We present a modification to the adaptive Goldstein radar interferogram filter which improves the quality of interferometry products. The proposed approach makes the Goldstein filter parameter alpha dependent on coherence, suchthat incoherent areas are filtered more than coherent areas. This modification minimizes loss of signal while still reducing the level of noise

Baran, Ireneusz

Kampes, B.

Lilly, Peter

Perski, Z.

Stewart, Michael

English

coherence adaptive spectral interferometric phase

Baran, Irek

Stewart, Marc P.

Kampes, Bert

Perski, Zbegnew

Institute of Transport Research:Publications

A Modification to the Goldstein Radar Interferogram Filter

espace@Curtin

2114 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 41, NO. 9, SEPTEMBER 2003A Modification to the Goldstein RadarInterferogram FilterIreneusz Baran, Mike P. Stewart, Bert M. Kampes, Zbigniew Perski, and Peter LillyAbstract—We present a modification to the adaptive Gold-stein radar interferogram filter which improves the quality ofinterferometry products. The proposed approach makes theGoldstein filter parameter alpha dependent on coherence, suchthat incoherent areas are filtered more than coherent areas. Thismodification minimizes loss of signal while still reducing the levelof noise.Index Terms—Phase filtering, phase statistics, radar interferom-etry, synthetic aperture radar (SAR).I. INTRODUCTIONA LTHOUGH interferometric synthetic aperture radar(InSAR) applications are well established, the improve-ment of the interferometry technique and the quality of itsproducts are desirable to further enhance its capabilities. Sincethe late 80s, many applications of radar interferometry havebeen developed, including the observation of ground motionover agricultural areas [1], creating high-accuracy digitalterrain models (DTMs) [2], and deformation monitoring of theearth’s crust with millimeter accuracy at very dense spatialsampling [3]. In addition, InSAR has proven to be a viablemethod for measuring surface deformation associated withsubsidence due to underground mining activities [4].The application of radar interferometry to DTM productionand deformation monitoring encounters problems due to noisein the interferogram phase measurement, which is caused bywater vapor in the atmosphere and decorrelation effects that canbe categorized as follows: 1) thermal, 2) temporal, 3) geomet-rical, 4) Doppler centroid, and 5) processing-induced decorre-lation [5].The quality of DTM and displacement maps can be improvedby many methods at different processing levels. One of themis filtering of the interferometric phase, as demonstrated in[6]–[10]. However, while filtering reduces noise in the interfer-ogram, it does not necessarily enhance or recover the signal.Manuscript received March 18, 2003; revised June 12, 2003. The work of I.Baran was supported by the IPRS Scholarship from Curtin University of Tech-nology, Western Australian School of Mines.I. Baran and P. Lilly are with The Western Australian School of Mines,Curtin University of Technology, Perth, WA 6845, Australia (e-mail:I.Baran@curtin.edu.au; lillyp@wasm.curtin.edu.au).M. P. Stewart is with The Western Australian Centre for Geodesy,Curtin University of Technology, Perth, WA 6845, Australia (e-mail:stewart@vesta.curtin.edu.au).B. M. Kampes is with the German Aerospace Center (DLR), Oberpfaffen-hofen 82234, Germany (e-mail: Bert.Kampes@dlr.de).Z. Perski is with the Department of Geological Mapping, University ofSilesia, Sosnowiec 41-200, Poland (e-mail: Perski@us.edu.pl).Digital Object Identifier 10.1109/TGRS.2003.817212Furthermore, the impact of the filter can significantly changethe structure of the interferogram.II. GOLDSTEIN RADAR INTERFEROGRAMFILTERGoldstein and Werner [9], [10] proposed an adaptive radarinterferogram filter based on the concept of multiplication ofthe Fourier spectrum of a small interferogram patch byits smoothed absolute value to the power of anexponent(1)where is the filter response (the spectrum of the fil-tered interferogram); is a smoothing operator; and arespatial frequencies; andis the filter parameter. Patches are de-fined as a small part of the interferogram and are overlapped toprevent discontinuities at the boundaries. The filter parameteris an arbitrarily chosen value between zero and one and has thebiggest impact on the filter performance. For the value of ,the multiplication factor becomes one, and no filtering occurs.However, for large values of, the filtering is significant.A problem using the Goldstein radar interferogram filter oc-curs when a high value of the parameteris applied. By sub-tracting the filtered from the unfiltered interferogram, a residualsystematic phase trend