Coherence Linearity and SKP-Structured Matrices in Multi-Baseline PolInSAR Stefano Tebaldini and Fabio Rocca Politecnico di Milano Dipartimento di Elettronica e Informazione IGARSS 2, Vancouer
Introduction The aailability of Multi-baseline PolInSAR data maes it possible to decompose the sinal into round-only and olume-only contributions y n i Trac n Polarization i Re{y n ( )} Re{y n ( 2 )} Re{y n ( 3 )} Trac Trac n V V VV V V VV Trac N V V VV Im{y n ( )} Im{y n ( 2 )} Im{y n ( 3 )} Decomposition 6 5 4 3 2-6 5 4 3 2 - Volume-only contributions 2 6 4 8 22 Ground-only contributions 2 6 4 8 22 Properties of the eetation layer Vertical structure Polarimetry Phase calibration Diital Terrain Model Ground properties
Introduction Polarimetric SAR Interferometry (PolInSAR) The coherence locus is assumed to be a straiht line in the complex plane G/V decomposition is carried out by fittin a straiht line in each interferometric pair Volume coherence Ground coherence Measured coherences real part real part real part Alebraic Synthesis The data coariance matrix is assumed to be structured as a Sum of 2 Kronecer Products G/V dec is carried out by tain the first 2 terms of the SKP decomposition of the data coariance matrix Scope of this or: Compare the to approaches from the alebraic and statistical points of ie
Model of the acquisitions We consider a multi-polarimetric and multi-baseline (MPMB) data Monostatic acquisitions: up to 3 independent SLC imaes per trac y n i Trac n Polarization i Trac N Re{y n ( )} Re{y n ( 2 )} Re{y n ( 3 )} Trac n V V VV Im{y n ( )} Im{y n ( 2 )} Im{y n ( 3 )} Trac V V VV V V VV
Coherence linearity PolInSAR is based on the ariation of the interferometric coherence.r.t. polarization i j, Σ Ey y n m i j Coherence linearity (*): i Σ nn Σ RVoG model => ESM coherences describe a straiht line in the complex plane i j Σ mm i j Multiple Scatterin Mechanisms (MSM) i = j Equalized Scatterin Mechanisms (ESM) j y n y y n n VV VV 2y V n, Volume coherence Ground coherence ESM coherences Polarization dependin factor Volume coherence Ground coherence real part (*) Papathanassiou and Cloude, Sinle Baseline Polarimetric SAR Interferometry
Coherence linearity PolInSAR is based on the ariation of the interferometric coherence.r.t. polarization i j, Σ Ey y n m i j Coherence linearity (*): i Σ nn Σ RVoG model => ESM coherences describe a straiht line in the complex plane i j Σ mm i j Multiple Scatterin Mechanisms (MSM) i = j Equalized Scatterin Mechanisms (ESM) j y n y y n n VV VV 2y V n Multiple baselines: one line per interferometric pair n = m = 2 n = m = 3 n = m = 4 Volume coherence Ground coherence ESM coherences real part real part real part (*) Papathanassiou and Cloude, Sinle Baseline Polarimetric SAR Interferometry
The SKP structure Without loss of enerality, the receied sinal can be assumed to be contributed by K distinct Scatterin Mechanisms (SMs), representin round, olume, round-trun scatterin, or other y n K s n; i i s (n, i ) : contribution of the -th SM in Trac n, Polarization i p: the data coariance is structured as a Sum of Kronecer Products y n K s n; i i -th Scatterin Mechanism K W K Eyy C R Each SM is represented by a Kronecer Product Coariance matrix amon polarizations: EM properties Coariance matrix amon tracs: Vertical structure R : interferometric coherences of the -th SM alone [NxN] C : polarimetric correlation of the -th SM alone [3x3] Note that R, C are positie definite
The ey to the exploitation of the SKP structure is the existence of a decomposition of any matrix into a SKP W SKP Dec P p p p V U W To sets of matrices U p, V p such that: then, the matrices U, V are related to the matrices C, R ia a linear, inertible transformation defined by exactly K(K ) real numbers 2 2 V V R V V R b b a a 2 2 U U C U U C a a b a b b b a R C R C W Corollary: If only round and olume scatterin occurs, i.e: then, there exist to real numbers (a,b) such that: K R C W Theorem: Let W be contributed by K SMs accordin to,2,3, i.e.: The SKP decomposition
eiht [m] heiht eiht [m] heiht [m] eiht [m] heiht [m] eiht [m] BIOSAR 27 Southern Seden P-Band TROPISAR French Guyana P-Band Courtesy of ONERA 6 6 5 4 3 2-6 5 4 3 2 - Forested areas: ho many KPs? 2 6 4 8 22 LIDAR Terrain eiht LIDAR Forest eiht V 2 6 4 8 22 slant rane [m] BIOSAR 28 Northern Seden P Band and L- Band 3 2-3 3 2 2 P-Band - V 2 25 3 35 4 45 5 L-Band - V - 2 25 3 35 4 Ground rane [m] 45 5 4 2 6 4 6 8 2 4 V 4 2 4 6 8 2 4 Slant rane [m]
