FR1.L09.3 - SAR TOMOGRAPHIC FOCUSING BY COMPRESSIVE SAMPLING: EXPERIMENTS ON REAL DATA

SAR TOMOGRAPHIC FOCUSING BY COMPRESSIVE SAMPLING: EXPERIMENTS ON REAL DATA

Università di Cassino Italy

Alessandra Alessandra Budillon Budillon

Gilda SchirinziGilda Schirinzi

IGARSS’10, Honolulu

Annarita Annarita EvangelistaEvangelista

Università di Napoli Parthenope Italy

IGARSS10, Honolulu July 25-30

Outline

Introduction Tomographic data model Compressive Sampling Tomography (CST) Height resolution enhancement Results on ERS-1/2 data Conclusions


Single-pass systems are not able to discriminate different scatterers lying in the same range-azimuth resolution cell and located at different elevations.

Multi-pass systems allow their discrimination, and the estimation of their reflectivity and height.

SAR Tomography

x

z

y


CST processing goals Compensation of the effects due to non-inform orbits

spacing.

Reduction of the number of acquisitions required to

achieve a given elevation resolution.

Increase of the elevation resolution achievable with a

given overall orthogonal baseline extent.


Tomographic Data Model

z

s

s’

P1

P2

P3

P4

P5

r

S1

ST

y

AB

C

r

S1 elevation extent of the sceneST overall baseline extentPi i-th orbit position(s) reflectivity functionR0 distance between the center of

the scene and the antennas ground elevations’ in-orbit elevationr range resolution

€

u s'm( ) ≅ γ s( ) e− j

4π

λR0

s'm s

−S1 /2

S1 /2

∫ ds

with − ST /2 ≤ s'm ≤ ST /2

m =1,....., M

€

s =λR0

2ST

Elevation resolution


Discrete data acquisition model

Discretizing elevation s with step s/ with ≥

The problem is under-determined

A solution can be found using truncated SVD

€

u = Φγ

€

u = [u(1),.....,u(M )]T

γ = [γ(1),.....,γ(N)]T

Φ{ }kn=

1

Ne

− j2πηST

s'k n

M×1M×N

N×1

€

N = S1η ρ s⎣ ⎦

M<N


Compressive Sampling Tomography (CST)

Assume that there exist a basis in which s has a K-sparse representation:

with only K<N coefficients i different from zero.

Considering the discrete elevation values sn=ns/ we can write in vector form:

with a K-sparse vector

:

Abd

€

s( ) = α iψ i s( )i=1

N

∑

€

=Ψ


Compressive Sampling Tomography

€

ˆ α = arg min α1 subject to u = ΦΨα

A solution can be found by solving the linear progamming problem:

€

=Ψ

€

u = Φγ

The norm minimization has the role of enforcing sparsity.

€

l 1

Measurement Matrix Sparsity Matrix



CS theory ensures that when it is satisfied an incoherency property between Φ and Ψ, it is indeed possible to recover with overwhelming probability the K largest αi from a number of measurements M satisfying the inequality :

where€

M ≥ C μ 2 Φ,Ψ( ) K N log N( )4

€

μ Φ,Ψ( ) = maxk , i

< ϕk ,ψi >

ϕk 2ψi 2



A solution can be found by solving the linear progamming problem:

€

=Ψ

€

u = Φγ + w

Given N and K, for obtaining a reliable solution the parameters and have to satisfy the relations: €

ˆ α = arg min α1

+1

2εΦΨα − u

2

2

€

≤sup = expM

CK

⎛

⎝ ⎜

⎞

⎠ ⎟1 4 ⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥

λR0

2S1ST

€

= 2 log Nσ W


Numerical results. Simulated data

ERS-1/2 parameters

Wavelength 0.0562 m

View angle 23°

Range distance R0 846 Km

Pulse bandwidth f 16 MHz

Range resolution r 9.37 m

Elevation resolution S 26.60 m

Height resolution z 10.39 m

S1 500 m

ST 893 m

z

s

s’

P1

P2

P3

P4

P5

r

S1

ST

y

AB

C

r

sA =-23.61 m

sB =0 m

sC =111.11 m

A:

B:

C:

AjAe

BjAe

CjAe

15 orbits 9 orbits


15 orbits 9 orbits

1

€

's =ρ s

η= 26.60 m

€

z = ρ 's sin(ϑ ) =10.39 m

---- SVD__ CS


€

For N = 9 and K = 3

→ η sup ≅ 3


15 orbits 9 orbits

€

=3

€

's =ρ s

η= 8.87 m

€

z = ρ 's sin(ϑ ) = 3.46 m

---- SVD__ CS




K=5

K=3K=4

€

=3


Stadium San Paolo (Naples –Italy)

290 m × 230 m

Height 35 m

Numerical results - Real data


CST TSVD TSVDCST


Conclusions

CST has proved to be very effective in reducing the number of acquisitions required for an accurate focusing in elevations.

It allows to noticeably improve the height resolutions attainable with a given overall orthogonal baseline span.

Results on ERS-1/2 data have showed the applicability of the method.

A wider experimentation on high resolution images is in progress.

FR1.L09.3 - SAR TOMOGRAPHIC FOCUSING BY COMPRESSIVE SAMPLING: EXPERIMENTS ON REAL DATA

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Transcript of FR1.L09.3 - SAR TOMOGRAPHIC FOCUSING BY COMPRESSIVE SAMPLING: EXPERIMENTS ON REAL DATA