1 F. Rocca SAR images and interferometry

Phase unwrapping

Fabio RoccaPolitecnico di Milano

e.mail rocca@elet.polimi.it

2 F. Rocca SAR images and interferometry

α

2π phase ambiguity

-π

+π

Phases are represented as angles within the range -π ~ +π.

+3π

+5π

α+2π

α+4π

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1D Phase unwrapping (1)Problem: Wrapped phase value of sample n are represented as angles within the range -π ∼ +π. ( ) ( ){ }nnn jangleW ϕϕψ exp==

Given a sequence of wrapped phase values we want to recover the unwrapped phase values

nψnϕ

0 5 10 15 20-4

0

4

8

12

nψnϕα

α+2nπ

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Solution:

We assume that the phase difference between 2 samples (adjacent) is not aliased, hence the absolute value of discrete gradient Δφ= φi- φi-1 should always be < π.

The discrete gradient Δφ is estimated it by wrapping the difference of the wrapped phase Δeφ= W(Δ ψ).

For example ψi-1 = 3π/4; ψi =- 3π/4

Δψ= -6/4 π

Δeφ= W(Δ ψ)= -6/4π+2π =π/2

W(Δ ψ)

ψi-1=3π/4

ψi=-3π/4

The unwrapped phase φ is then computed (but for a constant) by integrating the gradient Δeφ.

1D Phase unwrapping (2)

If the absolute value of the discrete gradient Δφ= φi- φi-1 is > π, then unwrapping errors will occur.

5 F. Rocca SAR images and interferometry

1D Phase unwrapping errors

original phase values φ

wrapped phases ψ

difference of the wrapped phase Δ ψ

Δeφ= W(Δ ψ)

unwrapped phase values

unwrapping phase errors

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2D Phase unwrapping

In the 2D we have different possible integration paths and we want to ensure at least the independence of the solution on the path.

At least, for a close path (a curl), we want to be back at the same height !

The unwrapped phase field should be conservative (rotation free): ∇×Δφ = 0where the gradient is now a two component vector Δφ= [Δφi Δφk] T

In absence of vortex, all fringes are well formed: they are closed (sometimes to the image edges) and unwrapping is an easy task: just counting them.

In presence of vortex, fringes appears and disappear suddenly and “closing”them is a hard task.

This does not hold when the curl embrace a “vortex” (residue), where phase is not defined:

Amplitude should be zero, and this happens only in the presence of noise. 0))(())(( == PsimagPsreal

rr

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Example: constant slope

We assume the phase rotating counter clockwise. Real parts zero for -π/2 or π /2. Imaginary parts zero for 0 or π.

In absence of noise real and imaginary parts never crosses. However, the steeper the slope, the closest the lines.

-π

+π

+π

-π

re=0

im=0

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Example: constant slope + noiseNoise shifts re=0 and im=0 lines. A crossing may happen.

re=0

im=0 For small noise, crossings are paired and nearby.

When the summation path contours the vortex, the line integral increases by +2π (or - 2π) every circuitation.

In fact every time we cross a blue or an orange line, we change to the adjacent quadrant. In the example, the contour integral adds -2π for every circuitation of the vortex.

im>0re>0

im=0

im<0 re<0re=0

im<0re>0

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Finding vortices (residuals)

+ -Residuals should never be encircled by integration paths, since solution will be non conservative. Residuals are always present in couples of opposite sign (yet one of the two can be outside the image). If we connect each couple by a “ghost” line, never to be crossed, then we avoid adding 2π to the contour integral.

Residuals cannot be found by inspection, since data are known on the sampled grid. However, it is enough to compute the shortest line integral, ∇×Δφ , on a square of 4 samples around the vortex.

Residuals can be due to topography (elevation aliasing) or noise (the sensitivity increases in areas sloping toward the satellite, since the phase contour lines are closer). Low noise creates short dipoles, that can be easily handled. Situation improves with SNR; however, smoothing the phase field is mandatory !

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Residuals0.0 0.0

-0.2 0.2

0.0 0.0

0.0 0.4

00

.4

-.4

0 -10

-.2

-.4

-.4 00

.2

-.4

.2

A single -1 charge is created.

