ressive - math.ncku.edu.tw

Com

pre

ssive

Imagin

g

Alb

ertFannjia

ng,UC

Davis

Outlin

e

•Review

:in

versesca

ttering

•Review

:co

mpressed

sensin

g

•Random

incid

ent

and

scatterin

gdirectio

ns:

SIM

O,SIS

O

•Random

illum

inatio

n

•Reso

lutio

nand

superreso

lutio

n

•M

USIC

:th

reshold

ing,noise

tolera

nce.

Inverse

scatte

ring

•Pla

ne

wave

incid

enceu

i(r)=

eiω

r·d,

r∈

Rd

where

d∈

Sd−

1,d

=2,3

isth

ein

ciden

tdirectio

n.

c0=

1:

ω=

frequen

cy/waven

um

ber.

•T

he

scattered

field

us

=u−

ui

then

satisfi

esth

eLip

pm

ann-

Sch

win

ger

equatio

n:

us(r)

=ω

2∫R

dν(r ′)

(ui(r ′)

+us(r ′) )

G(r,r ′)dr ′

where

Gis

the

Green

functio

nfo

rth

eopera

tor−(∆

+ω

2).

•M

easu

remen

t:sca

tteredfield

(near

field

)or

the

scatterin

gam

-plitu

de

(farfield

).

•Far-fi

eldasym

pto

tic:d

=3

eiω|r−

r ′|

4π|r

−r ′| ≈

eiω|r|

4π|r| e −

iωr·r ′.

Hen

ce

us(r)

=eiω|r|

|r| (d−1)/2

(

A(r,d

)+

O(

1|r| ))

,r=

r/|r|,d

=2,3

where

the

scatterin

gam

plitu

de

Ais

determ

ined

by

the

form

ula

A(r,d

)=

ω2

4π

∫Rdν(r ′)u

(r ′)e −iω

r ′·rdr ′.

•Born

appro

ximatio

n

A(r,d

)=

ω2

4π

∫Rdν(r ′)u

i(r ′)e −iω

r ′·rdr ′.

•Goal:

determ

ine

νfro

mm

easu

remen

tdata

:A.

Sta

ndard

theo

ry(N

ach

man,Noviko

v,Ram

m,Sylvester-U

hlm

ann

etc)of

inverse

scatterin

gasserts

the

injectivity

of

the

mappin

g

from

ν∈

C1c

with

anonneg

ative

imagin

arypart

toth

eco

rre-sp

ondin

gsca

ttering

am

plitu

de

for

afixed

frequen

cyin

three

di-

men

sions.

Thatis,

the

refractive

index

can

inprin

ciple

be

deter-

min

eduniq

uely

by

the

full

knowled

ge

of

A(r,d

),∀d,r,

fora

fixed

ω.

•In

versepro

blem

:discrete

vs.co

ntin

uum

.

Fin

itedata

,finite

num

ber

ofpixels

inco

mputa

tion

dom

ain

.

Issue

oferro

rs(extern

alorm

odel-m

ismatch

).

Com

pre

ssed

sensin

gwith

RIP

•Lin

earin

versepro

blem

:Y

=Φ

X+

Ewhere

Φis

an

n×

mm

atrix

with

n(#

rows)'

m(#

colu

mns),

i.e.severely

underd

etermin

ed.

•Prio

rin

form

atio

n:

the

target

vectoris

sparse,

‖X‖0

=s∼

n.

Diffi

culty:

toid

entify

the

low

dim

ensio

nalsu

bsp

ace

(the

support

space)

out

of

(mn

)ofth

emin

ahig

hdim

ensio

nalvecto

rssp

ace.

•Basis

pursu

itden

oisin

gorLasso

min

Z∈C

m‖Z‖1,

s.t.‖Y−

ΦZ‖2≤

ε

where

εis

the

sizeoferro

r,i.e.

‖E‖2≤

ε.

Reco

verydep

endson

RIP

/in

coheren

cepro

perty

ofΦ

and

sparsity

of

X.

•Restricted

isom

etrypro

perty

(RIP

):D

efine

the

restrictediso

m-

etryco

nsta

nt

(RIC

)δs

<1,s∈

Nto

be

the

smallest

positive

num

ber

such

that

the

ineq

uality

(1−

δs )‖

Z‖22≤‖Φ

Z‖22≤

(1+

δs )‖

Z‖22

hold

sfo

rall

Z∈

Cm

ofsp

arsityat

most

s.

