Davison PhaseUnwrapp

7/29/2019 Davison PhaseUnwrapp

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IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 37, NO. 1, JANUARY 1999 163

Multiresolution Phase Unwrappingfor SAR Interferometry

Gordon W. Davidson, Member, IEEE, and Richard Bamler, Member, IEEE

Abstract An approach to two-dimensional (2-D) phase un-wrapping for synthetic aperture radar (SAR) interferometry ispresented, based on separate steps of coarse phase and finephase estimation. A technique called adaptive multiresolutionis introduced for local fringe frequency estimation, in whichdifference frequencies between resolution levels are estimatedand summed such that a sufficiently conservative phase gradientfield is maintained. A coarse unwrapped phase of the full terrainheight is then constructed using weighted least-squares based oncoherence weighting. This coarse phase is used in a novel ap-proach to slope-adaptive spectral shift filtering and to reduce thephase variation of the interferogram. The resulting interferogramcan be more accurately multilooked and unwrapped with any

algorithm. In this paper, fine phase construction is done withweighted least-squares and with weights determined by simplemorphological operations on residues. The approach is verifiedon a simulated complex interferogram and real SAR data.

Index Terms Multiresolution spectral estimation, phase un-wrapping, synthetic aperture radar (SAR) interferometry.

I. INTRODUCTION

TWO-DIMENSIONAL (2-D) phase unwrapping is a crit-ical step in the generation of digital elevation models(DEMs) using synthetic aperture radar (SAR) interferometry.

Given a complex-valued interferogram created from two reg-

istered complex SAR images of the same scene, the terrain

height is related to the absolute phase of the interferogram,

whereas the measured phase is known only modulo 2

or is wrapped. Phase unwrapping consists of two steps:

the first step is the estimation of phase gradients from the

interferogrameither by multilooking and taking wrapped

phase differences [1] or by local fringe frequency estimation

[2]. The second step is the integration of the gradient estimates

to obtain an unwrapped phase surface. The presence of noise

and undersampling causes gradient estimates to be aliased so

that the measured gradient field is unconservative, leading to a

path-dependent integration and errors in the constructed phase

surface. Aliasing errors occur between points of nonzero curl

of the phase gradient field, or residues, and phase unwrappingalgorithms attempt to exclude the aliasing errors from the

integration process. This is done either by branch-cuts in a

path-following integration [1] or by zero-weights in weighted

least-squares or related algorithms [3][5]. In either case,

Manuscript received January 6, 1997; revised February 17, 1998.G. W. Davidson is with MacDonald Dettwiler, Richmond, B.C., Canada

V6V 2J3 (e-mail: [email protected]).R. Bamler is with the German Aerospace Center (DLR), German Re-

mote Sensing Data Center, Oberpfaffenhofen, D-82234 Wessling, Germany([email protected]).

Publisher Item Identifier S 0196-2892(99)00039-X.

in noisy areas, the integration of aliasing errors tends to

underestimate the terrain height [6]. Also when the residue

density is high, much of the unwrapped phase is not obtained

because much of the phase gradient information is excluded.

Phase unwrapping can be improved if a coarse resolu-

tion phase surface is available, such as from a DEM or

an unwrapped phase surface obtained at another baseline or

wavelength [7]. In this paper, we present a method to estimate

a coarse phase surface from the data itself. This can be thought

of as an extension of the common practice of removing the

flat earth phase to reduce the phase variation to make phase

unwrapping easier. Local estimation of fringe frequencies ofthe interferogram can provide the inputs to a least-squares

construction of a coarse phase [2]. However, in the presence

of terrain slope, the aliasing errors in the local frequency

estimates have a nonzero mean, resulting in a slope bias that

prevents construction of the full height of the phase surface

[6], [8]. To solve this problem, we introduce a technique,

called adaptive multiresolution frequency estimation, in which

difference frequencies between resolution levels are estimated

and summed such that a conservative phase gradient field

is maintained. A least-squares construction gives a smooth

unwrapped phase that is used for coherence estimation. Then

using weighted least-squares based on coherence weighting,

a coarse unwrapped phase surface that attains most of theterrain height is constructed.

The coarse phase is used to preprocess the data to improve

the quality of phase unwrapping. It provides information about

local slope for use in slope-adaptive spectral shift filtering

to improve coherence [9]. It is also used to reduce the

interferogram phase variation, or flatten the interferogram.

