SAR PHASE UNWRAPPING USING ADAPTIVE RECURSIVE SMOOTHING Olaf Hellwich Chair for Photogrammetry and Remote Sensing ¨ Munchen, Technische Universitat ¨ D-80290 Munich, Germany E-mail:
[email protected] URL: http://www.photo.verm.tu-muenchen.de Commission III, Working Group III/6
KEY WORDS: SAR, Interferometry, Phase Unwrapping
ABSTRACT Regarding the interferometric evaluation of Synthetic Aperture Radar (SAR) data, phase unwrapping is one of the most difficult processing steps. In the past several methods have been developed, such as the least squares or the branch cut methods. In this paper the influences of terrain shape and phase noise on phase unwrapping are discussed. A new approach to phase unwrapping is taken which consists of preprocessing the interferogram before the phase information is unwrapped: the interferogram is recursively smoothed, and the differences of consecutive smoothed versions of the interferogram are computed. The phase data of the difference interferograms and the final smoothed interferogram are unwrapped, and the unwrapped phases are added. The result of the process is the unwrapped interferometric phase. It is shown that the approach largely avoids the problem of ambiguous integration paths.
1 INTRODUCTION The phase difference of two SAR scenes is called interferometric phase. Relative interferometric phase is ambiguous, because it can only take values in the interval ; which is equivalent to a change of the range difference between an object point and the first and the second SAR antenna of half a wave length (in dual-pass interferometry). As range differences have to be evaluated which differ as much as many wave lengths, the ambiguity has to be solved. This is done by integration of the phase differences between neighbouring pixels, an operation usually called phase unwrapping. The resulting unambiguous data is called absolute interferometric phase. The phase difference between interferometric phase values of two is computed by wrapping neighbouring pixels i and i the difference i+1 i into the interval ; using the wrapping operator W
[0 2 [
+1
?
]?
]
= W ( i+1 ? i ) = i+1 ? i + 2 k; k 2 I such that 2] ? ; ]
In section 2 reasons for the occurance of residues in SAR interferograms and strategies towards a solution of the phase unwrapping problem are discussed. In section 3 a new phase unwrapping method using recursive smoothing is developed. In section 3.1 the theoretic background of the method is explained, and in section 3.2 a computer implementation of the method is shown. In section 4 experimental results are compared with the results of a standard branch-cut phase unwrapping method. Section 5 presents conclusions and recommendations.
(1)
where i+1 , i are either wrapped or unwrapped phase values, and I is the set of integers. Ambiguous integration paths in phase difference data have a manifestation in socalled residues. A residue exists when the integration of phase differences on a squared closed loop including four directly neighbouring pixels does not re. According to the sign of the sult in zero, but or integration result residues are called positive or negative. The handling of the residues is the core problem in SAR phase unwrapping.
2
phase data can be unwrapped without any ambiguities if a line connecting the positive and the negative residue is not crossed during the integration of phase differences. This connecting line is frequently called a branch cut. Finding branch cuts has been shown to be a difficult task (Gold¨ stein et al., 1988, Schwabisch, 1995, Flynn, 1996, Costantini, 1996, Costantini, 1997). Figure 1 will be used to illustrate other strategies for phase unwrapping explained in later sections.
?2
Figure 1 a) shows an example of a pair of residues in interferometric phase data. It is caused by a large interferometric phase value in the center pixel of the data which could either be due to noise or due to terrain topography. The This research was partially funded by Deutsches Zentrum f¨ur Luft- und Raumfahrt DLR e.V. under contract 50EE9423.