Theore

m1

(Candes

08)

Suppose

δ2s

<√

2−

1.

Then

the

solu

tion

XofLasso

satisfi

es

‖X−

X‖2≤

C1s −

1/2‖

X−

X(s)‖

1+

C2ε

where

X(s)

isth

ebest

s−sp

arseappro

ximatio

nof

X.

Exa

mples:

random

i.i.d.

matrices

(no

structu

re),ra

ndom

partia

lFourier

matrices

(i.e.ra

ndom

row

selections

from

DFT

).

Theore

m2

(Rauhut

2008)

If

n

lnn≥

Cδ −

2sln

2sln

mln

1γ

for

γ∈

(0,1

)and

som

eabso

lute

consta

nt

C,th

enwith

pro

bability

atlea

st1−

γth

era

ndom

partia

lFourier

matrix

satisfi

esth

ebound

δs≤

δ.

DFT

uses

unifo

rmsa

mplin

gover

the

full

Fourier

dom

ain

.

Our

scatterin

gm

atrix

issa

mplin

gonly

asm

all

part

ofit.

Mutu

alcohere

nce

•T

he

mutu

alco

heren

ce

µ(Φ

)=

max

i-=j

∣∣∣ ∑kΦ∗ik Φ

kj ∣∣∣

√∑

k |Φki | 2 √

∑k |Φ

kj | 2

.

Pro

positio

n1

δs≤

µ(s−

1).

Suffi

cient

conditio

nfo

rreco

very

µ(2

s−

1)≤√

2−

1

or

s≤

12

(

1+

√2−

1

µ

)

.

Lower

bound:

µ≥

√m−

n

n(m

−1)⇒

1µ=

O( √

n).

Hen

ce,by

mutu

alco

heren

cealo

ne,

we

can

recover

s=

O( √

n)

objects.

Opera

tornorm

Theore

m3

(Candes-P

lan

09)

Assu

me

that

E=

(Ej )∈

Cn

and

Ej ,j

=1,...,n

arei.i.d

.co

mplex

Gaussia

nr.v.s

with

variance

σ2

(ε=

O(σ √

n)).

Suppose

that

µ(Φ

)≤

A0/lo

gm

and

s≤

C0m

‖Φ‖2lo

gm

.

Assu

me

mini|X

i |>

8σ

√2lo

gm

.

Then

the

solu

tion

Xof

min

Z

12 ‖Y−

ΦZ‖22+

σ·2

√2lo

gm‖Z‖1

recovers

thesu

pport

ofX

with

hig

hpro

bability

atlea

st1−

2m−1((2

πlo

gm

) −1/2+

sm−1)−O

(m−2lo

g2).

Typ

ically,

‖Φ‖2∼

mn⇒

s=

O(n

/lo

gm

).

Poin

tsc

atte

rers

scattered waves

sourcesensor

incident waves

Recip

rocity:

SIM

O∼

multi-sh

ot

SIS

Om

easu

remen

t.

Assu

mptio

n:

poin

tsca

ttererssit

on

afinite

regular

grid

of

spacin

g).M

easu

remen

t:ra

ndom

lysa

mple

the

scatterin

gdirectio

ns

rl ,l

=1,...,n

.

SIM

O(sin

gle

-input-m

ultip

le-o

utp

ut)

The

scatterin

gam

plitu

de

isa

finite

sum

A(r

l ,d)

=ω

2

4π

m∑j=1

νj u

(rj )e −

iωrj ·r

l.

Excita

tion

field

u(r

i )sa

tisfies

the

Fold

y-Lax

equatio

n

u(r

i )=

ui(r

i )+

ω2

∑i-=j

G(r

i ,rj )ν

j u(r

j )

where

all

the

multip

lesca

ttering

effects

arein

cluded

but

the

selffield

isexclu

ded

toavo

idblo

w-u

p.

Let

X=

(νj u

(rj ))

∈C

m.

The

(l,j)-entry

ofth

esen

sing

matrix

is

e −iω

(zjsin

θl +

xjco

sθl )

where

θlis

the

sam

plin

gangle

and

rj=

(xj ,z

j )are

grid

poin

ts.

This

isnot

the

standard

random

partia

lFourier

matrix!

