ONE OF THE key processing steps in all SAR interferometry

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1 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 47, NO. 4, APRIL A Partcle Flter Approach for InSAR Phase Flterng and Unwrappng Juan J. Martnez-Espla, Tomás Martnez-Marn, and Juan M. Lopez-Sanchez, Senor Member, IEEE Abstract Ths paper presents a new phase-unwrappng (PU) algorthm for SAR nterferometry that makes use of a partcle flter (PF) to perform smultaneously nose flterng and PU. The formulaton of ths technque provdes ndependence from nose statstcs and s not constraned by the nonlnearty of the problem. In addton, an enhanced varant of ths method combnng a PF wth artfcal-ntellgence search strateges and an omndrectonal local phase estmator, based on the mode of the power spectral densty, s also presented. Results obtaned wth synthetc and real data show a sgnfcant mprovement wth respect to other conventonal unwrappng algorthms n some stuatons. Index Terms Artfcal ntellgence (AI), extended Kalman flter (EKF), grd flter, partcle flter (PF), path followng, phase unwrappng (PU), SAR nterferometry, state space. I. INTRODUCTION ONE OF THE key processng steps n all SAR nterferometry applcatons s phase unwrappng (PU). A lot of lterature works concernng dfferent technques to solve the PU problem have been publshed durng the last years. We refer to the book by Ghgla and Prtt [1] for an excellent overvew. Tradtonally, there have been two general types of conventonal PU methods. The algorthms of the frst group, generally named path-followng or regon-growng algorthms, solate and/or mask problematc zones contanng resdues (we refer to [1] for the defnton of resdues) and unwrap the nterferogram by avodng these zones. The network cost flow algorthm by Costantn et al. [2] can be regarded as an extenson of the classcal branch-cut methods contaned n ths frst group. The technques of the second group provde a global soluton whch mnmzes a cost functon over the whole nterferogram. Independently from ths tradtonal classfcaton, some technques make use of a preflterng stage before startng the unwrappng procedure wth the fltered phase (for nstance, [3] [8]). Consequences of all these strateges are the followng: Phase nformaton n nosy pxels s not recovered by the path-followng algorthms; nosy pxels dstort the soluton n global approaches, thus affectng the nose-free areas; and the nformaton contaned n nosy pxels s lost f a preflterng stage s carred out. Manuscrpt receved Aprl 2, 2008; revsed August 12, Frst publshed February 10, 2009; current verson publshed March 27, Ths work was supported n part by the Spansh Mnstry of Educaton and Scence (MEC) under Projects TEC C02-02 and TEC C02-02, and n part by the Generaltat Valencana under Project ACOMP07/087. The authors are wth the Department of Physcs, System Engneerng and Sgnal Theory, Unversty of Alcante, Alcante, Span (e-mal: jjme1@alu.ua.es; tomas@dfsts.ua.es; juanma-lopez@eee.org). Dgtal Object Identfer /TGRS Durng the last years, a complementary group of approaches has appeared, known as multchannel technques, whch requre multple acqustons (not only two mages) for beng appled. See, for nstance, solutons presented n [9] [13]. Multchannel technques have the advantage to elmnate or at least mtgate the ambguty problem related to the wrappng operator. Note that the approach proposed n ths paper s devsed for conventonal sngle-channel nterferometry,.e., for a sngle nterferogram obtaned wth only one par of mages. In a Bayesan framework, dfferent PU technques are analyzed n [14], and a stochastc nonlnear flterng approach s also presented n [15]. However, Wshart nature of the nose present n the model observaton s not consdered, and results nvolvng real data are not shown. From our pont of vew, the key dea s to recover as much nformaton as possble from the nterferogram, ncludng nosy pxels. Therefore, an deal method would consder every pxel to be smultaneously unwrapped and fltered. There exsts an algorthm that shares the same objectve and was already publshed some years ago n [16] [20]. Ths approach, recently revsted n [21], combnes a local phase estmator wth an extended Kalman flter (EKF) n an elegant fashon to smultaneously acheve nose reducton and PU. Unfortunately, that soluton s founded on the followng two assumptons. Frst, both evoluton model and measurement model are avalable and correspond to lnear functons. Second, the nose affectng both evoluton and measurement stages s Gaussan. When the constrant of Gaussan nose holds but the models are not lnear functons, the EKF has been used n many cases as a good soluton that approxmates the nonlneartes by local lnearzatons. The man problem arses when the nose present n the nterferogram s not Gaussan. The EKF cannot guarantee a successful PU, as we wll show later wth some examples. Note that a detaled tutoral about dfferent methods to broach nonlnear/non-gaussan problems can be consulted n [22]. In a prevous work, we proposed to substtute the EKF by a grd-based flter (GbF), whch s not subject to any lnear or Gaussan constrants, and a good performance was demonstrated wth both smulated [23] and real data [24]. In ths paper, we ntroduce a partcle-flter phase-unwrappng (PFPU) algorthm. We wll show, wth some examples from synthetc and real data, that ths soluton also performs as the GbF, but wth the great advantage of a lower computatonal cost. The proposed method ntegrates a partcle flter (PF) and a local phase estmator to smultaneously unwrap and flter the nterferometrc phase. In addton, ths approach can be combned wth path-followng strateges to enhance ther /$ IEEE