Then, noise filtering, or multilooking of the interferogram,

can be performed more accurately, and the effect of phase

slope on the aliasing error in phase gradient estimation is

reduced. Also, since most of the terrain height is attained

by the coarse phase, the overall phase unwrapping is more

robust to errors in the fine phase construction. In a sense,the coarse phase construction, through the use of frequency

estimation and coherence weighting, allows more information

to be extracted from the data.

Given the flattened, multilooked interferogram, the fine

phase can be constructed more reliably with any phase un-

wrapping algorithm. To complete the overall system here,

we use a weighted least-squares algorithm for fine phase

construction. The zero weights are determined by simple

morphological dilation and erosion operations on an initial

set of weights corresponding to residues and adjacent pixels.

01962892/99$10.00 1999 IEEE


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164 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 37, NO. 1, JANUARY 1999

The weighted least-squares approach has the advantage of

degrading more gracefully than branch cut methods in the

presence of undetected aliasing errors.

The multiresolution approach is verified on a simulated

interferogram. A realistic simulation is described, in which the

topographic phase is a fractal terrain model converted to slant

range. The simulation includes low coherence and a slope-

dependent coherence, according to the spectral shift. Finally,

the approach is verified on real SAR data.

The paper is organized as follows. Section II reviews the

problem of slope bias in least-squares phase unwrapping and

describes how adaptive multiresolution frequency estimation

can overcome this problem to construct the coarse phase.

Section III describes the applications of the coarse phase,

including a novel technique for slope-adaptive spectral shift

filtering. Section IV describes the fine phase construction,

and Section V describes the simulation and gives results

on simulated and real data. Finally, Section VI gives the

conclusions.

II. ADAPTIVE MULTIRESOLUTION FREQUENCYESTIMATION FOR COARSE PHASE CONSTRUCTION

A. Review of Slope Bias

Let be the unwrapped phase surface, the construction of

which is the goal of phase unwrapping, and let be its

gradient. Let the wrapped phase be In this paper, phase is

assumed to be measured in cycles, and phase gradient or fre-

quency is measured in cycles per sample. The phase gradient

estimated from the interferogram contains aliasing errors

such that the estimated phase gradient is not conservative

(1)

The aliasing errors arise from aliasing in frequency

estimation, or incorrect wrapping of phase differences, and

are equal to an integer number of cycles/sample. It is the

aliasing errors in the gradient estimate that disturb the phase

unwrapping and cause the slope bias in least-squares estima-

tion. To describe how the slope bias occurs, consider the phase

surface as a random process, due to decorrelation noise of

the interferogram, whose realizations may deviate from the

topographic phase Note that is a measure of signal

delay and is unambiguous, and can be defined as being

restricted to a 0.5 interval centered about

(2)

Fig. 1 shows the probability density functions (pdfs) of of

two adjacent interferogram samples with different topographic

phase and coherence of [10], [11]. It is important to

understand that, even if the imaging system guarantees that the

difference of from sample to sample is within cycles,

the noisy phase difference may exceed

these limits, as shown by the second plot in Fig. 1. If the phase

slope is greater than zero, the phase difference probability

distribution will exceed the 0.5 wrapping boundary with

higher probability than it will fall below the 0.5 boundary,

as seen in the figure. Hence, the field has a nonzero mean

Fig. 1. Probability density functions of the phase and the phase differenceof adjacent interferogram samples.

T

( i ; k ) = 0 : 1 2 5 , T

( i + 1 ; k ) = 0 : 3 7 5

cycles, and j j = 0 : 7 .

that tends to point in the opposite direction as the local phase

slope. Splitting into its mean and zero-mean components

(3)

a least-squares construction will not be able to recognize the

vector field as noise and will incorporate it into the

unwrapped phase. This gives the slope bias, which depends

on the local coherence and phase slope, such that the height

of the terrain is underestimated [6].

B. Adaptive Multiresolution

As an alternative to wrapped phase differences, local fringe

frequencies can be estimated over windows of the complex

interferogram and input to a phase unwrapping algorithm. That

is, the estimated phase gradient is Aliasing occurs

in local frequency estimation, and similar to the situation

described in Fig. 1, aliasing errors are more likely to occur in

the direction opposite to the local phase slope. To overcome

this problem, multiresolution frequency estimation uses the

benefits of frequency estimation over large windows to reduce

the effect of phase slope on frequency estimation over smaller

windows. Conceptually, this is illustrated in Fig. 2, in whichrepresents the interferogram. The largest window #3,

corresponding to the lowest resolution, is used to obtain an

estimate of an average frequency over the window, which

is then used to reduce the phase variation over the next smaller

window #2, or next higher resolution, by a complex multiply.