Cohere

nce

bound

Theore

m4

(AF

2009)

Suppose

m≤

δ8eK

2/2

,δ,K

>0.

Then

the

sensin

gm

atrix

satisfi

esth

eco

heren

cebound

µ(Φ

)<

χs+

√2K

√n

with

pro

bability

grea

terth

an

(1−

δ)2

where

χs≤

ct (1

+ω

)) −1/2‖

fs‖

t,∞,

where

‖·‖

t,∞is

the

Hold

ernorm

oford

ert

>1/2

and

the

consta

nt

ctdep

ends

only

on

t.

For

d=

3,

χs≤

c1 (1+

ω)) −

1‖fs‖

1,∞

If,however,

supp(f

s)does

not

conta

ins

any

Blin

dSpot,

then

χs

satisfi

esth

ebound

χs≤

ch (1

+ω

)) −h‖

fs‖

h,∞

where

the

consta

nt

ch

dep

ends

only

on

h.

•W

edo

not

need

full

viewm

easu

remen

t:th

esu

pport

of

fs

can

be

asm

all

portio

nof

Sd−

1,d

=2,3

.

We

need

som

esm

ooth

ness

infs:

anum

ber

ofexistin

gnum

erical

tests(b

yoth

ers)neg

lectth

is!

•To

have

µ'

1,need

ω)1

1and

n1

1.

•In

the

case

ofra

ndom

partia

lFourier

matrix,

χs=

0.

Pro

ofuses

concen

tratio

nin

equality

and

statio

nary

phase

analysis.

•T

he

pairw

iseco

heren

cehas

the

form

Sn

=1n

n∑

j=1

eiω

rj ·(r−

r ′)

•Hoeff

din

gin

equality

P[ |S

n−

ES

n |≥nt] ≤

2exp

[−nt 2

2

]

forall

positive

valu

esof

t.

•Exp

ectatio

nestim

atio

n:

1n E

n∑

j=1

eiω

rj ·(r−

r ′) =

∫2π

0eiω

r·(r−r ′)f

s(θ)dθ,

r=

(cosθ,sin

θ)

which

isth

eHerg

lotz

wave

functio

nwith

kernel

fs.

Opera

tornorm

bound

Theore

m5

(AF

2009)

Forth

eSIM

Om

easu

remen

twe

have

‖Φ‖2≤

2mn

with

pro

bability

larger

than

1−

c1 √n−

1

m

n(n−

1)

The

pro

bability

bound

ispro

bably

not

optim

al.

Multip

le-sc

atte

ring

wave

Lip

pm

ann-S

chwin

ger

equatio

n

u(r

i )=

ui(r

i )+

ω2

∑j-=i G

(ri ,x

j )νj u

(xj )

Let

ikbe

the

indices

for

which

ν(r

ik )-=

0.

Defi

ne

the

illum

inatio

nand

full

field

vectors

at

the

loca

tions

ofth

esca

tterers:

Ui

=(u

i(ri1 ),...,u

i(ris ))

T∈

Cs

U=

(u(r

i1 ),...,u(r

is ))T∈

Cs.

Let

Gbe

the

s×

sm

atrix

G=

[(1−

δjl )G

(rij ,r

il )]

andV

the

dia

gonalm

atrix

V=

dia

g(ν

i1 ,...,νis ).

Lip

pm

ann-S

chwin

ger

equatio

nca

nbe

written

as

U=

Ui+

ω2G

VU

or

U=

Ui+

ω2G

X

On

the

oth

erhand,

X=

V(I−

ω2G

V)−

1U

i.

Theore

m6

(AF

2009)

Suppose

ω−2

isnot

an

eigen

valu

eofth

em

atrix

GV

andU

iis

not

orth

ogonalto

any

row

vectorof

(I−ω

2GV

)−1

.

Then

the

true

target

Vis

given

by

V=

dia

g[

X

ω2G

X+

Ui ]

where

the

divisio

nis

inth

een

try-wise

sense

(Hadam

ardpro

duct).

Near-fi

eld

measu

rem

ents

incident wave

sensors

sD

-+

DD

z=0

scattered wave

sources

z=0

DD

+-

Ds

SIM

O∼

multi-sh

ot

SIS

Om

easu

remen

t.

min

ΔL

Theore

m7

(AF

2009)

Suppose

m≤

δ2e2K

2/r

20,δ

>0

where

c0dep

ends

on

the

min

imum

dista

nce

∆m

inbetw

eenz

=0

and

the

lattice

(For

d=

2,

r0

=O

(−lo

g∆

min );

for

d=

3,

r0

=O

(∆−1

min )).

