Phase Unwrapping for Interferometric SAR Using Helmholtz Equation. Eigenfunctions and the First Green's Identity. Igor Lyuboshenko and Henri Ma^tre

Transcription

1 1 Phase Unwrapping for Interferometric SAR Using Helmholtz Equation Eigenfunctions and the First Green's Identity Igor Lyuboshenko and Henri Ma^tre Ecole Nationale Superieure des Telecommunications, Centre National de Recherche Scientique, Unite de Recherche Associee 82 { Departement Traitement du Signal et des Images, 46, rue Barrault, F Paris cedex 13, France Tel: , Fax: , lyubo@ima.enst.fr, maitre@ima.enst.fr Abstract We develop a new algorithm for interferometric Synthetic Aperture Radar (SAR) phase unwrapping based on the rst Green's identity with the Green's function representing a series in the eigenfunctions of the two-dimensional Helmholtz homogeneous dierential equation. This provides closed-form solutions using one- or two-dimensional Fast Fourier Transforms. The algorithm is elaborated using adaptive regularization of the interferometric phase gradient estimation. To diminish underestimation of the unwrapped phase typical of the linear phase unwrapping algorithms, the bias in the measured interferometric SAR phase is calculated in terms of the probability density function of the error in the processed interferometric SAR phase. Key words: SAR interferometry, phase unwrapping, gradient, bias, phase inconsistency, Green's identity, Helmholtz equation, eigenfunction, probability density function.

2 2 1 INTRODUCTION Phase unwrapping is the problem common to many elds of research such as optics [1], MRI imaging [2], and Synthetic Aperture Radar (SAR) interferometry [3]. It consists of the restoration of a threedimensional continuous phase surface from its principal value contained in the interval [ ; ). In the SAR interferometry, phase unwrapping is a key step to obtain high-precision digital elevation models (DEM) [4] from the measured interferometric SAR (InSAR) phase. Recently, the Green's Formulation (GF) was developed for phase unwrapping that uses the rst Green's identity and the Green's function for an unbounded two-dimensional (2D) domain [5]. The GF [6] is robust and is equivalent to the least-squares unwrapping algorithm [7]. The unwrapped phase represents an averaged sum of contour integrations along straight lines from a boundary of an interferogram to an observation point [8]. However, to unwrap the phase, this algorithm necessitates the preliminary estimation of the unwrapped phase on the boundary of the interferogram by solving the Fredholm equation of the second kind [5], by mirror reection of the data and the Green's function or by their periodization [6]. Furthermore, the unwrapped phase is biased since GF unwraps the phase by means of the linear transform of the phase gradient. This produces a biased result in the presence of noise and slopes in initial data [9], [1]. Moreover, the phase unwrapped by using GF exhibits strong erroneous impulses in the vicinity of degradation sources [5]. The ultimate goal of this paper is to develop a fast and numerically ecient algorithm for InSAR phase unwrapping based on GF and possessing of a high stability with respect to degradations in the initial InSAR phase with ability to diminish the bias in a resulting unwrapped phase. To this end in Section 2, we briey describe GF and emphasize its assets and deciencies. In Section 3, we develop a method for InSAR phase unwrapping which overcomes one of the major shortcomings of GF, namely, the necessity to implement time-consuming procedures needed for estimation of the unwrapped phase on the boundary of an interferogram prior to unwrapping itself. In Section 4, we develop an adaptive regularization procedure for stable InSAR phase gradient estimation. In Section 5, we evaluate the bias of a noisy InSAR phase to invoke it when unwrapping the phase. We obtain the expression for the bias as a function of the correlation between two complex SAR images, exact InSAR phase estimate and interferometric number of looks. In Section 6, we apply the developed methods to unwrap a real InSAR phase.

3 3 2 GREEN'S FORMULATION: A REVIEW The GF developed in [5][6][8] is a global approach to phase unwrapping that outperforms the existing methods in computational eciency and robustness when inner regions of phase inconsistencies are to be accounted for. It is intended for solving the InSAR phase unwrapping problem that is generally stated as follows: nd an estimate of a continuous bivariate function serving as an argument of the complex correlation c between two complex radar returns S A and S B dened as: c = E[S A SB ] = exp(j) (1) fe[js A j 2 ]E[jS B j 2 ]g1=2 given a noisy measurement of its principal value p known modulo 2: = [ p + e ] 2 ; (2) where e is the error of the InSAR phase estimation. In SAR interferometry, the common procedure for obtaining the InSAR phase is spatial or temporal averaging of responses from the same terrain patch (multilooking): = arg ( MX l=1 S A (l)s B(l) ) ; (3) where M is referred to as the number of looks. Green's Formulation uses the rst Green's identity, which can be written in terms of the unwrapped phase and a function g as: Z Z S ds(r 2 g + r rg) = I S ; (4) where a generally multi-connected region S with a boundary C is the support of the measured InSAR phase and the outward unit vector n S is perpendicular to the tangent of the contour C (Fig. 1(a)). Choosing g as the Laplace equation Green's function of the unbounded two-dimensional domain: g(r r ) = 1 2 ln jr r j; (5) i. e. setting g equal to the solution of the Poisson equation: r 2 g(r r ) = (r r ); r; r 2 S; (6)

4 4 the unwrapped phase (r ) is expressed in terms of the computed modulo 2 gradient r (r) of the wrapped phase (r) [5] as: (r ) = Z Z S I dsr (r) rg(r r ) r ) : (7) S The values of (r) at points r c on the boundary C are found in [5] by solving the Fredholm integral equation of the second kind: Z Z 1 2 (r c) = dsr (r) rg(r r c ) S I C r S ; (8) where the prime above the contour integral sign means that the point r c itself is excluded from the integration path. Successive iterations applied to solve (8) in [5] have led to a solution after 1{15 steps, the contour C being the outer boundary of the interferogram (Fig. 1(a)). The potential possibility of phase unwrapping on the boundary C of a multi-connected region S irrespective of its values in the interior of the surface S itself is useful to take out of consideration any inner region H (Fig. 1(b)) in the interferogram corresponding to lay-over or radar shadow areas or containing phase inconsistencies such as phase aliasing caused by noise and decorrelation [11]. The phase unwrapped by GF represents an average sum of all phases unwrapped by phase gradient integration along all radial paths starting at a point of interest r and ending at a boundary point r 2 C [8]. However, the unwrapped phase on the boundary is generally unknown. If the boundary value problem in (8) is solved for the surface S in Fig. 1(a) by applying simple iterations (as in [5]), then in order to obtain (N + 1) (N + 1)-pixel unwrapped phase image from phase gradient dened on a N N-point grid according to (7) using Fast Fourier Transforms (FFT), one implements: 1. three 2N 2N-point 2D FFT's (F 2D (2N)) for the surface integration that includes two FFT implementations to compute Discrete Fourier Transforms (DFT) of partial derivatives of the wrapped phase, and one inverse FFT to compute the linear convolution given by the rst integral in (7); 2. 2N-point 1D FFT (F 1D (2N)) 2N times to estimate DFT coecients r )=@n S along four boundaries of the domain S; 3. four 2N-point 1D FFT to obtain DFT coecients of (r c ) along all four sides of the domain S and 2N-point inverse 1D FFT 4N times to compute the linear convolution in the contour integral in (8) in each iteration of the iterative procedure.