(II) Due to noise(enhanced by topography)

0.3

0

-.3

0 0.3

.4

.1

0 1.3

0

.3

.4

00.3

0

-.3

0 -1-.3

0

-.3

-.40-.3

-.1

0 .4

-0.4 -0.1

0.2 -0.1

0.0 0.3

0.0 0.3

0.0 0.3 -0.4 -0.1added noise: -0.4

(I) Due to topography (horizontal shear)

A dipole is created.

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Connecting residualsResiduals due to topography are much more difficult to be connected, since ghost lines can be very long. A foreshortened slope gives a ghost line parallel to azimuthas long as the slope itself (maybe a valley).

In these cases, the phase unwrapping solution is not unique, and different strategies exists for connecting residuals.

+-

++

-

-

+

+

+-

-

-

Shortest path or azimuth parallel ghost lines ?

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Original phases

Connecting residuals

Unwrapped phases

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Connecting residuals

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Connecting residuals

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A real example (the original phase)

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A real example (the residuals)

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A real example (unwrapping L=2)

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A real example (unwrapping L=7)

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A real example (ghost lines L=7)

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A real example (unwrapping L=8)

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[ ] qBK ini ˆ Δ⋅=Δφ

Height differences Δq are estimated exploiting jointly all the available interferometric pairs.The problem can be seen as a LS fitting of the slope of phase differences vs. baselines

Scalar value to be estimated

Advantages with respect to single interferogram:• More reliable estimation of the height differences.• High normal baseline data ensure good vertical resolution.• High coherence (low geometrical decorrelation) of low baseline pairs helps to unwrap correctly high baseline interferograms.

Ferretti A., Monti Guarnieri A., Prati C., Rocca F., “Multibaseline Interferometric Techniques and Applications”- ESA SP-406 - Proceedings of the “FRINGE96” Workshop on ERS SAR Interferometry - 1997http://www.geo.unizh.ch/rsl/fringe96

Bn

ΔφMulti-baseline DEM estimation principle (1)

Unfortunately interferometric phases are not unwrapped

Po

P1Δq

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Multibaseline phase unwrapping

( ) ( ) nkBKnk

BRq

nn

++Δ=++Δ⋅=Δ πφπφθπλ 22sin4

0

A welcome advantage provided by MB interferometry is the practical solution of pu. If one baseline only is available, the height difference between two pixel is ambiguous of an interval that depends on the baseline.An height q (from two points) is related to the phase gradient (and a noise) by the following expression:

Given the pdf of noise fn(n), of the interferometric phase difference fDf(Df) and an a-priori for the height difference (slope) fa(a) we can derive the ML estimate of the unknown q.

Df

qambiguity

In single baseline interferometry the most likely q is the one corresponding to usual wrapped gradient. The error probability is however large even in the noiseless case if ambiguity elevation is small compared to the standard deviation of slopes.

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Two noiseless interferograms available with different baselines and thus different altitude of ambiguity qa

• Independent measurements of the same height difference between Po and P1• Periodic behavior due to phase ambiguity.

Multi-baseline DEM estimation principle (2)

-500 0 5000

0.5

1

1.5(qa)2

(qa)1

Δq

-500 0 5000

0.5

1

Δq

(qa)12

(qa)12 : Minimum common multiple of (qa)1 and (qa)2

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Multibaseline phase unwrapping

ambiguity

A priori not applied: ambiguity is close to the prime factor product (Chinese remainder). The probability of error vanishes when ambiguity is large (say >> 100 m)

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• Pdf of the height differences relative to neighboring pixels:

• Maximum “a posteriori” estimate (Bayesian approach)

∏=

ΔΔΦΔ=ΔΦΔΦΔΦΔNI

i

iw

NIwww qfqf

cqf ap

1

21 )()/(1),...,,/(

• “ A priori” pdf: (can improve strongly the quality of results)

(e.g. normal distribution with μ=0 and σ depending on the local coherence)

)( qfap Δ

)/( iwqf ΔΦΔ

The Multi-baseline approach to PhUW

γ(P0)γ(P)=0.4σ=34.25 m

γ(P0)γ(P)=0.8σ=2.079 m

Δq

Pdf

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Multi-baseline interferograms

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Statistical distribution of the height difference relative to neighboring pixel.

