Obstacle detection using

limited measurements of

scattered waves

Jacob D. Rezac

University of Delaware

NIST ACMD Seminar
What is inverse scattering theory?

Determine information about unknown obstacles based on how acoustic or

electromagnetic waves scatter off of them.

By “scattered” I mean reflected, transmitted, and absorbed.

Inverse scattering problems

N
Ultrasound

Acoustic waves can characterize human tissue.

Inverse scattering problems

N
Testing airplane canopies

Quickly finding flaws can save millions of dollars.

Inverse scattering problems

N
TechSat 21 Project

RF Imaging with distributed satellites.

Inverse scattering problems

N
Challenges

• Vastly different material properties (human body vs. state-of-the-art

materials)

• Orders-of-magnitude different sizes (landmasses vs. microscopic

cracks)

• Cost to probe (satellites vs. ultrasound machines)

Can we design general techniques which are useful at all of these scales?

How expensive are these techniques?

Are we confident in our results?

Inverse scattering problems

N
Different approaches to inverse problems

Iterative

Methods

Statistical

Methods

Direct

Methods

• “Guess” a solution

and check against

collected data

• Solve large

non-linear

optimization

problem

• Need to understand

physical system

• Use Bayes rule to

incorporate prior

information into

solution

• Typically uses an

iterative method

within the

algorithm

• Use mathematical

properties of system

non-iteratively

• Fast, but need lots

of data

Inverse scattering problems

N
Different approaches to inverse problems

Iterative

Methods

Statistical

Methods

Direct

Methods

• “Guess” a solution

and check against

collected data

• Solve large

non-linear

optimization

problem

• Need to understand

physical system

• Use Bayes rule to

incorporate prior

information into

solution

• Typically uses an

iterative method

within the

algorithm

• Use mathematical

properties of system

non-iteratively

• Fast, but need lots

of data

Inverse scattering problems

N
Today’s topics

• Introduction to wave scattering theory

• Ways to use understanding of system to reduce measurement

requirements

• Uncertainty in reconstructions

• Bayesian inversion

Inverse scattering problems

N
Chapter One

Scattering Theory
Simple physical setting

c(x), ρ0

c0, ρ0

incident

scattered

• Scattering from an infinite cylinder

• Acoustic speed of sound, c, and

pressure, ρ, are constant outside

object

• Speed of sound changes inside

object

Acoustic wave scattering of a time-harmonic incident field from an

penetrable object.
Simple physical setting

c(x), ρ0

c0, ρ0

incident

scattered

∆u

s

y

(x) + k

2

n(x)u

s

y

(x) = −k

2

n(x)u

i

y

(x)

∆u

i

y

(x) + k

2

u

i

y

(x) = 0

lim

|x|→∞

|x|

1/2



∂u

s

∂ν − ikus



= 0,

k is the wave number

n(x) is related to physical

constants

u

s

is the scattered acoustic field

u

i

is the incident plane wave

Acoustic wave scattering of a time-harmonic incident field from an

penetrable object.
Simple physical setting

c(x), ρ0

c0, ρ0

incident

scattered

∆u

s

y

(x) + k

2

n(x)u

s

y

(x) = −k

2

n(x)u

i

y

(x)

∆u

i

y

(x) + k

2

u

i

y

(x) = 0

lim

|x|→∞

|x|

1/2



∂u

s

∂ν − ikus



= 0,

k is the wave number

n(x) is related to physical

constants

u

s

is the scattered acoustic field

u

i

is the incident plane wave

Acoustic wave scattering of a time-harmonic incident field from an

penetrable object.

• Everything that follows can be done

with the full system of Maxwell’s

equations (for realistic electromagnetic

materials).
Multistatic Data collection

D

D

u

i

Determine material parameters by measuring how the material affects an

incident wave.
Multistatic Data collection

D

D

u

s

u

s

Determine material parameters by measuring how the material affects an

incident wave.
Multistatic Data collection

D

D

u

i

Determine material parameters by measuring how the material affects an

incident wave.
Multistatic Data collection

D

D

u

s

u

s

Determine material parameters by measuring how the material affects an

incident wave.
Multistatic Data collection

D

D

u

i

u

s

u

s • Useful in medical settings

or testing material

properties of small objects

• Lots of reconstruction

algorithms can be used

with this type of data

• Impractical for most

applications

Can we do better?
Direct inversion algorithms

• Assume only the location of object is needed (e.g., looking for a crack

or a tumor).