5 5 The overall amount of FFT's needed to implement GF is therefore: N GF = 3F 2D (2N) + 2N F 1D (2N) + 4(N + 1)N it F 1D (2N); (9) where N it is the number of iterations (normally 1{15 [6]) required for the convergence. Experiments showed that the iterative procedure for solving (8) took up 9% of the overall time needed to unwrap the phase. The contribution of the boundary phase can be also eliminated by simultaneous periodic continuation of the Green's function (5) and of the relevant InSAR phase, as in Periodic Green's Formulation (PGF) [6], that results in substitution of the circular convolution for the linear one in (7) and consequently, changing the number of points for FFT's in (9) from 2N to N. This leads to increased computational eciency of PGF with respect to conventional LS procedures, as the one implemented in [12], especially for large-sized interferograms [6]. The iterations are needed to estimate unknown impulsive boundary components of the phase gradient, which arise on account of the periodicity imposed on the phase surface [6]. Another method to avoid solving for the boundary phase or the phase gradient is to perform the mirror reection before the periodization, the computational cost of the procedure becoming equal to 3F 2D (2N) [6]. However, this leads both to dramatic growth of the processed data volume and to superimposition of errors caused by degraded areas in the mirrored and periodically continued regions [8]. Furthermore, an unwrapping error has its greatest absolute values proportional to a gradient error in the vicinity of InSAR phase inconsistency [5]. The gradient computation problem being ill-posed [13], innitely small perturbations in initial InSAR phase may cause innitely large impulsive errors in a computed gradient. The computation of the discrete derivatives, or instantaneous frequencies [14], can only mitigate, but in no means eliminate, the consequences of this instability. Thus, despite of its computational exibility and robustness, GF calls for applying time-consuming procedures to solve the outlined boundary value problem and is locally unstable in case, for instance, of compact areas of InSAR phase measurement errors. Moreover, since GF implements the linear transform of the modulo 2 phase gradient estimate, it produces a biased unwrapped phase surface in the presence of noise and phase slopes [1]. In the insuing sections, we show that these deciencies can be eliminated by modifying the Green's function, by adaptive regularization of the gradient search and by computation of a nonbiased estimate of the wrapped InSAR phase.

6 6 3 HELMHOLTZ EQUATION EIGENFUNCTIONS FORMULATION A. Expansion of a Green's Function in a Series of the Helmholtz Equation Eigenfunctions In order to omit the integration of an initially unknown unwrapped phase along a boundary S of an interferogram (Fig. 1(a)), we need to choose another Green's function to be used in the formulation of the ultimate solution (7). This can be implemented by using the mathematical theory developed for Green's functions that ensures an expansion of a Green's function in a series of the scalar Helmholtz equation eigenfunctions separable for an arbitrary closed region in a given coordinate system [15]. Let the Green's function for the two-dimensional Laplace equation be written in its general form as the innite series: (r; r ) = 1X m;k= 2 mk 6= F mk (r)f mk (r ) 2 mk (1) in eigenvalues 2 mk and eigenfunctions F mk(r) of the Helmholtz homogeneous equation forming the orthonormal set and verifying: r 2 F mk (r) + 2 mkf mk (r) = ; m; k 2 (; 1): (11) From the theory, it is known that, if the eigenfunctions F mk (r) verify some homogeneous boundary conditions, these will also be veried for the expansion (1) [15]. Let now the region of support S of the interferometric phase be a square with side length a and boundary C (Fig. 1(a)): S = f(x; y) : x; y 2 [; a[g: (12) Then it can be readily shown that the functions and values: F mk (x; y) = 2a 1 cos( m x) cos( k y); 2 mk = 2 m + 2 k; m; k 2 (; 1); (13) where m = m=a; k = k=a and F mk (x; y) form the orthonormal set, verify the equation (11) along with the imposed Newmann homogeneous boundary conditions along the contour mk S r2c = : (14)

7 7 Therefore, for the Green's function: ~g(r; r ) = X m;k= m 2 +k 2 6= cos( m x) cos( k y) cos( m x ) cos( k y ) m 2 + k 2 (15) which is a particular form of the series (1), the same Newmann homogeneous boundary conditions r S r2c = ; r 2 S: (16) Note that contrary to the function g(r r ) in (7), ~g(r; r ) represents the Laplace equation Green's function of the bounded square domain S. The contour integration in (7) can now be omitted altogether to yield: (r ) = Z Z S dsr (r) r~g(r; r ): (17) This new Helmholtz formulation (HF) for the phase restoration from its gradient estimate gives rise to two numerical methods which dier by various strategies adopted when computing the expression (17) and which we discuss later in this section together with the consideration of an error in the result due to a streak-like perturbation in the phase gradient eld. Given the N N initial phase gradient vector eld, one of these methods uses three two-dimensional 2N 2N-point FFT's, while the other computes the unwrapped phase after implementation of 2N one-dimensional FFT's. The doubled number of points (2N) of the Discrete Fourier Transforms in both cases stems from the fact that on the discrete grid x(l) = lx; l = [; N 1] where a = Nx, the argument m x(l) = (=a)mlx of the cosine in the series (15) above becomes m x(l) = (=(Nx))mlx = (=N)ml. In order that FFT be implemented, one needs with this argument a doubled number of points in DFT, i. e. 2N [16]. B. Solution Using 2D DFT's For numerical implementation of the algorithm, we restrict the number of elements in the series (15): ~g(r; r ) = 4 2 N 1 X m;k= m 2 +k 2 6= cos( m x) cos( k y) cos( m x ) cos( k y ) m 2 + k 2 ; (18) and let a coincide with the number N of rows and columns for each of square N N interferometric phase partial derivative matrices so that the horizontal and vertical grid spacings x and y be equal to unity.