2 1198 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 47, NO. 4, APRIL 2009 performance wth the embedded nose flterng. We present frst the algorthm wthout any condtoned scheme for choosng the unwrappng path, showng ts good performance when dealng wth resdues. In partcular, the basc algorthm ntroduced n ths paper runs the nterferogram row by row, passng through zones of low coherence and contanng resdues, unwrappng and flterng the phase at the same tme. There exst also scenes that nclude zones of hgh densty of resdues or low coherence, whch would be preferable not to unwrap untl more favorable areas have already been unwrapped. Consequently, a better approach would ncorporate some strategy to unwrap the best pxels frst and the worst ones at the end. To broach ths knd of stuatons, we have also developed a second mproved verson of the PFPU algorthm, the so-called 2PFPU, by combnng a PF wth an enhanced omndrectonal varant of the local phase estmator and a pathfollowng method, based on artfcal-ntellgence (AI) search strateges. We refer to the book by Nlsson [25] for an excellent overvew on AI search strateges. Ths paper s organzed as follows. Secton II ntroduces the basc formulaton of the PU problem. The PF formulaton s presented n Secton III. The applcaton of PFPU and ts mproved verson (2PFPU) s presented n Secton IV. Then, Secton V llustrates the performance of the new methods wth some examples from both a synthetc data set extracted from [1] and real nterferograms. A comparson wth some conventonal algorthms, namely, Goldsten branch-cut, Flynn s, and precondtoned conjugate gradent (PCG) algorthms [1], the cted EKF approach [16] [21] and a GbF [23], [24] soluton, s also dscussed. Fnally, conclusons are drawn n Secton VI. II. PU BASIC FORMULATION Assumng a 1-D notaton, the complex coherence between two SAR mages γ s, at pxel k γ(k) =a(k) exp [j ϕ(k)] (1) where a(k) s the observed nterferometrc coherence (between 0 and 1) and ϕ(k) s the modulo 2π mapped nterferometrc phase, also called the wrapped phase. Ths phase mappng can be expressed by ϕ(k)= [ϕ(k)+ẽ ϕ (k)] 2π = ϕ(k)+ẽ ϕ (k) ± n 2π ( π, π] (2) where ϕ(k) s the true unambguous absolute phase at pxel k and ẽ ϕ (k) s the mapped phase error at pxel k. The fnal objectve s to obtan the unwrapped or absolute phase ϕ(k) from the nosy 2π mapped phases ϕ(k) contaned n the nterferogram. At ths step, we wll be nterested n the calculus of the phase dfference from one pxel to the next Δ ϕ (k) =[ ϕ(k +1) ϕ(k)] 2π (3) that yelds ] Δ ϕ (k) = [δ ϕ (k)+[ẽ ϕ (k +1) ẽ ϕ (k)] 2π 2π (4) where δ ϕ (k) s the true dscrete phase dervatve and ts modulus s always supposed to be smaller than π (.e., there s no alasng). Fnally, the unwrapped phase ϕ can be obtaned through the followng recursve expresson: ϕ(k +1)=ϕ(k)+ Δ ϕ (k). (5) III. PARTICLE FILTER A. Introducton As mentoned earler, most of the tradtonal PU methods suffer from the lack of flterng capacty. Kalman-flter-based methods provde flterng capacty, but they are subject to lnear and Gaussan constrants. The EKF s sutable to address the lnear restrcton, provded that the model presents weak nonlneartes whch can be lnearzed n a proper manner. Instead, f the model has dscontnuous nonlneartes, the EKF should not be employed [26]. In the case of PU, the observaton model s nonlnear wth non-gaussan statstcs. Moreover, f the unwrapped phase s chosen as state varable (.e., we employ a 1-D EKF), then the observaton model s nonlnear and dscontnuous (wrapped phase). In ths case, the applcaton of the EKF s not straghtforward. The correct applcaton of the EKF requres the 2-D formulaton proposed n [16] [21], where the observaton model contans real and magnary parts of the complex nterferogram [21]. Thus, the model remans nonlnear but contnuous. A more complete ntroducton to the use of ths type of approaches for PU can be consulted n [21]. In contrast to EKF, the applcaton of a PF soluton s more straghtforward, snce t exhbts the advantage of usng only one dmenson per pxel (drectly the phase), hence brngng down complexty to ts smplest case, and t s not subject to any lnear or Gaussan constrants. B. PF Formulaton Herenafter, to be coherent wth the notaton of state-space methods, the unwrapped phase at pxel k, ϕ(k), wll be referred to as the correspondng state x k at pxel k. The suboptmal soluton of a PF [22] can be adopted, snce the contnuous state space can be dvded nto N cells or states, {x k ; =1,...,N}, whch correspond to all possble values of the phase. The resultng grd must be suffcently dense to get a good approxmaton to the contnuous state space. Conceptually, the PF s smply a soluton that represents the posteror probablty functon (pdf) p(x k z 1:k ), by a dstrbuton of N s partcles, defned herenafter, extracted from the state space. The objectve s to calculate estmates of the state x k based on the set of all avalable observatons z 1:k = {z,= 1,...,k} up to pxel k. The partcles are defned as the set of states whose weghts are the hghest for each step. These partcles, or selected group of samples, and ther weghts become the startng pont for the next step of the algorthm. The basc stages at each step of the algorthm are represented n Fg. 1.

3 MARTINEZ-ESPLA et al.: PARTICLE FILTER APPROACH FOR InSAR PHASE FILTERING AND UNWRAPPING 1199 Fg. 1. Illustraton of the stages of the PF algorthm for a pxel. Consderng the Markovan assumpton, the pdf p(x k z 1:k ) can be obtaned recursvely from the pdf p(x k 1 z 1:k 1 ) calculated at prevous pxel k 1. It s supposed that the ntal pdf p(x 0 z 0 ) p(x 0 ) s known. If t s not the case, the samples can be ntally dstrbuted unformly over the state space. Next, at the present pxel k, then s samples are updated accordng to the acton taken and the current observaton z k. At frst sght, ths sequental problem could be dvded nto two stages: predcton and update. However, as t wll be explaned n the followng, a thrd stage, named resamplng, needs to be added. To broach predcton and update stages, t s assumed that evoluton model and observaton model exst and are known. The predcton stage s solved by applyng the evoluton model p(x k x k 1 ) to each one of the N s partcles, generatng a new set of samples. These samples represent the predcton of the state varable x k wthout bearng n mnd the observaton at pxel k. To consder the current observaton z k, the update stage s performed. The weghts assocated wth each sample, wk ; = 1,...,N s, are obtaned accordng to the observaton model p(z k x k ). As t was ntroduced before, a thrd stage named resamplng needs to be added n order to mprove the performance of the flter. Let {x 0:k,w k }Ns =1 denote a random measure to characterze the posteror pdf p(x 0:k z 1:k ), where {x 0:k ; I = 1,...,N s } s a set of support ponts wth assocated weghts {wk ; =1,...,N s}, and x 0:k = {x j ; j =0,...,k} s the set of all states up to pxel k. The weghts are normalzed, w k = 1. Then, the posteror pdf at pxel k can be approxmated as N s p (x 0:k z 1:k ) wkδ ( x 0:k x ) 0:k =1 where δ() represents the Drac delta. (6) Let x,=1,...,n s be samples easly generated from a functon q( ) called mportance densty. The weghts are chosen by means of the mportance resamplng prncple, whch reles on the followng. Suppose that p(x) π(x) s a probablty densty from whch t s not easy to draw samples but for whch π(x) can be evaluated (as well as p(x) up to proportonalty). In ths context, a weghted approxmaton to p( ) s where N S p(x) w δ(x x ) (7) =1 w k π(x ) q(x ) represents the normalzed weght of the th partcle. If the samples x 0:k are drawn by usng an mportance densty, the weghts are then defned by wk p ( x 0:k z ) 1:k q ( x 0:k z ). (9) 1:k As presented n [22], f the mportance densty s chosen to factorze such that q(x 0:k z 1:k )=q(x k x 0:k 1,z 1:k )q(x 0:k 1 z 1:k 1 ) (10) then samples x 0:k q(x 0:k z 1:k ) can be obtaned by augmentng each of the exstng samples x 0:k 1 q(x 0:k 1 z 1:k 1 ) wth the new state x k q(x k x 0:k 1,z 1:k ). Concernng p(x 0:k z 1:k ), after a few operatons [22], t can be shown that p(x 0:k z 1:k ) p(z k x k )p(x k x k 1 )p(x 0:k 1 z 1:k 1 ). (11) (8)