The

mutu

alco

heren

ceobeys

µ(Φ

)≤

|G(∆

max )| −

2(√

2K

√n

+c

√ω

L

)

,d

=2

µ(Φ

)≤

|G(∆

max )| −

2(√

2K

√n

+cωL

)

,d

=3

for

som

eco

nsta

nt

c(in

dep

enden

tof

ω>

0fo

rd

=2

and

ω>

1fo

rd

=3),

with

pro

bability

grea

terth

an

(1−

δ)2,

where

∆m

ax

isth

elarg

estdista

nce

betw

eenth

earray

and

the

lattice.

Need

ωL1

1and

n1

1.

Reso

lutio

n

ω)∼

1?

Multi-sh

ot

SIS

Osch

emes

pro

vide

more

info

rmatio

n

Multi-sh

ot

SIS

Osc

hem

es

The

(l,j)-entry

of

Φ∈

Cn×

mis

e −iω

l rl ·r

jeiω

l dl ·r

j=

eiω

l )(j2 (sinθl −

sinθl )+

j1 (cosθl −

cosθl )),

j=

(j1−

1)+

j2.

•Let

(ρl ,φ

l ),i=

1,..,n

be

the

polar

coord

inates

of

i.i.d.

unifo

rmr.v.s

(ξl ,η

l )∈

[0,2

π] 2.

•Schem

eI.

This

schem

eem

plo

ysΩ−band

limited

pro

bes,

i.e.ω

l ∈[−

Ω,Ω

].Set

θl

=θl +

π=

φl

(back

ward

sam

plin

g)

ωl

=Ω

ρl

√2

l=

1,...,n

.In

this

case

the

scatterin

gam

plitu

de

isalw

ayssa

m-

pled

inth

eback

-scatterin

gdirectio

nanalo

gous

toSAR.

•Schem

eII.

This

schem

eem

plo

yssin

gle

frequen

cypro

bes

no

lessth

an

Ω:

ωl=

γΩ

,γ≥

1,

l=

1,...,n

.

Set

θl=

φl +

arcsinρl

γ √2

θl=

φl −

arcsinρl

γ √2

.

The

diff

erence

betw

eenth

ein

ciden

tangle

and

the

sam

plin

gangle

is

θl −

θl=

2arcsin

ρl

γ √2

(scatterin

gangles)

which

dim

inish

esas

γ→∞

.In

oth

erword

s,in

the

hig

hfreq

uen

cylim

it,th

esa

mplin

gangle

appro

ach

esth

ein

ciden

tangle.

This

resembles

the

setting

ofth

eX-ray

tom

ogra

phy.

•T

heore

m8

(AF

2009)

Suppose

Ω)=

π/ √

2.

Then

schem

eIand

IIsa

tisfyRIP

with

hig

hpro

bability

and

the

errorbound

‖X−

X‖2≤

C1s −

1/2‖

X−

X(s)‖

1+

C2ε.

Num

eric

alte

sts

−200−150

−100−50

050

100150

200

−200

−150

−100

−50 0 50

100

150

200

0.2

0.4

0.6

0.8

1 1.2

1.4

1.6

1.8

−200−150

−100−50

050

100150

200

−200

−150

−100

−50 0 50

100

150

200

0.2

0.4

0.6

0.8

1 1.2

1.4

1.6

1.8

2 x 10−4

(left)Source

inversio

nwith

the

para

xialsen

sing

matrix

40

source

poin

tsand

121

anten

nas.

The

resultin

gerro

ris

0.0

164

while

the

error

with

exact

Green

functio

nis

7×

10−16

(not

shown).

(right)

MFP

image

pro

duced

on

the

sam

egrid

.T

he

redcircles

represen

tth

etru

elo

catio

ns

ofth

etarg

etsin

both

plo

ts.

0100

200300

400500

600380

390

400

410

420BP

0100

200300

400500

60015 20 25 30

MF w

. thresholding

Com

pressed

imagin

gby

MFP

(botto

m)

versus

BP

(top).

