RADAR imaging from multiply scattered waves · 2005-10-28 · Introduction Traditional SAR...

IntroductionTraditional SAR

Geometric Structure of Λ

Scattering from Environment

RADAR imaging from multiply scattered waves

C. Nolan

M. Cheney∗, T. Dowling, R. Gaburro

University of Limerick, Ireland∗ Rensselaer Polytechnic Institute

Imaging from Wave Propagation

IMA, October, 2005

Cheney, Dowling, Gaburro, Nolan RADAR imaging from multiply scattered waves




Outline

1 Introduction

2 Traditional SAR

3 Geometric Structure of Λ

4 Scattering from Environment





Introduction

Traditional SAR cannot distinguish scatterers on left offlight path from those on right, unless we operate inside-scan mode (beam-form to one side)

Problem is compounded in a waveguide situation, yieldingmany more artifacts from multiple scattering between wallsand target (Cirencester, 2005, Cheney & Nolan)

Even in side-scan mode, latter artifacts persist

In this talk, we consider a single wall

In side-scan mode, we show that we can rid ourselves ofboth types of artifacts at once!





SAR - Low Directivity ... for now

��

��

w inγ(s)

v wsc

Emit low-frequency (30-90 Mhz) radio waves from antenna

Goal: Construct image of the ground from scattered waves





SAR

Treat Electric Field as though its components satisfy ascalar wave equation:

(

1c2(x)

∂2t −∇2

)

u(t , x) = 0,

where c(x) is the wave speed of the field

Linearize about constant background (air):

c−2(x) − c−20 := V (x1, x2)δ(x3)

where c is speed of wave propagation





SAR

Suppose antenna is flown on a flight path

Γ := {γ(s) : s ∈ (smin, smax) }

Possible to show (as suggested by earlier figure): signalrecorded by antenna at location γ(s) at time t isapproximately

d(s, t) =

∫

dωdx e−iω(t−2|(x,0)−γ(s)|/c0) W (x , s, t , ω)V (x)

where W is a weighting function that incorporates sourceradiation pattern, bandwidth, etcThis defines a scattering operator:

F : V 7→ d





SAR

V has singularities at boundaries of objects on ground -e.g. walls of buildings

w inγ(s)

wscv

These singularities will be mapped to singularities in thedata





Singularities, with directions

E.G. f (x) = H(v · x) has a singularity at each point on lineperpendicular to v through origin in the direction v

v

1

2

singular set

f=1

f=0





SAR

DefineX ≡ R2, Y := (smin, smax) × (0, T )

⇒ F : E ′(X ) → E ′(Y )

F is an oscillatory integral operator with homogeneousphase and non-vanishing gradient

Amplitude of F behaves as an approximate symbol(decays in ω when differentiated w.r.t. ω)

Thus F is a Fourier integral operator (FIO)





SAR

Recall phase of F :

φ(s, t , x , ω) := −iω(t − 2|(x , 0) − γ(s)|/c0)

According to theory of FIOs, singularities in V at a pointx ∈ X in a direction ξ produce singularities in the data d at(s, t) in a direction (σ, τ) whenever they belong to the setrelation (hat means unit vector):

Λ = { ((s, t , σ, τ), (x , ξ)) |

t = 2|γ(s) − (x , 0)|/c0,

σ = 2τ ̂(γ(s) − x) · γ̇(s)/c0,

ξ = 2τ ̂((x , 0) − γ(s))H/c0

W (x , s, t , ω) 6= 0 }





SAR - Graphical Illustration of Wavefront Relation Λ

γ(s)

(x,0)

t(s)γ

σ

ξ

terrain





Lagrangian Submanifolds

Lagrangian submanifolds can be reduced (locally) to onelike the following model form ...

Think of a particle at x0 on a hypersurface H, withmomentum p0 at t = 0

At a later time t > 0, Hamiltonian∗ mechanics tells us thatthe particle will now be located at(x1, p1) = (C1(x0, p0), C2(x0, p0)), where C is a canonicaltransformation (that ensures conservation of energy)

The set of such pairs {(x0, p0), (x1, p1)} is the canonicalLagrangian submanifold

* This year is Hamilton’s bi-centenary





SAR

Λ is a Lagrangian submanifold with the properties

π : Λ → T ∗Y , ρ : Λ → T ∗X

are local diffeomorphisms except at points in

Σ := { ((s, t , σ, τ), (x , ξ)) ∈ Λ |

horizontal component of (x−γ(s)) and γ̇(s) are co-linear }





Artifact Analysis

Imaging methods often consist of application of a(weighted) adjoint scattering operator F∗ to the data

F∗ : E ′(Y ) → E ′(X )

Singularities in data are mapped to singularities in resultingimage by Λ∗ – the transpose relation of Λ

In summary, our proposed image transforms thesingularities of V via the composite relation Λ∗ ◦ Λ

Goal: arrange Λ∗ ◦ Λ ⊆ I (identity relation), so that werecover the visible singularities of the model V





Artifact Analysis

Problem: π is not injective, in fact it is a 2:1 map

γ(s)

z

xξ

(x,s)

ζ (x,s)

So a singularity (x , ξ) could be correctly imaged along withartifact (z(x , s), ζ(x , s)) artifact





Artifact Analysis

Artifact avoidance:1 Side-scan mode2 High-curvature for flight-track weakens artifact by smearing

it, enhanced by long (dwell) integration times

−30−20

−100

1020

30

−30

−20

−10

0

10

20

300

2

4

6

8

10

12

14

x (m)