appears, indicating a loss of resolution inthe filtered phase [10].III. FILTER MODIFICATIONThe quality of the interferometric phase can be estimated di-rectly from the interferometric data as a function of the complexcoherence () between two complex SAR images and(the interferometric pair) [11](2)or derived theoretically based on the sensor characteristics,signal processing algorithms, and acquisition conditions.Bamler and Hartl [12] define the phase standard deviation fora single-look ( ) interferogram as a function of the absolutevalue of the complex coherence given by(3)where is the Euler’s dilogarithm, defined as(4)0196-2892/03$17.00 © 2003 IEEEBARAN et al.: MODIFICATION TO THE GOLDSTEIN RADAR INTERFEROGRAM FILTER 2115Fig. 1. Phase standard deviation ( ) versus absolute coherence (L indicatesthe multilook number).Fig. 2. Simulated coherence map (multilook factorL = 1).Fig. 1 shows the relationship between the absolute coherenceand phase standard deviation. It shows that the phase stan-dard deviation is reduced as either the number of looks or co-herence increases. From the relationship expressed in (3), thecoherence can be recognized as a direct measure of the inter-ferogram phase noise. Therefore, it can be used to modify theGoldstein interferogram filter. The filter is modified by makingthe parameter in (1) dependent on the value of absolute co-herence(5)Equation (5) shows the modified Goldstein radar interferogramfilter, where the exponent has been replaced by . The pa-rameter is the mean value of the absolute coherence computedover the effective corresponding patch (patch minus overlap) onthe coherence map. This approach insures that the coherenceand filter response are of the same size and in the same place.Furthermore, it preventsbeing affected by the coherence fromthe overlapped part of the patch.TABLE IFILTER PARAMETERSFig. 3. Cross section over the simulated interferogram without noise. Eachline shows the phase after filtering for different value of the filter exponent. Thephase offsets (' ) are clearly visible.Fig. 4. Mean absolute coherence versusmax phase offset caused by theGoldstein (horizontal lines) and modified Goldstein filter (dashed line).The new filter parameter is automatically set up based on thecoherence value and without any user input. Such a modifica-tion adapts the Goldstein interferogram filter by preventing theareas of high coherence (less noise) being overfiltered, but al-lows stronger filtering in areas where there is a low coherence(high noise). Thus, the loss of resolution in the interferogramdu to filtering can be reduced in the areas where the coherenceis high.The interferogram and the coherence map are divided intooverlapping rectangular patches. Patches are defined as a smallpart of the interferogram of even-sized pixels (power of two),which ensures fast Fourier transform (FFT) efficiency. Theoverlap between patches is required to prevent discontinuitiesat the boundaries. The maximum overlap size is described asthe half of the patch size minus one. The choice of overlap sizeof 14 pixels in the subsequent section and Table I represents atradeoff between computation time and minimization boundaryffects.For each patch of the interferogram, the spectrumis computed (using two-dimensional FFT). At the same time,from the effective corresponding patch on the coherence mapthe mean value is computed. Finally, the spectrum of the in-terferogram patch is weighted by multiplying it by its smoothed2116 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 41, NO. 9, SEPTEMBER 2003(a) (b)Fig. 5. (a) Differences between filtered and unfiltered simulated interferogram for Goldstein and (b) for modified Goldstein filter.and scaled absolute spectrum to the power of an exponent.Smoothing of the absolute spectrum is performed by spatial con-volution with a block kernel ( ). As the block kernel, theFourier spectrum of the spatial moving average has been ap-plied. The size of the smoothing kernel must be small to preventthe spectrum being oversmoothed.IV. NUMERICAL VALIDATION USING A SIMULATED DATASETThe performance of the filter modification was analyzedusing simulated and real interferometry data. The wrappedphase interferogram was simulated by radar-coding a givenDTM and wrapping its phase. Although the Goldstein filteruses amplitude (the complex signal is filtered), the amplitudewas not simulated, as it is not required for reliable results. Thecoherence map was simulated based on three decorrelationeffects [5]: 1) geometrical (based on DTM), 2) thermal (basedon European Remote Sensing 1 and 2 (ERS-1/2) satellitesensor specifications), and 3) temporal (fractal surface), sothe final coherence was estimated as the product of each ofthe decorrelation factors (Fig. 2). Based on the relationshipthat links phase variance () with the absolute value of thecomplex coherence (3), a matrix of standard deviations wasgenerated from the modeled coherence map.A Gaussian noise matrix was computed by pointwise multi-plication of the phase standard deviation matrix with a matrixof normally distributed random numbers with mean zero andvariance one. Finally, the noise was added to the simulated in-terferogram.To this simulated dataset the Goldstein and modified Gold-stein interferogram filters were applied and tested. The filter pa-rameters for all tests are listed in Table I. The idea of choosingthe filter parameter was that this value demonstratesTABLE IIUSED DATASETS OFRADAR IMAGESrelatively large impact on the filter performance and the loss ofthe signal.In the first test, the modified Goldstein interferogram filterwas used to filter the interferogram without any noise at all toassess the filter impact on the wrapped phase itself in terms ofthe filter parameter . The uniform coherence value is as-sumed to be over the entire interferogram each time filtering isapplied. Filtering was performed for coherences from 0.0 up to0.9 with a step of 0.1. Fig. 3 shows an example cross sectionover the simulated interferogram without noise, which showsthe sample of a number of cross sections analyzed. Vertical off-s ts ( ) between the original and filtered phase due to impactof the filter can be seen. The magnitude of the offset increasesas the filter parameter increases. Furthermore, the biggestvertical offsets are experienced in areas of large curvature andthey always occur in the direction of concavity.Fig. 4 shows the characteristics of the Goldstein and modi-fied Goldstein interferogram filter in terms of the phaseoffset caused by the filters, versus the mean absolute coherence( ). The graph is constructed based on the phase differences be-tween the interferogram without noise and its filtered versions.The dashed line on the graph shows the phase offsetsfor the modified Goldstein filter. It appears that for high coher-ence values, the phase offsets become small. After analyzingthe whole interferogram, the maximum differences between theoriginal and filtered phase were found to be98 for and14 for . On the other hand, the two horizontal linesBARAN et al.: MODIFICATION TO THE GOLDSTEIN RADAR INTERFEROGRAM FILTER 2117(a) (b)Fig. 6. (a) Unfiltered and (b) filtered wrapped phase interferogram using modified Goldstein interferogram filter. The arrow indicates cross-section location (colorbar in radians).show the expected phase offset for the Goldstein filterand two different values of parameter 80 forand 59 for .In this case, the differences between the original and filteredphase are high, constant and strongly dependent on the chosenvalue.In the next test, both filters were applied to the simulatedphase interferogram with noise and the simulated coherencemap. The differences between the filtered interferogram andits original version were determined and are shown in Fig. 5.The visible patterns show the systematic phase trend indicatingsome loss of resolution in the filtered interferogram. Both filterslead to some loss of resolution. However, the modified filter re-duces loss of resolution over the part of the interferogram thatis characterized by higher coherence. The rectangular windowsin Fig. 5 show the area where the major differences between thefilters occur.V. NUMERICAL VALIDATION USING A REAL DATASETIn the final test both filters were applied to real data. Tworadar images acquired by satellites ERS-1 and ERS-2 overthe Western Australia Goldfields mining region have beenprocessed using the Delft University public domain InSARsoftware “Doris” [13]. The perpendicular baseline is 195 m.The radar image details are listed in Table II. The precise orbitsfor the satellites provided by Delft were used [14].After filtering the interferogram, an improvement in the phasenoise is clearly visible (Fig. 6). A closer look at the examplecross section (Fig. 7) over the wrapped phase interferogramshows the same characteristics of the Goldstein filter and itsmodified version observed in the simulated dataset. The hori-zontal line on Fig. 6(b) indicates the cross-section location. OnFig. 7, the circle “A,” large vertical phase offset can be ob-served for the Goldstein filter. At the same time, due to the highcoherence ( ) in this region, the modified Goldstein filterdoes not create the high phase offset.Fig. 7. Cross section over the unfiltered and filtered wrapped phaseinterferogram.Fig. 8. Cross section over the surface heights.To check how the phase offset affects terrain heights,phase unwrapping and height computation were performedusing SNAPHU, phase unwrapping software developed bythe Interferometry Group at Stanford University. SNAPHUimplements the Statistical-cost, Network-flow algorithm forphase unwrapping proposed by Chen and Zebker [15]. Fig. 8shows the cross section over the surface heights along the samelocation as Fig. 7.As expected, the vertical phase offset propagates directly intothe height of the DTM introducing additional systematic error( ). The magnitude of such error depends strongly onand is proportional to the height ambiguity. In the same way thephase offset will propagate into the deformation map leading to2118 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 41, NO. 9, SEPTEMBER 2003some loss of resolution. The improvement of the modified filterwhen applied to deformation monitoring may also be significantif the deformation signal is very small.VI. CONCLUSIONModification to the Goldstein interferogram filter has beenproposed. The filter parameteris replaced by the new param-eter that is dependent on the value of absolute coherence.This modification adapts the filter to the wrapped phase inter-ferogram more effectively by preventing the areas characterizedby high coherence being overfiltered. On the other hand, it al-lows stronger filtering on the areas with low coherence. How-ever, both filters perform almost identically in the areas wherecoherence is low (less than 0.3).In summary, interferograms that are characterized by high co-herence will benefit from the modified Goldstein radar interfer-ogram filter, as it decreases loss of resolution. In particular, in-terferograms of regions of rough terrain should benefit when themodified Goldstein filter is used.ACKNOWLEDGMENTThe European Space Agency (ESA) supplied SAR data underProject 1123. The authors would like to thank the four anony-mons reviewers who provided valuable and constructive com-ments on the earlier version of this letter.REFERENCES[1] A. K. Gabriel, R. M. Goldstein, and H. A. Zebker, “Mapping small el-evation changes over large areas: Differential radar interferometry,”J.Geophys. Res., vol. 94, no. B7, pp. 9183–9191, 1989.[2] H. A. Zebker and R. M. Goldstein, “Topographic mapping from inter-ferometric synthetic aperture radar observations,”J. Geophys. Res., vol.91, no. B5, pp. 4993–4999, 1986.[3] T. Strozzi, U. Wegmüller, Ch. Werner, and A. Wiesmann, “Measurementof slow uniform surface displacement with mm/year accuracy,” inProc.IGARSS, July 2000, pp. 2239–2241.[4] Z. Perski, “The test of applicability of land subsidence monitoring byInSAR ERS-1 and ERS-2 in the coal mine damaged region (UpperSilesia),” Int. Arch. Photogramm. Remote Sens., vol. 32, no. 7, pp.555–558, 1998.[5] H. A. Zebker and J. Villasenor, “Decorrelation in interferometric radarechoes,”IEEE Trans. Geosci. Remote Sensing, vol. 30, pp. 950–959,Sept. 1992.[6] B. Reeves, J. Homer, G. Stickley, D. Noon, and I. D. Longstaff, “Spatialvector filtering to reduce noise in interferometric phase images,” inProc.IGARSS, Hamburg, Germany, June 1999, pp. 260–263.[7] G. Bo, S. Dellepiane, and G. Beneventano, “A locally adaptive approachfor interferometric phase noise reduction,” inProc. IGARSS, Hamburg,Germany, 1999, pp. 264–266.[8] J.-S. Lee, P. Papathanassiou, T. L. Ainsworth, R. Grunes, and A. Reigber,“A new technique for noise filtering of SAR interferometric phase im-ages,”IEEE Trans. Geosci. Remote Sensing, vol. 36, pp. 1456–1464,Sept. 1998.[9] R. M. Goldstein and C. L. Werner, “Radar ice motion interferometry,”in Proc. 3rd ERS Symp., vol. 2, Florence, Italy, 1997, pp. 969–972.[10] , “Radar interferogram filtering for geophysical applications,”Geo-phys. Res. Lett., vol. 25, no. 21, pp. 4035–4038, 1998.[11] R. Bamler and D. Just, “Phase statistics and decorrelation in SAR inter-ferometry,” inProc. IGARSS, 1993, pp. 980–984.[12] R. Bamler and P. Hartl, “Synthetic aperture radar interferometry,”Inv.Prob., vol. 14, pp. R1–R54, 1998.[13] B. Kampes and S. Usai, “Doris: The Delft object-oriented radar interfer-ometric software,” inProc. 2nd Int. Symp. Operationalization of RemoteSensing, Enschede, The Netherlands, Aug. 16–20, 1999.[14] R. Scharroo and P. Visser, “Precise orbit determination and gravity fieldimprovement for the ERS satellites,”J. Geophys. Res., vol. 103, no. C4,pp. 8113–8127, 1998.[15] C. W. Chen and H. A. Zebker, “Two-dimensional phase unwrapping withuse of statistical models for cost functions in nonlinear optimization,”J.Opt. Soc. Amer. A, vol. 18, pp. 338–351, 2001.

A Modification to the Goldstein RadarInterferogram Filter

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A Modification to the Goldstein RadarInterferogram Filter

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