The

num

-ber

of

recovera

ble

objects

as

afu

nctio

nof

the

num

ber

of

senso

rsn

=1,2

,3,4

,5,6,8,10,12,15,20,24,25,30,40,50,60,75,100,

120,150,200,300,600

with

np=

600

fixed

.

Schem

eI:

success

pro

bability

05

1015

2025

3035

400

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9 1

11/21/41/8

Success

pro

babilities

forSch

eme

I.Asth

eback

ward

sam

plin

gco

ndi-

tion

isin

creasin

gly

viola

ted,th

eperfo

rmance

deg

rades

acco

rdin

gly.

Schem

eII:

success

pro

bability

1015

2025

3035

400

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9 1

gam

ma = 1, eta tilde = 1

gamm

a = 1, eta tilde = 1/2gam

ma = 1, eta tilde = 1/4

gamm

a = 20, eta tilde = 1gam

ma = 20, eta tilde = 1/2

gamm

a = 20, eta tilde = 1/4

Success

pro

babilities

forSch

eme

IIwith

γ=

1,2

0and

the

scatterin

gangle

conditio

nvio

lated

invario

us

deg

rees.

Com

pariso

nofSIM

Oand

SIS

O

1015

2025

3035

400

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9 1

S

IMO

gamm

a=1S

IMO

gamm

a=20S

IMO

gamm

a=200S

ISO

gamm

a=1

Solid

curves

areth

esu

ccesspro

babilities

forth

eSIM

Om

easu

remen

tat

γ=

1,2

0,2

00

and

the

dash

edcu

rveis

the

SIS

OSch

eme

IIat

γ=

1.

Distrib

ute

dexte

nded

targ

ets

•T

he

wavelet

expansio

n

ν(x

,z)=

∑

p,q∈Z

2

νp,q

ψp,q (x

,z)

where

ψp,q (r)

=2−(p

1 +p2 )/2

ψ(2−

pr−

q),

p,q∈

Z2

with

2−

pr=

(2−

p1x

,2−

p2z)

form

an

ONB

inL

2(R2).

•Littlew

ood-P

aley

basis

ψ(r)

=(π

2xz) −

1(sin(2

πx)−

sin(π

x))

·(sin(2

πz)−

sin(π

z))

which

isband-lim

ited

ψ(ξ,ζ)

=

(2

π) −

1.

π≤|ξ|,|ζ|≤

2π

0,

oth

erwise

.

•W

ithth

ein

ciden

tfield

s

uik (r)

=eiω

k r·dk,

k=

1,...,n

we

have

Yk=

2π

∑

p,q∈Z

2

2(p

1 +p2 )/2

νp,q

eiω

k 2p(d

k −rk )·q

ψ(ω

k 2p(r

k−

dk ))

with

cuto

ffs

|q|∞≤

mp,

|p|∞≤

p∗ ,|q′|∞

≤np′ ,

|p′|∞

≤p∗ .

•Let

l=

p1 −

1∑

j1 =−

p∗

p2 −

1∑

j2 =−

p∗ (2m

j +1)2+

(q1+

mp)(2

mp+

1)+

(q2+

mp+

1),

|q|∞≤

mp,

|p|∞≤

p∗ ,

k=

p ′1 −1

∑

j1 =−

p∗

p ′2 −1

∑

j2 =−

p∗ (2nj +

1)2+

(q ′1+

np′ )(2

np′ +

1)+

(q ′2+

np′ +

1),

|q′|∞

≤np′ ,

|p′|∞

≤p∗ .

Defi

ne

the

sensin

gm

atrix

elemen

tsto

be

Φk,l

=1

2np+

1ψ(ω

k 2p(r

k−

dk ))e

iωk 2

p(d

k −rk )·q

and

letΦ

=[Φ

k,l ],

where

dk ,r

k ,ωk

aregiven

belo

w.

Let

X=

(Xl )

withX

l=

2π(2

np+

1)2

(p1 +

p2 )/2

νp,q

be

the

target

vector.

•Sam

plin

gsch

eme:

Let

ξk ,ζ

kbe

indep

enden

t,unifo

rmra

ndom

variables

on

[−1,1

]and

defi

ne

αk=

π

ωk 2

p ′1·

1+

ξk ,

ξk∈

[0,1

]−1

+ξk ,

ξk∈

[−1,0

]

βk=

π

ωk 2

p ′2·

1+

ζk ,

ζk∈

[0,1

]−1

+ζk ,

ζk∈

[−1,0

].