The frequency then estimated from window #2 is the difference

frequency

(4)

where is the frequency that would have been estimated from

window #2, without the complex multiply. The frequency


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DAVIDSON AND BAMLER: MULTIRESOLUTION PHASE UNWRAPPING FOR SAR INTERFEROMETRY 165

Fig. 2. Multiresolution frequency estimation.

is calculated and used to reduce the phase variation in the next

higher resolution, window #1, giving the difference frequency

estimate

(5)and so on.

In general, given resolution levels, with the zeroth

level being the highest resolution and letting ,

the result is a series of difference frequency estimates

The sum of difference frequencies, in the

absence of aliasing, is the frequency estimate at the zeroth

level A difference frequency measurement depends on

the particular terrain shape and the change in window size

between resolution levels. However, on average, given a

small enough change in window size, from one level to the

next, the difference frequency being estimated is relatively

small, so the effect of slope on estimation is approximately

removed. This reduces the probability of aliasing for the

given coherence and window size. Also, since the frequency

is built as a sum of differences, an unaliased estimate can

be achieved even if the instantaneous frequency is near

0.5 cycles per sample. This is illustrated in Fig. 3, which

shows a histogram of highest resolution level frequency

estimates, with and without multiresolution. The histogram

was generated by frequency measurements on a simulated

complex interferogram corresponding to a phase ramp, with

added complex noise for a coherence of 0.7. Due to noise,

some instantaneous frequencies are greater than 0.5 cycles

per sample, for which unaliased estimates are obtained with

multiresolution, whereas the wraparound of the histogramwithout multiresolution can be seen in the figure.

If difference frequencies are summed over all resolution

levels, at some level, the window size becomes small enough

that the variance of the difference frequency estimate is large

and aliasing is likely. For example, say the estimate of is

aliased, adding an error of cycles

(6)

However, in the reduction of phase variation in window #1,

the result is seen to be

(7)

Fig. 3. Histogram of frequency estimates with and without multiresolution,terrain phase slope = 0 : 2 5 cycles per sample, coherence = 0 : 7 .

which is unaffected by the aliasing error. Thus, the estimation

at all resolution levels is still of a small frequency, so the

effect of terrain slope is still approximately removed. Let

be the aliasing error at resolution level

The final frequency estimate with total aliasing error is

(8)

Because of multiresolution, the aliasing errors are more

equally likely to be positive or negative, greatly reducing themean of the total aliasing error. The variance of the aliasing

error with multiresolution, compared to that without multires-

olution, depends on the tradeoff of a smaller probability of

aliasing at each level versus the fact that the aliasing error

from several levels are added together. Thus, for windows

larger than some critical size, depending on the coherence,

the variance of the aliasing error with multiresolution is

smaller than that without multiresolution, whereas for smaller

windows, the variance of the aliasing error is increased with

multiresolution. A frequency estimate at a certain resolution

level is found by accumulating the difference frequencies,


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starting at the lowest resolution up to level

(9)

In the adaptive multiresolution approach, the curl of the

2-D frequency field is computed at each resolution level to

determine if the difference frequencies from the current and

higher resolutions should be accumulated. At the pointin the th resolution level, let and be the

frequency components in the and directions, and the curl,

is computed as

(10)

If one or more of the frequency estimates are aliased,

a relatively large value of close to one, results.

Also, with local frequencies estimated over windows, the curl

is affected by noise in the frequency estimates and is not

necessarily an integer, as it is in the calculation of residues

from adjacent phase samples. In the algorithm, at points inwhich the magnitude of the curl exceeds a threshold, the

frequency differences are only accumulated up to the lower

resolution level. In practice, it was found that by limiting the

curl to value less than about 0.1, a consistently smooth coarse

phase was obtained. Since the coherence varies across the in-

terferogram, in some areas, the frequency should be estimated

at a lower resolution than at others. Thus, in the adaptive

multiresolution resolution method, the required window size

for frequency estimation is determined automatically at each

point. The resulting frequency field is then used in a least-

squares construction to obtain a smooth, unwrapped phase

without slope distortion.

C. Local Frequency Estimation

Although many methods of local frequency estimation could

be used [2], it is desired to have a method that is not too

computationally expensive and that gives a robust estimate in

the presence of phase variation over a large window. For this

reason, the spectral centroid is used, as is commonly estimated

by the phase of the first lag of the autocorrelation function [12].

