SAR interferometric phase statistics in wavelet domain

Re e ences 1 MARTiN-NEIRA, M., and GOUTOULE, J.M.: 'MIUS - a wo-dimensional ape u e-syn hesis adiome e o soil-mois u e and ocean salini y obse a ions', ESA Bull., 1997, (92), pp. 95-104 2 SIVESTRIN, p., BERGER, M., ERR, Y., and FONT, J.: 'ESA's Second Ea h Explo e Oppo uni y Mission: The Soil Mois u e and Ocean Salini y Mission - SMOS', IEEE Geosci. Remo e Sens. Newsl., 2001, (1 18), pp. 11-14 CAMPS, A., BAe, J., CORBELLA, I., and TORRES, F.: 'The p ocessing o hexagonally sampled signals wi h s anda d ec angula echniques: applica ion o 2D la ge ape u e syn hesis in e e ome ic adiome e s', IEEE T ans. Geosci. Remo e Sens. GRS-35, 1997, pp. 183-190 4 BA~, J., CAMPS, A., TORRES, F., and CORBELLA, I.: 'Angula esolu ion o wo-dimensional hexagonally sampled in e e ome ic adiome e ', Radio Sci., 1998, 33, (5), pp. 1459-1473 BA~, J.: 'Reliabili y analysis in ape u e syn hesis in e e ome ic adiome e s: applica ion o L-band MIRAS ins umen ', Radio Sci., 3 5 VALL-LLOSSERA, M., DUFFO, N., CAMPS, A., CORBELLA, I., TORRES, E, and 2001,36, (I), pp. 107-1 17 ~, Ku osis KS signi icance le el (%) SAR in e e ome ic phase s a is ics in wa ele domain ~, ., Ku osis KS Ku osis KS signi icance signi icance le el (%) le el (%) C. Lopez and X. Fab egas Scale 1 1 2.98 Syn he ic ape u e ada (SAR) in e e ome y is employed o ob ain opog aphic in o ma ion. Owing o noise, in e e ome ic in o ma ion has o be il e ed. The wa ele ans o m can be employed o il e he in e e ome ic phase, main aining he spa ial esolu ion, bu new signal models ha e o be s udied in his domain o u he p ocessing. 87.93 1 2.98 1 79.83 2.94 1 98.34 In oduc ion: Syn he ic ape u e ada in e e ome y (InSAR) is an es ablished echnique o ob ain in o ma ion abou he ea h's su ace opog aphy. The in e e ome ic phase is calcula ed as he phase di e ence be ween wo complex SAR images om he same a ea, bu aken om sligh ly di e en posi ions. Owing o he lack o in e e ome ic cohe ence (y I be ween bo h SAR images, he in e - e ome ic phase is noisy. In addi ion, he in e e ome ic phase is only known wi hin he in e al [-z, n), i being necessa y o unw ap i o eco e unambiguously he heigh in o ma ion. The unw apping p ocess is also a ec ed by phase noise, since i induces phase esidues. Phase il e ing is hus necessa y o educe noise e ec s. In he las decade, he wa ele ans o m (WT) has shown a big po en ial o image p ocessing applica ions. In he ield o SAR da a p ocessing, he use o he WT is eme ging since i allows p ocessing o SAR image y, keeping he spa ial esolu ion and image de ails. Since he physics behind SAR da a is comple ely di e en om ha o op ical images, any da a p ocessing has o ake his in o accoun . Thus, i is necessa y o e iew o e en o de ine new noise models adap ed o his p oblem. In his Le e , we p o ide a s udy o a signal model o he in e e ome ic phase in he wa ele domain. This model is alida ed wi h eal in e e ome ic da a. equency plane. N, has a one- o-one ela ion wi h he cohe ence IyJ p o iding, hus, he same in o ma ion [3,4]. Fo a cons an in e e ome ic phase and homogeneous noise (i.e. cons an Iyl), he pa ame e Nc, as well as he e ms COS(C$;) and sin (4:) a e cons an . The e o e, signal andomness is only due o ," and ,". The disc e e wa ele ans o m (DWT) can be seen as he addi ion o (weigh ed) andom a iables. By he cen al limi heo em, he weigh ed sum o iden ically dis ibu ed andom a iables can be app oxima ed by a Gaussian dis ibu ion. The e o e, ," and ," a e app oxima ely Gaussian dis ibu ed. To es i , a oiding any in e e ence om he phase d;, a cons an slope p oducing 20 pixel inges, co up ed wi h a noise equi alen o a cohe ence IyI = 0.6, has been simula ed. Table 1 shows a s a is ical es applied o e he eal pa o he in e e ome ic complex phase in he wa ele domain. As shown, since he use ul signal is concen a ed in he low equency band (LL), he wa ele bands (HL, LH and HH) p esen a ku osis close o 3 and he signi icance le els o he Kolmogo o -Smimo (KS) es , assuming a Gaussian dis ibu ion, a e high. These esul s demons a e ha ," and ," can be desc ibed by a Gaussian dis ibu ion. The same ag eemen is obse ed o any o he alue o Iyl, and o he imagina y pa o he complex in e e ome ic phase in he wa ele domain. The LL band dese es special a en ion. In his case, as he e is signal con en , he signal model will be ep esen ed by he eal and imagina y pa s o exp (j&') plus a Gaussian noise. The e o e, in his case, he ampli ude lpl +jp21 has a Rice dis ibu ion. This esul is equally alid o he es o he wa ele bands. The ampli ude in he wa ele domain will be Rayleigh dis ibu ed o IyI = 0, in a pa icula space- equency egion, and Rice dis ibu ed o IyJ > 0 Table 1: Ku osis and KS signi icance le els (Gaussian assump- ion) o eal pa o simula ed complex in e e ome ic phase amp in wa ele domain I Ho izon al band (HW I Ve ical band (LH) I Diaaonal band (LW I I I I I I I Scale 3 I 3.08 I 82.87 I 2.95 I 85.32 I 3.00 I 82.87 Complex in e e ome icphase signal model: The opog aphic model assumed p e iously (i.e. cons an slope) does no ake in o accoun spa ial de ails, which a e impo an , o ins ance, in u ban a eas. Since no in o ma ion is a ailable abou he dis ibu ion o he ' ue' opog aphic phase 4x, an a p io i model o he spa ial de ails is no a ailable in he spa ial domain. This d awback can be o e come in he wa ele domain. As men ioned in he p eceding Sec ion, he DWT can be in e p e ed as a weigh ed sum o andom a iables. The e o e, DWT2D{N, cos(&)} and DWT2D{Nc sin(&)} can be supposed o be de e mined as a i s app oxima ion, by a Gaussian dis ibu ion. Tes s wi h eal da a (see Tables 2 and 3) show ha wa ele s a is ics ha e ku osis highe han 3. To ake in o accoun his de ia ion om Gaussian beha iou , a double s ochas ic model is p oposed o he wa ele coe icien s x: Complex in e e ome icphase noise model: Ea h opog aphy can be ep esen ed locally by a cons an slope [I], hus he in e e ome ic phase 4, can be assumed o be a cons an phase amp. In he spa ial domain he measu ed in e e ome ic phase complies wi h he model dZ = q5x + [2], whe e is a phase noise e m. The eal and imagina y (5) pa s o he measu ed phase 4z coded in he uni ci cle, de ined as he complex in e e ome ic phase, can be modelled by [3, 41: whe e P,(X~~~) ep esen s he Gaussian dis ibu ion o he wa ele coe icien s and ~,d( ~) is a gene alised gamma dis ibu ion (GGD) (1) (2) modelling he ahabili6 o he- a iance h ough he phase image. px(x) canno be ob ained in a gene al o m. Nume ical in eg a ion o (9, see Fig. 1, indica es ha px(x) can be assumed o be a GGD model. To es he wa ele ans o ms o which a e [3]: he alidi y o his model, wo eal in e e ome ic phase images aken wi h he Ge man senso E-SAR om DLR ha e been employed. The i s image is an X-band in e e og am o Moun E na (I aly) and he second one is an L-band in e e og am o he Obe p a enho en es si e (Ge many), we e man-made s uc u es a e p esen . These wo in e e ome ic phases we e il e ed wi h he algo i hm p esen ed in [3], which is based on main aining he spa ial esolu ion. Tables 2 and 3 p esen he ku osis and he KS es signi icance le els, assuming a GGD model, applied o he eal pa o he il e ed complex in e e o- /I, = DWT,,{COS(~,)] = 2'Nc COS(^:) + V: p2 = DWT2D{sin(&)] = 2'Nc sin(+,") + y (3) (4) whe e i ep esen s he wa ele scale. ," and ," a e noise e ms independen om he wa ele scale. The phase e m 4: ep esen s he in e e ome ic phase in he wa ele domain, which con ains he same in o ma ion as q5= The WT is able o localise dx in he space- ELECTRONICS LETTERS 26 h Sep embe 2002 Vol. 38 No. 20 1207 me ic phase in he wa ele domain. These esul s show ha he GGD model is well adap ed o eal da a. Tables 2 and 3 also show ha he Obe p a enho en image p esen s highe ku osis alues. This esul is explained by he ac ha his image con ains mo e spa ial de ails (as buildings o oads) han he Moun E na image, which con ains only opog aphic in o ma ion. Ho izon al band (HL) Ku osis KS signi icance -1 I I I I -0.4 -0.2 0 0.2 0.4 a 20.0 Ve ical band (LH) Diagonal band (HH) Ku osis KS Ku osis KS simi icance signi icance I I I I , I I I I -0.100-0.075 -0.050-0.025 0 0.025 0.050 0.075 0.100 b Fig. 1 Dis ibu ion compa ison be ween GGD model and nume ical in eg a ion o double s ochas ic model a Low ku osis b High ku osis __~ GGD model ~ double s ochas ic model Ho izon al band (HL) Ku osis KS signi icance le el (“A) Ve ical band (LH) Diagonal band (LL)? Ku osis KS Ku osis KS signi icance signi icance le el (“A) le el (“A) I l; el(%) I I G el(%) I I I; el(%) Scale 1 I 10.51 I 70.61 I 11.30 I 77.16 I 12.88 I 56.61 Scale 2 I 6.39 I 86.51 I 6.42 I 79.89 I 6.12 I 98.33 Scale 3 I 4.42 I 97.34 I 5.62 I 60.95 I 3.92 I 80.70 Table 3: Ku osis and KS signi icance le els (GGD assump ion) o eal pa o Obe p a enho en complex in e e o- me ic phase in wa ele domain. Scale 1 I 22.50 I 10.53 1 21.44 I 31.14 I 26.95 I 14.76 Scale 2 I 13.82 I 16.24 I 11.51 I 53.28 I 12.68 I 91.26 I I I Scale 3 I 9.25 I 85.52 I 13.09 I 37.32 I 8.56 I 11.75 Conclusions: We ha e de eloped a s ochas ic model o he complex in e e ome ic phase in he wa ele domain. In he i s pa o his Le e , assuming he opog aphy locally as a cons an slope, a e e ence signal model is p oposed and alida ed. To inc ease i s lexibili y, a double s ochas ic signal model, which leads o a GGD model, is es ed o he wa ele coe icien s. In he wa ele domain, complex phase noise is Gaussian dis ibu ed, whe eas he use ul signal complies wi h a GGD model. The GGD assump ion o wa ele coe icien s ag ees wi h obse a ions epo ed by o he esea che s. Finally, his ep esen a ion can be employed as a p io i in o ma ion o u he p ocessing inside he wa ele domain. Acknowledgmen s: This wo k was unded by he Spanish go e nmen in he CICYT p ojec TIC1999-1050-C03-01 and he Ca alonian go e nmen in a DURSI ellowship. 0 IEE 2002 Elec onics Le e s Online No: 20020820 DOI: 10.1049/e1:20020820 C. Lopez and X. Fab egas (Signal Theo y and Communica ions Depa men , Elec omagne ics & Pho onics Enginee ing G oup, Uni e si a Poli kcnica de Ca alunya (UPC), Building 03 Room 11 8, Jo di Gi ona 1-3, E-08034 Ba celona, Spain) E-mail: [email p o ec ed] 27 July 2002 Re e ences GATELLI, E, GUARNERI, A.M., PARIZZI, E, PASQUALI, P, PRATI, C., and ROCCA, F.: ‘The wa enumbe shi in SAR in e e ome y’, IEEE T ans. Geosci. Remo e Sens., 1994,32, (4), pp. 855-865 LEE, J.s., PAPATHANASSIOU, K.P., AMSWORTH, T.L., GRUNES, M.R., and REIGBER, A,: ‘A new echnique o noise il e ing o SAR in e e ome ic phase images’, IEEE T ans. Geosci. Remo e Sens., LOPEZ-MART~EZ, C., and F~REGAS-CANOVAS, EX.: ‘Modeling and educ ion o SAR in e e ome ic phase noise in he wa ele domain’, Accep ed o publica ion IEEE T ans. Geosci. Remo e Sens., 2001 LOPEZ MART~EZ, C., and FABREGAS CANOVAS, X.: ‘Resul s on SAR in e e ome ic phase noise educ ion using wa ele ans o m’. EUSAR’02, Cologne, Ge many, 2002, Vol. 1 1998,36, (5), pp. 1456-1465 Adap i e a ay an enna based on adial basis unc ion ne wo k as mul iuse de ec ion o WCDMA Chang-Jun Ahn and Iwao Sasase An adap i e a ay an enna is p oposed based on he adial basis unc ion (RBF) ne wo k as a mul iuse de ec o o a WCDMA sys em. The p oposed sys em calcula es he op imal combining weigh coe icien s using sample ma ix in e sion wi h a common co ela ion ma ix algo i hm and ob ains he channel esponse ec o using he RBF ou pu signal. In oduc ion: Wideband code di ision mul iple access (WCDMA) communica ion sys ems ha e ecen ly a ac ed conside able a en ion as mobile cellula and IMT-2000 communica ion sys ems due o hei abili y o supp ess a wide a ie y o in e e ing signals including na owband in e e ence, mul iple access in e e ence (MAI), and mul ipa h in e e ence (MPI). One o he app oaches o imp o ing WCDMA sys em pe o mance is he use o spa ial il e ing a a base s a ion wi h an adap i e an enna a ay. The adap i e a ay an enna is widely accep ed, since i p o ides many p omising ea u es such as high capaci y, high spec um e iciency, and mo e deg ees o eedom o adjus cell co e age cha ac e is ics, leading o mo e e icien use o adio esou ces. To ob ain he op imal combining weigh coe icien s used in his echnique, se e al adap i e algo i hms such as he sample ma ix in e sion (SMI) and leas mean squa e (LMS) algo i hm ha e been p oposed. In a ay an ennas using hese con en ional algo i hms, weigh p ocessing is indi idually ca ied ou o each use . The e o e, he base s a ion mus ha e many independen weigh p ocesso s, which inc ease compu a ional complexi y. To sol e his p oblem, a sample ma ix in e sion wi h a common co ela ion ma ix (CCM-SMI) has been p oposed. This algo i hm has low compu a- ional complexi y, as weigh con e gence, and good BER pe o - mance a he base s a ion in a mul iuse en i onmen . Howe e , he CCM-SMI-based adap i e a ay an enna sys em has a poo channel esponse ec o (as does SMI) due o he ac i e mul iuse . Since he MF ou pu signal includes MA1 wi h inc easing use s, he sys em 1208 ELECTRONICS LETTERS 26 h Sep embe 2002 Vol. 38 No. 20