SAR Interferometry: a Quick and Dirty Coherence Estimator for Data Browsing

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1 SAR Interferometry: a Quick and Dirty Coherence Estimator for Data Browsing A. Monti Guarnieri, C. Prati Dipartimento di Elettronica - Politecnico di Milano Piazza. da Vinci, Milano - Italy Ph.: Fax: e.mail: monti@elet.polimi.it; prati@elet.polimi.it Abstract Usual coherence estimation in SAR interferometry is a time consuming task since an accurate estimation of the local frequency of the interferometric fringes is required. This paper presents a fast algorithm for generating coherence maps, mainly intended for data browsing. The proposed estimator is based on the speckle similarity of coherent SAR data, and is thus independent of fringe frequency. The following advantages, with respect to the usual estimates, are achieved: i- The estimator is more than 100 times faster, achieved at the cost of a reduced statistical confidence. ii- The estimator is not affected by possible local frequency estimation errors. iii- The estimator can be directly applied to single look detected images. The theoretical derivation of the statistical properties of the frequency independent estimator is carried out in the stationary case. The non stationary case is then analyzed on real ERS SAR images. 1 Introduction Interferometric SAR coherence is an indicator of achievable interferogram quality (the higher the coherence the better the quality) [1,, 3, 4, 5, 6, 7]. Thus, to select SAR pairs that actually can be exploited for interferometric applications, the analysis of coherence should be carried out routinely on those pairs that show a suitable perpendicular baseline [8]. With the huge amount of interferometric SAR data that are now and potentially will be available from satellite SAR missions (ERS-1, ERS-, RADARSAT, SIR-C/-SAR III), the computing time to generate interferograms and coherence maps in a semioperative scenario should be kept as short as possible. It is well known that the usual coherence estimation is a very time consuming task as an accurate estimation of the local frequency of the interferometric fringes is required. In this paper a fast algorithm for generating coherence maps (e.g. to be exploited for interferometric data browsing and to speed up the image registration step) is presented. It will be shown that the proposed technique is much faster than the usual techniques (at the cost of a reduced statistical confidence) as it is independent of the local frequency of the fringes. Moreover it is more robust as it is not affected by possible local frequency estimation errors. The usual coherence estimator The coherence between two SAR images v 1 (n; m) and v (n; m) (here regarded as random processes) can be theoretically defined on a pixel basis as the correlation coefficient between two zero-mean, complex gaussian random variates v 1 = v 1r + jv 1i ; and v = v r + jv i [10]: = je [v 1 v r i ]j i (1) E hjv 1 j E hjv j In general the coherence in the imaged scene changes from area to area (or even from pixel to pixel). However it cannot be estimated on the basis of a single pixel from two SAR images. Nevertheless, coherence in stationary regions can be estimated as ensemble averages can be substituted by spatial averages (i.e. by assuming mean ergodicity). In practice, can be estimated from (1) by substituting the ensemble averages with spatial sampled averages (i.e. by assuming process ergodicity in a small estimation area of say N M pixels). It should be noted, however, that even if the ergodicity 1