8 8 Let the real discrete image ^(m; k) be dened as: ^(m; k) = =fmx(m; k) + ky (m; k)g ; 4N(m 2 + k 2 ) X(m; k) = ^X( m ; k ) + ^X( m ; k ); Y (m; k) = ^Y ( m ; k ) ^Y ( m ; k ); (19) where m; k = ; : : :; N 1, =f g denotes the imaginary part, and DFT coecients of the InSAR phase partial (x(l); y(n))=@x (x(l); y(n))=@y; x(l) = l; y(n) = n; l; n = [; N 1] corresponding to grid positions with coordinates m = (=N)m and k = (=N)k in the frequency domain are denoted as ^X(m ; k ) and ^Y (m ; k ), respectively. Then, the unwrapped phase is shown in Appendix A to verify: (l; n) = 4N 2 <f(l; n) + (l; n)g; l; n = [; N 1]; (2) where stands for the inverse DFT of ^ and <f g denotes the real part. Hence, the computation of the unwrapped phase calls for implementation of the two 2D FFT's to obtain ^ and one inverse 2D FFT to compute, so iterations and related time-consuming and complicated operations such as the contour integration are no longer needed. The number of computations (in terms of FFT's) is reduced by 2N F 1D (2N) + 4(N + 1)N it F 1D (2N) compared to that needed for GF in [5]. Furthermore, if the presented method is to be compared with that described in [12], where after data doubling by mirror reection one needs 4(N + 1) implementations of 2N-point 1D FFT's to obtain a solution, that is computationally equivalent to roughly one implementation of the 2N 2N-point 2D FFT, then it should be emphasized that obtaining the solution in (2) does not require any initial data doubling although all the computations are performed using the data grid the linear dimensions of which are twice as those of the initial data. Instead, it does require padding the initial N N data with zeros to ensure that the grid spacing in the frequency domain equals =N. C. Error in the Unwrapped Phase The inuence of a streak-like perturbation in the InSAR phase gradient eld can be now investigated. Let the gradient of the measured InSAR phase dier from its exact value by 2 along a line of length 2L centered at a point (; u) 2 S in the plane. Then, the error in the unwrapped phase is shown in Appendix A to be expressed by a series: e(r ) = 8L N N 1 X N 1 X m= k= F mk (r ) k sin( k u) cos( m ) m 2 + sinc( m L); (21) 2 k

9 9 where sinc(x) = sin(x)=x, and the case where m = k = is excluded from the summation. From (21), it can be seen that the discrete 2D frequency response of the error e(x ; y ) will be proportional to the 2D discrete sequence sinc( m L), multiplied by the magnitude of the frequency response = :5y=jrj 2 and modulated by harmonic functions along axes and in the frequency domain with spatial frequencies and u. Therefore, the error given by (21) has a duration along the x -axis, comparable with L with strong peaks in the vicinity of the point (; u) where there is a degradation source and decays inversely proportionally to the distance from the point (; u) in the y -direction and to the squared distance in the x -direction. The error surface described by expression (21) is plotted in Fig. 2 for = u = 32 with L = 4 and a = 64. From Fig. 2, it is seen that the phase unwrapping error (21) exhibits the same behavior as that of the algorithm developed in [5], i. e. has strong local peaks in points where there are degradations in the InSAR phase gradient. It can be inferred that the developed algorithm is robust since the unwrapping error is conned to a vicinity of a degradation source. D. Solution Using 1D DFT's and Numerical Implementation Issues Consider again the presentation (15) of the Green's function and rewrite it in a slightly dierent form, consisting of two successive summations of eigenfunctions, pertaining to pairs of identical coordinates of a potential source r and an observation point r : Using the identity [17]: ~g(r; r ) = 4 2 1X m=1 1X k= m 2 +k 2 6= cos( k y) cos( k y ) cos(mx) m 2 + k = cosh(k( x)) 2 2k sinh(k) 1X m= cos( m x) cos( m x ) m 2 + k 2 : (22) 1 ; x 2; (23) 2k2 and supposing that k 6=, the second sum in (22) can be reduced to its closed form written as: 2 1X m=1 cos( m x) cos( m x ) m 2 + k k 2 = k f k(x; x ); (24) where f k (x; x ) = 1 cosh(k k x) cosh( k x ); x > x ; sinh(k) cosh(k k x ) cosh( k x); x < x : (25) Compute now the derivative of ~g(r; r ) with respect to y. The requirement m 2 + k 2 6= can be replaced by k 6=, since the term corresponding to k = in the series (22) does not depend on y. Restricting the

10 1 number of elements in the rst series in (22) by N 1 we obtain: X ~g(r; r ) = 2N 1 cos( k y )f k (x; x ) sin( k y); (26) k=1 where by convention N = a. By the same reasoning, the derivative of ~g(r; r ) with respect to x is: X ~g(r; r ) = 2N 1 cos( m x )f m (y; y ) sin( m x): (27) m=1 Substitution of (26) and (27) into (17) and integration yield the sought-for expression for the unwrapped phase: N 1 X (x ; y ) = 2N 1 k=1 X cos( k y ) N 1 + 2N 1 cos( m x ) m=1 Z a Z a f k (x; x )r x ( k )dx f m (y; y )r y ( m )dy; (28) where r x ( k ) and r y ( m ) represent imaginary parts of the Fourier Transform (FT) of the corresponding derivatives of the wrapped phase. They are calculated from real parts of FT's U x () and U y () of u x (y) = (x; y)j x=const and u y (x) = (x; y)j y=const, respectively: r x ( k ) = k <fu x ( k )g; r y ( m ) = m <fu y ( m )g; m; k = 1; : : :; N 1: (29) The round-o error of unwrapping using (28) will be less than that when (2) is used since the 2D series (22) is approximated in (28) by a nite sum only once. The truncation of (22) that yielded (28) enabled to make optimal use of the available degrees of freedom of the image for subsequent use of DFT. From (28, 29), the InSAR phase with gradient dened on a N N-point grid with a = N is unwrapped after successive implementations of 1D FFT's according to the following scheme: 1. implementation of a 2N-point 1D FFT to estimate <fu x ( k )g and <fu y ( m )g, m; k = 1; : : :; N 1. For instance, to obtain <fu x ( k )g for some xed x, we unwrap (x; y) along a line x = const using the standard 1D phase unwrapping algorithm to avoid the inuence of phase discontinuities caused by wrapped phase transitions from to, then we subtract the linear and constant components from the result to avoid strong erroneous impulses on both ends of the resulting derivative curve,

