Multibas e line gradient ambiguity res olution to s upport Minimum Cos t Flow phas e unwrapping

, , Marie Lachais e Richard Bamler Fernando Rodriguez Gonzalez

Remote Sensing Technology Institute, DLR

2 IGARSS 2010 – 30/07/2010

Outline

TanDEM-X mission

Multibaseline gradient-based phase unwrapping

Maximum A Posteriori with the zero curl constraint as prior

Message passing

Results

3 IGARSS 2010 – 30/07/2010

- TanDEMXmis s ion requirement &proces s ingconcepts

RequirementRequirement: Spatial resolution: 12m x 12mAbsolute vertical accuracy: <10m

Relative vertical accuracy < 2m

Year 1: full coverage with smaller baseline

height ambiguity ~ 45 m good for moderate terrain

Year 2: repeat with larger baseline

height ambiguity ~ 30 m gives full accuracy

robust phase unwrapping by combining both years

multi baseline PU

4 IGARSS 2010 – 30/07/2010

interferometric phase modeled as:

gradient estimate:

gradient pdf:

in i-direction:

in k-direction:

Multibaseline gradient-based phase unwrapping

( ) ( ) ( ){ }),(,,, kinkiWakiaki ccc +== φψψ ),[,1 ππψ −∈= cc

c B

Ba

( ) ( ){ }( ) ( ){ }kikiWki

kikiWki

ccck

ccci

,1,),(

,,1),(

,

,

ψψδψψδ

−+=−+=

],)[,;(],1)[,;(],)[(,

kiLpdfkiLpdfkipdf ccccciγφγφδφ −∗+=

],)[,;(]1,)[,;(],)[(,

kiLpdfkiLpdfkipdf ccccckγφγφδφ −∗+=

5 IGARSS 2010 – 30/07/2010

Some gradient pdfs

0 2π-2π

1

2

3

Gradient in azimuth (rad)

pdf

0 2π-2π

0.5

1

1.5

Gradient in range (rad)

pdf

6 IGARSS 2010 – 30/07/2010

curl(i,k) = δφi(i,k)+δφk(i+1,k)-δφi(i,k+1)-δφk(i,k)=0

(x,y+1)

?Which additional information can we us e

Phase pixels

(i,k) (i+1,k)

(i+1,k+1)Gradient estimates

δφk(i+1,k)

δφi(i,k)+δ φk(i+1,k)-δφi(i,k+1)-δφk(i,k)=0

(i,k+1)

Zero-curl constraint

δφi(i,k)

δφi(i,k+1)

δφk(x,y)

7 IGARSS 2010 – 30/07/2010

∏∏=C

c

P

kic kicurlPPPP

ckci,

)),(()()(),(,, φφ δδ

ki φφ δδ

The zero curl cons traint

The constraint or the prior is: the zero curl constraint

Joint probability:

],)[,;(]1,)[,;()(

],)[,;(],1)[,;()(

,

,

kiLpdfkiLpdfP

kiLpdfkiLpdfP

cccc

cccc

c

c

γφγφ

γφγφ

−∗+=

−∗+=

k

i

φ

φ

δ

δ

=

=otherwise 0

0),( if 1)),((

kicurlkicurlP c

c

8 IGARSS 2010 – 30/07/2010

( )∑∑ ++=C

c

P

kic kicurlEEE

ckci,

)),(()()(minargˆ,ˆ,, φφφφ δδ

kiδδ

( )∑∑ −−−=C

c

P

kic kicurlPPP

ckci,

)),((log)(log)(logminargˆ,ˆ,, φφφφ δδ

kiδδ

Data energy or data penalty

∏∏=C

c

P

kic kicurlPPP

ki,

)),(()()(maxargˆ,ˆφφφφ δδ

kiδδ

Maximum A Pos teriori and energy minimization

Compatibility between neighboring variables

∞+=

=otherwise

0),( if 0)),((

kicurlkicurlE c

c

9 IGARSS 2010 – 30/07/2010

Gradient estimates Observable variable node

Function nodePdf (hidden | observation)