Parabolic inversion: a=0.04, Ns=50, ds=0.8, Nt=50, dct=0.54

y (m)Cheney, Dowling, Gaburro, Nolan RADAR imaging from multiply scattered waves




Scattering from a nearby wall

We now switch attention to a target near a wall (or ground)

Waves can scatter in three ways (plus a fourth way whichreverses path 3):

��

��

��

��

γ (s)

x

γ (s)−

1

2

3





Scattering from a target with a nearby wall

We represent the wall (x1 = 0) using the method ofimages, placing a virtual source at (−γ1(s), γ2(s), γ3(s))

The data d now becomes

d =

3∑

i=1

FiV

and we enforce a Dirichlet condition at the wallWhere Fi is an operator of the same form as F with amodified amplitude and phase

φ1 = −iω(t − 2|x − γ(s)|)

φ2 = −iω(t − |x − γ(s)| + |x − γ−(s)|)

φ3 = −iω(t − 2|x − γ−(s)|)






We assume scatterers are located in x1 > 0

Idea: Examine singularities in data (s, t , σi , τ) due to thethree different paths

We’ll show how to arrange σ1 6= σ3

Observe σ2 = (σ1 + σ2)/2, which means none of the σi ’scan ever have common values!

Latter point is key to avoiding further artifacts inbackprojection (see later)






That we might arrange σ1 6= σ3 is readily seen graphicallyfor the case when RADAR flies perpendicularly away fromwall ...

σ1, σ3 are proportional to the direction cosines of the rangevectors from the source and virtual source respectively

Thus σ1, σ3 are monotonically increasing and decreasingrespectively, and only agree for RADAR located on wall(dis-allowed)






Significance of distinct σi ’s ...

F∗d =

3∑

i=1

3∑

j=1

F∗i Fj V

Singularities are reconstructed properly from the diagonalterms (i = j) as we’ve seen from first part of talk ...provided we employ side-scan mode, or other methods

Once we establish the how to arrange distinct σi ’s, we willhave arranged Λ∗

i ◦ Λj = ∅

So, no artifacts arise from off-diagonal terms (i 6= j)





Conditions for disjoint σi ’s

As noted, we just need to ensure σ1 6= σ3

��

��

x v

x’’

γγ +−

x’+v−

1

2

Figure shows that x-scatterer produces a σ3 equal to σ1

due to x ′′-scatterer in experiment 1.Thus data could be backprojected along experiment 1 toproduce x ′′-scatterer, which could be an artifactBUT x ′′ artifact is avoided due to Side-scan mode!






Algebraically, we see this from (σ1 = σ3):

(γ1 + x1)v1 + (γ2 − x2)v2 + (γ3 − x3)v3

|γ− − x |=

(γ1 − x ′′1 )v1 + (γ2 − x ′′

2 )v2 + (γ3 − x ′′3 )v3

|γ+ − x |

In composing Λ∗1 ◦ Λ3 we are composing points

((x ′′, ξ′′), (s, t , σ1, τ)) and ((s, t , σ3, τ), (x , ξ))

Where t = |γ− − x | = |γ+ − x ′′|, which allows us to canceldenominators

As x and x ′′ are at same depth beneath respectivesources, we cancel last term in denominator, yielding ...






Condition for σ1 = σ3

V (2) ·

[

x1 + x ′′1

x ′′2 − x2

]

= 0

whereV (2) = [v1 v2]

T

If we fly perpendicular to the wall (v2 = 0), then x = x ′′ andx1 = 0 (x is on wall), which is vacuous, since scatterersassumed to have x1 > 0

So no artifact-scatterers away from wall






If we fly parallel to wall (v1 = 0), then x2 = X ′′2 , so

artifact-scatterers lie on intersection of a line perpendicularto the wall and a circle centred at γ+

In generalv1x1 + v2x2 = f (x ′′)

which gives a series of parallel lines (perpendicular to γ̇) tobe intersected with a circle centred at γ+

This confirms graphical assertion that side-scan mode willavoid mixed backprojection artifacts as well as the originalleft-right ambiguity artifacts simultaneously





Data Set

2 4 6 8 10 12 14 16 18 20

5

10

15

20

25

30

fast

tim

e

slow time

Experiment 1

Experiment 2

Experiment 3

−0.3

−0.2

−0.1

0

0.1

0.2





Angular Resolution Increased

Virtual source has benefit of making aspects of scatterersvisible that would not otherwise be visibleIn effect, for scatterers located close to the wall, we doublethe ‘opening angle’

��

γγ − +





Intersecting Lagrangians

Allowing scatterers to be located on wall, the following isthe situation ...

Without the wall, the operator F∗F is associated to theclass of singular FIO’s (Ip,l(Λ1,Λ2)) with cleanlyintersecting Lagrangians

Insertion of the wall results in composition of operatorsassociated to triply intersecting Lagrangians (seeNeumann version of previous data set).

No calculus for singular FIO’s associated totripple-intersecting Lagrangians

So, if you want to consider scatterers on the wall, you willhave to carry out explicit analysis to see if inversion can bearranged





Summary & Acknowledgement

Simple message is that side-scan SAR (Sonar) avoidsleft-right ambiguity artifacts as well as those that mighthave been expected due to multiple scattering with the wall

Acknowledgement: Thanks to Science Foundation Ireland(SFI) and NSF for supporting this research.

Invitation: Go see ‘Praire Home Companion’ rehearsalshow in St. Paul Friday evening (Lake Wobegon: Wherethe women are strong, the men are good-looking, and allthe children are above average)


RADAR imaging from multiply scattered waves · 2005-10-28 · Introduction Traditional SAR...

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Transcript of RADAR imaging from multiply scattered waves · 2005-10-28 · Introduction Traditional SAR...