11 11 and implement (29). Then the linear phase component is restored by adding to (29) corresponding DFT coecients. This step results in estimates of r x ( k ) and r y ( m ), m; k = 1; : : :; N 1, for each discrete value of x; y 2 [; a[; 2. calculation of the functions f k in (25) and numerical evaluation of the integrals in (28). To avoid the overow and round-o errors when m or k is large, the following approximations were used which stem from letting k be close to innity: cosh(k k x)= sinh(k) exp( k x); cosh( k x)= sinh(k) exp( k + k x); (3) 3. implementation of the inverse 2N-point 1D FFT 2N times according to (28) to obtain the unwrapped phase (x ; y ) on the square N N grid. The second and third steps represent computation of the unwrapped phase from the gradient of the wrapped phase, estimated at the rst step, and call for numeric integration and implementation of 2Npoint 1D FFT's 2N times. The gain of the numerical implementation of the algorithm with respect to GF in [5] in terms of FFT's is therefore 3F 2D (2N) + 4(N + 1)N it F 1D (2N). The approximate numbers of multiplications associated with all relevant methods are incorporated into Table 1, where HF1 and HF2 denote the methods developed in Subsection 3.B and this Subsection, respectively, and PS stands for the method developed in [12]. The derivation of the computational costs of HF and GF is provided in Appendix B, and the corresponding quantities for PGF and PS are taken from [6]. The third column of Table 1 shows the approximate numbers of multiplications given N = 512 and N it = 1. From Table 1, it is seen that HF1 and HF2 have the same order of the computational cost as PGF and PS algorithms. Nonetheless, the number of required multiplications for HF2 given N = 512 is reduced by 3% and 4% with respect to PGF and PS, respectively. The computational cost for HF2 is reduced by 9% with respect to GF, mainly because HF needs no iterations to obtain a solution. In contrast to PGF and PS, the developed algorithm does not impose mirror symmetry or periodicity constraints onto the interferogram and sought-for unwrapped phase. Absence of mirror reection and periodization steps in HF is a consequence of the special form of the Green's function (15), which allows omitting the formulation of the boundary conditions imposed on phase values, as in [12], where the explicit formulation of boundary conditions is obviated due to the periodicity constraint imposed on the reconstructed surface.

12 12 Absence of the mirror reection and periodicity constraints precludes propagating the unwrapping errors from adjacent mirror symmetric or periodically continued portions of the initial interferogram [8]. Thus, the major advantage of the developed method is, besides its computational exibility and robustness, the possibility of omitting the contour integration in (7) along the exterior boundary of the support region S thus taking into consideration only interior areas of phase inconsistencies (Fig. 1(b)). This can be done, for example, by solving a Fredholm integral equation similar to (8), but with the contour integration solely along the inner boundaries B of the interferogram corresponding to the regions H to be excluded from the integration domain S. In Appendix C, we outline another way in which the contour integration of an initially unknown unwrapped phase could be avoided. We show that if the boundary B encircles closely enough a phase discontinuity region or an area in the interferogram corresponding to a terrain surface patch with weak radiometry, the following equation holds: Z Z I (r ) = U(r ) dsr (r) r~g(r; r ) + dc~g(r; r ; (31) S H where S = S H B and: U(r ) = 8 >< >: 1=2; r 2 H; 2=3; r 2 B; 1; r 2 S : (32) The normal derivative of the continuous phase (r) may be readily estimated from that of the wrapped phase (r). Due to separability of the Helmholtz equation eigenfunctions (13), the contour integration can be reduced to the form independent on r. For a rectangular domain H this allows, for example, to reduce the number of FFT's implemented for its evaluation from 4N (as is required for the contour integration in (7)) to only four. These latter expressions explicitly use the wrapped phase gradient in the region S of reliable phase estimation and implicitly dene the phase gradient in the other regions by continuing its values subject to condition of the harmonic phase surface (see Appendix C). Therefore, the algorithm described by (31) and (32) is hereinafter referred to as the phase gradient continuation algorithm. It will be tested by numerical simulations in Section 6. It should be noted that the rst Green's identity (4) holds solely for continuous functions with continuous partial derivatives. To meet these requirements of continuity, the unwrapped phase surface becomes smoothed out in regions corresponding to true discontinuities in the terrain (with respect to the radar-

13 13 ground geometry). Given the chosen Green's function ~g(r; r ), the identity: F [(r )] = Z Z S dsrf [(r)] r~g(r; r ); (33) will hold, where F is some continuous transform (e.g. continuous weighting function). This weighting function could be connected to the amplitude of radar responses, since there is an increased radar response intensity in regions adjoining the steep slopes on the terrain, the phenomenon being widely used in radarclinometry [18]. We can impose some general conditions (for ex.: invertibility) on F and write an alternative solution in the form: (r ) = F 1 Z Z S dsrf [(r)] r~g(r; r ) : (34) Provided that the F is chosen so that to render F [(r)] continuous given generally discontinuous function (r), the use of (34) would permit to avoid contour integration in the interior of the region S and potentially smoothing out the unwrapped phase in the vicinity of steep slopes, since the solution (r ) would not be necessarily continuous. The rigorous mathematical background for this promising formulation is currently under construction. 4 REGULARIZATION OF THE INSAR PHASE GRADIENT Because of temporal decorrelations and processing artifacts [19], regions with unreliable InSAR phase estimate will appear in any measured interferogram. In order to get rid of local erroneous perturbations in the unwrapped phase given by (21), a regularized InSAR phase gradient should be employed. We make use of classical adaptive regularization means that, to the best of our knowledge, has not yet been applied to phase unwrapping. In the frequency domain, FT's of regularized derivatives of u x and u y are written [13] as: x () = iu x ()R x (; ); y () = iu y ()R y (; ); (35) where R x (; ) and R y (; ) are stabilization factors and is a regularization parameter. The regularization parameter is found either by trial and error or according to the orthogonality principle [13]. Let the stabilization factor R x (; ) be dened in terms of the regularization parameter and the frequency response U x () as: R x (; ) = h ju x ()j 2i 1 ; (36)