• Based on model, derive an indicator function I(z), depending on

coordinates, so that

I(z) = (

0 z ∈/ object

1 z ∈ object.

• I(z) must be easy and fast to compute from scattered field data.
Intuition behind reconstruction algorithm

z

u

i us

x

z

u

i

x

The scattered field from a small object at point z is proportional to an

incident field emitted from the point z.
Reconstruction from one obstacle

• Collect u

s

at receivers

around obstacle.

• Generate a grid Z which

contains the obstacle.

• For each point ζ ∈ Z test

if measured scattered field

is proportional to an

incident field coming from

point ζ.

• This is possible if and only

if ζ is the center of the

obstacle.

ζ

obstacle

u

s

y

(x)gζ = u

i

ζ

(x)
Intuition behind reconstruction algorithm

z1

u

i

u

s

x

z2

u

i

u

s

x

z1

z2

u

i

u

s

x

u

s

Scattered fields from multiple small objects behave nearly linearly.
Reconstruction of multiple obstacles

• Collect u

s

at receivers

around obstacle.

• Generate a grid Z which

contains the obstacle.

• For each point ζ ∈ Z try

to match measured

scattered field to an

incident field coming from

point ζ.

• The is possible if and only

if ζ is the center of the

obstacle.

ζ

obstacle

obstacle

Z

Γi

u

s

y

(x)g(y) ds(y) = u

i

ζ

(x)
Reconstruction of multiple obstacles

• Collect u

s

at receivers

around obstacle.

• Generate a grid Z which

contains the obstacle.

• For each point ζ ∈ Z try

to match measured

scattered field to an

incident field coming from

point ζ.

• The is possible if and only

if ζ is the center of the

obstacle.

ζ

obstacle

obstacle

Z

Γi

u

s

y

(x)g(y) ds(y) = u

i

ζ

(x)

Exactly the same algorithm as for one obstacle.
Range test

At the discrete level, we want to find g so that

Ng = u

i

ζ

, (1)

where N is the matrix which approximates R

Γi

u

s

y

(x)g(y) ds(y).

Let N = UΣV

H be a singular value decomposition of N.

Equation (1) holds if and only if

I(z) =

number incident fields

X

r=number obstacles+1

[U]ru

i

ζ ≈ 0.
Reconstruction algorithm

• Collect u

s

at receivers

around obstacle

• Generate a grid Z which

contains the obstacle

• For each point ζ ∈ Z, plot

 number incident fields

X

number obstacles+1

[U]ru

i

ζ

!−1

• Large values on grid

indicate object

ζ

obstacle

obstacle
Simulated Reconstruction
Measurement geometry

Reconstruction methods are independent of receiver geometry.

• More receivers leads to more stable reconstructions

• Theoretically, transmitter locations need to surround obstacles -

experimentally we can use fewer

• Physical and financial constraints can be used to select measurement

geometry
Quasi-backscattering measurements

D

D

u

i

u

s

u

s

• Measure scattered field

only near transmitter

location

• Easier data collection

when obstacles are large

(e.g., plane canopies and

TechSat 21)
Simulated results
Simulated results
Extensions

• Large objects

• Requires deeper mathematical theory, but idea is the same

• Limited aperture transmitter locations

• Time domain data

• Multi-frequency data

• Couple with nonlinear optimization routines

• Different physical models

• Similar methods have been shown to work for time dependent acoustic

data and time harmonic elastic data

• The necessary components (nearly linear scattering and knowledge of

how waves propagate in free space) are available in many physical

situations
Chapter Two

Uncertainty Propagation and

Bayesian Inverse Problems
The problem with noisy measurements

For each ζ ∈ Z, we solve

Z

Γi

u

s

y

(x)g(y) ds(y) = u

i

ζ

(x)

• Uncertainty in measurements of u

s

• Uncertainty in shape of u

i

• Uncertainty in location of transmitters and receivers

• Uncertainty in model (constant background parameters?)