2 PHASE UNWRAPPING PROBLEMS There are several reasons for problems, i.e. the occurance of residues, in phase unwrapping. The following list gives an overview:
In shadow regions the received SAR-signal is not related to the terrain, but only due to the dark signal of the sensor. It can be considered a form of noise, and phase values of neighbouring pixels do not depend on each other. Thus, residues occur very often. In layover regions something similar happens. The signal of a pixel is composed from backscatter of spatially disconnected parts of the terrain. The resulting phase contains influences of geometric and backscatter properties of all these parts. It is presumably something like a weighted average. Therefore, the interfer-
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Figure 2: Pixels neighbouring in range direction separated by a difference of interferometric phases of
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Figure 1: Residue problem in phase unwrapping and treatment by smoothing. The interferometric phase and phase differences of neighbouring pixels are shown. Residues are found by clockwise integration of phase differences on a squared closed loop including four pixels. Positive and negative residues are indicated by a “plus” or a “minus” sign. a) wrapped phase, b) phase of terrain-approximating plane, c) difference of wrapped phase and phase of terrainapproximating plane, d) unwrapped phase.
r2 r’ 2 θ
ometric phase does not decribe a point on the surface of the terrain. As the combination of backscatter from spatially diconnected parts of the terrain will vary from pixel to pixel, the occurance of residues is rather probable. Furthermore, the terrain shape extracted from the interferometric phase information would deviate strongly from the true terrain shape.
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The shape of the terrain can cause phase differences between neighbouring pixels which have a magnitude larger than . In those cases the occurance of close-by residues is a consequence (except from special cases which would not allow to recover the terrain shape by SAR interferometry). Phase noise due to lacking coherence will cause residues.
In this work, SAR interferometry in regions of shadow or layover is not considered useful and will therefore not be discussed. Terrain shape- and noise-related residues will be treated more carefully. 2.1 Implications of Terrain Shape As can be seen in the example of Figure 1, residues occur if there is a single absolute phase difference between neighbouring pixels which is larger than in magnitude. Those phase differences can be caused by the shape of the terrain. Here, two cases of terrain shape are investigated for which phase differences larger than can occur: slope in range and slope in azimuth direction. Figure 2 shows two neighbouring pixels p and p0 separated by a range pixel spacing r located on a slope towards
p ∆r p’ Figure 3: Pixels neighbouring in range direction with a maximum slope away from the sensor which can be imaged the illuminating SAR sensor such that they have a phase difference . Then the ranges fulfill the equations
=
r10 = r1 + r r20 = r2 + r ? 4 ;
where is the wave length and 4 causes a difference of interferometric phases of . From this configuration expressions for the ground range distance x, the height difference z and the elevation angle x between p and p0 can be derived.
780 23 8 56 = 100 121 43 19 5 139 50 19 9
, For ERS-SAR with H km, r m, : cm and B m the following values result: x m, z m and x : . If the influence of the “flat earth” is taken into account before phase unwrapping, the values change to x m, z m and x : , i.e. a steeper and longer terrain slope could be treated without the occurance of residues.
Figure 3 shows two pixels p and p0 neighbouring in range direction which are located on a slope tilted away from the sensor. The slope is extreme in the sense that any steeper slope angle would result in shadow. For this case the difference of interferometric phases of p and p0 can be computed. For ERS-SAR the phase difference would be . This means that residues due to slope tilted away from the sensor cannot occur as the required phase differ cannot be reached. ence
0 =?
The results show that residues caused by terrain slope in range direction only occur in regions of foreshortening. Due to the steep incidence angle these are relatively frequent in case of ERS-SAR.
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2.3 Strategies Towards a Solution
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∆z p ∆x Figure 4: Pixels neighbouring in azimuth direction separated by a difference of interferometric phases of Figure 4 shows two neighbouring pixels p and p0 separated by an azimuth pixel spacing y which have a phase difference of . In this case the ranges fulfill
r10 = r1 r20 = r2 ? 4 :
42
For ERS-SAR with y : m a ground range distance of x m and a height difference of z m between pixels neighbouring in azimuth direction cause a phase difference of . If the slope direction strictly corresponds with azimuth direction the elevation angle in azimuth direction , i.e. the slope could become relatively would be y steep before residues occur.