14 14 where ju x ()j denotes the normalized FT magnitude and let the expression similar to (36) hold also for R y (; ). It can be readily shown that R x (; ) given by (36) veries all the requirements imposed on stabilization factors for the ill-posed problem of the gradient search [13]. It is well known [13] that using conventional stabilization functionals decreases the resolution of the result. The major advantage of the adaptive regularization dened by expressions (35) and (36) is its increased resolving ability compared to conventional regularization techniques [2]. In Fig. 3, we demonstrate the improved resolution obtained by adaptive regularization in comparison with the conventional regularization using quadratic functionals: ^R x (; ) = [1 + 2 ] 1 : (37) In Fig. 3, the regularization parameter was adjusted so that the mean-square error of the derivative computation be the same for both regularization types. From Fig. 3, it is seen that the resolution of the regularized solution (35) with R x (; ) given by (36) is much higher than that with ^R x (; ) given by (37). All the peaks of the simulated function derivative are well resolved by the adaptive regularization algorithm. We see from this example that the use of R x (; ) from (36) allows to preserve high resolution when processing noisy interferometric data irrespective of the noise level. Introducing adaptive regularization factors (36) enables to rewrite the expressions (29) as: r x ( k ) = k <fu x ( k )gr x ( k ; ); r y ( m ) = m <fu y ( m )gr y ( m ; ); m; k = 1; : : :; N 1; (38) which will be used later in Section 6 while unwrapping a real InSAR phase. 5 BIAS SUPPRESSION PROCEDURE The underestimation of the unwrapped phase tilt by linear techniques such as weighted and unweighted least squares (WLS and LS) phase unwrapping algorithms is a well-established phenomenon. These methods tend to underestimate the overall phase surface steepness in the presence of slopes and noise in the initial data. This is because when there is a nonzero slope in the wrapped phase and when the gradient of the true phase is estimated as a wrapped gradient of the InSAR phase, the resulting conservative mean gradient noise eld is also nonzero and tends to have an opposite polarity with respect to the true phase gradient eld [1]. This result is general for all linear (with respect to r ) algorithms, including GF [5] and PGF [6]. It holds also for the method presented in Section 3 of this paper. The conventional iterative means to reduce the underestimation eect consists of unwrapping the

15 15 original phase, subtracting modulo 2 the resulting estimate from the initial InSAR phase and applying the unwrapping procedure to the obtained dierence [21] [22]. The reconstructed phase surface is updated in each iteration by adding the unwrapped dierence to its previous estimate. The underestimation problem is also partially solved when the weights are chosen in WLS according to residue positions and resulting discontinuities in the InSAR phase [23]. Another option is to correct the reconstructed phase slope using the expression for the slope bias [1]. In this section, we present a novel approach that seeks to diminish the tilt loss by one computation step embedded in the unwrapping procedure described in Sections 3 and 4. It invokes the known phase noise probability density function derived in [24] from the interferometric number of looks and coherence to compute a mean value and a corresponding variance of the dierence = p between measured and exact InSAR phases. From theory on probability density distribution of the coherence parameters of complex SAR responses [24], it is known that the probability density function p(; M; ) of e can be expressed in terms of the absolute value of the complex correlation (1) and the number of looks M as: p(; M; ) = (M + :5)(1 2 ) M 2 1=2 (M)(1 2 ) + (1 2 ) M M F (M; 1; :5; 2 ); (39) +:5 2 where = cos and F (M; 1; :5; 2 ) is the Gauss hypergeometric function. In Appendix D, it is shown that the mean value and variance of are conditioned by p and given by: Efj p g = 2sign( p ) Z +jpj Dfj p g = Df; Mg Efj p g 2 2Efj j p jg + 4 p(; M; )d; (4) Z +jpj p(; M; )d; (41) where signf g denotes the sign of the argument, Df; Mg represents the error variance of the estimate of the InSAR phase. To avoid unwieldy computation of the variance from the expression (39), one can use the Cramer-Rao lower bound [25] on Df; Mg: Df; Mg CR = 1 2 2M 2 : (42) The plots of Efj p g and Dfj p g versus positive exact InSAR phase p and correlation are depicted for M = 1 in Fig. 4(a) and Fig. 4(b), respectively. As M increases, the surface corresponding to the bias (4) becomes more concave, and that corresponding to variance (41) { more convex, that is consistent with

16 16 the decrease of the noise in the InSAR phase when increasing the number of looks M. From (4) and (41), it can be inferred that the bias Efj p g and the variance Dfj p g verify the following properties: 1. Efj p g is the probability of the event j p j + e >, multiplied by 2 and calculated with the sign opposite to that of p. Hence, for the measured InSAR phase and corresponding exact phase p, the bias Efj p g has the higher absolute value the closer j p j is to ; 2. Efj p g = Efj p g, Dfj p g = Dfj p g; 3. p = : Efjg =, Dfjg = Df; Mg; p = : Efjg =, Dfjg = Df; Mg R p(; M; )d; 4. = ; j p j 2 (; ): p(; M; ) = 1=2, Efj p g = p, Dfj p g = 2 =3; = 1; j p j 2 (; ): p(1; M; ) = (), Efj p g =, Dfj p g =. From the property 1, it can be concluded that the estimate of the gradient r obtained by computing the principal value nite dierence (PVFD) [14] will be biased towards its smaller absolute value: if for successive InSAR phase samples p1 > and p2 >, the inequality < p2 p1 < holds (that is, if there is a positive slope), then given the same correlation coecients and number of looks corresponding to these samples, the inequality Efj p2 g Efj p1 g < will hold since the probability of the event j p2 j + e > is greater than that for the event j p1 j + e >. In general, the bias for a positive value of the PVFD estimate is negative, and vice versa. Hence, the PVFD absolute value is underestimated and the image of the principal value of the phase unwrapped from noisy data by using linear unwrapping algorithms contains less fringes than does the original interferogram. Therefore, an increased amount of ground tie points will be necessary to generate a reliable DEM. Having computed the bias, we perform the correction of the initial wrapped phase to obtain its unbiased estimate ^: ^ = Efjp g; (43) which is not necessarily conned to the basic interval [ ; ) and changes rather between 2 and 2. By so doing, we change the dynamic range of phase and phase derivative variations from the interval [ ; ) to [ 2; 2). From the above discussion on the property 1, it can be inferred that implementing (43) will have a reconstructing eect onto underestimated absolute values of phase derivatives. The unwrapped phase is obtained by applying (17) to r ^. The gradient of ^ is the dierence between the wrapped phase gradient r and the gradient of the bias refj p g. The latter will contain the impulses in points where p changes the sign. These impulses can be got rid of by numerical interpolation of the