In the case of frequency estimation from an interferogram, it

was found that the estimate using only the first autocorrelation

lag was biased. This is because of the particular shape of the

interferogram spectrum, as illustrated in Fig. 4. The spectrum

consists of a narrow peak at the desired fringe frequencyand a broad component centered on zero that is due to the

correlation of the noise power spectra of the complex SAR

images [13]. (Note that wavenumber shift filtering centers

the triangular noise pedestal at the spectral peak; in practice,

however, this is never perfectly true due to unmodeled terrain

slope.) The problem can be overcome by using the second

lag of the autocorrelation function. However the second lag

alone cannot be used because its phase is proportional to

twice the desired frequency estimate, so only frequencies in

half of the normalized frequency interval can be estimated

unambiguously.

Fig. 4. Illustration of interferogram power spectrum (without spectral shiftfiltering).

Thus, a frequency estimator was developed as a function of

the first and second lags of the autocorrelation function. To

estimate frequencies in the - and -directions, the following

summations are formed over an estimation window at the th

resolution level:

(11)

(12)

(13)

(14)

where the superscript indicates direction and the subscripts 1

and 2 refer to the first and second lags. The asterisk denotes

complex conjugation. Considering, e.g., the -direction, the

local frequency estimate at the th resolution level is denoted

functionally as

(15)

where the function of the first and second autocorrelation lagsis defined as

(16)

Here, the second autocorrelation lag is used to correct for

the bias that occurs using only the first lag. In the function

, the phase of is reduced by the argument of

to prevent aliasing and the argument of the result is scaled

and added to the argument of An aliasing error in

does not affect the complex multiply of but it is added as

an aliasing error in the estimate The argument involving

is small, being only a correction, but if an aliasing error

occurs, it is halved and increases the variance of the frequency

estimate.

The difference frequency is estimated using the auto-

correlation functions calculated at resolution level with

the phase reduced by

(17)

An aliasing error in does not affect the difference

frequency estimate.


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Fig. 5. Multilevel frequency estimation.

In order to estimate the spatially varying frequencies at each

of the resolution levels, the estimation windows for computing

the sums in (11)(14) must be centered at points throughout

the interferogram. An advantage of the frequency estimation

method described above is that the autocorrelations for a given

resolution level can be computed from those of the next higher

level, giving an efficient, hierarchical implementation.

Consider the computation of in (11). The other sums

are computed similarly. Let the whole interferogram corre-

spond to the lowest resolution level , and let the sizeof the interferogram be in each direction. To start the

calculation of let the highest level be formed

as an array of products of adjacent interferogram samples

(18)

and

(19)

where (19) is chosen to satisfy a zero-frequency boundary con-dition. Given this array, the values of for resolution

levels to are computed by summing the values

of , that is, by filtering and possibly subsampling of

the previous resolution level, as illustrated in one dimension

in Fig. 5.

In order to have some overlap in the estimation windows for

smooth estimates, the resolution levels from to some

level are computed using a sliding window, without

subsampling. This can be represented as

(20)

where denotes 2-D convolution. The actual window at

level is the result of the convolution of the filter responsesup to this level

(21)

To obtain a rectangular window at each resolution level, the

set of filter responses defined by

(22)

can be used. In one dimension, this is illustrated as

and so on. Convolving these responses up to level gives a

3 3 point rectangular window.

The number of resolution levels computed with a sliding

window determines the amount of overlap of windowsat lower levels. In the results below, a value of

was found to given sufficiently smooth estimates. Then, for

efficiency, at lower resolution levels to

each point of is the sum of the values of with

subsampling by a factor of two in each direction

(23)

At each resolution level the sums

are computed and the difference

frequencies are estimated and stored. Once the

difference frequencies for all resolutions have been found, they

are summed from the lowest to highest resolutions using the

adaptive multiresolution approach to get the final frequency

estimates.