2 hypothesis holds (i.e. all the scatterers within the estimation area are independent with identical statistical properties) the two images differ by a deterministic phase (i.e. the interferometric phase (n; m)) that should be estimated and compensated. Thus, the following usual coherence estimation ^ holds: ^ = P N P M v n=0 m=0 1(n; m) v (n; m) e,j(n;m) q PN n=0 P M m=0 jv 1(n; m)j P N n=0 N M being the size of the estimation window (in complex samples)..1 Statistical properties P N m=0 jv (n; m)j () The mean value and variance of the estimator () have been computed in reference [11]. It can be shown that the estimate ^ is unbiased for high values of independent complex samples (i.e. ' N M for full bandwidth images), and/or (i.e. 1 ). The estimator s variance can be approximated for high values of and/or as follows [1]: b = (1, ) (3) The estimator s mean value carried out from 400 simulations is plotted in Fig. 1.a. The estimator s variance is plotted in Fig. 1.b and compared with the approximation (3). It will be used in the following as a reference to evaluate the variance of the proposed estimator.. Computational efficiency From a computational point of view the estimation of the interferometric phase (n; m) (even if assumed linear in both azimuth and slant range directions) is very time consuming. On the other hand, if it is not compensated, the coherence estimate ^ would be biased by the local frequency of the fringes that is proportional to the topography (i.e. with large baselines even a rolling topography would have a strong biasing effect). Computational efficiency can be greatly improved if a phase independent estimator is adopted, as shown in detail in section A frequency independent coherence estimator It is well known that detected SAR images are affected by so-called speckle noise. However, in SAR interferometry, speckle should be considered a useful signal rather than noise [15]. Thus, speckle differences between two interferometric images can be used to estimate their coherence, as will be shown [1]. et us consider the normalized cross-correlation of the detected SAR images jv 1 j and jv j :, E jv 1 j, jv j = p E [jv1 j 4 ] E [jv j 4 ] The cross-correlation can be expressed as a function of the complex coherence [9] by exploiting the following well known property that holds for circular, zero mean, normal random variables [10]: E [x 1 x x 3 x 4 ]=E [x 1 x ] E [x 3 x 4 ]+E [x 1 x 3 ] E [x x 4 ]+E [x 1 x 4 ] E [x x 3 ] (5) If we pose x 1 = v 1 ;x = v ;x 3 = v 1 ;x 4 = v ; then: The expression (6), for v 1 v becomes: E jv 1 j jv j = E [v 1 v ] E [v 1v ]+E jv 1 j E jv j + E [v 1 v ] E [v 1v ]= = E jv 1 j E jv j + E [v 1 v ] E [v 1v ] (6) (4) E jv 1 j 4 =E jv 1 j (7) Eventually combining (4,6,7) we get the following relation between and : = E jv 1 j jv j + je [v 1 v ]j E [jv 1 j ] E [jv j ] = 1+ (8)

3 Thus, the absolute value of the complex coherence can be estimated 1 as follows: ^ M = p ^, 1 ^ > 1 0 ^ 1 (9) where b = 3.1 Computational efficiency r PN P N P M jv n=0 m=0 1(n; m)j jv (n; m)j P M PN P jv n=0 m=0 1(n; m)j 4 M (10) jv n=0 m=0 (n; m)j 4 SAR coherence interferometric maps are carried out by space averaging and cross-correlating the complex images on small overlapped windows. The coherence estimator (10) is stationary, thus its implementation on overlapped windows corresponds to a D FIR filtering. In appendix A it is shown that 50 flops (floating point operations) per image sample are required to compute a coherence map, regardless the size of the estimation window. On the other hand, if the usual coherence estimate b is performedaccording to (), the computingtime is much higher due to the necessity of frequency estimation (in both directions). Even if the simplest (and cheapest) frequency estimation technique [14] is exploited (i.e. two covariance lags are computed in each estimation window), it can be shown that the computation of b is more than 9 times that of b M. Moreover, the accuracy of the technique is poor, as shown in Fig. 3, having a strong negative impact on the coherence estimation. As an example, let us assume an estimation window of N r =8;N az =16samples (range, azimuth), and a range fringe frequency f r =0:5 f s (f s being the sampling frequency). The frequency estimation error fr should be much smaller than 1=N r =0:1, otherwise the estimated coherence would be meaningless (e.g. if fr =1=N r then b =0for whatever value of the actual coherence ). However from Fig. 3 we see that the required frequency estimation accuracy can be achieved for coherence values > 0.7. For smaller values a more accurate frequency estimator (e.g. MUSIC [13]) should be adopted, even though this involves a much greater computational complexity (e.g. the computation of b is about 100 that of b M ). 3. Statistical properties As in the case of the estimator ^ also the estimator ^ M is not biased for large values of and/or. Moreover it is shown in appendix B that its variance can be approximated for large values of : j^m j = 1 8, , The mean values and the standard deviation of the estimator ^ M are shown in Fig.. 4 A comparison of the two estimators The relative accuracy of the usual estimator ^ is higher than that of the proposed estimator ^ M. Namely they are: b = 1 (1, ) p p ^M = , p As a consequence the number of independent samples that should be used to get an accurate coherence estimate is larger for ^ M than for b: If 1 and indicate respectively the number of independent samples used to estimate the coherence with the estimators b and ^ M then the accuracy of the two estimators is the same provided that the following ratio is maintained: 1 = , (1, ) (11) 1 The coherence estimator ^M has been preferred to other possible frequency independent estimators for two reasons: i- the computation of the squared absolute value is faster than the computation of the simple absolute value (the calculation of a square root would be required) with no substantial loss of performance. ii- no local averages are subtracted from the cross-correlation since they would increase the bias of the estimate. 3

