IGARSS 2011, 24-29 July 2011, Vancouver, Canada

Multi-Component Nonlinear Motion Estimation in Differential SAR Tomography – The Time-Warp Method

Xiao Xiang Zhu and Richard Bamler

Remote Sensing Technology Institute, DLR/TUM

TomoSAR System Model

b

b

elevationaperture

sz

yx

r

, ,x r s

3-D reflectivity distribution

reference surface s = 0

s

TomoSAR System Model

b

b

elevationaperture

2 nn

b

r

Complex pixel value in acquisition n (after some phase corrections):

exp 2

[ ( )] | , 1,...,n

n n

s

g s j s ds

FT s n N

sz

yx

r

, ,x r s

3-D reflectivity distribution

reference surface s = 0

s

TomoSAR System Model

b

b

elevationaperture

2 nn

b

r

Complex pixel value in acquisition n (after some phase corrections):

exp 2

[ ( )] | , 1,...,n

n n

s

g s j s ds

FT s n N

sz

yx

r

, ,x r s

3-D reflectivity distribution

reference surface s = 0

s

TomoSAR = spectral estimation

• irregular sampling

• small N

• motion must be considered

Test Site

Bellagio Hotel, Las Vegas

Optical image, © Google Earth TerraSAR-X spotlight mode

SL1MMER: 30% double scatterers

Number of Scatterers

Blue: no scatterer per pixel

Green: single

Red: double

Bellagio Hotel in 3D

What about motion?

D-TomoSAR System Model

General form

exp 2

2 ,n n

s

nd s tg s j s ds

displacementpossibly nonlinear & multi-component

D-TomoSAR System Model

General form

PSI:▫ displancement model directly fitted to phase, e.g. by LAMBDA

▫ only single scatterers

exp 2

2 ,n n

s

nd s tg s j s ds

displacement

D-TomoSAR System Model

General form

PSI:▫ displancement model directly fitted to phase, e.g. by LAMBDA

▫ only single scatterers

D-TomoSAR:▫ displacement term in exponent no longer FT (except for linear motion)

▫ for moderately non-linear motion: velocity spectrum

▫ multiple scatterers

exp 2

2 ,n n

s

nd s tg s j s ds

displacement

D-TomoSAR System Model

General form

The time warp method

▫ Linear motion:

▫ Seasonal motion:

exp 2

2 ,n n

s

nd s tg s j s ds

n

p s

artificial temporal baseline

motion parameter along s

n nt p s V s

0sin 2n nt t

p s a s

amplitude of seasonal motion

displacement

2exp 2 , n

n n n n

p s

g s p p s j s p dsdp

Let’s Do the Time Warp …

nd t

0sin 2 nn t t

nt

nn

nt

nd

n nd a

0sin 2n nd t a t t

Before warp:

After warp:

Single + DoubleAmplitude of Seasonal Motion [mm]

Bellagio Hotel in 4D

Thanks to Y. Wang for visualization

exaggerated 1000

Multi-component Motion

Deformation pattern, Las Vegas Linear + seasonal motion

Multi-component Motion

Deformation pattern, Las Vegas Linear + seasonal motion

Let’s Do the Time Warp Again

Generalized time warp

Single-component motion M-component motion

2D spectral estimation M+1-D spectral estimation

1

1 1

1, 1 , 1

... ,...,

exp 2 ... ...

M

n M M

p p s

n n M n M M

g s p p s p p s

j s p p dsdp dp

5D City Mapping: Linear + Seasonal Motion

P

P1

P2

P1 P2epicenter

Elevation [m]

Linear Subsidence [mm/a]

Amplitude of Seasonal Motion [mm]

Conclusions

Non-linear motion is everywhere in urban environment

Non-linear motion estimation is necessary and possible for TomoSAR

Single- and multi-component motion models can be accommodated by the time warp method