17 17 refj p g values from nearby points. By implementing the computation of the corrected phase gradient in this way, we obviate the conventional modulo 2 gradient computation, as was recently suggested in [9], [26]. In addition, the conventional iterative procedure mentioned earlier in this section that forms a residual phase and upgrades the unwrapped result to reduce the bias in each iteration is avoided. 6 EXPERIMENTAL RESULTS In this section, we show and analyze experimental results on unwrapping real interferograms by using the methods developed in Sections 3 through 5. Our test site is the region of Bern in Switzerland, a mountainous area with many lay-over regions. Therefore, in the interferogram (Fig. 5(a)), singular points (where the rotational component of the InSAR phase gradient is nonzero) account for 15% of all points. The corresponding correlation is shown in Fig. 5(b). We analyze the quality of unwrapping by applying the developed algorithms and by comparing results with those obtained with GF developed in [5]. All the results on unwrapping in this section correspond to the case where no iterations to reduce the bias were employed. Since the exact DEM is unavailable to us, we test the performance of all the procedures by comparing the principal values of the unwrapped phase with the initial InSAR phase of Fig. 5(a). For better visual evaluation of the obtained results, only a small portion (of size ) of the unwrapped phases corresponding to the square region bordered by white dashed line in Fig. 5(a) is displayed while for the rewrapped and residual phase images the original size ( ) is preserved. The quality of unwrapping is also assessed by following three chosen reference lines (RF). Their positions are specied in Fig. 5(a) as RF 1, RF 2 and RF 3. The results produced by GF are depicted in Fig. 6. From Fig. 6(a), it is seen that erroneous impulses exist everywhere in the image of the unwrapped phase and render the discrimination of the terrain relief features rather dicult. In Fig. 6(b), we note the reduction of the density of fringes in comparison with that in the original InSAR phase. If we roughly measure the fringe density in principal phase images by the number of fringes along chosen reference lines, then for RF 1 we obtain 11 fringes for the unwrapped phase principal value in Fig. 6(b) while there are 24 fringes in the initial interferogram along this line. If we follow reference lines 2 and 3, then we obtain the number of fringes in Fig. 6(b) equal to 7 and 6, respectively, although the corresponding quantities for the initial InSAR phase are equal to 15 and 13. There are plenty of residual fringes in Fig. 6(c). Hence, the terrain altitude will be underestimated if a DEM is generated by using only the unwrapped phase in Fig. 6(a). In that case, the use of an increased number of ground tie points would be necessary to produce a reliable DEM.

18 18 To apply the bias suppression procedure, the interferometric number of looks M for the processed interferogram and the exact InSAR phase p must be known. The interferogram in Fig. 5(a) was obtained after cross-multiplication of two complex SAR images with averaging 2 pixels in range and 1 pixels in azimuth. As adjacent pixels are partly correlated in the images registered by ERS-1, the interferogram in Fig. 5(a) exhibits statistical properties of 1-look images [19]. Therefore, the number of looks M is assumed for the interferogram in Fig. 5(a) equal 1. In the absence of any a priori information about the exact InSAR phase p, we calculated its estimate by median ltration (with window size 3 3) of the complex image exp(j (r)) and then applied (4) to compute the bias using this estimate. The unwrapped phase, its principal value, and the residual phase obtained by applying the developed algorithms, are presented in Fig. 7(a), Fig. 7(b), and Fig. 7(c), respectively. The computation time is 4 seconds on a Sun Ultra-2 workstation that is 1 times less compared to that needed for GF due to the choice of the relevant Green's function as shown in Section 3. From Fig. 7(a), it is seen that the quality of unwrapping allows better discrimination of the features of the terrain relief. Moreover, from comparison of the initial interferometric phase in Fig. 5(a) and the principal value of the unwrapped phase in Fig. 7(b) it follows that both the location of fringes and their spatial density are approximately the same except for some disagreement in the lower right corner of the image due to very low correlation (on the order of.1) between complex SAR returns. It can be directly veried that the number of fringes along all chosen reference lines in Fig. 7(b) is the same as that in the initial interferogram. From the comparison between residual phases in Fig.6(c) and in Fig. 7(c) it can be inferred that the latter has a smoother appearance and hence corresponds more adequately to initial InSAR data. Results presented in this section hold also for larger data volumes. The point interferogram shown in Fig. 5(a) is an upper right corner portion of a larger point one presented in Fig. 8(a) and accompanied by a corresponding coherence image (Fig. 8(b)). The phase unwrapped with our method and its principal value are presented in Fig. 9(a) and in Fig. 9(b), respectively. For the sake of presentation clarity, the image of the unwrapped phase is superimposed in Fig. 9(a) with the corresponding image of amplitude of radar returns. We now investigate the potential gain from using the phase gradient continuation algorithm the detailed formulation of which was given in Appendix C and the major result was reproduced in expressions (31) and (32). In Fig. 1(a), a simulated continuous phase is presented with a region of phase discontinuity located in an interferogram region corresponding to a steep slope on the terrain. The InSAR phase is undersampled in this region so that the dierence between neighboring phase samples is greater than. The result of applying GF is depicted in Fig. 1(b). It is seen that the discontinuity region is not discriminated properly.

19 19 Since GF produces an averaged phase over all possible integrations from the boundary of the interferogram to the point of interest [6], it tends to smooth out this rupture with a pronounced underestimation eect on the peak next to it. Like any least squares approach, it lacks proper determination of rupture (cut) lines and cannot prevent the integration paths to pass across the zones of phase inconsistencies [23]. In contrast, the result of applying the phase gradient continuation algorithm (Fig. 1(c)) is much closer to the initial continuous phase. The unwrapped phase is seen to allow the phase dierences to be greater than. It is also seen that both the placement and the form of the rupture are correct and that the underestimation eect is reduced. To be implemented on real data, the algorithm necessitates a proper determination of the regions with weak radiometric response and (or) coherence of radar returns. This question, however, is beyond the scope of the present paper and will be addressed in future studies. 7 CONCLUSIONS A closed-form solution was obtained for the Green's Formulation for InSAR phase unwrapping. Fast and stable algorithms were proposed that allow to unwrap the InSAR phase after a succession of computations of Discrete Fourier Transforms. The major advantages of the developed methods are their computational exibility and the possibility to omit the contour integration which is mandatory in (7) along the exterior boundary of the interferogram thus taking into consideration only interior areas of InSAR phase inconsistencies. The experiments on real InSAR data conrmed both high computation speed and stability of the developed algorithms. A procedure for excluding phase inconsistency areas was outlined and tested in this paper. It gave excellent results on simulated data that allows to use it while processing real InSAR phase. ACKNOWLEDGMENTS This work was supported by Centre National d'etudes Spatiales (French Space Agency). The authors express their gratitude to anonymous rewievers whose comments were benecial for improving the contents of the paper.