Independence - Product of the different pdf

Maximum “a posteriori”estimate (Bayesian approach)

Bn=60 m, γeff=0.2 Bn=120 m, γeff=0.25 Bn=150 m, γeff=0.3

• Independent measurements of the same height difference.• Different noise power.• Periodic behavior due to phase ambiguity.

The Multi-baseline approach to PhUW

29 F. Rocca SAR images and interferometry

qdqfTq

Tq

NIwww Δ⋅ΔΦΔΦΔΦΔ= ∫

+Δ

−Δ

ˆ

ˆ

21 ),...,,/(ρ

The Multi-baseline approach to PhUWMany different measures of confidence can be considered for the MAP estimation.We can define the reliability ρ of the elevation estimate of each point as follows:

The reliability is then the probability that the correct value of the elevation difference lies inside the interval:

[ ]TqTq +Δ÷−Δ ˆˆ

It is a positive value less than 1 and can be considered a measure of the multi-image topographic coherence.

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The Multi-baseline approach to PhUW

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SAT ORB DATE Bn

ERS-1 20794 07/07/95ERS-2 1121 08/07/95 39ERS-1 21295 11/08/95ERS-2 1622 12/08/95 57ERS-1 22297 20/10/95ERS-2 2624 21/10/95 135ERS-1 22798 24/11/95ERS-2 3125 25/11/95 220ERS-1 23299 29/12/95ERS-2 3626 30/12/95 253ERS-1 23800 02/02/96ERS-2 4127 03/02/96 146ERS-1 24802 12/04/96ERS-2 5129 13/04/96 106

Mt. Vesuvius data set

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Vesuvius (MB) elevation map

DEM generation: MultiBaseline

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SAT ORB DATE BnERS-1 21159 01/08/95ERS-2 1486 02/08/95 59ERS-1 21660 05/09/95ERS-2 1987 06/09/95 106ERS-1 22662 14/11/95ERS-2 2989 15/11/95 176ERS-1 23163 19/12/95ERS-2 3490 20/12/95 337ERS-1 24666 02/04/96ERS-2 4993 03/04/96 125ERS-1 25167 07/05/96ERS-2 5494 08/05/96 129

Mt. Etna data set

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Etna elevation map + relflectivity

DEM generation: MultiBaseline

35 F. Rocca SAR images and interferometry

Courtesy of M. Eineder, DLR

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Courtesy of M. Eineder, DLR

37 F. Rocca SAR images and interferometry

Courtesy of M. Eineder, DLR

38 F. Rocca SAR images and interferometry

Courtesy of M. Eineder, DLR

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DEM generation

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SAR Interferometry: DEM generation (1)

Focusing and co-registration

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SAR Interferometry: DEM generation (2)

Fringes generation and baseline estimation

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SAR Interferometry: DEM generation (3)

Interferogram flattening

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SAR Interferometry: DEM generation (4)

Unwrapping 1

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SAR Interferometry: DEM generation (5)

Identification of disconnected zones

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SAR Interferometry: DEM generation (6)

Unwrapping 2

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SAR Interferometry: DEM generation (7)

Interpolation

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y

O

Geocoding

θ

ΔrP

Δq P’

Δyθθ tansin

qry Δ+

Δ=Δ

The ground range position Δy of the scatterer depends both on its slant range position Δr and elevation Δq:

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SAR Interferometry: DEM generation (8)

qBsinR nΔ=Δ

θλπϕ

0

4

phase to height conversion&

geocoding

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SAR Interferometry: DEM generation (9)

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Roma: InSAR Digital Elevation ModelRoma: InSAR Digital Elevation Model

Frascati

ESRIN

Ciampino

ROMAROMA

Grottaferrata

Lago diAlbano

EUR

GR

A

51 F. Rocca SAR images and interferometry

Etna (MB) elevation map + relflectivity (ascending + descending)

52 F. Rocca SAR images and interferometry

Etna elevation map + countour lines (ascending + descending)