D. Comparison of Frequency Estimator with Multilooking

In the case of an approximately constant topographic phase,

multilooking provides the maximum likelihood phase estimate

and the phase gradient for unwrapping is then obtained by

wrapped phase differences. Thus, the accuracy of the local

frequency estimator described above should be compared tothe accuracy of phase gradient estimation from multilooked

data, for the same number of samples and assuming a flat

phase. This was done by simulation, and the results are shown

in Fig. 6. An interferogram was simulated with a flat phase

and constant coherence, and frequencies were estimated over

windows of samples using multilooking and the

modified correlation centroid estimator. With multilooking,

each half of the window was used to estimate the phase and

then the wrapped phase differences were taken. Estimations

from many windows were used to calculate the standard

deviation for each algorithm, as a function of coherence, for

and . From the figure, it is seen that compared to

multilooking for a given window size, the standard deviation ofthe modified centroid estimator is lower for higher coherence

because using two lags of the autocorrelation provides more

information. The standard deviation of the modified centroid

estimator increases quickly as coherence decreases below a

certain value, depending on the window size, because of the

aliasing involved in the second autocorrelation lag in (16).

Thus, although multilooking is the maximum likelihood phase

estimator for phase itself, the method of taking wrapped

phase differences from multilooked data is not optimum for

phase gradient estimation, even on a flat phase. In addition,

it is known that multilooking degrades fringe visibility in the


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Fig. 6. Frequency standard deviation for multilooking and modified corre-lation centroid estimator, flat phase.

Fig. 7. Slope-adaptive spectral shift filtering.

interferogram in the presence of phase variation, especially

over larger windows, whereas the modified centroid estimator

is unaffected by phase slope.

The results support the two-step approach of coarse phase

and fine phase estimation. The multiresolution approach with

the modified centroid estimator gives more accurate and un-

biased local frequency estimates, in the presence of phase

variation, up to a certain coarse resolution or window size

determined by the coherence. The coarse unwrapped phase

is constructed from these estimates and used to flatten the

interferogram. Then, for the relatively small windows used to

obtain the desired spatial resolution of the final unwrapped

phase, multilooking provides more accurate phase gradientsfor the fine phase construction.

E. Coarse Phase Construction

The multiresolution frequency estimates are input to a least-

squares construction to obtain a smooth phase without slope

bias. However, in areas of very low coherence, such as water

and steep slope, this smooth phase can show large distortions

due to inaccurate, although smoothly varying, frequency es-

timates. In the fine phase construction, there areas generate

residues and are zero weighted, so that when the coarse and

Fig. 8. Structure elements in morphological operations to connect zero

weights.

Fig. 9. Interferogram simulation.

fine phase are added together, the distortions show up in the

final unwrapped phase.

Thus, it is useful to introduce another step in the construc-

tion of the coarse phase. The first smooth phase is used to

flatten the interferogram for accurate estimation of coherence.

Areas of very low coherence, e.g., less than 0.3, are detected,

and the detected low coherence areas are then assigned zero

weights in a weighted least-squares coarse phase construction.

In the low coherence areas, or in areas of very steep slope,

the flattened interferogram still has substantial phase variation

because of error in the smooth phase, but this only leads

to a lower estimation of coherence such that the areas will

be assigned zero weights in the reconstruction, which is the

desired result anyway. With this method for coarse phase

construction, the influence of low coherence areas on the

unwrapped phase is limited. Also, terrain height is even more

fully recovered in the coarse phase in areas of very steep

slope, which are overly smoothed in the first smooth phase

construction.

III. APPLICATION OF COARSE PHASE

A coarse estimate of the unwrapped phase provides infor-

mation about terrain slope for slope-adaptive spectral shiftfiltering of the SAR images to improve coherence [9]. Rather

than using a spatially varying, time-domain filter kernel, we

introduce a method for slope-adaptive filtering that involves

only a multiplication by a complex exponential of the coarse

phase and a low-pass filtering in the frequency domain. This

is illustrated in Fig. 7. The local spectral shift is equal to

the local fringe frequency. Thus, multiplying the complex

SAR images by complex exponentials of the coarse phase,

performs a spatially varying range frequency shift of

the images such that spectral components in each image are

shifted according to the local (slope-dependent) wavenumber


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Fig. 10. Topographic phase in simulation.

Fig. 11. Coarse phase constructed from multiresolution frequency estimates and coherence weighting.

shift. That is, the frequency shifted SAR images are

(24)

(25)

where the shift for is in the opposite direction as the

shift for so that the uncorrelated parts of each range

spectrum are shifted outside of the nominal range bandwidth.

Then, assuming the images are interpolated in range so that

the range bandwidth is less than 0.5 in normalized frequency,

a simple low-pass filter implemented in the frequency domain

removes the uncorrelated parts of the spectrum for each image.