eiht [m] heiht eiht [m] heiht [m] eiht [m] heiht [m] eiht [m] BIOSAR 27 Southern Seden P-Band TROPISAR French Guyana P-Band Courtesy of ONERA 6 6 5 4 3 2-6 5 4 3 2 - Forested areas: ho many KPs? 2 6 4 8 22 V BIOSAR 28 Northern Seden P Band and L- Band 3 2 - P-Band - V 2 25 3 35 4 45 5 L-Band - V LIDAR Terrain eiht 3 3 LIDAR Forest eiht 2 2 2 KPs account for about 9% of the information carried by the data in all inestiated cases - 2 25 3 35 4 2 6 4 8 22 Ground rane [m] slant rane [m] 2 Layered-models (Ground + Volume) are ell suited for forestry inestiations 45 5 4 2 6 4 6 8 2 4 V 4 2 4 6 8 2 4 Slant rane [m]
eiht [m] heiht eiht [m] heiht [m] eiht [m] heiht [m] eiht [m] BIOSAR 27 Southern Seden P-Band TROPISAR French Guyana P-Band Courtesy of ONERA 6 4 2 6 5 4 3 2-6 5 4 3 2 - Forested areas: ho many KPs? 2 6 4 8 22 LIDAR Terrain eiht LIDAR Forest eiht V 2 6 4 8 22 slant rane [m] BIOSAR 28 Northern Seden P Band and L- Band 3 2-3 3 2 2 Oerie tal: P-Band - V 2 25 3 35 4 45 5 P-Band penetration in tropical and boreal forests: Tomoraphical results Friday 4:4 Room L-Band - V - 2 25 3 35 4 Ground rane [m] 45 5 6 4 6 8 2 4 V 4 2 4 6 8 2 4 Slant rane [m]
Polarimetric Stationarity (PS): Coherence linearity and 2KPs: Alebraic connections Introduced by Ferro-Famil et al. to formalize the idely considered RVoG consistent case here the scene polarimetric properties are inariant to the choice of the passae Σ Alays alid after hitenin the polarimetric information of each imae in such a ay as: Alays retained in the remainder nn E y y Σ Ey y n n Σ mm I n nn 3 3 Under the PS condition the ESM coherence can be decomposed into a eihted sum: m m W K C R Σ E K y y C n m R (PS) K, K C C
Alebraic connections SKP decomposition W K C R (PS) K, ESM coherence decomposition 2 KPs Coherence Linearity W C R C R,, C C, C Coherence Linearity N(N-)/2 + KPs Im, a Re, b a Im Re b Each of the matrices R is fully specified by the real parts ( N(N-)/2 ) plus one constant that multiplies the affine term b There are at most N(N-)/2 + linearly independent KPs Poor physical interpretation: N(N-)/2 + Scatterin Mechanisms hose interferometric coherences are constrained to belon to the same line
Alebraic connections SKP decomposition (PS) ESM coherence decomposition Sinle baseline (N=2): perfect equialence 2KPs Coherence Linearity Coherence Multi-baseline Linearity (N>2): N(N-)/2 assumin KPs 2KPs entails more alebraic constraints than assumin coherence linearity: Each of the matrices R is fully specified by the real parts ( N(N-)/2 ) plus one constant that multiplies the affine term b There are at most N(N-)/2 + linearly independent KPs Poor physical interpretation: 2KPs Coherence Linearity N(N-)/2 + Scatterin Mechanisms hose interferometric coherences are constrained to belon to the same line
Alebraic connections In the multi-baseline case assumin 2KPs brin to adantaes oer coherence linearity:. Determination of physically alid solutions: W C R C R ith C, C, R, R positie definite Assumin coherence linearity: Imposin pair-ise positie definitieness results in physically alid round and olume coherences to be in manitude Assumin 2KPs: The positie definitieness constraint results in the reions of physical alidity to shrin from the outer boundaries toards the true round and olume coherences The hiher the number of tracs, the easier it is to pic the correct solution
Alebraic connections In the multi-baseline case assumin 2KPs brin to adantaes oer coherence linearity:. Determination of physically alid solutions: W C R C R ith C, C, R, R positie definite Assumin coherence linearity: Imposin pair-ise positie definitieness results in physically alid round and olume coherences to be in manitude Assumin 2KPs: The positie definitieness constraint results in the reions of physical alidity to shrin from the outer boundaries toards the true round and olume coherences The hiher the number of tracs, the easier it is to pic the correct solution 2. Coherence identification: Assumin coherence linearity: Independent identification in each interferometric pair 2 N(N-)/2 possibilities Ground Volume OR Ground Volume??? Assumin coherence 2KPs: Joint identification on all interferometric pairs 2 possibilities real part