4 1200 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 47, NO. 4, APRIL 2009 Therefore, by substtutng (10) and (11) nto (9) ) p ( x k x k 1) p ( x 0:k 1 z 1:k 1 ) wk p ( z k x k q ( x k x 0:k 1,z ) ( ) 1:k q x 0:k 1 z 1:k 1 = wk 1 p ( ( ) z k xk) p x k x k 1 q ( x k x 0:k 1,z ). (12) 1:k Furthermore, f we consder that q(x k x 0:k 1,z 1:k )= q(x k x k 1,z k ), then only x k and the current observaton z k need to be stored. The weght s then w k w k 1 p ( z k xk) ( p x x ) k k 1 q ( ) x. (13) k x k 1,z k The posteror densty of probablty s gven by N s p (x k z 1:k ) wkδ ( x k x ) k =1 (14) where the weghts are defned n (13). It can be shown that the larger the N s, the more (14) approaches the true pdf p(x k z 1:k ). Fnally, t s convenent to choose the mportance densty to be q ( x k x k 1,z k ) = p ( xk x k 1 ). (15) Ths selecton smplfes the calculus of the weghts as follows: w k w k 1 p ( z k x k ). (16) The graphcal evoluton of the samples at each stage of a standard PF algorthm s also shown n Fg. 1. It s assumed that the samples estmate only one parameter, dstrbuted over the horzontal axs. The samples are represented by crcles centered on the value of the parameter that they represent. The area of the crcles represents the weght of the correspondng sample. In ths example, the algorthm starts at pxel k 1 wth an unweghted dstrbuton { x k 1,N 1 s }. It provdes an approxmaton of p(x k 1 z 1:k 2 ). For each partcle, ts assocated weght s computed by usng the nformaton provded by the observaton model p(z k 1 x k 1 ) at pxel k 1. It results n the weghted measure { x 1 k 1, w1 k 1 }, whch yelds an approxmaton of p(x k 1 z 1:k 1 ). Fnally, the resamplng step, whch wll be explaned herenafter, selects only the fttest partcles to obtan the unweghted measure {x 1 k 1,N 1 s }, whch s stll an approxmaton of p(x k 1 z 1:k 1 ). The next step of the algorthm starts wth the evoluton stage, and nformaton from the local phase estmator p( x 1 k x1 k 1 ) s added at ths moment. The evoluton stage ntroduces a predcton and ts uncertanty, yeldng the measure { x 1 k,n 1 s }, whch s an approxmaton of p(x k z 1:k 1 ).The sequence of the algorthm s the same for the rest of subsequent pxels. An aprordrawback of PF methods could be the constrant of a fnte state space, but ths s not the case n the nterferometrc PU problem. The PF soluton that we propose here makes Fg. 2. Pseudocode of a generc PF. use of a sldng wndow Wk s to cover the complete state space. Ths wndow s 2π wde and s centered n the prevous value for the unwrapped phase, gven by PF mx, ntroduced n the followng. It s based on the assumpton that the phase dfference between two contguous pxels must be contaned n the nterval ( π, +π]. In ths way, ths sldng state space allows us to afford the nterferometrc PU problem ndependently of the wdth of the complete state space. Note that computatonal cost ncreases lnearly wth N s. A common and crtcal problem affectng PFs s the degeneracy phenomenon. It means that after a few steps of the algorthm, all but one partcle wll have neglgble weght. Ths problem mples that a consderable computatonal work wll be dedcated to updatng partcles whose contrbuton to the approxmaton of the posteror (pdf) p(x k z 1:k ) s nearly zero. Therefore, degeneracy s an undesrable ssue that should be avoded. The frst dea to remove ths effect could be the brute force,.e., the use of a very large N s, whch s normally nfeasble and neffcent. However, there exst some other solutons, lke resamplng or good choce of the mportance densty [22]. Resamplng wll be the one to be used wth the PF presented n ths paper due to ts smplcty and excellent results. More detals of the resamplng are provded n next secton. The pseudocode of a generc PF s shown n Fg. 2. Note that the sldng wndow update and resamplng stages have been omtted and wll be ntroduced as part of the PU algorthm. C. Resamplng The pseudocode of the resamplng algorthm s shown n Fg. 3. Ths method allows the reducton of the effects of degeneracy. The basc dea of resamplng s to drop partcles that have small weghts and to concentrate on those wth large weghts. A new set of samples s generated by resamplng the set of samples and takng out, wth replacement, N s samples from the current set, proportonally to ther weghts. In ths new set, for nstance, the samples wth the lowest probabltes wll dsappear. Next, the weghts assocated wth the samples are scaled n order to represent the probablty assocated wth each sample. The resultng set of samples s n fact an..d. sample from the dscrete posteror probablty functon p(x k z 1:k ). Therefore, the weghts can now be reset to w k =1/N s.

5 MARTINEZ-ESPLA et al.: PARTICLE FILTER APPROACH FOR InSAR PHASE FILTERING AND UNWRAPPING 1201 Fg. 4. Pseudocode of the proposed 2-D PU algorthm usng a PF (PFPU). Fg. 3. Pseudocode of the resamplng algorthm. n ths paper). It has been modeled as a Gaussan dscrete dstrbuton for the sake of comparson wth [16] [21]. IV. PARTICLE FILTER PHASE UNWRAPPING A. Introducton The undmensonal PU problem wll be addressed frst for a better process understandng. In ths case, the probabltes p(x k x k 1 ) and p(z k x k ) n (13) are defned by the evoluton and observaton models at pxel k, gven by x k = x k 1 +Δˆϕ k 1 + n k 1 (17) z k = x k + v k (18) where Δˆϕ k 1 represents a local phase-slope estmate, whle n k and v k are supposed to be approxmate models for the nose present n both evoluton and observaton stages, respectvely. Several technques can be used to obtan an estmaton of the phase slope. For nstance, the mode of the power spectral densty was used n [16] and [17] due to ts supposed nsenstveness and robustness to supermposed whte nose, whereas a phase-average-based method was used n [27], and a matrx pencl approach was proposed n [28]. All of them are perfectly applcable. The frst one has been selected n ths case for the sake of comparson. It s well known that the estmated covarance matrx for correlated SAR mages s statstcally descrbed by the complex Wshart dstrbuton, wth the dstrbuton of the nterferometrc phase beng a margnal densty dstrbuton of the Wshart one [29], [30]. Therefore, nose v k has been modeled as a dscrete margnal Wshart dstrbuton. It s mportant to note that ths modelzaton s not strctly necessary, snce ths approach can work wth any dstrbuton or probablty densty functon, known or not, thus provdng ndependence from nose statstcs. When the statstcs s known apror, the algorthm can use ts analytcal expresson, thus makng the computatons easer. When the statstcs s not avalable, the algorthm can calculate t. Concernng n k, t models the uncertanty ntroduced by the local phase estmator (based on the power spectral densty B. PFPU Algorthm To apply the PFPU to 2-D nterferograms, any predcton estmate wll be calculated, dependng on two neghbors. Hence, the stage concernng the evoluton model (17) needs to be modfed as follows: x k 1 2 [p ( ) ( )] H x k x k 1 + pv x k x k m (19) where p H (x k x k 1 ) s the evoluton model n the horzontal drecton at pxel k 1, gven by x k = x k 1 +Δˆϕ H k 1 + n H k 1 (20) where x k 1 refers to the unwrapped phase at pxel k 1, Δˆϕ H k 1 refers to the horzontal local phase estmaton at pxel k 1, and n H k 1 refers to the Gaussan nose sample at pxel k 1 that models the uncertanty ntroduced by the local phase estmator. Note that pxel k 1 s the prevously unwrapped one, placed n the same row and adjacent to the current pxel k. On the other hand, p V (x k x k m ) s the evoluton model n the vertcal drecton, at pxel k m, gven by x k = x k m +Δˆϕ V k m + r V k m (21) where x k m refers to the unwrapped phase at pxel k m, Δˆϕ V k m refers to the vertcal local phase estmaton at pxel k m, and rk m V refers to the Gaussan nose sample at pxel k m that models the uncertanty ntroduced by the local phase estmator. Note that pxel k m s the prevously unwrapped one, placed n the same column and adjacent to the current pxel k. The pseudocode of the 2-D PFPU algorthm s shown n Fg. 4. C. 2PFPU Algorthm As t wll be shown later wth some examples, the row-byrow PFPU soluton presented n the prevous secton (see Fg. 5)

6 1202 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 47, NO. 4, APRIL 2009 Fg. 5. Path of the row-by-row PFPU. performs generally better than conventonal solutons when workng wth sparse dstrbutons of resdues, showng a good recovery from errors nduced by them. However, n many occasons, zones contanng hgh densty of resdues or low coherence have to be processed and unwrapped. In ths knd of stuatons, some added strategy to know the best path to follow for unwrappng would be much apprecated. As outlned n Secton I, the PFPU algorthm can be combned wth path-followng approaches to enhance the performance of these well-known technques wth the nose flterng ncorporated by the PF. The so-called 2PFPU soluton presented here combnes a PF wth a path-followng method, makng use of AI search strateges based on the use of cost functons and an enhanced omndrectonal varant of the local phase estmator. Both enhanced features are descrbed n the followng. AI search strateges are effcent and provde the optmal soluton. In ths paper, the optmal soluton wll be the path that maxmzes ts assocated cost functon. It works as follows: At each step, the pxel to be unwrapped wll be the one wth the hghest assocated cost functon. The cost functon wll be computed as the weghted summaton of the coherence of pxel k and the number of unwrapped neghbors. To provde the soluton wth at least the same relablty as the PF method presented n Secton IV-B, two or more unwrapped neghbors should be avalable. The cost functon f c for pxel k s then expressed as f c = c 1 COH + c 2 UNW (22) where c 1 and c 2 are adjustable weghts, COH represents the coherence of pxel k, and UNW denotes the number of unwrapped neghbors. Note that cost functon s updated per pxel and per step. Coherence for each pxel has been estmated wth a 5-5-pxel wndow. The local phase estmator presented so far, based on the mode of the power spectral densty, provdes slope estmatons n horzontal and vertcal drectons. Ths method has been mproved n ths paper to provde slope estmatons also n oblque drectons. To do so, two types of 5 5 wndows around the pxel under study have been used smultaneously. These two wndows are shown n Fg. 6. They have been separated n the pcture for the sake of clarty. However, actually, they work overlapped and centered on the same pxel. The concepts explaned n [16] and [17] can now be appled over both wndows. As a result, omndrectonal slope nformaton s suppled. We have named ths enhanced varant as the omndrectonal local phase estmator. The PFPU presented n Secton IV-B only uses Fg. 6. Wndows used by the omndrectonal local phase estmator for computng the spectral mode. Fg. 7. Example of omndrectonal local phase estmaton. the slope estmatons from two contguous neghbors. On the contrary, the 2PFPU soluton presented here s able to collect slope nformaton from up to the eght neghbors, provdng estmatons wth hgher relablty. As a consequence of the new nformaton suppled to the algorthm, the stage concernng the evoluton model (17) needs to be modfed as follows: x k 1 N n N n m=1 ( p m x k x ) m (23) where N n s the total number of unwrapped neghbors used for the predcton and p j (x k x m ) s the evoluton model to be appled n the drecton from pxel m toward pxel k, gven by x k = x m +Δˆϕ j m + n j m (24) where x m refers to the unwrapped phase at pxel m, Δˆϕ j m refers to the slope estmaton at pxel m n the j-drecton (the one toward pxel k), and n j m refers to the Gaussan nose sample at pxel m that models the uncertanty ntroduced by the slope estmaton at pxel m n the j-drecton. One example of such a stuaton s shown n Fg. 7. In ths case, wth N n =6, the evoluton model from each unwrapped neghbor (pxel m) to the pxel k wll be appled accordng to (24). Fnally, the composton of all the estmatons s computed by usng (23). The necessty of savng and updatng a lst contanng the fnal partcle dstrbuton at every unwrapped pxel that stll has a neghbor to unwrap entals an addtonal computatonal cost. However, as we wll show later wth some examples, ths path selecton and nose flterng combnaton mproves mportantly the performance of the algorthm and enables the soluton of more complcated stuatons. The pseudocode shown n Fg. 4 also apples to 2PFPU wth the dfference that phase slopes are estmated accordng to (24).

7 MARTINEZ-ESPLA et al.: PARTICLE FILTER APPROACH FOR InSAR PHASE FILTERING AND UNWRAPPING 1203 Fg. 8. Longs Peak synthetc nterferogram. (a) Coherence. (b) Resdues. (c) True absolute unwrapped phase. (d) Nosy nput sgnal (wrapped phase). V. R ESULTS The results n ths secton llustrate the performance of dfferent PU solutons as well as the nfluence of the nosy nput sgnal. The followng PU solutons are compared: the branchcut algorthm proposed by Goldsten [1], Flynn s algorthm [1], the PCG [1], the EKF algorthm descrbed n [16] [21], two approaches of the GbF proposed n [23] and [24], the row-by-row PF soluton (PFPU) proposed n ths paper (see Fgs. 4 and 5), and the mproved 2PFPU algorthm. The two GbF optons consst of selectng the state whose weght s the maxmum GbF mx or selectng the average state of the dstrbuton GbF av. Algorthms have been appled to dfferent synthetc and real scenaros. Note that all the phase results are presented n radans. 1) Synthetc Data Over Longs Peak (Colorado) [1]: We have appled all, but the 2PFPU algorthm, to one example that most of the methods ntroduced n [1] fal to unwrap: a porton of a synthetc nterferogram from a mountanous terran around Longs Peak, Colorado (Fg. 8). The data set, as detaled n [1], was generated wth a hgh-fdelty InSAR smulator by employng dgtal terran elevaton models from the U.S. Geologcal Survey. We have also chosen ths synthetc example because the true unwrapped phase s avalable, so one can compare t wth the phase delvered by the PU algorthms. In the study area, there exst zones where the coherence (.e., the qualty) of the data s very poor. For nstance, n the surroundngs of rows 3 5 and columns 25 35, t reaches values lower than 0.3 [see Fg. 8(a)]. As a consequence, resdues appear [see Fg. 8(b)]. We wll llustrate the followng examples by showng the fnal unwrapped phase provded by the PU algorthms, whch can be compared aganst the true absolute unwrapped phase represented n Fg. 8(c). In addton, the result of wrappng the soluton (rewrapped phase) wll be analyzed for detectng potental errors ntroduced by the PU algorthms wth respect to the orgnal wrapped phase (.e., the nput sgnal), whch s shown n Fg. 8(d). The PU algorthm proposed by Goldsten [1] dentfes the resdues and defnes the so-called branch cuts between them, whch wll not be crossed by the unwrappng paths. Ths method attempts to mnmze the branch-cut lengths. As a result, many areas can be completely solated and, consequently, not unwrapped consstently wth the rest of the nterferogram. Ths s the case shown n Fg. 9. The results obtaned by applyng Flynn s algorthm and the PCG method [1] are shown n Fgs. 10 and 11, respectvely. It can be observed how both methods also fal to unwrap correctly ths synthetc data set. As observed n Fg. 12, the EKF [16] [21] fals as a consequence of a concentraton of resdues around the pxel (3, 27). It produces errors n the slope estmates and n the unwrapped phase n ths area, thus generatng an error whch propagates n a dagonal fashon and producng an erroneous phase dscontnuty. In contrast, as shown n Fg. 13 for the same test nterferogram, the PFPU soluton does not suffer from ths error propagaton, snce the PF s able to fx ths problem. Although more expensve than other technques, the computatonal cost of the PFPU was only a few seconds to unwrap ths area, whch s much shorter than the analogous GbF solutons as a consequence of usng partcles nstead of a complete grd of states.

8 1204 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 47, NO. 4, APRIL 2009 Fg. 9. Goldsten soluton for Longs Peak synthetc nterferogram. (a) Unwrapped phase. (b) Rewrapped phase. Fg. 10. Flynn soluton for Longs Peak synthetc nterferogram. (a) Unwrapped phase. (b) Rewrapped phase. Fg. 11. PCG soluton for Longs Peak synthetc nterferogram. (a) Unwrapped phase. (b) Rewrapped phase. Fg. 12. EKF soluton for Longs Peak synthetc nterferogram. (a) Unwrapped phase. (b) Rewrapped phase.

9 MARTINEZ-ESPLA et al.: PARTICLE FILTER APPROACH FOR InSAR PHASE FILTERING AND UNWRAPPING 1205 Fg. 13. PFPU soluton for Longs Peak synthetc nterferogram. (a) PFPU mx unwrapped phase. (b) PFPU mx rewrapped phase. The contnuous space contaned nsde the 2π sldng wndow has been translated nto a dscrete state space composed of N = 100 cells, and a maxmum of N s =50partcles wll be used. Fg. 14. GbF soluton for Longs Peak synthetc nterferogram. (a) GbF mx unwrapped phase. (b) GbF mx rewrapped phase. (c) GbF av unwrapped phase. (d) GbF av rewrapped phase. The contnuous space contaned nsde the 2π sldng wndow has been translated nto a dscrete state space composed of N = 100 cells. The results provded by both GbF solutons, GbF mx and GbF av, are also shown n Fg. 14. As expected, the results are qute smlar to the ones obtaned by the PFPU, but they have been obtaned wth a 35% hgher computatonal effort. 2) Small Area of a Real ERS-1 and ERS-2 Tandem Interferogram Over Alcante Provnce: The followng results correspond to a small crop extracted from a real nterferogram obtaned wth mages acqured by the ERS-1 and ERS-2 satelltes n tandem confguraton. The same algorthms as n the prevous example have been compared. Coherence and nput wrapped phase are shown n Fg. 15. As observed n Fg. 16, the EKF fals n two dfferent areas. Frst, a concentraton of resdues around the pxel (40, 80) produces errors n the slope estmates and n the unwrapped phase n ths area. Second, the conjuncton of low coherence and hgh varance of the local phase estmators n pxel (7, 70) produces an error whch propagates n a dagonal fashon, generatng an erroneous phase dscontnuty. For ths example, the rest of tested algorthms show smlar results (see Fgs ). The man dfferences appear for Goldsten s and the PCG soluton. Goldsten s algorthm faled to unwrap a small group of pxels around (40, 80), snce they appear completely solated as a result of the combnaton of the branch cuts [see Fg. 17(a)]. In the case of PCG, by nspectng the rewrapped phase n Fg. 19(b), t can be observed that part of a phase cycle has been lost for a group of pxels around (40, 80). Ths problem s common n mnmum-norm methods. On the other hand, the soluton provded by the PFPU

10 1206 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 47, NO. 4, APRIL 2009 Fg. 15. Small real nterferogram wth ERS mages. (a) Coherence. (b) Input wrapped phase. Fg. 16. EKF soluton for the small ERS nterferogram. (a) Unwrapped phase. (b) Rewrapped phase. (c) Zoom of resdues. (d) Zoom of rewrapped phase. Fg. 17. Goldsten soluton for the small ERS nterferogram. (a) Unwrapped phase. (b) Rewrapped phase.

11 MARTINEZ-ESPLA et al.: PARTICLE FILTER APPROACH FOR InSAR PHASE FILTERING AND UNWRAPPING 1207 Fg. 18. Flynn soluton for the small ERS nterferogram. (a) Unwrapped phase. (b) Rewrapped phase. Fg. 19. PCG soluton for the small ERS nterferogram. (a) Unwrapped phase. (b) Rewrapped phase. Fg. 20. PFPU soluton (PFPU mx) for the small ERS nterferogram. (a) Unwrapped phase. (b) Rewrapped phase. The contnuous space contaned nsde the 2π sldng wndow has been translated nto a dscrete state space composed of N = 100 cells, and a maxmum of N s =50partcles wll be used. s smoother than that of Goldsten and Flynn solutons n the upper part of the nterferogram due to the flterng capablty of ths approach. 3) X-SAR Interferogram Over Mount Etna: Asmplerowby-row PF algorthm (PFPU) has been used so far for computatonal cost reasons. It has been shown how t can deal wth resdues and even recover from errors. However, there exst many stuatons n whch zones of very low coherence or wth a hgh concentraton of resdues are present. In these cases, t would be preferable to unwrap them the last. To do so, an enhanced algorthm (2PFPU) has been ntroduced n Secton IV-C, combnng a path-followng technque wth the PF soluton. The evdent drawback of ths method s that the computatonal cost grows from the necessty of storng the partcle dstrbuton at each unwrapped pxel that stll has any unwrapped neghbors. However, flterng and PU capactes are more powerful than n the row-by-row approach, snce the phase s unwrapped movng from the hgher qualty zones to the lower ones. Just to gve an approxmate comparson about the relatve computatonal cost of these approaches, t took only a few seconds to process the nterferogram by usng the Goldsten algorthm, 1 mn by usng the PCG, nearly half an hour by means of Flynn s algorthm, and nearly 8 h by means of the 2PFPU method. The data shown n Fgs. 21 and 22 are the coherence and nput wrapped phase, respectvely, extracted from a real nterferogram formed by a par of SLC mages wth pxels, acqured by the X-SAR msson over Mount Etna. There exst

12 1208 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 47, NO. 4, APRIL 2009 Fg. 21. Coherence of the X-SAR nterferogram over Mount Etna. Fg. 23. Mask employed for unwrappng the X-SAR nterferogram over Mount Etna. Black areas wll not be unwrapped. Fg. 22. Input wrapped phase of the X-SAR nterferogram over Mount Etna. Fg. 24. Goldsten soluton for the X-SAR nterferogram over Mount Etna: Unwrapped phase. some areas where the phase nformaton s ncoherent, one of them correspondng to the sea (at the top) and others probably due to the presence of vegetaton and to shadow areas (as n the volcano crater). We have masked out the sea regon but not the other low-qualty areas. The mask employed n the subsequent unwrappng s shown n Fg. 23. Black areas wll not be unwrapped. The 2PFPU algorthm wll be compared to Goldsten and Flynn solutons, snce they share the path-followng concept, and also to the PCG soluton, as a comparson wth mnmumnorm methods. In some cases, an addtonal test has been ncluded here. The dfference between the unwrapped phase by each algorthm and the orgnal nterferogram has been computed, and the result has been wrapped agan. Ths process should result n a nosy constant, and no bas should appear even for large-sze mages. Tradtonal path-followng technques, Goldsten and Flynn (see Fgs. 24 and 25), fal partally to unwrap ths nterferogram because there appear some solated regons n the fnal soluton. Fg. 25. Flynn soluton for the X-SAR nterferogram over Mount Etna: Unwrapped phase.

