Phase unwrapping has been long considered one of thekey operations to enhance measurement accuracy insuch areas as interferometric synthetic aperture radar~SAR!,1 magnetic resonance imaging,2 phase-shiftingnterferometry,3 signal processing,4 optics,5 and
speckle interferometry.6 Phase unwrapping consistsof obtaining the phase surface f from the principalphase values u, defined on a two-dimensional grid ofpoints. The consistency relation between the un-wrapped and the principal phase values should hold,
@f~r! 1 c#mod 2p 5 u~r!, u [ ~2p, p#,
r 5 ~x, y! [ S, (1)
here c is an arbitrary real constant, @ z #mod 2p is mod2p operation that subtracts or adds a multiple of 2prad to ensure that the result lies in the interval ~2p,p!, and S is a support region where the wrappedphase u~r! is known. Numerous techniques havebeen developed to accomplish phase unwrapping.
When this research was performed, the author was with theDepartement Traitement du Signal et des Images, Centre Nationalde Recherche Scientifique, Ecole Nationale Superieure des Tele-communications, Unite de Recherche Associee 820, 46, rue Bar-rault, F-75634 Paris cedex 13, France. He is now with PhilipsResearch Laboratories, Prof. Holstlaan 4, 5656 AA Eindhoven,The Netherlands. His e-mail address is [email protected].
Received 3 January 2000; revised manuscript received 12 June2000.
Generally, these methods either belong to a family ofleast-mean-squares ~LMS! algorithms or representlocal approaches. Whereas LMS methods aim atachieving a minimum of a quadratic norm in thedifference between the wrapped and the unwrappedphase gradients,7 local methods integrate the wrappedphase gradient by following neatly chosen integrationpaths.8 Various extensions of these two strategieshave been developed, such as region growing algo-rithms9 and network flow programming methods.10
By use of the first Green’s identity in the LMSframework, the Green’s formulation11 has been de-veloped in which the unwrapped phase is obtainedthrough the convolution of the Green’s function for anunbounded domain and the wrapped phase gradient,with addition of the integrated unwrapped phase val-ues ~estimated by an iterative procedure! along theoundary of a given bounded support region. In Ref.2 the Green’s formulation was modified to excludehe latter step and to reduce computational cost,hereas in Ref. 13 the influence of wrapped phase-radient inconsistencies ~residues! on the solutionrovided by the method in Ref. 12 was taken intoccount for a finite rectangular support region.The majority of the phase-unwrapping methods de-
eloped to date consider rectangular support regionsor interferograms. However, the scope of problemsequiring phase unwrapping suggests that otherorms of support regions should be considered. Spe-ifically, the use of a circular support region can be ofmportance for measuring the blood flow in magneticesonance imaging,14 or for unwrapping of interfero-
metric fringes, given the circular aperture of the op-tical measurement system.3,15 The consideration of
0 September 2000 y Vol. 39, No. 26 y APPLIED OPTICS 4817
f
i
td@otGt
r
4
the circular support region can also be of potentialuse when one has an interferogram with a curvilinearboundary so that the radii of curvature of the bound-ary are measurable and a steplike unwrapping is pos-sible. This is often the case, for example, with theinterferometric SAR phase data, when regions of noisyphase that are due to low correlation of radar returns~e.g., over sea areas! form a natural sinuous boundaryor the interferometric phase-support region ~Fig. 1!.
Although some existing algorithms, such as thenetwork flow method in Ref. 10 and the mean-fieldannealing approach in Ref. 16, can be easily adaptedto process fringe patterns on circular domains, to finda closed-form solution to the problem our goal in thisstudy. To generalize the LMS phase unwrappingfor the case of circular support regions, the phase-unwrapping method based on the Green–Helmholtzformulation ~GHF! from Ref. 12 is modified in thepresent study, with consideration of the influence ofresidues on phase unwrapping, given the circularboundary of the interferogram. Unlike the Green’sformula from Ref. 11, the preliminary estimation ofthe unwrapped phase on the boundary is not manda-tory in our method.
In Section 2 the GHF is extended to include thecase in which the wrapped phase is known in a cir-cular support region, and in Section 3 expressions arederived for two terms in the unwrapped phase—aharmonic term and a hidden phase term—that aredue to the presence of wrapped phase-gradient incon-sistencies, called residues. The developed method isthen applied to unwrap simulated and real interfero-grams. Finally, in Section 4, conclusions are made,and future research is outlined.
Fig. 1. SAR image of New Caledonia area ~image size is 512 3 512pixels!. Dashed circles, boundaries of support regions for the cor-responding interferograms shown in Fig. 6, below. Image spansthe area of ;20 km ~in ground range! 320.5 km ~in azimuth!.Resolutions in the radar range, ground range, and azimuth are 10,25, and 6 m, respectively.
818 APPLIED OPTICS y Vol. 39, No. 26 y 10 September 2000
2. Closed-Form Phase Unwrapping from the PhaseGradient
A. Green’s Function Solution
The gradient of the wrapped phase u~r! for r 5 ~x, y![ S ~S is the support region for an interferogram! isdefined as a vector consisting of principal-value de-rivatives of the wrapped phase,
F~r! 5 limd30
H@u~r 1 dx! 2 u~r!#mod 2p
dx
1@u~r 1 dy! 2 u~r!#mod 2p
dyJ , (2)
x and y being the unit vectors in the x and the ydirections, respectively, on a Cartesian grid. Ac-cording to the consequence of Helmholtz’s theorem,17
the vector field F~r! can always be presented as a sumof its irrotational ~potential! component ¹c~r! and ofts rotational ~transversal! component ¹ 3 A~r!.
To consider a circular support region, it is naturalo choose the polar coordinate system. In this coor-inate system the support region S 5 $r 5 ~r, w! : r [0, a@, w [ @0, 2p@% is considered, where a is the radiusf the region ~Fig. 2!. Let g~r, r9! be a Green’s func-ion for the Laplace equation. Provided that thereen’s function satisfies Neumann boundary condi-
ions along the boundary C of the support region S,
]g~r, r9!
]nSU
r[C
5 0, (3)
Fig. 2. Sample configuration of vectors a, r, and r9 for derivationof the Green’s function in Eq. ~6! satisfying Neumann boundaryconditions on the boundary C of the support region S ~filled withgray!, example of integration path C0 used for computation of aesidue sign in Eq. ~13!, and example of integration path C9
~dashed curve in the second quarter! to obtain the hidden phase atpoint P9 [ S according to Eq. ~25!.
b
a~
e
s
~af
in~f
~
w
i
where nS is the outward unit vector normal to theoundary C ~Fig. 2!, the unwrapped phase f~r9! can
be related to wrapped phase gradient ~2! through theexpression12
f~r9! 5 * *S
dSF~r! z ¹g~r, r9!. (4)
Equation ~4! results after we relate the phase f~r9!,the gradient F~r!, and the Green’s function g~r, r9!through the first Green’s identity, expressing the un-wrapped phase f~r9! as a sum of the integral over thesupport region S itself and the integral over itsboundary C,
f~r9! 5 * *S
dSF~r! z ¹g~r, r9! 2 *C
C dcf~r!]g~r, r9!
]nS,
(5)
nd omitting the latter integral, owing to property3!.
By using the method of images ~e.g., in Ref. 17! wecan readily show that the function
g~r, r9! 5 2p21 ln~ir 2 r9i i2a 2 r 2 r9i! (6)
is the desired Green’s function for the circular do-main S. In Eq. ~6! the vector a has the magnitude aand points in the direction of r9, i.e., a 5 ar9yir9i. Asample configuration of vectors a, r, and r9, illustrat-ing the validity of expression ~6!, is shown in Fig. 2.
Equation ~4! also holds if the Green’s function g~r,r9! for a support region S is written as a series ex-pressed in terms of separable eigenfunctions fmn~r! ofthe Helmholtz equation, which form an orthonormalsystem,
g~r, r9! 5 (m,n50vmn
2 Þ0
` fmn~r! fmn~r9!
vmn2 , (7)
where vmn2 are associated eigenvalues.17 For the
function g~r, r9! from Eq. ~7! to satisfy Eq. ~4! theigenfunctions fmn~r! should satisfy homogeneous
Neumann boundary conditions along the boundary Cof the support region S:
]fmn~r!
]nSU
r[C
5 0. (8)
Owing to property ~8!, the homogeneous Neumannboundary conditions are also satisfied by the Green’s
f~r, w! 5 A000
1 ~1yÎ2!A040~6r4 2 6r
1 A051~10r5 2 12r3 1
1 A022r2~2 cos2 ~w! 2 1
1 ~1yÎ2!A120~2r2 2 1!
1 A111r cos~w!
10
function in the form of expansion ~7! for the boundedupport region S.17 In polar coordinates the Helm-
holtz equation separates into two equations, the so-lutions of which are cosine and sine functions of mwm integer! and Bessel functions. To satisfy bound-ry conditions ~8! along C, the argument of the Besselunctions should have the form pamnrya, where
J9m~pamn! 5 0, m, n 5 0, . . . , `. Then the eigen-functions of the Helmholtz equation are
fmn~r, w! 5 @cos~mw! 1 sin~mw!#Jm~pamnrya!, (9)
whereas their associated eigenvalues have valuesvmn 5 pamnya @and so the condition vmn
2 Þ 0 in Eq. ~7!mplies that m and n should not equal zero simulta-eously#. The squared norm of eigenfunction ~9! is
with orthogonality properties of Bessel functionsrom Ref. 18!
i fmni2 5 *0
2p
dw *0
a
rdrf mn2 ~r, w!
5 ~a2ypamn2 !~p2amn
2 2 m2!Jm2 ~pamn!. (10)
By combining Eqs. ~7!, ~9!, and ~10!, Green’s function7! can be written as a series:
g~r, r9! 5 ~1yp! (m,n50
`
em
3cos@m~w 2 w9!#Jm~pamnrya!Jm~pamnr9ya!
~p2amn2 2 m2!Jm
2 ~pamn!,
(11)
here e0 5 1, em 5 2, m . 0. Both Green’s functions~6! and ~11! will be needed later in Section 3 to derivethe components of the phase unwrapped by use of Eq.~4! that are due to inconsistencies ~residues! inwrapped phase gradient ~2!. Their choice will de-pend on the associated computational cost.
B. Practical Example: Reconstruction of the Wave FrontAffected by Phase Aberrations
To examine the quality of the phase unwrapping pro-vided by the method, we consider the reconstructionof a wave front from mod 2p phase, measured in acircular domain ~with radius defined by the apertureof the optical system!, which is affected by aberra-tions and partly undersampled during the acquisi-tion. The wave front is modeled by use of Zernikepolynomials19 to ensure that the phase surface f~r, w!ncludes all primary aberrations,
piston term
1! spherical aberration
os~w! high-order coma
astigmatism
curvature of the field
distortion, (12)
2 1
3r!c
!
September 2000 y Vol. 39, No. 26 y APPLIED OPTICS 4819
u
pIpf
sor
4
where the coefficients Alnm have the following val-es: A000 5 10, A040 5 8=2, A051 5 216, A022 5 12,
A120 5 26=2, and A111 5 6. Wave front ~12! wassampled on a 128 3 128 point grid. Within the sup-
ort region of radius a 5 1, phase surface ~12! has arms equal to 8.8868 rad and a maximum value equalto 37.8591 rad. The asymmetric wave front and themeasured phase u~r, w! 5 @f~r, w!#mod 2p are shown inFigs. 3~a! and 3~b!, respectively. No additional in-formation ~e.g., intensity! is available in our model.
Fig. 3. ~a! Simulated wave front, ~b! corresponding phase surface.Image size, 128 3 128 pixels. The phase is affected by all primaryaberrations, which render problematic its unambiguous unwrap-ping in the presence of undersampling.
820 APPLIED OPTICS y Vol. 39, No. 26 y 10 September 2000
t is seen from Fig. 3~b! that undersampling and theresence of severe phase aberrations result in a highringe density in the interferogram.
The wave front reconstructed by use of Eq. ~4! ishown in Fig. 4~a!. The computation time was 7 minn a SUN Sparc 5 workstation. For comparison theesult of a reconstruction of the same wave front
Fig. 4. Comparison between wave fronts reconstructed by ~a!developed Green’s function method and ~b! conventional LMS re-construction algorithms proposed in Refs. 7 and 20. Reconstruc-tion error of Green’s function solution remains localized in theundersampled area ~left-hand side!, whereas the LMS-unwrappingerror has also a marked destructive effect on the other areas of thereconstructed wave front.
aattu
siabie
aio
w~enha
bf
fiRE
obtained by conventional LMS methods ~e.g., devel-oped in Refs. 7 and 20! formulated for circular do-mains, is shown in Fig. 4~b!. These methods assumethat the unwrapped phase derivatives in each pixelare less than p in absolute value. Therefore theunwrapped phase is produced only from the potentialcomponent ¹c~r! of vector field ~2!.21
Visual inspection of Fig. 4 reveals to the superiorquality of unwrapping given by the Green’s functionsolution in Eq. ~4! when compared with that providedby the other LMS methods, yielding a completelycorrupted wave front. In spite of a considerable re-construction error of the Green’s function solution inthe vicinity of the undersampled area, this error re-mains localized and does not propagate to the otherparts of the support region.
The reason behind this marked difference betweenthe results is in the fact that the solution given by Eq.~4! partly accounts for the rotational component ¹ 3A~r! of wrapped phase-gradient field ~2!. Whereasthe divergence of the gradient ¹ z F~r! 5 ¹2c~r! servess the initial information for LMS methods in Refs. 7nd 20, the gradient field F~r! itself enters Eq. ~4!,hereby guaranteeing that an attempt to reconstructhe phase that is due to the rotational component isndertaken.In the undersampled regions the quality of recon-
truction by the Green’s function solution remainsnferior to that of wave-front reconstruction whenccurate additional information ~e.g., the well-ehaved, zero-free accurate modulating intensity, asn Ref. 3! is available. The most pronounced differ-nce between the result shown in Fig. 4~a! and that
reported in Ref. 3 can be seen in the ~undersampled!regions where phase-gradient inconsistencies ~resi-dues! appear, thus locally affecting the quality of un-wrapping. The influence of the residues on thephase unwrapped by use of Eq. ~4! is discussed inSection 3.
3. Residue Influence on the Unwrapped Phase
A. Residue-Induced Terms in the Unwrapped Phase
Expression ~4! is the basis for the GHF developed inRef. 12. It was shown in Ref. 13 that GHF, as aparticular LMS estimator, adds a harmonic surfaceto the phase unwrapped only from the potential partof wrapped phase gradient ~2!. The harmonic sur-face was shown in Ref. 13 to be due to the presence ofresidues, which are commonly defined in the litera-ture ~e.g., in Ref. 22! as isolated singular points ri [S around which the circulation vi~ri! of the wrappedphase gradient is nonzero,
vi~ri! 5 *C0
C F~r! z dr 5 62p, (13)
where C0 is a closed path of an arbitrary small radiusround ri ~Fig. 2!. Whenever vi~ri! 5 62p, the res-due at ri [ S is referred to as a positive or a negativene, depending on the sign of the 2p term. The har-
10
monic term was shown in Ref. 13 to be of the form
x~r9! 5 2 (i
vi~ri! * *S
dS@¹g~r, ri! 3 ¹g~r, r9!#z,
(14)
here @ z #z denotes the z component @perpendicular tox, y! or ~r, w! planes# of the vector product in brack-ts. When we rewrite this result for polar coordi-ates ~r, w!, the relationship is obtained between thearmonic term and the derivatives, with respect to rnd w, of the Green’s functions ~6! and ~11!,
x~r9! 5 2 (i
vi~ri! *0
2p
dw *0
a
dr@2gw~r, ri!gr~r, r9!
1 gr~r, ri!gw~r, r9!#, (15)
where the sum is over all positive and negative res-idues. The harmonic term in Eq. ~15! is rewritten as
x~r9! 5 *0
2p
dw *0
a
drr¹x~r! z ¹g~r, r9!, (16)
where the derivatives of yet unknown function x~r!satisfy equations,
xr~r, ri! 5 2r21 (i
@2vi~ri!#gw~r, ri!,
r21xw~r, ri! 5 (i
@2vi~ri!#gr~r, ri!, (17)
so that the harmonic term can be written as
x~r9! 5 *0
2p
dw *0
a
dr@rxr~r, ri!gr~r, r9!
1 r21xw~r, ri!gw~r, r9!#. (18)
By use of the Green’s function defined in Eq. ~11! andorthogonality properties of Bessel functions,18 it cane shown that the harmonic term can be written asollows,
x~r9! 5 4p21 (i
vi~ri! (m51
`
m sin@m~wi 2 w9!#
3 sm~ri!sm~r9!, (19)
where
sm~r! 5 (n50
` Jm~pamnrya!
~p2amn2 2 m2!Jm~pamn!
. (20)
The use of the Green’s function ~11! for the computa-tion of the harmonic term is expedient here, sincethere does not exist a simple analytic form for thisterm that could be derived from the derivatives of theGreen’s function ~6!.
Referring to the previous example of the wave-ront reconstruction, we note that harmonic term ~19!s exactly the difference between the LMS solution inefs. 7 and 20 and the Green’s function solution ofq. ~4!. Given the residue map in Fig. 5~a!, contain-
September 2000 y Vol. 39, No. 26 y APPLIED OPTICS 4821
i
i
w
Ib
s
4
ing 28 residues corresponding to the interferogram inFig. 3~b!, we can calculate the harmonic term by us-ng Eq. ~19! as shown in Fig. 5~b!.
When we compare expression ~16! for the harmonicterm with the formula for the potential part c~r! ofthe GHF-unwrapped phase, written in terms of thepolar coordinates
c~r9! 5 *0
2p
dw *0
a
drr¹c~r! z ¹g~r, r9!, (21)
Fig. 5. ~a! Residue map associated with the phase in Fig. 3~b!.Bright pixels denote positions of positive residues, whereas darkpixels indicate positions of negative ones. ~b! Corresponding har-monic term computed with Eq. ~19!. Image size, 128 3 128 pixels.
822 APPLIED OPTICS y Vol. 39, No. 26 y 10 September 2000
it is inferred that the harmonic term described byEqs. ~14!–~20! can be interpreted as a result of anattempt undertaken by the GHF to restore the un-known function x~r!. The function x~r! can thereforebe referred to as hidden phase, since the GHF re-stores it only partially. The components of the gra-dient vector of the hidden phase are related to thederivatives of the Green’s functions ~6! or ~11!through the Cauchy–Riemann conditions, formu-lated in Eq. ~17! for polar coordinates.
The hidden phase cannot be fully restored by theGHF, and therefore the technique of integrationalong a path in the ~r, w! plane is applied. To obtainthe hidden phase at point P9 5 ~r9, w9!, a point P0 5~r0, w0! is chosen as the starting point of the path ofntegration C9 5 ~P03 P9! for the hidden phase gra-
dient defined in Eq. ~17!, and ¹x~r! is integratedalong C9 ~Fig. 2!,
x~r9! 5 *C9
¹x~r! z dc9, (22)
where dc9 5 drr 1 rdww and r, w denote unit vectorsin the radial and the angular directions, respectively.By substituting hidden phase derivatives from Eq.~17! for ¹x~r! in Eq. ~22!, we obtain the followingexpression:
x~r9! 5 (i
@2vi~ri!# F*C9
rgr~r, ri!dw 2 r21gw~r, ri!drG .
(23)
The path of integration C9 for a particular point r9 5~r9, w9! is chosen so that it starts at the point r0 5 ~a,
0! ~w0 arbitrary! and goes to the point ~a, w9! alongthe boundary C of S. Then, from the point ~a, w9!,the path continues along the radial direction inwardto the point r9 5 ~r9, w9!. The reason for choosingsuch a path can be clearly understood when we notethat, since the Green’s function satisfies Neumannboundary conditions on the boundary C, the first in-tegral in Eq. ~23! can be omitted. Therefore the hid-den phase is equal ~to within a constant! to the secondintegral in Eq. ~23!; that is,
x~r9! 5 (i
vi~ri! *a
r9
r21gw~r, w9; ri!dr. (24)
f the path described in this way crosses a branch cutetween matched residues, the vi~ri! term is added to
the integral’s values for points lying in the sectorsubtended by the branch cut ~the notion of the branchcut is explained in detail, e.g., in Ref. 1!. Figure 2hows an example of such an integration path.Applying the integration of Eq. ~24! to the Green’s
function in Eq. ~6! yields the hidden phase given by
x~r9! 5 2~2p!21 (i
vi~ri!@xi~r9, w9! 2 xi~a, w9!#, (25)
f
rigt
w~
where the auxiliary function xi~r9, w9! is defined asollows:
xi~r9, w9! 5 tan21Fr9 2 ri cos~w9 2 wi!
ri sin~w9 2 wi!G
1ri
2 2 2ari cos~w9 2 wi!
4a2 1 ri2 2 4ari cos~w9 2 wi!
tan21
3 F2a 2 r9 2 ri cos~w9 2 wi!
ri sin~w9 2 wi!G
1ari sin~w9 2 wi!
4a2 1 ri2 2 4ari cos~w9 2 wi!
3 ln@~r9!2@~r9 2 2a!2 1 ri2 1 2~r9
2 2a!ri cos~w9 2 wi!#21#. (26)
The use of the Green’s function ~6! is more practicalfor computation of the hidden phase than the use ofthe one in Eq. ~11!, since the latter choice would
Fig. 6. Interferograms of New Caledonia and associated residurespectively, in Fig. 1. The boundaries of the support regions ar
10
entail summing a series of integrals of Bessel func-tions ~expressed with Struve functions!, which, inturn, would result in an inappropriately high compu-tational cost.
If there is a priori information as to how the branchcuts should be placed, the total unwrapped phaseequals the sum of the separate components that aredue to the potential and the rotational parts ofwrapped phase gradient ~2!. As noted above, theharmonic term @which can be regarded as an incom-plete reconstruction of the hidden phase that is due tothe rotational part of Eq. ~2!# is already present in theesult yielded by Eq. ~4!, together with the phase thats due to the potential part of the wrapped phaseradient. Keeping this in mind, we can obtain theotal unwrapped phase as follows,
F~r9! 5 f~r9! 2 x~r9! 1 x~r9!, (27)
here f~r9!, x~r9!, and x~r9! are obtained from Eqs.4!, ~19!, and ~26!, respectively.
ps corresponding to Regions 1 @~a! and ~b!# and 2 @~c! and ~d!#,wn as dashed white circles. Image size, 128 3 128 pixels.
e mae sho
September 2000 y Vol. 39, No. 26 y APPLIED OPTICS 4823
sr
ir~cqggFw
4
B. Practical Example: Interferometric Synthetic ApertureRadar Phase Unwrapping
In synthetic aperture radar ~SAR! interferometry theinterferometric phase is the mod 2p measure of theheight of a terrain area irradiated by the radar beam.Apart from wrapped phase-gradient inconsistenciescaused by noise, residues appear, since the radarbeam incidence angle can be superior to the inclina-tion angle of the terrain. This inconsistency in theradar geometry is often referred to as layover andhadow effects in the image of amplitude of the radareturns.23 The availability of this a priori informa-
tion allows us to reconstruct the hidden phase in Eq.~25! by placing connecting curves ~branch cuts! be-tween pairs of oppositely signed residues along thepaths in the interferogram corresponding to bright
Fig. 7. ~a! Phase unwrapped by use of the developed method @Eqs.~4!, ~19!, and ~25!# and ~b! mod 2p unwrapping error image corre-sponding to the interferogram in Fig. 6~a!. Unwrapped phase issuperimposed with the corresponding amplitude values from Fig.1. Remaining shears in the unwrapping error image are due tounmatched residues. Image size, 128 3 128 pixels.
824 APPLIED OPTICS y Vol. 39, No. 26 y 10 September 2000
stripes in the amplitude image. An example of aSAR amplitude image is shown in Fig. 1.
In our experiments, two interferograms corre-sponding to two regions in the SAR amplitude image~Regions 1 and 2 in Fig. 1! were unwrapped. Themage in Fig. 1 and the interferograms in Fig. 6 wereegistered by the European Remote-Sensing SatelliteERS-1; bandwidth 15.55 MHz, wavelength l 5 5.6m!. The two interferograms contain differentuantities of residues: 174 residues in the interfero-ram in Fig. 6~a! and 232 residues in the interfero-ram in Fig. 6~c!. In Fig. 1, and in interferograms inig. 6, the SAR range and azimuth directions coincideith the horizontal ~x! and vertical ~y! axes, respec-
tively. One pixel covers a region of 31.64 m in therange direction from the radar by 42 m in the azi-
Fig. 8. ~a! Phase unwrapped by use of the developed method @Eqs.~4!, ~19!, and ~25!# and ~b! mod 2p unwrapping error image corre-sponding to the interferogram in Fig. 6~c!. Unwrapped phase issuperimposed with the corresponding amplitude values from Fig.1. Remaining shears in the unwrapping error image are due tounmatched residues. Image size, 128 3 128 pixels.
@
2. G. H. Glover and E. Schneider, “Three-point Dixon technique
muth direction. The ground area spanned by thedata in Fig. 6 is approximately 5.0 km in groundrange by 5.12 km in azimuth.
Unwrapping was accomplished by use of Eq. ~4!with the Green’s function from Eq. ~6!# to compute
the Green’s function solution. The harmonic termand the hidden phase term were computed by use ofEq. ~19! and Eqs. ~25! and ~26!, respectively. Theresults of unwrapping the three interferograms inFig. 6 are shown in Figs. 7 and 8, along with the mod2p unwrapping error corresponding to each experi-ment. The shears in the unwrapping error imagesare due to unpaired residues. The quality of un-wrapping is satisfactory with approximately constantunwrapping error over the scene except in the com-pact regions with unmatched residues.
4. Conclusions
The Green’s function solution for two-dimensionalphase unwrapping proposed in Refs. 11 and 12 hasbeen extended to circular regions of support. Thenovelty of the approach lies in the use of two types ofGreen’s function for the Neumann boundary condi-tions: One Green’s function is written in a closedform, whereas the other is a series expressed in termsof Helmholtz equation eigenfunctions for a circulardomain. The use of both Green’s functions hasproved necessary for developing efficient algorithmsfor computing unwrapped phase components, owingto the rotational and the irrotational vector fields inthe wrapped phase gradient.
We have examined the quality of unwrapping byconsidering the reconstruction of a wave front in thepresence of severe phase aberrations and reconstruc-tion of a continuous phase surface from syntheticaperture radar ~SAR! interferograms. Residue in-fluence on phase unwrapping in both cases has beendescribed in terms of a harmonic term and a hiddenphase term, for which closed-form expressions havebeen obtained from the knowledge of residue loca-tions and residue signs.
To generalize the developed technique further, gen-eral forms for the interferograms are currently beingconsidered to allow for the application of the Green’sfunction formalism to unwrapping interferogramswith general ~e.g., annular, hexagonal, elliptic, and soon! apertures.
This study was funded by the Centre Nationald’Etudes Spatiales ~French Space Agency!. The au-thor gratefully acknowledges Henri Maıtre for hiscontinuous support of this research and for manyfruitful discussions, Benedicte Fruneau from the Uni-versite de Marne-la-Vallee for providing SAR imagesof New Caledonia, and Phillip Christie ~Philips Re-search Laboratories! for proofreading the manu-script. In addition, the contributions of anonymousreviewers are acknowledged.
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10
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