as λ, elevation angle as θ, distances from ground to the master and the slave satellite as R m and R s , distance between the master and the slave as B CT , relative angle of the master and the slave as γ CT . The phase value Φ of the interferogram corresponds to R m and R s as

InSAR and SP problem
Geometrically, R m − R s corresponds to B CT and γ CT as From (1) and (2), the relationship between Φ and θ is expressed as If there is a height increment between the neighbor pixels, the increment of Φ is From (4) and (5), the relationship between the interferogram phase increment ΔΦ and the height increment of the observation point ΔH can be expressed as which indicates that the phase gradient of 2π corresponds to the height gradient of λR m sin(θ) 2B CT cos(θ−γ CT ) .
In order to analyze the ground topography, we have to unwrap, line integrate, the phase information. As the ground topography is the conservative field, its contour integral should be zero.
c ΔΦds ′ = 0( 7 ) However, there are many non-zero rotational points, namely singular points (SPs), in the interferogram. As shown in Fig.2, if there is a rotational point in the interferogram, phase unwrapping will fail. We assume that there are three origins of the SP emergence.
1. Low SNR caused by low scattering reflectance 2. Sharp cliff and layover 3. Local distortion in the co-registration of the master and the slave Origin (i) is generally thought as the main reason of SPs which should be erased. SPs generated by origin (ii) should remain. The third reason (iii) has been conventionally ignored. A pixel in the master representing a small local area should completely correspond to a pixel in the slave that represents the same area. However, we assume that the slight difference of the radar incidence direction between the master and the slave distorts this correspondence in sub-pixel order, and that interferograms show these local distortions as massive SPs. In the next section, we introduce the local-o-registration method to solve the local distortion. To create an interferogram with master and slave, we have to co-register them in advance as they observe the same place from slightly different angle. The typical co-registration process is explained as follows Tobita et al. (1999). First, we affine-transform the slave map adaptively to maximize the cross correlation between the master and slave amplitude maps in a macro scale, e.g., 64×64 pixels. Next, we maximize the complex-valued cross correlation locally in 1/32 subpixels with interpolation, e.g., 8×8 pixels. Figure 3(a) shows the interferogram of Mt. Fuji created by this method, and Fig.3(b) gives its SP plot. This interferogram has 304×304 pixels and contains 11,518 SPs. In Fig.3(a), the phase value is shown in gray scale, in which a white dot stands for π as the phase value and a black dot means −π. In all figures in this article, the up-down direction is the azimuth and the left-right direction is the range. In Fig.3(b), a white dot indicates a clockwise SP, while a black dot shows a counterclockwise one.

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Local, Fine Co-Registration of SAR Interferometry Using the Number of Singular Points for the Evaluation www.intechopen.com Figure 4 represents how the interferogram and its SP plot change when the slave shifts leftward or rightward by integral multiple of 1/8 pixel from the maximum cross-correlation position for the right half area in the black square in Fig.3(a). It is obvious that many SPs move, emerge, or disappear, but the fringes in the interferogram show only small changes.
The emergence and the disappearance occur rather locally than the scale of correlation calculation. This fact suggests that we need a more local, and consequently nonlinear, co-registration process in addition to the conventional one. The changes in the fringes is not so large, which means that the rough landscape is not changed by this additional process, but the local precise co-registration is expected to improve the local landscape since, basically, no SPs are expected in non-distorted interferogram.
The aim of our proposal is to improve the accuracy of DEM by removing the local distortion.
In this removal process, we pay attention to the number of SPs as explained in the following section. Based on this idea which came from the result of the above preliminary experiment, we introduce our new method below.

Proposal of the SPEC method
Interferogram ( Fig. 6. Relationship between the blocks and the interferogram pixels: (a)Blocks for making regular 16-look mean-filtered interferogram in 1 pixel coordinate system, (b)blocks equal to (a) in 1/8-pixel coordinate system, (c)a SP in the interferogram made in 1/8-pixel coordinate system, and (d)local movement of the interpolated slave to delete the SP.
16 (=8×2) interferogram pixels. That is, to make an interferogram, in this paper, 8 times azimuth compression and 2 times range compression are required (i.e., to make a 304×304 pixels interferogram, 2432×608 pixels master and slave are required). Next, we find SPs in it and co-register interpolated master with interpolated slave locally and nonlinearly as follows.