13 MARTINEZ-ESPLA et al.: PARTICLE FILTER APPROACH FOR InSAR PHASE FILTERING AND UNWRAPPING 1209 Fg. 26. PCG soluton for the X-SAR nterferogram over Mount Etna: Unwrapped phase. Fg PFPU soluton for the X-SAR nterferogram over Mount Etna: Unwrapped phase. The contnuous space contaned nsde the 2π sldng wndow has been translated nto a dscrete state space composed of N = 100 cells, and a maxmum of N s =50partcles wll be used. Fg. 27. PCG soluton for the X-SAR nterferogram over Mount Etna: Wrapped dfference between the unwrapped phase and the nput wrapped phase. On the other hand, the PCG approach results n a smooth soluton (see Fg. 26) wthout solated regons, as expected. In ths case, the qualty of the soluton can be tested by observng the dfference of the unwrapped and the orgnal phase, whch s shown n Fg. 27. Other than the expected nosy areas, there appear several dscontnutes or frnges n ths mage, so the fnal soluton s not satsfactory. Fnally, Fgs. 28 and 29 show the capactes for the enhanced algorthm 2PFPU, combnng a path-followng technque wth a PF and the omndrectonal phase estmator. Fg. 28 shows the unwrapped phase. The qualty of ths soluton s shown by Fg. 29, snce the wrapped dfference between the soluton and the orgnal nterferogram exhbts a nosy constant aspect, and no bas has appeared even for ths large nterferogram. Note that the nosy aspect s due to the flterng ncorporated n ths technque. Fg PFPU soluton for the X-SAR nterferogram over Mount Etna: Wrapped dfference between the unwrapped phase and the nput wrapped phase. VI. CONCLUSION A new soluton that smultaneously flters and unwraps the phase contaned n an nterferogram s presented n ths paper. Ths soluton s based on a PF. Two varants of ths method (PFPU and 2PFPU) have been mplemented: The frst one s a smple row-by-row unwrappng algorthm, and the second one s the result of combnng a PF wth AI search strateges and an enhanced verson of the local phase estmator ntroduced n [16] [21]. These algorthms have been compared to some conventonal methods lke the branch-cut algorthm proposed by Goldsten, Flynn s algorthm, and PCG. Ths soluton performs better n some cases, snce PF can deal wth zones contanng resdues. PF-based solutons have also been compared to the EKF method and the GbF algorthm, whch share the same

14 1210 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 47, NO. 4, APRIL 2009 phlosophy of smultaneously flterng and unwrappng. For some stuatons whch the EKF fals to unwrap, a PF soluton has shown a better performance and the capacty to recover from errors. It has also been shown that PF solutons perform better than the grd-based ones, snce they obtan the same results wth the advantage of a lower computatonal cost. Our research lnes at present and n the near future are focused on the followng two man ssues: frst, utlzaton of other local phase estmators, for nstance, the matrx pencl approach proposed n [28], and second, the combnaton of the 2PFPU approach wth regon-growng technques wll be analyzed, snce a better performance s expected from the synergy between both strateges. ACKNOWLEDGMENT The authors would lke to thank Dr. K. P. Papathanassou (DLR) and Dr. C. Lopez-Martnez (UPC) for provdng access to the X-SAR mages of Mount Etna. The ERS mages used n ths paper have been provded by the European Space Agency (ESA) n the framework of the ESA EO Project Cat REFERENCES [1] D. C. Ghgla and M. D. Prtt, Two-Dmensonal Phase Unwrappng. Theory, Algorthms and Software. New York: Wley, [2] M. Costantn, A. Farna, and F. Zrll, A fast phase unwrappng algorthm for SAR nterferometry, IEEE Trans. Geosc. Remote Sens., vol. 37, no. 1, pp , Jan [3] D. Meng, V. Sethu, E. Ambkarajah, and L. Ge, A novel technque for nose reducton n InSAR mages, IEEE Geosc. Remote Sens. Lett., vol. 4, no. 2, pp , Apr [4] R. Yamak and A. Hrose, Sngularty-spreadng phase unwrappng, IEEE Trans. Geosc. Remote Sens., vol. 45, no. 10, pp , Oct [5] J.-S. Lee, K. P. Papathanassou, T. L. Answorth, M. R. Grunes, and A. Regber, A new technque for nose flterng of SAR nterferometrc phase mages, IEEE Trans. Geosc. Remote Sens., vol. 36, no. 5, pp , Sep [6] N. Wu, D.-Z. Feng, and J. L, A locally adaptve flter of nterferometrc phase mages, IEEE Geosc. Remote Sens. Lett., vol. 3, no. 1, pp , Jan [7] A. B. Suksmono and A. Hrose, Adaptve nose reducton of InSAR mages based on a complex-valued MRF model and ts applcatons to phase unwrappng problem, IEEE Trans. Geosc. Remote Sens., vol. 40, no. 3, pp , Mar [8] Q. Yu, X. Yang, S. Fu, X. Lu, and X. Sun, An adaptve contoured wndow flter for nterferometrc synthetc aperture radar, IEEE Geosc. Remote Sens. Lett., vol. 4, no. 1, pp , Jan [9] R. Lanar, G. Fornaro, D. Rcco, M. Mglacco, K. Papathanassou, J. Morera, M. Schwäbsch, L. Dutra, G. Pugls, G. Franceschett, and M. Coltell, Generaton of dgtal elevaton models by usng SIR-C/X-SAR multfrequency two-pass nterferometry: The Etna case study, IEEE Trans. Geosc. Remote Sens., vol. 34, no. 5, pp , Sep [10] M. G. Km and H. D. Grffths, Phase unwrappng of multbaselne nterferometry usng Kalman flterng, n Proc. Image Process. Appl., Conf. Publcaton No. 465, 1999, pp [11] A. Ferrett, A. Mont Guarner, C. Prat, and F. Rocca, Mult baselne nterferometrc technques and applcatons, n Proc. ESA Workshop Appl. ERS SAR Interferometry (FRINGE), Zurch, Swtzerland, Oct [Onlne]. Avalable: papers/ferrett-et-al/ [12] G. Ferrauolo, V. Pascazo, and G. Schrnz, Maxmum a posteror estmaton of heght profles n InSAR magng, IEEE Geosc. Remote Sens. Lett., vol. 1, no. 2, pp , Apr [13] G. Fornaro, A. Paucullo, and E. Sansost, Phase dfference-based multchannel phase unwrappng, IEEE Trans. Image Process., vol. 14, no. 7, pp , Jul [14] G. Nco, G. Palubnskas, and M. Datcu, Bayesan approaches to phase unwrappng: Theoretcal study, IEEE Trans. Sgnal Process., vol. 48, no. 9, pp , Sep [15] J. Letão and M. Fgueredo, Absolute phase mage reconstructon: A stochastc nonlnear flterng approach, IEEE Trans. Image Process., vol. 7, no. 6, pp , Jun [16] R. Krämer and O. Loffeld, Presentaton of an mproved Phase Unwrappng Algorthm based on Kalman flters combned wth local slope estmaton, n Proc. ESA Workshop Appl. ERS SAR Interferometry (FRINGE), Zurch, Swtzerland, Oct [Onlne]. Avalable: [17] O. Loffeld and R. Krämer, Phase unwrappng for SAR nterferometry A data fuson approach by Kalman flterng, n Proc. IEEE IGARSS, Hamburg, Germany, 1999, pp [18] O. Loffeld and R. Krämer, Phase unwrappng for SAR nterferometry, n Proc. IEEE IGARSS, Pasadena, CA, 1994, pp [19] O. Loffeld, C. Arndt, and A. Hen, Estmatng the dervatve of modulo-mapped phases, n Proc. ESA Workshop Appl. ERS SAR Interferometry (FRINGE), Zurch, Swtzerland, Oct [Onlne]. Avalable: [20] O. Loffeld, Demodulaton of nosy phase or frequency modulated sgnals wth Kalman flters, n Proc. IEEE Int. Conf. Acoust., Speech, Sgnal Process., Adelada, Australa, 1994, pp. IV_177 IV_180. [21] O. Loffeld, H. Nes, S. Knedlk, and W. Yu, Phase unwrappng for SAR nterferometry A data fuson approach by Kalman flterng, IEEE Trans. Geosc. Remote Sens., vol. 46, no. 1, pp , Jan [22] M. S. Arulampalam, S. Maskell, N. Gordon, and T. Clapp, A tutoral on partcle flters for onlne nonlnear/non-gaussan Bayesan trackng, IEEE Trans. Sgnal Process., vol. 50, no. 2, pp , Feb [23] J. J. Martnez-Espla, T. Martnez-Marn, and J. M. Lopez-Sanchez, Usng a grd-based flter to solve InSAR phase unwrappng, IEEE Geosc. Remote Sens. Lett., vol. 5, no. 2, pp , Apr [24] J. J. Martnez-Espla, T. Martnez-Marn, and J. M. Lopez-Sanchez, Introducton of a grd-based flter approach for InSAR phase flterng and unwrappng, n Proc. IEEE IGARSS, Barcelona, Span, Jul. 2007, pp [25] N. J. Nlsson, Artfcal Intellgence: A New Synthess. San Mateo, CA: Morgan Kaufmann, [26] B. Rstc, S. Arulampalam, and N. Gordon, Beyond the Kalman Flter: Partcle Flters for Trackng Applcatons. Norwood, MA: Artech House, [27] W. Xu and I. Cummng, A regon-growng algorthm for InSAR phase unwrappng, IEEE Trans. Geosc. Remote Sens., vol. 37, no. 1, pp , Jan [28] G. Nco and J. Fortuny, Usng the matrx pencl method to solve phase unwrappng, IEEE Trans. Sgnal Process., vol. 51, no. 3, pp , Mar [29] J. S. Lee, K. W. Hoppel, S. Mango, and A. Mller, Intensty and phase statstcs of multlook polarmetrc and nterferometrc SAR magery, IEEE Trans. Geosc. Remote Sens., vol. 32, no. 5, pp , Sep [30] R. J. A. Tough, D. Blacknell, and S. Quegan, A statstcal descrpton of polarmetrc and nterferometrc synthetc aperture radar data, Proc. R. Soc. Lond. A, Math. Phys. Sc., vol. 449, pp , Juan J. Martnez-Espla was born n Alcante, Span, n He receved the M.S. degree n telecommuncaton engneerng from the Techncal Unversty of Valenca (UPV), Valenca, Span, n He s currently workng toward the Ph.D. degree n the Department of Physcs, System Engneerng and Sgnal Theory, Unversty of Alcante, Alcante, Span. Snce 2003, he has been brngng hs nternatonal experence as a Project and Product Manager n telecommuncatons busness, n addton to hs research at the Sgnals, Systems and Telecommuncaton Group, Unversty of Alcante. Hs research topcs nclude mcrowave remote sensng for dgtal elevaton models and advanced state-space technques for nose reducton, flterng, and phase unwrappng wth applcaton n SAR nterferometry.