20 2 APPENDIX A We derive a closed-form solution for the phase unwrapping problem in terms of the Green's function (15) and compute the error induced by incorrect InSAR phase gradient estimation. The partial derivatives of (18) are given by: N 1 X ~g x (r; r ) = 2 1 N 1 X m= k= N 1 X ~g y (r; r ) = 2 1 N 1 X m= k= F mk (r ) cos( k y) m 2 + k 2 m sin( m x); (A1) F mk (r ) cos( m x) m 2 + k 2 k sin( k y); (A2) where m = m=n; k = k=n; m; k = ; : : :; N 1. Scalar multiplication by r (r) and surface integration yield: N 1 X (x ; y ) = 2 1 Z Z + k S N 1 X Z Z F mk (r ) m m 2 + k (r) cos( mx) sin( k y)ds (r) sin( mx) cos( k y)ds : (A3) In the sums (A1), (A2) and (A3) above, the case m = k = is excluded from summation. We denote by R(m; k) the expression in brackets in (A3) and rewrite it: R(m; k) = Z Z Z Z S (r) (r) sin( m x + k (r) sin( m x k y)ds: (A4) Approximating the integrals in (A4) by nite sums and using the denition of the 2D DFT [16], we obtain: R(m; k) ^(m; k) = 2N(m 2 + k 2 ) = =fmx(m; k) + ky (m; k)g 4N(m 2 + k 2 ) (A5) for m; k = ; : : :; N 1, where =f g denotes the imaginary part of the expression in parentheses. Complex matrices X(m; k) and Y (m; k) are given by: X(m; k) = ^X(m ; k ) + ^X(m ; k ); Y (m; k) = ^Y (m ; k ) ^Y (m ; k ); (A6) where, in turn, ^X(; ) and ^Y (; ) denote the 2N 2N-point 2D DFT's of the corresponding partial derivatives of the wrapped (x(l); y(n))=@x (x(l); y(n))=@y; x(l) = l; y(n) = n; l; n =

21 21 [; N 1] padded with zeros for l; n N. Using the inverse 2N 2N-point 2D DFT and standard trigonometric identities, we obtain the ultimate result: (l; n) = 4N 2 < [(l; n) + (l; n)]; l; n = [; N 1]; (A7) where (l; n) stands for the inverse 2D DFT of ^(m; k) and <f g denotes the real part of the expression in parentheses. Let the y-component of the InSAR phase gradient be perturbed along a line of length 2L centered at point (; u) 2 S of @ (x; y) = (x; (x; (x; y) 2(y u)rect x 2L : (A8) Then it follows from (A3) that the phase unwrapping error is given by a series: e(r ) = 4 = 4 N 1 X = 8L N N 1 X m= k= X X N 1 N 1 m= k= X X N 1 N 1 m= k= Z kf mk (r ) (y u)rect m 2 + k ZS 2 kf mk (r ) m 2 + k 2 sin( ku) Z +L L x 2L cos( m x)dx F mk (r ) k sin( k u) cos( m ) sinc( m 2 + k 2 m L); cos( m x) sin( k y)ds (A9) where sinc(x) = sin(x)=x. APPENDIX B The number of multiplications involved in computation of 1D N-point FFT equals :5N log 2 N, and this number for processing N N-pixel image using N N-point 2D FFT is N 2 log 2 N [16]. Hence, the computational cost for GF can be derived from (9) to yield: N GF = 14N 2 log 2 (2N) + N it (4N 2 + 4N) log 2 (2N) + 2(2N) 2 N 2 (f14 + 4N it g log 2 N N it ); (B1) where the rst-order term in N was neglected and 2(2N) 2 multiplications for ltering phase derivatives were added. The computational cost for HF1 was derived as follows. To compute ^X( m ; k ) and ^Y ( m ; k ) from (19), two 2N 2N-point 2D FFT's should be implemented, calling for 4N 2 log 2 (2N) 2 multiplications.

22 22 For ltering ^X and ^Y and calculating ^(m; k) from (19), 4(2N) 2 multiplications are needed. To obtain (l; n) from (2), one additional FFT is used. The number of multiplications is therefore: N HF 1 = 4N 2 log 2 (2N) 2 + 4(2N) 2 + 2N 2 log 2 (2N) 2 = N 2 (12 log 2 N + 28): (B2) In order to implement HF2, two DFT's to obtain (x ; y ) in (28) call for implementation of 2N-point 1D FFT 2N times, yielding the number of multiplications 2N 2 (log 2 N + 1). There are 8N 2 multiplications in integrals in (28). To show this, consider the rst term in (28): where N 1 X (1) (x ; y ) = 2N 1 k=1 X + 2N 1 N 1 k=1 Z x f (1) k (x ) cos( k y ) f (2) k (x ) cos( k y ) Z a f (2) k (x)r x( k )dx x f (1) k (x)r x( k )dx; (B3) f (1) k (x) = [sinh(k)] 1=2 cosh(k k x); f (2) k (x) = [sinh(k)] 1=2 cosh( k x): (B4) To obtain two integrals in (B3), it suces to implement 2N 2 multiplications, since the integral bounds are x -dependent. The same number of multiplications (2N 2 ) is needed to multiply N N-point images f (1) k (x ) and f (2) k (x ) by images evaluated after integrations. Taking into account the other part of (28), the number of multiplication doubles, yielding 8N 2. To obtain r x () and r y () directly from phase gradient (as in PGF [6]), one should implement 2N-point 1D FFT 2N times. If the N N-pixel real images r x () and r y () are to be ltered, 2N 2 multiplications should be implemented. The overall number of multiplications is given by: N HF 2 = 2N 2 (log 2 N + 1) + 8N 2 + 2N 2 (log 2 N + 1) + 2N 2 = N 2 (4 log 2 N + 14): (B5) APPENDIX C One way to decrease the inuence of areas of phase inconsistencies during the unwrapping procedure, is to exclude them from the integration domain S. Let the overall boundary of the interferogram consist now of the outer boundary C of the support region S and the boundary B of an area H where the InSAR phase is degraded by phase inconsistency (Fig. 1(b)). Denote S = S H B the domain where the phase

23 23 is consistent. Then expression (17) may be written as: (r ) = Z Z S dsr (r) r~g(r; r ) I B r S ; (C1) where the unit vector n S is perpendicular to the tangent to S and directed inside the region H at the points of the contour B. r S r H ; r 2 B; r 2 S ; (C2) where n H stands for the unit vector perpendicular to the tangent to B and directed outside the region H (Fig. 1(b)), the expression (C1) can be rewritten as: (r ) = Z Z S dsr (r) r~g(r; r ) + I B r H : (C3) Using the second Green's identity, the equality r 2 ~g(r; r ) = (r r ) and the basic property of the Dirac -function: Z Z H ds(r)r 2 ~g(r; r ) = 8 >< >: (r ); r 2 H; :5(r ); r 2 B; ; r 2 S ; we obtain the alternative expression for the contour integral in (C3) in which the unwrapped phase is unknown: I B r H = 8 >< >: G(r ) (r ); r 2 H; G(r ) :5(r ); r 2 B; G(r ); r 2 S ; (C4) (C5) where G(r ) = I B dc~g(r; H Z Z ds~g(r; r )r 2 (r): H (C6) The equality (C5), when applied along with the Green's function for unbounded domain (5), is usually referred to as the third Green's identity [27]. The Laplacian r 2 (r) can be obtained either by dierentiating the elements of the InSAR phase gradient (in reliable areas of H), or by using some a priori information (presence of a known slope on the terrain, or a water surface). An acceptable solution is also to let r 2 (r) = ; r 2 H, since this is consistent with the presence in H of a rotational (turbulent) component in the InSAR phase gradient eld. This hypothesis is veried in the regions of continuity rupture in the