Because of the spatially varying frequency shift implied in (24)

and (25), the filtering of uncorrelated parts of the spectrum is

adapted to the local slope.

Before forming the interferogram, half of the coarse phase

that was removed in (24) and (25) is added back to the SAR


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Fig. 12. Unwrapped phase using coarse phase subtraction and weighted least-squares construction of fine phase.

images by a complex multiply, so that when the interferogram

is formed, it is already flattened by the coarse phase. The

filtered SAR images are described by

(26)

(27)

where is the low-pass filter impulse response and

means convolution in the range direction. The flattened

interferogram is formed by

(28)

and the result is multilooked by the desired window size and

possibly subsampled. The fine phase is obtained by

unwrapping the flattened, multilooked interferogram, and the

total unwrapped phase is then the sum of the coarse and fine

phase. If the interferogram is subsampled after multilooking,

the coarse phase needs to be subsampled by the same amount

and should then be filtered before being used in filtering and

flattening of the interferogram.

IV. FINE PHASE CONSTRUCTIONGiven the flattened, multilooked interferogram, range and

azimuth frequency estimates are obtained from wrapped phase

differences, and the fine phase construction could proceed with

any phase unwrapping algorithm. Here, in order to complete

the phase unwrapping system, we use a weighted least-squares

algorithm. The use of the coarse phase to improve coherence

and flatten the interferogram decreases the residue density, thus

facilitating the determination of zero-weights without excess

weighting. Obviously, the better the method for finding zero

weights (or branch cuts), the better the fine phase construction

and overall quality of the unwrapped phase, but in any case, the

phase unwrapping method should be as insensitive as possible

to errors in weighting.

In this work, a simple approach to weight determination is

used, in which zero weights are initially set at residue locations

and adjacent pixels, and morphological dilation and erosion

operations are used to connect zero weights that are relatively

close together. A series of dilationerosion operations with

different structure elements, as shown in Fig. 8, is used. The

structure elements are applied in succession in the order shownin the figure, but each element only operates on the ungrouped

zero weights that remain from the previous dilationerosion

operation. The grouped zero weights that result from each of

the operations are then combined to get the final weighting.

V. RESULTS FROM SIMULATED AND REAL DATA

A. Simulation

To verify the method on controlled data, a realistic method

of interferogram simulation was developed, as shown in Fig. 9.

Interferograms are generated from simulated complex SAR

images, where each image is composed of a common com-plex signal containing the topographic phase, and

mutually independent complex noises and

The initial signal and noise sequences are independent with

circularly Gaussian statistics. The signal and noise are scaled

to correspond to the maximum desired coherence,

The noise is not necessarily system noise, but also models

the uncorrelated scatterer contribution. Also, the intensity of

the images is modulated according to the terrain

slope as determined by the topographic phase to simulate

the effect of slope on radar brightness. The topographic

phase is generated by a fractal terrain model, which


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Fig. 13. Unwrapping with coarse phase subtraction: unwrapped phase error.

Fig. 14. Weighted least-squares without coarse phase subtraction: unwrapped phase error.

is converted to an interferometric phase difference in slant

range according to ERS parameters and a baseline of 200 m.

The complex images are then low-pass filtered in the range

direction, with a bandwidth of half the sample rate minusthe flat earth spectral shift Thus, the interferogram

is simulated as if flat earth phase removal and flat earth

spectral shift filtering were already done. However, a slope-

dependent frequency shift is introduced in the images by the

multiplication of the complex exponential of the topographic

phase. Since the spectral components are shifted differently

in each image, low-pass filtering keeps some uncorrelated

spectral components in each image, thus introducing a realistic

slope-dependent coherent loss according to the spectral shift

in real interferograms. The complex SAR images are then

conjugate-multiplied to obtain the interferogram.

Fig. 10 shows the topographic phase for the simulation, with

phase measured in fringe cycles. A maximum terrain height of

280 m was used over the 256 256 pixel (about 5 5 km)

image, and the conversion to slant range created very steepslopes and small amounts of layover. In the low flat areas, the

coherence was set to zero to simulate water. Elsewhere, the

coherence depended on phase slope up to a maximum of 0.7.

First, the coarse phase was constructed from multiresolution

frequency estimates and coherence weighting, as described

above, and this is shown in Fig. 11. The coarse phase was

used to flatten the interferogram, which was then multilooked

using a window of four samples in azimuth and two in

range. The fine phase was constructed using weighted least

squares as described above, and the coarse and fine phase

were added to get the unwrapped phase shown in Fig. 12. The


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Fig. 15. Histogram of unwrapped phase errors.