How can we quantify this lack of knowledge in our reconstructions?
Monte Carlo approach

Seek the probability law of gζ and an estimate of its statistics.

Assume errors can be separated so that

Z

Γi

u

s

y

(x)g(y) ds(y) = u

i

ζ

(x) + .

A simple Monte Carlo-type method (or, e.g., spectral expansion method)

can be used to find statistics of gζ .

Ignores modeling assumptions which validate reconstruction algorithm.
Bayesian Inverse Problems

∆u

s

y

(x) + k

2

(1 − m(x))u

s

y

(x) = −k

2

(1 − m(x))u

i

y

(x)

∆u

i

y

(x) + k

2

u

i

y

(x) = 0

lim

|x|→∞

|x|

1/2



∂u

s

∂ν − ikus



= 0,

• Calculate probability law of m, given the data we collected

• Requires a priori information about how errors are distributed

• Bayesian approach helps to incorporate lack of information in a

principled fashion

πpost(m|u

s

obs) ∝ πlike(u

s

obs|m)πprior(m)
Bayesian Inverse Problems

∆u

s

y

(x) + k

2

(1 − m(x))u

s

y

(x) = −k

2

(1 − m(x))u

i

y

(x)

∆u

i

y

(x) + k

2

u

i

y

(x) = 0

lim

|x|→∞

|x|

1/2



∂u

s

∂ν − ikus



= 0,

• Calculate probability law of m, given the data we collected

• Requires a priori information about how errors are distributed

• Bayesian approach helps to incorporate lack of information in a

principled fashion

πpost(m|u

s

obs) ∝ πlike(u

s

obs|m)πprior(m)
Components needed for Bayesian inversion

• Function, f (m), mapping from m to u

s

obs

• Numerical approximation to scattering PDE

• Assume

u

s

obs = f (m) + η

where η ∼ N (0, Γnoise )

• Allows us to write

πlike(u

s

obs|m) ∝ exp 

−

1

2

(f (m) − u

s

obs)

T

Γ

−1

noise(f (m) − u

s

obs)



.

• A priori distribution for m

• Assume normal distribution for simplicity,

m ∼ N (m0, C0)

• For technical reasons, C0 = A−2 must be related to the inverse of a

solution map for an elliptic PDE
Linearized case

Assume that errors are small. Then,

u

s

obs ≈ f ( ˆm) + F(m − mˆ ) + η

is a good approximation where F is the Frechet derivative of f and

mˆ = min kf (m) − u

s

obsk

2 + kA(m − m0)k2



.

In this case,

πpost(m|u

s

obs) ∼ N 

mˆ ,



F

∗Γ

−1

noiseF + C

−1

0

−1



Statistics on such a distribution and samples from it can be obtained easily

(but maybe slowly).
Sample reconstructions

Estimate of mean
Extensions

• Non-linear assumptions

• Harder (impossible?) to write closed-form solution

• Monte Carlo-type sampling can be used, but requires many PDE solves

• Lots of research on how to address these issues, but problems is not

solved

• Speed of solution

• PDE needs to be solved for each parameter (e.g., m on every point of a

finite element grid)

• Need fast solution techniques, particularly for non-linear problem

• A priori assumptions

• Infinite dimensional Bayesian inverse problems use prior distribution to

ensure that solution exists

• We would like to use physical reasoning or bootstrap based on

deterministic reconstructions
Answers to challenges

Can we design general techniques which are useful at all of these scales?

• Many problems have similar characteristics which lead to useful

reconstruction techniques

• Better understanding of physical situation can be incorporated into

model uncertainties

How expensive are these techniques?

• Cost of data collection can be reduced by better understanding model

• Fast reconstructions available when looking for limited data

Are we confident in our results?

• Different uncertainties can be included in reconstruction algorithms

• Speed of reconstructions is decreased, but we gain extra information

Thanks!