4 The ratio between and 1 is plotted in Fig. 4 as a function of. For values of coherence greater than 0:8, ' 3 1,for =0:5, =5 1 and for 0:3, On the other hand, the frequency independent estimator ^ M is faster, more robust and simpler than ^ in many respects: i- no stationarity of the local interferometric phase is required ii- it is independent of any focusing phase error not affecting speckle amplitude iii- it can be directly applied to detected single look iv- it is independent of any fringe frequency or phase estimation error. 4.1 Experimental results The almost stationary ERS-1 scene of the area of Bonn (Fig. 5) was used for a visual comparison of the coherence maps carried out with the estimators ^ and ^ M. The perpendicular baseline of the interferometric pair is about 190 meters. In Fig. 6 the interferometric fringes flattened with the phase term derived from the available orbital data are shown. The coherence maps generated with the estimators ^ and ^ M are shown in Fig. 7 and Fig. 8 respectively. In both cases the estimation window size is pixels. The higher variance of the second estimator is clearly visible as comparing the two images. Nonetheless the coherence structure is well identified in both images. On the other hand, it should be pointed out that on the same workstation the first coherence map was generated in about 30 minutes whereas the second one took about 9 seconds. Note also that the coherence map of Fig. 7 might be affected by possible errors of the estimated local frequency. Fig. 9 shows the coherence map achieved with the estimator ^ M by increasing the estimation window size to 1 1 pixels. It can be seen that the shape of the Rhein river is well defined as the estimated coherence is very low (dark in the image). It can be noted, however, that the coherence ^ M is higher than ^ almost everywhere. Moreover, some structures of moderate coherence (up to 0:6) appear in the upper and lower parts of the image where one expects low coherence. It can be seen that the structures closely resemble the shape of the detected image (see Fig. 5). These effects can be explained by the non stationarity of the absolute values envelope within the estimation window. In this case, some correlation between the envelopes of the detected images is always found even in the case of totally uncorrelated speckles, as explained in the next section. 5 Non stationary scenes Up to now we have discussed the stationary case for the sake of simplicity. In practice, however, the effect of non stationary absolute values within the estimation window should be taken into consideration [1]. As it was shown in the Bonn case, whenever a SAR image contains non stationary absolute values (e.g. corresponding to mountainous areas) the proposed coherence estimator ^ M is strongly biased by the absolute values envelope. et us consider the case of two stationary detected interferometric signals x and y with a given spatial correlation coefficient xy, the same power and mean. As an example the signal x is shown in Fig. 10. et us now multiply both signals by a coefficient a 1 in the first half of the estimation window (here assumed one dimensional for the sake of simplicity) and a in the second half as shown in Fig. 10. It can be shown that the spatial correlation coefficient of the resulting signals u and v has the following expression: uv = k + xy 1+k (1) where k = (a 1, a ) (a 1 + a ) From (1) it can be seen that even if the signals x and y are uncorrelated (i.e. xy = 0) the correlation coefficient of the signals u and v is greater than zero whenever a 1 6= a. Thus, in order to reduce this effect on non-stationary SAR images a sort of automatic gain control (AGC) should be applied to both detected images before computing the coherence estimate ^ M. A possible way to perform the AGC in practical cases is the following: i - The two detected images are multi-look averaged (e.g. summed together and low-pass filtered, - a 3 3 or 4 4 average has been seen to be enough in all the examined ERS images). ii - A small constant (white light) is added to the filtered image. iii - Both detected images are divided by the image generated in the first two steps. The AGC effect can be observed in Fig. 11 that shows the detected image of the area of Bonn after compensation: the speckle is still alive, whereas the absolute value non stationarities have been almost completely eliminated. The new coherence image As a matter of fact the correlation between detected images is usually exploited to get a rough estimate of the images registration parameters. 4