15 MARTINEZ-ESPLA et al.: PARTICLE FILTER APPROACH FOR InSAR PHASE FILTERING AND UNWRAPPING 1211 Tomás Martnez-Marn receved the B.S. degree n telecommuncaton engneerng from the Unversty of Alcalá, Alcalá de Henares, Span, n 1990 and the M.S. and Ph.D. degrees n telecommuncaton engneerng from the Techncal Unversty of Madrd (UPM), Madrd, Span, n 1995 and 1999, respectvely. He was wth the Unversty of Alcalá as an Assstant Professor n In 1997, he was wth the European Unversty of Madrd (UEM), Madrd, Span, as an Assstant Professor. Snce 2000, he has been wth the Department of Physcs, System Engneerng and Sgnal Theory, Unversty of Alcante, Alcante, Span, where he s currently an Assocate Professor. Hs research nterests nclude renforcement learnng, optmal control, ntellgent vehcles, and SAR mage processng. Juan M. Lopez-Sanchez (S 94 M 00 SM 05) was born n Alcante, Span, n He receved the M.S. and Ph.D. degrees n telecommuncaton engneerng from the Techncal Unversty of Valenca (UPV), Valenca, Span, n 1996 and 2000, respectvely. From 1998 to 1999, he was a Predoctoral Grant Holder wth the Space Applcatons Insttute, Jont Research Centre, European Commsson, Ispra, Italy. Snce 2000, he has been leadng the Sgnals, Systems and Telecommuncaton Group, Unversty of Alcante, Span, where he s currently an Assocate Professor wth the Department of Physcs, System Engneerng and Sgnal Theory. Hs research topcs nclude analytcal and numercal models for electromagnetc scatterng problems, mcrowave remote sensng for nverson of bophyscal parameters, advanced polarmetrc and nterferometrc technques, and SAR magng algorthms. He has coauthored more than 20 papers n nternatonal journals and 50 contrbutons to scentfc conferences and symposa. Dr. Lopez-Sanchez receved the INDRA Award for the best Ph.D. thess about radar n Span n 2001.