Fig. 16. Interferogram wrapped phase.

error surface is shown in Fig. 13, and the standard deviation

of phase error is 0.17 cycles. Then, for comparison, the

simulated interferogram was phase unwrapped without coarse

phase subtraction by multilooking with the same window sizeand using weighted least squares with the same method for

determining zero weights from residues as for the fine phase

construction. The resulting error surface is shown in Fig. 14,

in which the jump in error due to loss of terrain height

can be seen. The standard deviation of phase error is 0.42

cycles.

In addition, histograms of the unwrapped phase errors

are shown in Fig. 15. For the case without coarse phase

subtraction, the histogram shows a larger number of errors

between zero and one cycle, due to the loss of terrain height

rather than just random noise. Finally, as another comparison

Fig. 17. Rewrapped coarse phase.

Fig. 18. Histogram of coherence of Sarajevo scene before and afterslope-adaptive spectral shift filtering.

of the results with and without coarse phase subtraction,

the residue density was computed for each case. Without

coarse phase subtraction, the residue density of the multilooked

interferogram, outside the areas of water, was 0.0134. Withcoarse phase subtraction, the multilooked fine phase outside

the areas of water had a residue density of 0.0067, for a 50%

reduction.

B. Real Data

The approach was used to phase unwrap a part of an inter-

ferogram over Sarajevo, obtained by the ERS-1/2 sensor with

a normal baseline of 88 m. Complex images, 1024 samples

in range by 4096 samples in azimuth, were extracted, and

flat earth spectral shift filtering and phase removal were


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Fig. 19. Rewrapped unwrapped phase.

performed. The interferogram was formed, and a preliminary

multilooking of four samples in azimuth was done to reduce

the data size. The interferogram wrapped phase is shown in

Fig. 16. The scene is fairly mountainous, with some areas of

fairly low coherence, making it a difficult scene for phase

unwrapping.

The resulting interferogram was input to the multiresolution

frequency estimation and coarse phase construction, and the

resulting coarse unwrapped phase, rewrapped, is shown inFig. 17. The coarse phase gives the general shape of the

terrain and attains most of the height. The coarse phase was

used for slope-adaptive spectral shift filtering, and histograms

of coherence before and after filtering are given in Fig. 18

to show the improvement in coherence. Also, the residue

density of the interferogram before spectral shift filtering

was 0.075, and after filtering, it was 0.067, for an 11%

reduction.

Finally, the interferogram was further multilooked by two

independent samples in both range azimuth, so the overall

multilooking of the original interferogram is two samples in

range by eight samples in azimuth. The residue density after

multilooking was 0.0151 residues per sample. For comparison,the residue density after multilooking, without using the coarse

phase for spectral shift filtering or interferogram flattening,

was 0.0186. Thus, the use of the coarse phase gave an

overall improvement of about 20%. Then, the fine phase

was constructed from the flattened, multilooked interferogram

using weighted least squares, and the coarse and fine phase

were added to give the unwrapped phase, which is shown

rewrapped in Fig. 19. A comparison of the unwrapped phase

with the interferogram wrapped phased shows a very good

preservation of detail and no loss of fringes. Considering

the difficulty of the scene and the simplicity of the method

for finding weights in the weighted least-squares fine phase

construction, the results indicate a robust approach to phase

unwrapping.

VI. CONCLUSION

An approach for phase unwrapping has been described,based on separate steps of coarse phase and fine phase es-

timation. The core of the technique is the multiresolution

instantaneous frequency estimation for construction of a coarse

phase that attains most of the terrain height. The use of

the coarse phase in slope-adaptive spectral shift filtering and

interferogram flattening for multilooking gives a noticeable

improvement in the data for phase unwrapping, as shown by

the results using simulated and real data. Also, the coarse

phase provides information about the terrain shape and the

full terrain height, so that the approach is relatively robust in

the presence of errors in the fine phase construction. While

a simple method for fine phase construction has been used

here, the multiresolution approach offers improved results inconjunction with any phase unwrapping algorithm.

REFERENCES

[1] R. M. Goldstein, H. A. Zebker, and C. L. Werner, Satellite radarinterferometry: Two-dimensional phase unwrapping, Radio Sci., vol.23, pp. 713720, 1988.