5 generated with the proposed estimator b M of the area of Bonn shown in Fig. 1 should be compared with that shown in Fig. 9: note that most of the effects cited in the previous section have been compensated. A more dramatic effect of the AGC can be observed in the coherence map that was made from an ERS-1 interferometric pair of the area of Bonn with a baseline of about 900 meters (quite close to the decorrelation baseline of the ERS interferometric system). The interferometric fringes are shown in Fig. 13. Fig. 14 shows the coherence map ^ M carried out with no AGC and a 1 1 window. From this image it is clearly seen that the coherence image closely resembles the structures of the detected image shown in Fig. 5 even where, as shown by the fringes of Fig. 13, almost no coherence is expected. Instead, the coherence map with the AGC (Fig. 15), does not show such a strong dependence on the absolute value envelope. From a computational point of view, the cost of the AGC is not negligible compared to the coherence map generation (it takes about 0% of the total computing time). Nevertheless the overall computing time is still much shorter than that required by the usual coherence estimator ^. 6 Conclusions A new coherence estimator based on the similarity of the speckle signal of SAR coherent images (slightly different from that derived in [1]) has been presented. The estimator s variance has been computed and compared to that of the usual coherence estimator (i.e. derived from complex single look images) in order to determine the window dimension to be used in practice. As in [1], the bias introduced by scenes non-stationarities has been recognized, however a cure has been found (the Automatic Gain Control) that is essential to make the proposed estimator work with non-stationary scenes. The new coherence estimator is much more efficient, in terms of computational costs, than the usual estimators, despite a reduced statistical confidence. It appears to be a good compromise between quality and efficiency and could be usefully exploited for routine data browsing to generate catalogs of usable interferometric SAR images. Finally it should be pointed out that in some cases when the estimation of the local frequency of the interferograms could be problematic (e.g. in mountainous areas with large baselines), the proposed estimator might be more reliable than the usual one. 7 Acknowledgments The authors would like to thank Mr. F. Gatelli and Mr. G. Quario for the generation of the experimental results. Our gratitude goes to Prof. F. Rocca who reviewed the paper and helped us with useful hints. Finally, we would like to thank ESA-ESRIN for the constant support and ACS for partly financing this work. 5

6 A Efficient implementation of the estimator b M et ^ M (k; l) be a coherence map, obtained by using the estimator ^ M on overlapped boxcar windows of the two interferometric images: ^ M (k; l) = The three D sequences in (14) are defined as follows: p ^(k; l), 1 ^ > 1 0 ^ 1 Pn;m ^(k; x 1(k, n; l, m) l) = (14) n;m x (k, n; l, m) rp P n;m x 11(k, n; l, m) x 11 (k; l) =jv 1 (k; l)j 4 ; x (k; l) =jv (k; l)j 4 ; x 1 (k; l) =jv 1 (k; l)j jv (k; l)j (15) v 1 ;v being the two complex images. An efficient implementation of the three summation in (14) can be achieved by means of the following recursions (i.e. frequency sampling implementation of boxcar 3 filtering [16]): z(k; l) = y(k; l) = M,1 m=0 N,1 n=0 m=0 x(k; l, m) =z(k; l, 1) + x(k; l), x(k; l, M ) (16) M,1 x(k, n; l, m) =y(k, 1;l)+z(k; l), z(k; l, M ) where x(k; l) is any one of the three sequences defined in (15), y(k; l) the filtered output, and the sequence z(k; l) is used to store temporaryresults. The scanning order in (16) assumesk as the fastest varying index; the initialization of the two recursions can be obtained by assuming x(k;,m:::, 1) = z(k;,:::, 1) = 0. If (16) is used to carry out one of the three summations in (14), a computational cost of 3 4 real additions for each output sample results. This cost does not depend on the window size. It can be verified that the total computational cost of ^ M (k; l), as defined in (13,14), is 50 flops per image sample. (13) B Statistics of the estimator b M The expected value of ^ has the following expression: E [^] =E 4 q Pk jx kj jy k j 5 (17) Pk jx kj 4 1 Pk jy kj 4 where all the sums go from 1 to, being the number of independent samples. For large values of, both the numerator and denominator of (17) can be approximated with gaussian random variates A and B, with very sharp probability density functions centered around their mean values 4 A and B: A A + a E [^] ' E = E B B + b A = E " 1 k jx k j jy k j # 3 A similar scheme can be derived also for Bartlett windows. 4 As an example, (0) and () show that, for = 00 samples, the ratios A=A and B =B are less than

7 B = E "s 1 k jx k j 4 1 Since a and b are zero mean gaussian random variates, small compared with A and B, E [^] can be further approximated as follows: A + a A + a E [^] ' E ' E 1, b = B + b B B Thus, the variance of ^ can be approximated as follows: ^ = E " A + a B 1, b B # k, E jy k j 4 # E a, =, A B +, B E b A, 4, B E 3 [ab] A B, E [ab], B (18) " # A B, E, [ab] (19) B All the terms that appear in (19) can be expressed as a function of the SAR image power (i.e. P x and P y ) and of the correlation. i A = E hjx k j jy k j =P x P y E a = E A,, A 1 = E 4 i h = 1 E jx i j 4 jy i j 4i + = P x P y k j (, 1) E jx i j jy i j jx j j jy j j 3 5, Px P y h jx i j 4 jy i j 4 jx j j jy j j i, P x P y, 1 +16, 8 (0) The term B can be conveniently computed by approximating the square root up to the second term of its Taylor series expansion: "s # 1 B = E jx k j 4 1 hp i p p jy k j 4 ' E C + c ' C C, k 8, C E c (1) where: C = E B E B = E " 1 k E c = E C,, C = E B 4,, C jx k j 4 1 =4P x P y + P x P y E B 4 = E " 1 4 k = 4, 6 3 E E h k jy k j 4 # = 1 E h jx k j 4 jy k j 4 i + (, i 1) E hjx k j 4 jy h j 4, , 1 () h m n jx k j 4 jy h j 4 jx m j 4 jy n j 4 # hjx k j 4 jy h j 4 jx m j 4 jy n j 4i h + 1 E jx k j 8 jy h j 4 jy n j 4i + jx k j 4 jy k j 4 jx m j 4 jy n j 4i jx k j 4 jy h j 8 jx m j 4i h + 4 E, , (3) =16P 4 x P 4 y

8 From (1) and () the expression of E b = E B,, B can be derived. Also the last term E [ab] can be usefully approximated by using the Taylor series expansion of the square root: where E [AB] = " = E E AB = 1 k h Ap C + c 1 3 E " i = 3, 3 E E h =8P 3 x P 3 y E [ab] =E [AB], A B (4)!s jx k j jy k j 1 i k p ' A C E + p [Ac] C k h jx k j 4 1 jx i j jy i j jx k j 4 jy h j 4 # k jy k j 4 # = A p E [B ]+ E AB p E [B ] hjx i j jy i j jx k j 4 jy h j 4i h + 1 E jx i j 6 jy i j jy h j 4i + jx i j jy i j jx k j 4 jy k j 4i jx i j jy i j 6 jx k j 4i + 1 E h + 1, , 6 p Then, by approximating B with E [B ] in (4) (i.e. the same approximation used for E [AB]) the following expression of E [ab] holds: E [ab] ' 1 p, A E [B E AB! ]+ p E [B ] Finally it can be shown that for very large values of (i.e. >500) all the terms that form (19) can be further approximated giving the following simple expression: ^ = The variance of the coherence estimator ^ M can thus be approximated: 0 p 1 ^M B d b, 1 C db A b b=e[^m ] , = 1+, 4 + 3, 6 +5, 1 (5), 1 = 1 8, , (6) 8