Details of the local and nonlinear co-registration based on the number of SPs
Figures 6 and 7 are schematic diagrams of our local and nonlinear co-registration based on the number of SPs. We call the 8-times interpolated master and slave maps as "1/8-pixel  Fig. 7. The process of moving the slave: (a)Move S /8 (m, n) by 1/8 pixel as S new /8 (p, q; m, n) ← S /8 (p − 1, q − 1; m, n) in order to erase the SP made by the 4 blocks, (b)replace back S new /8 (p, q; m, n) and do the same process to S /8 (m, n + 1),(c)move S new /8 (p, q; m, n) by 1/8 pixel to a different direction, and (d)move S new /8 (p, q; m, n) by 2/8 pixels.
coordinate-system" maps, and the original ones as "1-pixel coordinate-system" maps. Table  1 lists the definitions used here, in which I /8 (m, n) denotes the pixel value of interferogram in the 1/8-pixel coordinate system, and I(m, n) denotes the value in the 1-pixel coordinate system simply. Coordinate (m, n) stands for global position in the (16-look) mean-filtered interferogram, while (p, q) represents the local position in the 1/8-pixel coordinate system.
Simultaneously, in this square I /8 (m, n)-I /8 (m + 1, n + 1), there is local distortion in the master and / or the slave. Then we therefore move one or some of the blocks among the four S /8 (m, n)-S /8 (m + 1, n + 1) blocks to modify the interferogram. For example, we move S /8 (m, n) locally in such a manner that S new /8 (p, q; m, n) ← S /8 (p + 1, q + 1; m, n) to try to erase the SP as shown in Fig.6 There are nesting stages in our method. First, we move the slave by 1/8 pixel and replace S /8 (m, n) as S new /8 (p, q; m, n) ← S /8 (p − 1, q − 1; m, n) a ss h o w ni nF i g . 7 ( a ) . T h e nw ec h e c k whether the SP disappears with this operation or not. If it does not, we move S /8 (m, n + 1), S /8 (m + 1, n),a n dS /8 (m + 1, n + 1) in the slave in turn, in the same direction ( Fig.7(b)). If it is impossible to erase the SP with the above up-leftward movements, we employ other seven directions (Fig.7(c)). Then for the remaining SPs, we try 2/8 shifts in the same way ( Fig.7(d)). If the SP cannot be removed with up to 8/8(=1) shifts, we abandon the elimination of the SP there, and try to erase the next one in the interferogram.
We apply the above process to all of the SPs in the interferogram iteratively. If we find that there is no erasable SP any more, we apply a similar shifting process for 4 (=2 × 2) big blocks. For example, we shift S /8 (m, n), S /8 (m, n + 1), S /8 (m + 1, n),a n dS /8 (m + 1, n + 1) simultaneously as a single large block.
We can also apply a 9 (=3 × 3) bigger block movement afterward, if needed. Figure 8 presents the changes of the phase map and corresponding SP distributions when we apply the SPEC method. The shown area is the black-squared part in Fig.3(a). As shown in Figs.8(a) and (b), there were 1,014 SPs in the original interferogram. The first iteration in our proposed method erased more than 60 percent of them, resulting in 396 SPs (Figs.8(c) and (d)). In the second iteration, 324 points left (Figs.8(e) and (f)). For the present data, with 1-block movement, no SP was erased anymore. With the additional 4-block move, our proposed method decreased the SP number to 184 (Figs.8(g) and (h)). With the 9-block move, our method decreased the SP number to 171 (Figs.8(i) and (j)), where our method finally erased about 83% of the SPs in the original interferogram. Figures 9(a) and 9(b) show the resulting interferogram and its SP plot for the data in Fig.3(a). The number of the SPs decreased from 11,518 to 1,865. The decreasing ratio was about 83% again for the whole interferogram.  method. We find improvement in some regions in the SPEC result in (b). For example, the dotted-square region in Fig.10(b) (zoomed in Fig. 11 ) shows more accurate ridges than those in Fig.10(c). It is obvious that the valleys and edges in Fig.11(b) are more distinct than those in Fig.11(c). We compared the mean signal-to-noise ratio (MSNR) (≡ squared height range / mean squared error) and the peak signal-to-noise ratio (PSNR) (≡ squared height range / peak squared error) based on the true data. We intend that our SPEC method compensates the local phase distortions in the interferogram. As this ability is similar to filtering (i.e., phase estimation), we calculated whether the SNR of the filtered interferogram changes if we co-register the interferogram with our proposed method. We used the iterative LS technique for unwrapping and the complex-valued Markov random field model (CMRF) filter Yamaki & Hirose (2009)      before the CMRF filter works. That is why the CMRF filter could estimate the fringes of the interferogram more precisely.

Conclusion
We proposed a new method of co-registration, namely, the SPEC method. This method uses the number of SPs in the temporary interferogram as the evaluation criterion to co-register the master and slave maps locally and nonlinearly. By applying our method to real data, we found that the SPEC method successfully improves the quality of the DEM in many cases.At the same time, we found that the SPEC method make ambiguous fringes clearer in its appearance. Our present method uses only the number of singular points as the evaluation criterion. In the future, we have the possibility to use other information, in addition to the SP number, to improve the performance further.

Recent Interferometry Applications in Topography and Astronomy
Edited by Dr Ivan Padron ISBN 978-953-51-0404-9 Hard cover, 220 pages This book provides a current overview of the theoretical and experimental aspects of some interferometry techniques applied to Topography and Astronomy. The first two chapters comprise interferometry techniques used for precise measurement of surface topography in engineering applications; while chapters three through eight are dedicated to interferometry applications related to Earth's topography. The last chapter is an application of interferometry in Astronomy, directed specifically to detection of planets outside our solar system. Each chapter offers an opportunity to expand the knowledge about interferometry techniques and encourage researchers in development of new interferometry applications.

How to reference
In order to correctly reference this scholarly work, feel free to copy and paste the following: Ryo Natsuaki and Akira Hirose (2012). Local, Fine Co-Registration of SAR Interferometry Using the Number of Singular Points for the Evaluation, Recent Interferometry Applications in Topography and Astronomy, Dr Ivan Padron (Ed.), ISBN: 978-953-51-0404-9, InTech, Available from: http://www.intechopen.com/books/recentinterferometry-applications-in-topography-and-astronomy/local-fine-co-registration-of-sar-interferometry-usingthe-number-of-singular-points-for-the-evaluat