[2] U. Spagnolini, 2-D phase unwrapping and instantaneous frequencyestimation, IEEE Trans. Geosci. Remote Sensing, vol. 33, pp. 579589,Mar. 1995.

[3] D. C. Ghiglia and L. A. Romero, Robust two-dimensional weightedand unweighted phase unwrapping that uses fast transforms and iterativemethods, J. Opt. Soc. Amer. A, vol. 11, pp. 107117, 1994.

[4] G. Fornaro et al., Interferometric SAR phase unwrapping using Greensfunctions, IEEE Trans. Geosci. Remote Sensing, vol. 34, pp. 720727,May 1996.

[5] M. Pritt, Phase unwrapping by means of multigrid techniques ininterferometric SAR, IEEE Trans. Geosci. Remote Sensing, vol. 34,pp. 728728, May 1996.

[6] R. Bamler, N. Adam, G. W. Davidson, and D. Just, Noise-inducedslope distortion in 2-D phase unwrapping by linear estimators with ap-plications to SAR interferometry, IEEE Trans. Geosci. Remote Sensing,vol. 36, pp. 913921, May 1998.

[7] R. Lanari, G. Fornaro, D. Riccio, M. Migliaccio, K. P. Papathanassiou, J.R. Moreira, M. Schwabisch, L. Dutra, G. Puglishi, G. Franceschetti, andM. Coltelli, Generation of digital elevation model by using SIR-C/X-SAR multi-frequency two-pass interferometry: The Etna case study,

IEEE Trans. Geosci. Remote Sensing, vol. 34, pp. 10971114, Sept.1996.

[8] R. Bamler and G. W. Davidson, On the nature of noise in 2-Dphase unwrapping, in Proc. Eur. Symp. Satellite Remote Sensing III,Taormina, Italy, Sept. 1996.

[9] F. Gatelli et al., The wavenumber shift in SAR interferometry, IEEETrans. Geosci. Remote Sensing, vol. 32, pp. 855865, July 1994.

[10] D. Just and R. Bamler, Phase statistics of interferograms with applica-tions to synthetic aperture radar, Appl. Opt., vol. 33, pp. 43616368,1994.

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Gordon W. Davidson (S83M86) received theB.Sc. degree in electrical engineering from the Uni-versity of Calgary, Calgary, Alta., Canada, in 1984,the M.S. degree in systems and computer engineer-ing from Carleton University, Ottawa, Ont., Canada,in 1986, and the Ph.D. degree in electrical engineer-ing from the University of British Columbia (UBC),Vancouver, B.C., Canada, in 1994, for which heinvestigated squint mode SAR signal properties anddeveloped high-squint SAR processing based on the

chirp scaling algorithm.He was with Bell Northern Research, Ottawa, in software development from

1986 to 1988, and in 1989, he was a Research Assistant in adaptive signalprocessing for communications at Carleton University. During his Ph.D. study,he consulted for MacDonald Dettwiler, Richmond, B.C., in the area of SARprocessing, and in 19941995, he was a Lecturer at UBC in digital signalprocessing. From 1995 to 1996, he was a Guest Scientist at the GermanAerospace Research Establishment (DLR), Oberpfaffenhofen, working inScanSAR and SAR interferometry. He is currently with MacDonald Dettwiler,working on autofocus for airborne SAR.

Richard Bamler (M95) received the Diploma de-gree in electrical engineering, the Eng.Dr. degree,and the habilitation degree in the field of signaland systems theory in 1980, 1986, and 1988, respec-tively, from the Technical University of Munich,Munich, Germany.

He was with the Technical University of Munichfrom 1981 to 1988, where he worked on opticalsignal processing, holography, wave propagation,and tomography. He is the author of a textbook

on multidimensional linear systems. He joined theGerman Remote Sensing Data Center (DFD), German Aerospace ResearchEstablishment (DLR), Oberpfaffenhofen, in 1989, where he is currently aSection Head for the development of algorithms for SAR and atmosphericsensors. He was involved in ERS processor and product validation, and he hasdesigned the signal processing algorithms for the German X-SAR processorand for interferometric SAR data processing systems. In early 1994, he wasa Visiting Scientist at Jet Propulsion Laboratory (JPL), California Instituteof Technology, Pasadena, where he worked on processing algorithms for theSIR-C ScanSAR, along-track interferometry, and azimuth tracking modes. Hiscurrent research interests include SAR signal processing, SAR interferometry,phase unwrapping, ScanSAR, estimation theory, and model based inversionmethods.

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