9 References [1] C. Prati, F. Rocca, Range resolution enhancement with multiple SAR surveys combination, IGARSS 9. pp , 199 [] H. Zebker, J. Villasenor, Decorrelation in interferometric radar echoes, IEEE Transactions Geoscience and Remote Sensing, vol 30 no. 5, pp , 199. [3] E. Rodriguez, J. M. Martin, Theory and design of interferometric synthetic aperture radars, IEE Proceedings-F, vol. 139, no., pp , 199. [4] D. Just, R. Bamler, Phase statistics of interferograms with applications to synthetic aperture radars, Applied Optics, vol. 33, no. 0, pp , [5] F. Gatelli, A. Monti Guarnieri, F. Parizzi, C. Prati, F. Rocca, The wavenumber shift in SAR interferometry, IEEE Transactions Geoscience and Remote Sensing, vol. 3, no. 4, pp , [6] F. Rocca, C. Prati, P. Pasquali, A. Monti Guarnieri, ERS-1 SAR Interferometry techniques and applications, European Space Agency report n /9/hge-i, [7] J. O. Hagberg,. M. H. Ulander, J. Askne, Repeat-pass SAR interferometry over forested terrains, IEEE Transactions Geoscience and Remote Sensing, vol. 33, no, pp , [8] G. A. Solaas, ERS-1 Interferometric Baseline Algorithm Verification, European Space Agency report no. ES-TN- DPE- OM-GS0, Aug [9] N. R. Goodman, Statistical Analysis Based on Certain Multivariate Complex Gaussian Distribution (An Introduction), Ann. Math. Stats, vol. 34, no. 15, pp , [10] A. Papoulis, Probability, Random Variables and Stochastic Processes, Mc. Graw-Hill, [11] R. Touzi, A. opes, Statistics of the Stokes parameters and of the complex coherence parameters in one-look and multi-look speckle fields, IEEE Transactions Geoscience and Remote Sensing, vol. 34, no., pp , [1] E. J. M. Rignot, J. V. Van Zyl, Change detection techniques for ERS-1 SAR data, IEEE Transactions Geoscience and Remote Sensing, vol. 31, no. 4, pp , [13] P. Stoica, A. Nehoray, MUSIC, Maximum ikelihood, and Cramer-Rao Bound, IEEE Transactions Acoust. Speech and Signal Processing, vol ASSP 34, pp , Apr [14] U. Spagnolini, -D Phase Unwrapping and Instantaneous Frequency Estimation, IEEE Transactions Geoscience and Remote Sensing, vol. 33, no. 3, pp , [15] C. Prati, F. Rocca, imits to the resolution of elevation maps from stereo SAR images, International Journal of Remote Sensing. vol. 11, no. 1, pp.15-35, [16] A. V. Oppenheim, R. W. Shafer, Digital Signal Processing, Prentice-Hall, Inc. New York,

10 =3 = COHERENCE 0.14 (a) =3 theoretical experimental = COHERENCE (b) Figure 1: Mean value, (a); and standard deviation, (b); of the estimator ^ for different numbers of independent samples. 10

11 =3 = COHERENCE (a) theoretical experimental =3 0.5 = COHERENCE (b) Figure : Mean value and standard deviation of the estimator ~ for different numbers of independent samples. 11

12 f=0 f=0.5 f=.5 f=0.75 CRB COHERENCE Figure 3: Mean squared error of the frequency estimate got by exploiting the AR model of order 1, for = 18 independent samples and different values of the actual frequency. The lower curve is the Cramer-Rao Bound. 1

13 10 / coherence Figure 4: Ratio between the number of real samples and complex samples 1 to get the same estimation accuracy with the estimators ^ M and ^. 13

14 Figure 5: Detected ERS-1 image of the Bonn area, 30 km (azimuth - horizontal) 100 km (range). The Rhein river is clearly visible crossing the image from left to right. 14

15 Figure 6: Interferometric fringes of the area of Bonn. Note the lower part of the figure where a horizontal strip of random phase (non coherent) corresponds to aliased data. Also the upper part of the image shows quite noisy fringes. 15

16 Figure 7: Coherence map carried out with the usual estimator ^. The estimation window size is =

17 Figure 8: Coherence map carried out with the proposed estimator ^ M. The estimation window size is = 11. The higher estimates variance with respect to the previous figure is clearly visible. 17

18 Figure 9: Coherence map carried out with the proposed estimator ^ M. The estimation window size is '

19 U 15 a a Figure 10: An example to show the coherence estimation bias generated by non-stationary absolute values within the estimation window. Upper: stationary signal x. ower: non-stationary signal u: 19

20 Figure 11: Detected image of the area of Bonn after the AGC application: the speckle is still alive, whereas the absolute value non stationarities have been almost completely eliminated. 0

21 Figure 1: Coherence map of the area of Bonn generated with the AGC. 1

22 Figure 13: Interferometric fringes (Bonn area) of the 900 meter baseline interferometric pair.

23 Figure 14: Coherence map ^ M carried out with no AGC and a 1 1 window on a 900 meter baseline interferometric pair. 3

24 Figure 15: Coherence map ^ M carried out with the AGC and a 1 1 window on a 900 meter baseline interferometric pair. 4