24 24 InSAR phase due to a terrain slope the elevation angle of which is incompatible with the radar beam incidence angle, or in the regions of the interferogram corresponding to a water surface or any at surface with weak radiometric response. Furthermore, constraining the phase Laplacian to be zero is commonly used in many algorithms for extrapolation or interpolation of various physical elds [27]. APPENDIX D Denote p the principal value of the argument of the complex correlation c dened by (1) and e the error of the InSAR phase estimation with pdf p(; M; ) given by (39). Let = [ p + e ] 2 be the measured modulo 2 InSAR phase. Denote = p + e. Then: = 8 >< >: + 2; 2 ( 2; ]; ; 2 ( ; ]; 2; 2 (; 2]: (D1) The pdf of the measured wrapped InSAR phase is given by: p () = Z 2 2 Since p () = p(; M; p ), we obtain: p j(ju)p (u)du = [p ( 2) + p () + p ( + 2)] rect h i : (D2) 2 p () = [p(; M; p 2) + p(; M; p ) h i + p(; M; p + 2)] rect : (D3) 2 Let = p be the dierence between measured and exact InSAR phases. Then the conditional probability p jp () is equal to p ( + p ). Hence, p jp () = [p(; M; 2) + p(; M; ) + p(; M; + 2)] rect + p : (D4) 2 By denition and from (D4), Efj p g = Z p+ p p jp ()d: (D5) Since p(; M; ) is zero outside the interval ( ; ], the expression (4) is obtained from (D4) after readily implemented calculations.

25 25 Denote for brevity E = Efj p g. Then the variance of is: Dfj p g = Let p >. Denote Z p+ p ( E) 2 p jp ()d: (D6) Dfj p g = D (1) fj p g + D (2) fj p g + D (3) fj p g; (D7) where D (1) fj p g = Z p+ p ( E) 2 p(; M; 2)d = ; (D8) D (2) fj p g = Z p+ p since p(; M; ) = outside the interval ( ; ], ( E) 2 p(; M; )d = Z p+ ( E) 2 p(; M; )d; (D9) D (3) fj p g = = = Z p+ Z p Z + 4 ( E) 2 p(; M; + 2)d p+ ( 2 E) 2 p(; M; )d Z +p p+ ( E) 2 p(; M; )d 2( + E)E p(; M; )d: (D1) Hence, summing D (2) fj p g and D (3) fj p g, we obtain: Dfj p g = Z + 4 2E ( E) 2 p(; M; )d 2( + E)E Z +p Z 2( + E)E + 4 p(; M; )d = Z p(; M; )d + E 2 Z Z +p 2 p(; M; )d p(; M; )d p(; M; )d: (D11) The expression for p < can be readily derived in a similar manner, but then D (3) fj p g would be zero so the result would be the sum of D (1) fj p g and D (2) fj p g. The general expression for Dfj p g is given by expression (41) for any phase value p 2 ( ; ].

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29 29 TABLE AND FIGURE CAPTIONS Table 1 Computational costs of phase unwrapping methods. Fig. 1 InSAR phase region of support: (a) without a region of InSAR phase inconsistency; (b) with a region H of InSAR phase inconsistency. Fig. 2 Unwrapping error caused by a local InSAR phase gradient degradation. Fig. 3 Improved resolution of dierentiation using adaptive regularization (solid line) compared to conventional regularization (dashed line). Dotted line is the exact derivative. The original signal was degraded by Gaussian noise with variance.1 prior to computing the derivatives. Fig. 4 Mean value (a) and variance (b) of the error in the measured InSAR phase versus the absolute value of the correlation coecient and the exact monolook InSAR phase. Fig. 5 Initial interferometric data: (a) interferogram of Bern region as registered by ERS-1 ccnes. The dashed lines are used to count the number of fringes per length unit; (b) coherence corresponding to the same region ccnes. The size of both images is Fig. 6 Unwrapping the InSAR phase by using GF [5]: (a) enlargement of the unwrapped phase corresponding to the lower left quarter of the InSAR phase image in Fig. 5(a); (b) rewrapped phase; (c) residual phase: there are many residual fringes. Fig. 7 Unwrapping the InSAR phase in Fig. 5(a) by using the developed method: (a) enlargement of the unwrapped phase corresponding to the lower left quarter of the InSAR phase image in Fig. 5(a); (b) rewrapped phase; (c) residual phase: the number of residual fringes is substantially diminished. Fig. 8 Initial interferometric data: (a) interferogram of Bern region as registered by ERS-1 ccnes; (b) coherence corresponding to the same region ccnes. The size of both images is InSAR phase and coherence in Fig. 5 correspond to upper right corners of these images. Fig. 9 Unwrapping the InSAR phase in Fig. 8(a) by using the developed method: (a) unwrapped phase; (b) rewrapped phase. Fig. 1 Gradient continuation algorithm performance: (a) initial phase; (b) phase unwrapped by using GF; (c) result of the gradient continuation algorithm. Initial and reconstructed proles along the thick solid lines through point y = 25 in the x direction and through point x = 49 in the y direction are shown in the background of the coordinate box.

30 3 Table 1: Method Number of multiplications, M(N) M(512) HF1 N 2 (12 log 2 N + 28) HF2 N 2 (4 log 2 N + 14) GF N 2 (f14 + 4N it g log 2 N N it ) PGF N 2 (6 log 2 N N it ) PS N 2 (8 log 2 N + 12)

31 31 Figures (a) (b) Fig. 1:

32 phase error, [rad] y 2 2 x 4 6 Fig. 2:

33 df(x)/dx x Fig. 3:

34 Bias, [rad] Correlation Phase, [rad] (a) 1 8 Variance, [rad 2 ] Correlation Phase, [rad] (b) Fig. 4:

35 35 (a) (b) Fig. 5:

36 phase, [rad] y x (a) Fig. 6:

37 37 (b) Fig. 6:

38 38 (c) Fig. 6:

39 phase, [rad] y x (a) Fig. 7:

40 4 (b) Fig. 7:

41 41 (c) Fig. 7:

42 42 (a) (b) Fig. 8:

43 43 (a) (b) Fig. 9: