Phase unwrapping through fringe-line detection insynthetic aperture radar interferometry

Qian Lin, John F. Vesecky, and Howard A. Zebker

Interferometric synthetic aperture radar presents a new technology for performing high-resolutiontopographic mapping. One of the factors critical to mapping accuracy is phase unwrapping. This paperpresents a new algorithm that unwraps phase through fringe-line detection. The algorithm is effectivein preventing error propagation, it preserves the image resolution, and, in the mean time, it producesnegligible local errors. An edge-segment linking approach and a curve-fitting approach are investigated,and their performances are compared. Test results of the algorithm are shown with the actualinterferometric SEASAT synthetic aperture radar images over large-relief terrain near YellowstoneNational Park with a short baseline, and results are shown over relatively flat terrain in Death Valley ofCalifornia with a long baseline.

1. Introduction

An interferometric synthetic aperture radar is asynthetic aperture radar (SAR) with two closely spacedreceiving antennas. One possible configuration con-sists of two satellite orbits from which terrain isimaged by a SAR successively.' Another configura-tion is the mounting of two receiving antennas on anairplane, separated in the cross-track direction.2When the complex SAR images formed from each ofthe SAR signals are properly registered, the phasedifference between the two images exhibits an inter-ference fringe pattern. This phase difference, ob-tained directly from the SAR images, is the principalvalue between 0 and 2r. When the phase differenceis properly unwrapped, such that the phase differencebecomes continuous, surface elevation can be com-puted. Hence the key to the elevation mappingprocess is phase unwrapping.

In the ideal case in which no noise is present in thedata, we can unwrap the phase difference by following

When this research was performed, Q. Lin and J. F. Veseckywere with the STAR Laboratory, Department of Electrical Engineer-ing, Stanford University, Stanford, California 94305. H. A. Zeb-ker is with the Jet Propulsion Laboratory, California Institute ofTechnology, Pasadena, California 91109. Q. Lin is now withHewlett-Packard Laboratories, 1501 Page Mill Road, Palo Alto,California 94304. J. F. Vesecky is also with the Department ofAtmospheric, Oceanic and Space Science, the University of Michi-gan, Ann Arbor, Michigan.

Received 1 April 1992; revised manuscript received 28 June 1993.0003-6935/94/020201-08$06.00/0.© 1994 Optical Society of America.

an integration path and adding multiples of 2 Tr to thephase difference whenever a sudden drop from 21r to 0occurs. In the presence of strong phase noise, how-ever, this simple phase-unwrapping method does notwork. Since a local error becomes a global one,Goldstein et al. 3 devised a scheme to establish localbranch cuts in noisy regions so that the integrationpath could avoid it. However, the branch cuts areusually so dense that phase unwrapping is impossible.To reduce the number of branch cuts, one filters thephase-difference image, thus reducing resolution.In addition, the algorithm may not work in thepresence of extended layover regions. Recently Pratiet al. proposed a technique in which ghost lines wereidentified and then avoided by the integration path.4Bone developed a phase-unwrapping algorithm byconstructing a bit-map mask with the second differ-ence of phase and by unwrapping the phase with aflood fill algorithm.5 The algorithms mentionedabove are all based on local phase properties.

This paper describes a new algorithm that per-forms phase unwrapping through fringe-line detec-tion. We seek to unwrap the phase by using verybroad contextual information. We show that in alow signal-to-noise ratio environment, phase unwrap-ping can be successfully implemented through thedetection of fringe-line positions. Knowing wherethe phase jump occurs, we can add multiples of 21 tothe phase difference such that the phase differenceincreases monotonically with slant range. Thismethod of phase unwrapping is based on the originalinterferogram and the detected fringe line, so there isno degradation in resolution. The algorithm uses an

10 January 1994 / Vol. 33, No. 2 / APPLIED OPTICS 201

edge-detection technique to find the location of thefringe lines in the phase-difference image and thenunwraps the phase difference by adding a multiple of27r each time an integration path crosses a fringe line.

In this paper we first point out the characteristicsof an interferometric SAR phase-difference imagethat lead to the development of a new phase-unwrapping algorithm. We then present the newalgorithm, including the edge-linking approach andthe curve-fitting approach. We also show results ofapplication of the algorithm to real SEASAT SARinterferometry data.

2. Phase Unwrapping Using Fringe-Line Detection

One of the critical issues in the elevation-mappingprocess is the recovery of the phase difference be-tween radar signals received from two nearly re-peated orbits so that the relative phase-differencevalue between any two pixels in the image is known.Phase errors incurred in the process are multiples of2ir, which translate to large elevation and positionerrors. Using spatial information, we can unwrapthe phase difference first by determining where thephase difference discontinuities are and then byadding appropriate multiples of 2 Tr. The discontinui-ties are referred to as fringe lines in this paper.

Since the main interest is in detection of the fringelines, their origin is described first. The phase differ-ence between radar signals received from a satellite intwo parallel orbits successively increases with slantrange. The phase difference obtained from a pair ofcomplex SAR images is, however, only its principalvalue between 0 and 2 7r. Suppose the vertical coordi-nate of the synthetic aperture radar image representsthe slant range and the horizontal coordinate repre-sents the azimuth position. For a flat region with noelevation variation the phase difference is constant inthe horizontal direction and increases linearly in thevertical direction. The vertical cross section of thephase difference image is a sawtooth pattern between0 and 27r. When the phase difference reaches 2'r, itreturns to zero, and it restarts. The points at whichthe transition occurs form the fringe lines, and in thiscase they are straight lines in the azimuth direction.The density of the fringe lines depends on the base-line length. The longer the baseline, the higher thedensity. We can obtain the actual phase differenceby integrating increments along a vertical path, add-ing 2 Tr to its value whenever a jump from 2 Tr to 0occurs. Figure 1 shows the fringe lines and theunwrapped phase for flat terrain.

When there are terrain variations, the fringe pat-tern is modulated by the elevation. As the elevationfluctuates, the phase difference deviates from theideal value, and the amount of deviation is roughlyproportional to its elevation, as shown in Ref. 6.The fringe lines are no longer straight. However, ifthe direction of a fringe line is assumed to be from leftto right, then in most cases the tangent of the line isoriented between -90° and 90° from the horizontaldirection. When phase noise is present, it becomes

Phase differenceof a flat area

Crosssection

Unwrapped phase

2,t phase 2 4 6n 8n phase

slant range slant range

Fig. 1. Fringe lines and unwrapped phase for flat terrain. Thefringe lines are parallel to the azimuth direction (along track).The cross-section view of the phase difference is a sawtoothwaveform.

difficult to decide where the 2 Tr jumps occur. Oncean error is made, it propagates throughout the integra-tion path, and it becomes a global error. To clarifythis, we show samples from the actual SEASAT SARinterferometry data over the Yellowstone NationalPark area and over the Death Valley area. Figures2(a) and 2(b) show the phase difference as a functionof slant range, taken from the Yellowstone data setand the Death Valley data set, respectively. It isclear from the figure that it is very difficult to identifythe transition from 2 7r to 0 from the line profile alone.The situation is further complicated by the layovereffect, which occurs when the slope of the terrainfacing the SAR is larger than the SAR look angle.The slant range in a layover region would decrease asthe ground range increases. In that case, even ifthere is no noise present, the phase difference is notwell behaved. More details about the layover effecton the phase difference are discussed in Ref. 7.

The phase-transition positions can be identified byuse of spatial information. The decision of whetheror not a pixel is at the phase transition is much easierto make by examination of its neighboring pixels; i.e.,we decide whether or not the pixel is on a fringe line.Detecting fringe lines thus becomes a critical step in

8

0 500 1000 1500 2000 2500 3000 351slant range (meters)

(a)

§a.5

Jo

slant range (meters)(b)

Fig. 2. Examples of the phase difference as a function of range,taken from the SEASAT SAR data: (a) Yellowstone area, (b)

Death Valley area.

202 APPLIED OPTICS / Vol. 33, No. 2 / 10 January 1994

---- ----

the phase-unwrapping process. A closely related butnot identical issue is the detection of edge segments.Fringe-line detection is different from conventionaledge detection. In the absence of large terrain discon-tinuities each of the fringe lines is connected, and theend points can only be on the boundaries of the image.Not every change in the pixel value, or edge in theusual sense, is desirable for detection; only thoselying on the fringe lines are. In addition, the imagehas a low signal-to-noise ratio. It is corrupted byspeckle noise caused by the coherent signals used toform the image, by volume scattering effect, and byadditive noise generated in the transmitter and re-ceiver. The positions of fringe lines in such regionsare not clear.

A similar problem is found in optics, in which aninterferomeric technique is used in surface inspec-tion.8 9 By analyzing the interference fringes, one isable to obtain measurements of microscopic surfaceheight or roughness. The main difference betweenthe optics problem and the problem we are investigat-ing is that the fringe patterns represent differentphysical quantities. With optical interferometry thefringe-pattern image is the intensity of two interfer-ing laser beams. There is no abrupt change inbrightness of the fringe. The fringe pattern is oftencharacterized by traces of extrema points.8 In radarinterferometry the brightness of the fringe-patternimage is the phase difference between two complexSAR images, and the phase difference can go frommaximum to minimum abruptly. The fringe pat-tern is characterized by the transition lines.

The problem of fringe recognition has been studiedin optical interferometry, although not for the pur-pose of phase unwrapping.810 The fringe-analysisalgorithms were different depending on the applica-tions. In the surface-inspection problem the posi-tions of the fringe lines were estimated and comparedwith the phase map of an idealized flat surface so thatsurface height could be calculated.8 However, sincethe surface to be examined was relatively smooth, thefringe lines did not deviate much from straight lines.In a study done by Crescentini,9 median points ofeach dark fringe were found, and a second-orderphase distribution was fitted to the points. Thefringe pattern was again quite simple, so fringe-linedetection was straightforward. In other applica-tions in optical metrology, different attributes such asthe direction of parallel fringe patterns, the center ofring fringe patterns, and the number of and locationof rings in a complicated interferogram were deter-mined by use of image-processing techniques.10

Subsections 2.A and 2.B present two approachestaken in this research for detection of fringe lines.One of the approaches is detection of the fringe linesby edge linking. The other approach is location offringe lines by fitting of the edge-enhanced phase-difference pattern with basis functions.

A. Edge-Segment Linking Approach

The fringe-line detection algorithm with the edge-segment linking approach consists of four steps.

First, we reduce the speckle noise by applying amedian filter. Then, we enhance the phase-transi-tion regions by using a bank of edge masks. Next,we obtain edge segments by locating the local maxi-mum points in the edge enhanced image. Finally,we connect the edge segments into fringe lines byusing an edge-linking algorithm.

Since SEASAT SAR images have a low signal-to-noise ratio, the fringe-line detection algorithm devel-oped in this research makes use of two levels of edgeimages, as in an image pyramid. The lower-leveledge image contains more details, while the upper-level edge image, being an average, is more reliable.Before locating fringe line segments, we enhance thephase-transition regions by applying a bank of edge-detection masks with different orientations. Theorientations of the edge masks are 00, 450, -45°, 90°,and -90°, as shown in Fig. 3. The maximum ofthe five outputs for each pixel is calculated and usedto form an image in which the fringes are enhanced.The pixels on or near a fringe line are bright.Furthermore, the orientation of the edge mask thatyields the maximum output at each pixel is recorded.The points that are local maxima in the enhancedimage normal to the recorded direction are thenfound, and they are stored in what we refer to as alower-level edge image. In the edge image, edgepoints have a pixel value of 255, while nonedge pointshave a pixel value of 0. Although the image containsdetails of the detected edges, it also has many undesir-able edge segments. To get more reliable fringesegments, we filter the phase-difference image by amedian filter. The selection of filter window sizedepends on the noise level and the density of thefringe lines. When the fringe density is large, asmaller window size needs to be used. The reasonfor choosing a median filter is that it preserves edgeswhile effectively reducing noise." The filtered im-age then goes through the edge-enhancement pro-cess, and the local maximum points are stored in anupper-level edge image.

The edge points that are on a fringe line are linkedsequentially, starting from a given initial fringe-linepoint. The linking algorithm is similar to a sequen-tial decoding algorithm,12 and it was first used in edge

1 I I

0 0 -1 -1

00

.1 .1 0

-45 o

-90

1 1

1 0 1.

0 .1

450

I I1771

900

Fig. 3. Masks used in the edge-enhancement process.

10 January 1994 / Vol. 33, No. 2 / APPLIED OPTICS 203

detection by Eichel et al. 13 However, the algorithmpresented in Ref. 13 adds three more paths at eachedge point, making the required memory space toolarge for a computer to handle in some cases. Thealgorithm is modified so that only when in doubt doesone add three more paths. Figure 4 shows the flowchart of the linking algorithm.

Figure 5 shows the sequence of neighborhoodpoints searched in the process of locating the nextlink point on a fringe line. As can be seen from thefigure, a neighborhood here is formed by layers ofpoints extending out from the center pixel. Theterm nth neighborhood is used to refer to the pointsof n innermost layers. For example, the first neigh-borhood consists of pixels 2, 3, 4, 5, and 6. Thesearch for the next link point begins with the upper-level image, since the edge points there are morelikely to be on the fringe line. If the next link pointis found in the tenth neighborhood, then the fringeline is extended to that point by linear interpolation.If, however, the next link is not found, then a searchis performed on the lower-level edge image. If thealgorithm still cannot find the next link, it goes backto the upper-level edge image and searches until threepossible links are found. The fringe line then splitsinto three, each following one link, while keeping apath metric that indicates the goodness of eachpossible fringe line. A path metric is defined here asthe average of the gray values of pixels along a fringeline in the edge image. Hence a good fringe line hasa large path metric, which means that it passesthrough only a few nonedge points.

We then unwrap the phase by adding a multiple of2 rr to each pixel, the multiple being dependent on howmany fringe lines lie between the pixel and a refer-ence pixel. In some places a fringe line is mislocatedby a pixel or two. After phase unwrapping, pixels insuch places have phase-difference values that differ

start -

Fig. 5. Sequencepoints are linked.

28

15

6

1

5

14

27

26 24 22

13 11 20

49 18

16

38 17 _2

!5l l

of neighborhood points searched when edge

from their neighbors' in a square window by 2w.Since these pixels are isolated in most cases, we detectthem by comparing their phase-difference values withthe means of their neighborhood. If the difference ismore than 7r, then we add or subtract 27r, whichever isappropriate.

This fringe-detection algorithm is applied to theYellowstone data set and the Death Valley data set.After applying the linking algorithm to the Yellow-stone data set, we superimpose the linked fringe lineson the original phase-difference image, shown in Fig.6. It is shown that in most cases the detected linesfollow the phase discontinuities very well. We canfurther reduce local errors by employing an alterna-tive fringe-line detection method, described in Subsec-tion 2.B.

Figure 7 shows the result of unwrapping the phasedifference shown in Fig. 2(a). The linear trend in theunwrapped phase difference is caused by the varyingground range positions of the pixels, while the fluctua-

Fig. 4. Flow chart of the proposed sequential linking algorithm.Fig. 6. Linked fringe lines superimposed on the original phase-difference image.

204 APPLIED OPTICS / Vol. 33, No. 2 / 10 January 1994

I

I

4

I

I

I

l

2 * * b

8

LI

slant range (meters)

Fig. 7. Result of phase unwrapping: the dotted curve shows theprincipal value of the phase difference, and the solid curve showsthe unwrapped phase difference.

tion over the linear part is caused by terrain elevation.The surface elevation is roughly proportional to theunwrapped phase difference after a linear trend in therange direction (the vertical direction in the image) istaken out. The mapping from phase difference toelevation was discussed in detail in Ref. 6.

Figure 8 shows the detected fringe lines superim-posed on the phase-difference image from a pair ofSEASAT SAR images (orbits 1226 and 1269) of theDeath Valley area. Since the baseline separation ismuch larger, 560 M as compared with 75 M in theYellowstone data set, the fringe pattern is muchdenser. However, since the imaged area is relativelyflat, and since phase noise is not severe, the algorithmtracked the fringe lines more closely than those forthe Yellowstone data set. There is no detectablelocal error present in the unwrapped phase-differenceimage.

Fig. 8. Detected fringe lines superimposed on the phase-differ-ence image of the Death Valley data set.

As shown in the detected fringe lines in the Yellow-stone data set, a problem with the edge-segmentlinking approach is that the linking algorithm doesnot follow the fringe-line positions very well in someregions in which radar return signals are weak as aresult of the receding terrain and in some regionswith large terrain variations. Local errors in thedetected fringe lines are sometimes conspicuous insuch regions, although global errors are prevented.To reduce local errors, we use a curve-fitting tech-nique, discussed in Subsection 2.B, to estimate theposition of the fringe lines. Each fringe line isapproximated by a series of basis functions. Thecoefficients of the basis functions are obtained througha least-squares algorithm.

B. Curve-Fitting Approach

The linking algorithm discussed previously workswell with phase-difference images of relatively smoothterrain, as shown by the Death Valley data set.When the terrain is rough, however, the fringe linesdeviate substantially from parallel. As is shown inthe Yellowstone data set, in some places, the fringelines even come very close together. It becomesdifficult to keep tracing the same fringe line in thissituation. The curve-fitting approach discussed inthis subsection improves the phase-unwrapping accu-racy in such cases.

To detect the fringe lines using the curve-fittingtechnique, we must first enhance the fringe lines.As with the edge-linking technique, the phase-difference image is processed with the median filterand directional masks. After enhancement, pixelson or near a fringe line have large gray values, whilethose in relatively smooth regions have small grayvalues. The pixels are fitted with a line by theweighted linear least-squares method.14

Assume the azimuth direction of the syntheticaperture radar image is the x axis and the rangedirection is the y axis. And for simplicity, assumethe image contains only one fringe line. The grayvalue of the image is large on or near the fringe lineand approximately zero in other parts of the image.We discuss below how we transform a phase-differ-ence image into this form. The experimental datapoints to be fitted are therefore image pixels on aninteger grid. There can be multiple points corre-sponding to the same x coordinate. A fringe line canbe represented as a function of x: y = f(x). f(x) isapproximated by M basis functions b(x), b2 (x) . . .bM(x) as

M

f(x) cibi(x),

where the c, i = 1, 2,. , M, are unknown coeffi-cients to be determined from the N data points (xi, yj).

To find the best set of parameters ci, i = 1, . . , M;

10 January 1994 / Vol. 33, No. 2 / APPLIED OPTICS 205

we minimize a distance E between data points and a basis functions arefringe line, defined as .

E(b) = N Wk[2 cibi(xk) - Yk]k=1 i=l

where wi is the weight of a data point (xk, yk). Theweight of a pixel is chosen to be proportional to itsgray value, since those with higher gray values aremore likely to be on a fringe line and hence shouldcarry more weight. The minimum of the distancefunction occurs where the partial derivatives areequal to zero:

aE(b)acj

N -M 1

= b WkJ cib - (Xk) = 0,k=1 L=1 J

j = 1,2, . . .,M.

X -bo(x) = to - ti1

bj(x) =

to < X t,

ti < X tM

0 to < X < tj-l

X - tj-'tj - tj-

X - t+ltj - tj+1

tj , < X < t

tj < x < tj+,

0 tj+ < X < tM

bM(x) = {X - tM-,

t - t -1

to < X < t-l.

tMi < X tM

The set of linear equations are called the normalequations. They can be rearranged into matrix formas

[ all

am,

... aiml[ 1 d

... aMM CM dM

where

N

aj = E Wkbj(Xk)bi(Xk),k=1

N

j = E Wkykbj(Xk),k=1

j=1,2,...,1.

j = 1, 2,...,M.

Singular value decomposition can therefore be usedto solve the above system of linear equations and tofind the parameters ci, i = 1,... , M. In dealingwith experimental data it may occur that two sets ofbasis functions fit the data equally well. The matrixtai I in this case would be singular, and direct matrixinversion would break down.14 Solving the normalequations through singular value decompositionavoids the problem. The algorithm uses singularvalue decomposition to invert the matrix {aji}. Ingeneral, the computation load increases as O(M 3 ),where M is the number of basis functions used in theapproximation.

The choice of basis functions is critical to thesuccess of the technique. From this experiment it isfound that a set of sinusoidal functions performsbetter than polynomial functions. This is consistentwith the observation that a fringe line tends tooscillate back and forth around a horizontal line.

When the basis functions are chosen as piecewiselinear functions, the solution becomes that of smooth-ing the data with linear splines.15 If the break pointsof the linear splines are to t < * t, then the

The matrix aji} is band diagonal. It requires onlyO(M) computations to invert the matrix. The com-putation load is thus greatly reduced. When thebreak points are selected as an integer grid, theleast-squares fit at each x would be just the centroidof the data points at that x coordinate.

This algorithm is applied to complex SEASAT SARimages over the Yellowstone area. Since the algo-rithm assumes that only one fringe line is present inthe image, the image needs to be segmented first.When the baseline separation is small, as in this case,there are relatively few fringe lines in the image, andit is reasonable to segment the phase-difference imagemanually in the UNIX X-window environment, sothat only one fringe line and its surrounding pixelsappear at one time. When the baseline separationbetween the two satellite orbits is large, the fringelines become denser, and an automatic segmentationalgorithm is needed to separate the fringe lines. Onepossibility would be the use of the linking algorithm

Fig. 9. Detected fringe lines superimposed on the phase-differ-ence image with the cosine series.

206 APPLIED OPTICS / Vol. 33, No. 2 / 10 January 1994

developed in Subsection 2.A. for detection of thefringe-line positions roughly and then the use of thecurrent algorithm to refine them.

Each fringe line is approximated by a series ofcosine functions. The data points are on an integergrid, with x in the range 0 to P - 1. If the data aresingle valued, i.e., there is only one data point corre-sponding to each x, then from Fourier-transformtheory the data can be represented by P/2 frequencycomponents equally spaced between 0 and 0.5.However, when the data are multiple valued, and weare interested in an approximation of the underlyingcurve, fewer frequencies are needed. The number offrequencies necessary to represent a fringe line accu-rately depends on its shape and smoothness. In thisexperiment, P is 498. It was found that 101 frequen-cies were enough for the first five curves, and 151frequencies were needed for the last two. The coeffi-cients of the cosine functions were determined fromthe weighted least-squares algorithm. The fringelines located with this method superimposed on thephase-difference image are shown in Fig. 9. Thecomputation involved in the least-squares estimationcan be drastically reduced by use of linear splines.The result of linear splines being used as basisfunctions is similar to that of the edge-segmentlinking approach being used. The algorithm per-formed well in areas with a clear fringe-line positionbut was not very stable in areas of uncertainty.

Figure 10 shows the slant range elevation mapobtained from the Yellowstone SAR data after thephase unwrapping and the subsequent cleaning pro-cess. The brightness of a pixel represents its eleva-tion. The vertical direction of the image is the SARrange direction (cross track), while the horizontaldirection is the azimuth direction (along track). Thedetails of derivation of the elevation from SAR inter-ferometric data were discussed in a previous paper.6Except for some local errors that are caused by phase

Fig. 10. Slant range elevation map obtained from the interferomet-ric SEASAT SAR data over the Yellowstone area after phaseunwrapping. The image covers an area of 8 km in the horizontal(azimuth) direction and 9 km in the vertical (range) direction.

noise or mislocation of the detected fringe lines, thereis no error propagation in the image.

3. Discussion

The phase-unwrapping problem is approachedthrough fringe-line detection. This permits the intro-duction of contextual information over a broad spa-tial area. Fringe lines in the phase difference imageare located first by enhancement of phase-transitionregions and then by use of edge-segment linking orcurve-fitting methods. Since the phase is unwrappedon the original interferogram without filtering, theresolution of the image is maintained.

The algorithm works well when fringe lines areclearly separated, as shown in the experimental datasets. It is especially suitable for images with largephase noise, which results in a large number ofresidue regions, making it difficult for the conven-tional phase-unwrapping algorithm to avoid residueregions.3 Further, the fringe-line detection algo-rithm is capable of bridging over small areas in whichthe interference fringe pattern is extremely noisy as aresult of phase incoherence or layover. The fringe-line position in these areas may not be accurate, butas long as the algorithm can link the right fringelines, there will not be a global error. When ex-tended layover regions occur near a fringe line,certain corrections have to be performed before thephase unwrapping can proceed. One semiautomaticcorrection method was discussed in Ref. 16.

In a case in which the baseline separation betweentwo satellite orbits is large, such that adjacent fringelines are squeezed together with a few pixels betweenadjacent fringe lines, oversampling might be neces-sary before application of the phase-unwrapping algo-rithm. If the imaged terrain has large elevationvariations while the baseline is long, the fringe linesmay mix together in layover regions, and it would bemuch more difficult to separate them. Phase un-wrapping under this condition requires further re-search.

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208 APPLIED OPTICS / Vol. 33, No. 2 / 10 January 1994