Zero-curl constraint Zero-curl constraint check nodescheck nodes

Hidden variable node = unknown true values of gradients

= gradient node

Partial derivative over rangePartial derivative over azimuth

Phase value

Gradient node

Measured gradient

Graphical model

Constraint node

10 IGARSS 2010 – 30/07/2010

Mes s age pas s ing

Sum-product algorithm (a.k.a. belief propagation) allows to find an approximate cost labeling (belief) of energy functions

inexact in graphs with cycles but produces excellent results

To obtain the MAP, a variant called max-sum is used (If used with negative log probabilities -> min-sum)

1. Initialize gradient nodes to their energy

2. Update messages iteratively

3. Receive messages and take the minimum

)),((or )),((),( kiEkiEkim kiδ δδ=

11 IGARSS 2010 – 30/07/2010

: Mes s age update from cons traint node to gradientnode

Gradient node

Constraint node

m4

m1

m2

m3

( )∑∑∑−= −= −=

+++←1

1

1

1

1

13214 )),((min

r s ttsrq mmmkicurlEm

12 IGARSS 2010 – 30/07/2010

: Mes s ages update from gradient node to cons traintnode

Gradient node

Constraint node

m4

m5

qiq mpdfm 45 )(log +−← φδ

13 IGARSS 2010 – 30/07/2010

MAP computation

Gradient node

Constraint node

m4

)min(),(ˆ54 mmyxk +=δ

m5

14 IGARSS 2010 – 30/07/2010

Gradient log pdf and s peed up

0 2π-2πGradient [rad]

0

20

40

60

pdf

0 2π-2πGradient [rad]

0

20

40

60

pdf

0 2π-2πGradient [rad]

0

20

40

60

pdf

0 2π-2πGradient [rad]

0

20

40

60

pdf

Range (i-direction) Azimuth (k-direction) Pixel 1

Pixel 2

15 IGARSS 2010 – 30/07/2010

-6.30

: Mes s age update Forward

-3.33

9.3

-6.10.26.5

-6.30

6.3

-9.4-3.13.2

6.30 E0=0.9+0.5+0.8=2.2

0.3

-2.7

0.2

-3.1-3.1

0.0

3.0

0.2

curl

0 E-1=0.9+0.5+0.0=1.4δφi(i,k) + δφk(i+1,k) - δφi(i,k+1) - δφk(i,k)

1.40.90.4

1.40.90.4

1.41.90.5

0.90.40

0.50

0.8

0.80

0.6

0.00.70.7

01

0.1

0.30

0.9

0.40

0.9

2.01.71.6

16 IGARSS 2010 – 30/07/2010

: Mes s age update MAP

0.3

-2.7

0.2

-3.1-3.1

0.0

3.0

0.2

0.90.40

0.50

0.8

0.80

0.6

0.00.70.7

01

0.1

0.30

0.9

0.40

0.9

1.40.31.0

2.01.71.6

3.43.02.72.7

17 IGARSS 2010 – 30/07/2010

: Res ults Tes t s ite

“footprint” south from Salar de Arizaro (Argentina)

18 IGARSS 2010 – 30/07/2010

: Res ults Unwrapped Gradients with MAP

Unwrapped gradient in Azimuth Unwrapped gradient in Range

19 IGARSS 2010 – 30/07/2010

: Res ults Remaining res idues and MCF res ults

Residues and branch-cuts from MCF Unwrapped phase

20 IGARSS 2010 – 30/07/2010

: , Res ults MCF res ults comparis on with s inglebas e line

Single baselinephase unwrapping

(MCF)

Multibaseline gradient-based

phase unwrapping

21 IGARSS 2010 – 30/07/2010

: Res ults The unwrapped phas e

22 IGARSS 2010 – 30/07/2010

Conclus ions

Multibaseline gradients unwrapping is used to disambiguate the gradients. The range of search is thus very restricted to 3 or 5 cycles.

The zero-curl constraint (in every interferogram) is used as a prior.

A graphical model which incorporates the gradients log pdf and this constraint is introduced.

We propose a method based on probability passing which is a good method for inferring the different gradients.

Since we just want to find out the right ambiguity, the messages can be reduced to the number of cycles and thus the processing is accelerated

The prior could be improved by incorporating at least a smoothness criterium

23 IGARSS 2010 – 30/07/2010

than

k you

!

than

k you

!