Estimation from sample data Simulated scenario: 2 KPs: n = m = 2 - - W C Number of tracs : N = 4 n = m = 3 - - real part real part real part n = m = 2 R C Number of independent loos: L = {9 69} R Case : ih round coherences Case 2: Lo round coherences n = m = 3 n = m = 4 - - n = m = 4 2KP Estimators: L2 norm minimization fast but NOT optimal. Pair-Wise Estimator Each pair is processed independently Equialent to assumin coherence linearity 2. Joint Estimator All pairs are processed jointly 3. Preconditioned Joint Estimator All pairs are processed jointly As aboe, but the retrieed coherence matrices are alloed to be slihtly neatie Criteria for coherence retrieal: Volume: existence of a round-free polarization Ground: coherence maximization - - - - real part real part real part - - Note: coherence are assined to round or olume basin on nolede of the true alues coherence identification is NOT considered
Estimation from sample data Case : ih round coherences L = 6 Remars: Pair ise: Joint: Ground coherence is alebraically bounded to belon to the unitary circle Good accuracy hen the true round coherence is close to Systematic bias for lo round coherences Ground coherence is NOT bounded to belon to the unitary circle ih coherence may be underestimated Improed accuracy oer the Pair Wise approach for loer olume coherences Preconditioned Joint: Ground coherence underestimation is partly recoered Pair ise Joint Prec. Joint n = m = 2 - - n = m = 2 - - True round True olume n = m = 2 - - n = m = 3 - - n = m = 3 - - n = m = 3 - - Estimated round Estimated olume n = m = 4 - - real part real part real part n = m = 4 - - real part real part real part n = m = 4 - - real part real part real part Estimated olume after round phase compensation
Estimation from sample data Case : ih round coherences L = 49 Remars: Pair ise: Joint: Ground coherence is alebraically bounded to belon to the unitary circle Good accuracy hen the true round coherence is close to Systematic bias for lo round coherences Ground coherence underestimation is mitiated by increasin the number of loos Not a systematic bias Improed accuracy oer the Pair Wise approach for loer olume coherences Preconditioned Joint: Underestimation of round coherence is recoered Pair ise Joint Prec. Joint True round True olume n = m = 2 - - - - real part real part real part n = m = 2 - - real part real part real part n = m = 2 - - real part real part real part Estimated round n = m = 3 n = m = 3 - - n = m = 3 - - Estimated olume n = m = 4 - - n = m = 4 - - n = m = 4 - - Estimated olume after round phase compensation
Estimation from sample data Case : ih round coherences L = Remars: Pair ise: Joint: Ground coherence is alebraically bounded to belon to the unitary circle Good accuracy hen the true round coherence is close to Systematic bias for lo round coherences Ground coherence underestimation is mitiated by increasin the number of loos Not a systematic bias Improed accuracy oer the Pair Wise approach for loer olume coherences Preconditioned Joint: Underestimation of round coherence is recoered Pair ise Joint Prec. Joint True round True olume n = m = 2 - - - - real part real part real part n = m = 2 - - real part real part real part n = m = 2 - - real part real part real part Estimated round n = m = 3 n = m = 3 - - n = m = 3 - - Estimated olume n = m = 4 - - n = m = 4 - - n = m = 4 - - Estimated olume after round phase compensation
Estimation from sample data - n = m = 2 2 - n = m = 3 2 2 - n = m = 4 2 Case : ih round coherences Error on olume coherence n = m = 2 n = m = 3 n = m = 4 - - - n = m = 2 Error on round coherence n = m = 3 n = m = 4 - - - Error on olume coherence after round phase compensation n = m = 2 n = m = 3 n = m = 4 Pair-ise Joint Preconditioned Joint - - -
Estimation from sample data - n = m = 2-2 - n = m = 2 2 n = m = 3-2 2 n = m = 3 - n = m = 4-2 n = m = 4 2 Case 2: Lo round coherences Error on olume coherence n = m = 2 n = m = 3 n = m = 4 - - - n = m = 2 Error on round coherence n = m = 3 n = m = 4 - - - n = m = 2 Error on olume coherence after round phase compensation n = m = 3 n = m = 4 Pair-ise Joint Preconditioned Joint - - -
Conclusions Sinle-baseline case: assumin Coherence Linearity is equialent to assumin 2 KPs Multi-baseline: assumin 2KPs entails more alebraic constraints than assumin coherence linearity More accurate estimation of lo-alued round and olume coherence Simplifies the coherence identification problem to a sinle choice ih round coherence are underestimated if fe loos (say < 5) are employed Underestimation is mitiated by pre-conditionin the problem Estimators operatin throuh L2 norm minimization fast but not optimal The need for a pre-conditionin operator suests that sinificant improements could be achieed from the inestiation of a statistically optimal multi-baseline estimator for the 2KP model