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2.2 Noise If interferometric phase noise is uncorrelated over longer distances – an assumption which seems to be reasonable – residues occur in short-distanced pairs of a positive and a negative residue. In real data this can be observed frequently. If pairs of residues are connected by branch cuts, phase unwrapping can be conducted without any difficulties. As long as phase noise is relatively low the identification of noise-related residue pairs should be easy. But as soon as noise reaches higher levels the positioning of branch cuts is difficult. Wrong connections of residues can lead to phase unwrapping errors. As can be seen from the example in Figure 1, the occurance of residues is not related to the ambiguity of relative interferometric phase; yet it is caused by wrapping the phase differences, i.e. by the ambiguity of phase differences between neighbouring pixels. The ambiguity of a phase difference is only relevant when its true magnitude is larger than . This observation has some consequences concerning the practical integration of phase differences. First, noise-free interferometric phase data from flat terrain does not contain any residues and can therefore be unwrapped without any difficulties. Second, if the phase difference caused by imaging geometry and terrain topography is close to , then a small amount of noise will be sufficient to slightly exceed which is equivalent to a wrapped difference close to . Then a residue occurs. Third, if the phase difference caused by imaging geometry and terrain topography is close to , then only a comparatively large noise contribution causes a phase difference change due to wrapping.
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Three strategies will be discussed. The first strategy uses approximate terrain information. If a simulated interferogram of an approximation of the terrain surface is subtracted from the measured interferogram before phase unwrapping, the occurance of terrain shape-related residues can be largely avoided. Then the resulting unwrapped interferogram is used to refine the approximate terrain information. In principle, this strategy is applied by all approaches which subtract an interferogram of the “flat earth” from the measured interferogram before phase unwrapping. Certainly, the “flat earth” is only a comparatively rough approximation to the terrain. A second strategy frequently being used to avoid residues is smoothing the interferogram, implicitly assuming that smoothing does not introduce any deviations with respect to the terrain surface or an approximation to it. Unfortunately, it has to be noted that a smoothed interferogram conceptually differs significantly from the simulated interferogram of an approximation to the terrain surface whose use was suggested in the preceeding paragraph. The simulated interferogram is based on the information of a smoothed terrain, whereas the smoothed interferogram is only based on the information of the measured interferogram, i.e. the wrapped phase inherent to the complex numbers of the interferogram which still contains the problems due to terrain shape-related residues. The first strategy reduces terrain shape-related problems, whereas the second one deals with noise-related residues. A third strategy tries to overcome the lack of approximate terrain information with the help of strongly smoothed results of phase unwrapping. According to this strategy, at the beginning a phase unwrapping result is generated, e.g. from a smoothed interferogram. In spite of the fact that the phase unwrapping method used to do this is not perfect or even weak and that there are deficits in its results it is assumed that these results contain some reasonable information about the shape of the terrain. This information could be used to ease phase unwrapping in a second trial. For this purpose the unwrapped phase data is strongly smoothed, e.g. with a Gaussian with a very large . With this measure the influence of the deficits is kept on a relatively small level. From the smoothed unwrapped phase, complex data of an interferogram with constant magnitude is generated. It is subtracted from the measured interferogram in the same way as the interferogram of the “flat earth”. The remaining interferogram does then contain less residues due to terrain shape. A following processing step using the second strategy would be eased as there is a smaller lack of information of the smoothed interferogram with respect to the terrain information it is supposed to infer. In a lucky case, the remaining interferogram would consist of data belonging to a single fringe interval, i.e. the unwrapping results would be in the interval ; . Hence, the third strategy can be applied interatively, each iteration improving the approximate interferogram subtracted from the measured interferogram.
[0 2 [
In section 3 a method is developed which applies smoothing to ease SAR phase unwrapping. In section 4 it is tried to illuminate the usefulness of smoothing for phase unwrapping by some simulations.
3
ADAPTIVE RECURSIVE SMOOTHING
The new approach to phase unwrapping is demonstrated using the example introduced before. If the terrain that gave origin to the phase data in Figure 1 a) is replaced by a terrain-approximating tilted plane, the phase data for this plane (cf. Figure 1 b) would have to be unwrapped. In this data no residues occur. If the interferogram of the tilted plane is subtracted from the interferogram of the terrain, the difference of interferograms has the phase data shown in Figure 1 c). This phase data also does not contain any residues. Adding the unwrapped phase values of the tilted plane and the difference of interferograms results in the absolute interferometric phase (Fig. 1 d). In contrary to the unwrapping shown in Figure 1 a), it has this time been achieved without any residue management. (Davidson and Bamler, 1996) proposes a method for phase unwrapping based on recursively smoothed interferograms. In this paper, a method is proposed which more rigorously exploits the advantages of recursive smoothing by adjusting the amount of smoothing to the data. It is therefore called adaptive recursive smoothing. 3.1 Theory The new phase unwrapping method is based on two observations. In an extremely smoothed version, an interferogram does not have any structure, but is a tilted plane without any detail. For this interferogram the first observation holds. Proposition 1 For a maximum smoothed interferogram equal to an interferogram of a tilted plane without any noise influences, residues do not occur. Proof: For a tilted plane, phase differences between neigbouring pixels are equal in range direction and in azimuth direction. Thus, integration of these differences on a closed loop including four neighbouring pixels always has a result equal to zero, and residues do not occur. The second observation is based on a smoothing method which allows a gradual adjustment of the amount of smoothing. Such a method is e.g. Gaussian smoothing where the parameter allows arbitrary amounts of smoothing. Proposition 2 There exists always a smoothed version of an interferogram whose difference with respect to the unsmoothed (original) interferogram does not contain any residues. Proof: For a Gaussian smoothing operation conducted with parameter the smoothed interferogram is equal to the unsmoothed one. Then the difference between smoothed and unsmoothed interferogram is zero in all pixels, and the differences between neighbouring pixels computed from this data are also equal to zero. Provided that the amount of smoothing is a continuous function of , continuous enlargement of allows to generate a non-trivial smoothed version of the interferogram which only slightly differs from the unsmoothed interferogram such that the difference interferogram does not contain residues.
=0
As a sequence of recursive smoothing operations finally results in the maximum smoothed tilted-plane version of the interferogram, phase unwrapping is possible without any residues. For each smoothing operation, a parameter has to be chosen which is larger than zero and which does not result in residues in the difference interferogram. Finally, the unwrapped difference interferograms and the unwrapped maximum-smoothed interferogram have to be combined. 3.2 Implementation The general strategy for adaptive recursive smoothing in SAR phase unwrapping is as follows: The original interferogram is smoothed and the difference between original and smoothed interferogram is computed. Then the difference interferogram is unwrapped. In a second step, the smoothed interferogram is smoothed again, and the difference between smoothed and repeatedly smoothed interferogram is computed. Also these differences are unwrapped. This procedure is repeated recursively. It comes to an end when the newly smoothed interferogram does not contain any residues. The phase data of this interferogram is also unwrapped. Finally, the unwrapping results, unwrapped phases from the final smoothed interferogram and the difference interferograms, are added. The result of this operation is the final unwrapped interferogram. For each smoothing step, the amount of smoothing - i.e. for Gaussian smoothing the parameter - is adjusted to the data. The problems of this strategy are that the amount of allowable smoothing has to be determined individually and that the number of smoothing steps might be very high. In order to limit the number of smoothing steps, an algorithm has been developed and implemented which allows some residues. The algorithm uses an initial amount of smoothing ini , a minimum amount of smoothing min conducted during one step of the recursion, and a smoothing increment to determine a maximum amount of smoothing imax during step i of the recursion. Regularly imax is the largest value of which does not result in residues in the unwrapping of the difference interferogram. If the largest value which does not result in residues is less than min , imax is set to min .
A pseudo-code version of the algorithm for phase unwrapping using adaptive recursive smoothing follows: 1. Set the current amount of smoothing to the initial amount of smoothing: i ini .
=
2. Set maximum amount of smoothing for current recursion step to the minimum amount of smoothing to be conducted in one recursion step: imax min .
=
3. Smooth the interferogram (smoothed version of preceeding recursion step) using i . 4. Compute the difference between input and output of the smoothing operation. 5. Check if difference interferogram contains residues. (a) Difference interferogram contains residues: i. Decrease current amount of smoothing: i i .
= ?
(b) Difference residues:
interferogram
does
not
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contain
i. Memorize amount of smoothing: imax i . ii. Increase current amount of smoothing: i i .
=
roof ridge
= +
6. If i
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> imax , go to 3. canopy
7. Smooth the interferogram using imax . 8. Compute difference interferogram. 9. Unwrap phases of difference interferogram. 10. If the interferogram resulting from the last smoothing operation does not contain any residues, or if a maximum number of recursion steps has been conducted, go to 13. 11. Prepare the next recursion step i imax .
+1 by setting i+1 =
12. Go to 3 and conduct next recursion step i
+ 1.
13. Unwrap phases of interferogram resulting from step 7. 14. Add result of step 13 and results of all preceeding steps 9. In contrary to the theory explained in section 3.1, the unwrapped phases resulting from step 14 do contain residues. This is to avoid smoothing steps with an unacceptably small amount of smoothing and to limit the recursive processing to a certain number of steps.
4
RESULTS
The new phase unwrapping method was tested on two simulated and two real data sets. The first simulated data set contains an absolute phase difference between neighbouring pixels greater than causing a pair of residues as discussed in section 2.1 (cf. Figure 2). The second simulated data set is a pyramid with a ledge which has been used previously by (Fornaro et al., 1996, Fornaro et al., 1997). It contains many fringes and cases of phase differences between neighbouring pixels greater than several fringes. The first real data set was evaluated to show how effectively adaptive recursive smoothing reduces residues and branch cuts. The second real data set is a larger example demonstrating a standard application scenario. The first simulated phase data set contains a saddle-roof structure with a tilted canopy (s. Fig. 5). Principally, phase differences between neighbouring pixels are smaller than . By adjusting the absolute phase difference p between the roof ridge and the upper end of the canopy residues due to “terrain shape” can be generated. If p < , then there are no residues. If, otherwise, p > , one or more pairs of a positive and a negative residue occur on the line separating the saddle roof from the canopy. By adding Gaussian noise to the phase data further residues can be caused. In reality such a data set could be generated when the SAR sensor is horizontally flown along the canopy, perpendicularly to the roof ridge. Then, the canopy is the equivalent of a horizontal plane imaged by the sensor; the wall between
Figure 5: Perspective view of the first simulated phase data set. The phase difference p between the roof ridge and the pixel in front of the almost vertical wall structure is adjustable. Values of p : and p : have been chosen.
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canopy and saddle roof is a hill slope tilted towards the sensor; and the saddle roof is the opposite hill slope tilted away from the sensor.
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First, p was set to : , i.e. a state without terrain shape-related residues, and pixel-wise independent Gaussian noise with : was added to the phase data. Then the phase data was wrapped using the wrapping operator (1). Figure 6 shows the results of unwrapping with adaptive recursive smoothing (I.a) and with a standard branch cut method (II.a) comparable to (Goldstein et al., 1988) as reference. The unwrapping operations of the adaptive recursive smoothing method are conducted with the same branch-cut algorithm. The left half of the data is the canopy, the right half the saddle roof. The roof ridge horizontally splits the right half into two parts. Using adaptive recursive smoothing the data is unwrapped correctly. There are only very few residues left in the final results. In Figure 7 a) the differences between unwrapped phase values and the noiseless simulated (i.e. true) phase values are shown as an empirical probability density function. It : . agrees very well with the normal distribution for This means that the phase unwrapping operation does not cause errors in addition to the Gaussian phase noise. In contrast to this, the reference method has to handle many residues. It is not able to connect the residue pairs properly. Consequently, longer branch cuts are generated resulting in three areas with larger phase unwrapping errors.
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In a second test, the phase difference p was set to : simulating two residues due to terrain shape. In the result of the adaptive recursive smoothing method (Fig. 6 I.b) a zone of erroneous phase values occurred between the canopy and the ridge (center of the data). This is due to the phase differences larger which cannot be treated correctly by smoothing the interferogram, as smoothing complex values of an interferogram implicitly uses wrapped-phase information. The erroneous phase data also show in the empirical probability density function as a higher probability of phase values close to (Fig. 7 b). The performance of the reference method (Fig. 6 II.b) was very similar to the first case.
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Finally, the simulation was repeated with p : , but without phase noise. In this case, the adaptive recursive smoothing method uses strong smoothing from the first it-
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Figure 7: Empirical probability density function of the difference between true (noiseless) absolute phase and unwrapped interferometric phase resulting from adaptive recursive smoothing. The normal distribution with : (dashed line) is the desired result for cases a) and b) including Gaussian noise. For case c) without noise all differences should be .
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Figure 6: Results of phase unwrapping using adaptive recursive smoothing (I) and a standard branch cut method (II). a) and b) with noise, c) without noise. Unwrapped phase is displayed by varying shades of grey. Residue and branch-cut pixels are shown in black. Except from residues and branch cuts, Fig. II.c is equivalent to the true phase data. eration onwards. This propagates the errors from the central zone to the whole data set such that phase unwrapping completely fails (Figs. 6 I.c and 7 c). The ridge zone has phase values lower than the upper part of the canopy indicating that phase differences where applied with there nominal wrapped value. At the margins of the data, absolute phase differences of occur between neighbouring pixels adjusting for the wrong interpretation in the center of the scene. In contrast to this negative result, the branch cut method finds the pair of residues and connects it correctly (Fig. 6 II.c). Therefore, its results agree perfectly with the true phase data.
2
The second simulated data set was the Fornaro pyramid with a ledge (Fornaro et al., 1996, Fornaro et al., 1997) shown in the perspective view of Figure 8. The size of the data set is pixels. The phase is at the border and in the center, i.e. the height of the pyramid is : fringes. The ledge is horizontal and goes to the inside of the pyramid such that there are vertical edges on three sides towards the center of the pyramid. The maximum height of the vertical edges is : . Figure 9 contains the wrapped phase data. It shows that the fringe to which the ledge belongs, i.e. its absolute phase, cannot be identified without additional assumptions. In case of vertical edges on each of its four sides the ledge could also be a raised ridge to the outside of the pyramid. Therefore, it is obvious that a phase
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Figure 8: Perspective view of the Fornaro pyramid
unwrapping method which does not have explicit information about absolute phase differences > between neighbouring pixels will have difficulties to acquire the correct phase unwrapping solution. This was also recognized by (Fornaro et al., 1997) who introduce special weights along the vertical edges around the ledge, an information equivalent to correct branch cuts. The data set was simulated twice with Gaussian noise of : and : . According to (Just and Bamler, 1994, Figure 3), these noise contributions are equivalent to
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Figure 9: Wrapped phase of the Fornaro pyramid
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Figure 11: Results of phase unwrapping for real SAR data: a) interferometric phase unwrapped with adaptive recursive smoothing, b) residues of adaptive recursive smoothing, c) interferometric phase unwrapped with branch cut method, d) residues of branch cut method
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Figure 10: Central cross-section of the Fornaro pyramid. The graph compares the true phase (continuous line), the the unsmoothed unwrapped phase of the data set with : (dashed line), and the smoothed unwrapped phase of the data set with : (dotted line).
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: and : , respectively, as coherence values used by (Fornaro et al., 1997). The first, more coherent data set was unwrapped using adaptive recursive smoothing with parameters min : and : , and recursion steps. In Figure 10 the resulting unwrapped phase (dashed line) is compared with the true phase (continuous line). Note that the unwrapped phase does not contain any influence of smoothing. As can be seen in Figure 10 true and unwrapped phase agree perfectly except from the ledge area where difficulties were expected as additional information about vertical phase edges was not introduced.
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The second, less coherent data set was unwrapped using parameters min : and : , and recursion steps. According to the third strategy to solve terrain shape and noise-related problems, proposed in section 2.3, strongly smoothed phase unwrapping results were subtracted twice from the measured interferogram before
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deriving the final unwrapped phase. Figure 10 compares it (dotted line) with the unwrapped phase of the more coherent data set and the true phase. Note that this time the unwrapped phase includes a certain amount of smoothing because the unsmoothed data contain too many residues. The smoothed unwrapped phase is generated in step 14 of the algorithm for adaptive recursive smoothing if the phase unwrapping results of the first difference interferograms (step 9) are not included in the addition. Again, the unwrapped phase and the true phase agree perfectly except from the ledge area and – to a lesser degree – the top of the pyramid where the smoothing caused some differences. The first test with real data was conducted for a small subsection of an ERS tandem data set. Figures 11 a) and b) show the interferometric phase data unwrapped with the adaptive recursive smoothing method and the corresponding residues and branch cuts. For comparison the same data was unwrapped using the branch-cut method of (Goldstein et al., 1988) without recursive smoothing. The results are shown in Figures 11 c) and d). With the recursive smoothing approach 24 residues occur which have to be connected by 1 branch cut pixel. The standard approach has to deal with 81 residues and uses 41 branch cut pixels. This means that the new method reduces residues by 70% and simplifies their connection with branch cuts considerably.
14 14
The second real data set is a km2 ERS-SAR tandem interferogram of the Weilheim area in Upper Bavaria acquired on Dec. 6 and 7, 1995, on a descending orbit. Figures 12 and 13 show magnitude and coherence, respectively. The average coherence is approximately : . The pixel data set was unwrapped using adap: , : and tive recursive smoothing with min recursion steps. After subtracting an interferogram of the strongly smoothed unwrapping results from the measured
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Figure 12: Frost-filtered histogram-equalized magnitude of a km2 ERS-SAR scene of Weilheim, Upper Bavaria
14 14
Figure 14: Unwrapped phase of the ERS-SAR tandem interferogram. Black: residues and branch cuts; grey: unwrapped phase method
branch cut method adaptive recursive smoothing
number of residues 2140 1660
number of branch cut pixels 3540 1018
1540
890
(“flat earth” subtraction)
adaptive recursive smoothing (approximate terrain subtraction)
Table 1: Comparison of the standard branch cut method, adaptive recursive smoothing after subtracting an interferogram of the “flat earth”, and adaptive recursive smoothing after subtracting an interferogram of the “flat earth” and an interferogram of strongly smoothed phase unwrapping results 5 CONCLUSIONS AND RECOMMENDATIONS
Figure 13: Histogram-equalized coherence of an ERS-SAR tandem interferogram of Weilheim, Upper Bavaria
interferogram the method was applied a second time (cf. section 2.3). The unwrapped phase is displayed in Figure 14. Table 1 compares the results of both adaptive recursive smoothing procedures with the results of the standard branch cut method. Adaptive recursive smoothing reduces in one iteration, and by althe number of residues by in two iterations. The length of branch cuts is together reduced by in one iteration, and by altogether in two iterations.
28% 71%
22%
75%
In a recent publication (Bamler et al., 1996) it is shown that the least squares method for phase unwrapping does not perform correctly as soon as residues occur in the data, because residues are not a result of additive, zero-mean, isotropic, and stationary noise which is implicitly assumed by least squares approaches. To avoid residues in order to be able to use a least squares method for phase unwrapping (Bamler et al., 1996) proposes a multi-resolution approach to phase unwrapping. Here such a multi-resolution method is developed based on adaptive recursive smoothing. The advantage of this method is that it - at least theoretically - completely avoids residues. In this case any simple integration of phase differences on arbitrary paths can be used for phase unwrapping. The most severe limitation of the new method lies in the fact that smoothing complex values of interferograms is always conducted based on wrapped phase values, and, therefore, does not directly approximate the terrain. This may result in phase unwrapping failures where the shape of the terrain causes phase differences between neighbouring pixels
larger than especially when pairs of residues are separated by a larger distance. For practical applications, the proposed method poses two more problems. First, each smoothing operation would result in a new version of the interferogram, and, therefore, the demand for storage capacity would be rather high. A solution to this problem is blockwise processing of the data. For this, the area of interest is subdivided into rectangular chips, and for each chip only the final unwrapping result is stored. Second, recursive smoothing is a time consuming operation. This problem could be simplified by filtering in frequency domain. Furthermore, the avoidance of residues accelerates the unwrapping operations, as optimal branch cuts between residues do not have to be found any longer.
ACKNOWLEDGMENTS The author would like to thank Richard Gawron and Werner Stempfhuber for programming and experimental computations.
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