Let

(ρk ,φ

k )be

the

polar

coord

inates

of

(αk ,β

k )used

todefi

ne

schem

esIand

II.

•Φ

k,l

arezero

ifp-=

p′.

Conseq

uen

tlyth

esen

sing

matrix

isth

eblo

ck-d

iagonalm

atrix

with

each

blo

ck(in

dexed

by

p=

p′)

inth

efo

rmofra

ndom

Fourier

matrix

Φk,l

=1

2np+

1eiπ

(q1ξk +

q2ζk ).

The

above

observa

tion

mea

nsth

atth

etarg

etstru

ctures

ofdiff

er-en

tdya

dic

scales

aredeco

upled

and

can

be

determ

ined

separa

telyby

ourappro

ach

usin

gco

mpressed

sensin

gtech

niq

ues.

Imagin

gofan

extended

target

ofth

esca

lesp∗

=0

with

m0

=32.

Localiz

ed

exte

nded

targ

ets

scatterers

scattered waveprobe wave

•In

terpola

tion

from

the

grid

ν) (r)

=) 2

∑q∈I

g( r)−

q)ν

()q),

I⊂

Z2.

Y=

ΦX

+E

where

Ein

cludes

the

discretiza

tion

error.

Theore

m9

(AF

2009)

Consid

erth

esa

mplin

gsch

emes

Iand

II(w

ithγ

=1).

Inadditio

nto

the

previo

us

assu

mptio

ns

assu

me

‖ν−

ν) ‖

1≤

2πε

‖g −

1‖L∞

([−π,π

] 2) .

Then

schem

esIand

IIsa

tisfyRIP

with

hig

hpro

bability

and

the

errorbound

‖X−

X‖2≤

C1s −

1/2‖

X−

X(s)‖

1+

C2ε.

Random

illum

inatio

n

objects0

A

z=0

z=z

sensors

•Rayleig

hreso

lutio

n:

A)

z0λ

=O

(1)

•Para

xialGreen

functio

nG

Gpar (r,a

)=

eiω

z0

4πz0eiω|x−

ξ| 2/(2

z0 )e

iω|y−

η| 2/(2

z0 ),

r=

(x,y

,z0 ),

a=

(ξ,η,0

)

•Random

illum

inatio

nui.

Assu

me

we

have

afu

llco

ntro

lof

the

source

poin

tsin

(x,y

,z):

x,y∈

[−L

/2,L

/2],z

=z1

and

write

the

incid

ent

wave

as

ui(r)

=∫

L/2

−L

/2

∫L

/2

−L

/2G

par (r,(ξ,η

,z1 ))f

(ξ,η)d

ξdη

•Let

the

source

distrib

utio

nf

be

aco

mplex-va

lued

,circu

larlysym

-m

etricGaussia

nwhite-n

oise

field

ofvaria

nce

κ2:

E[f

(ξ,η)f∗(ξ ′,η ′) ]

=κ2δ(ξ

−ξ ′,η

−η ′)

E[f

(ξ,η)f

(ξ ′,η ′) ]=

0,

∀ξ,ξ ′,η

,η ′.

•Fresn

eltra

nsfo

rmatio

nis

unitary

and

hen

ceuiis

also

aco

mplex-

valu

ed,circu

larlysym

metric

Gaussia

nra

ndom

field

.

The

random

incid

ent

field

takes

on

i.i.d.

random

valu

esat

grid

poin

ts.Sin

ceth

ein

ciden

tfield

hasth

esa

me

magnitu

de

thro

ugh-

out

the

object

pla

ne,

after

norm

aliza

tion

itseff

ectat

the

grid

poin

tsca

nbe

represen

tedby

aphase

facto

reiθ

j,j=

1,...,N

where

θj

arei.i.d

unifo

rmra

ndom

variables

in[0

,2π](i.e.

circu-

larlysym

metric).

•T

heore

m10

Suppose

aK√

2√

p+

2K

2

√np≤

a0

log

N

where

a=

max

j-=j ′ ∣∣∣∣ E

(eiξ

l ω(x

j ′ −x

j )/z0 )

E(eiη

l ω(y

j ′ −yj )/z

0 )∣∣∣∣ .

Assu

me

that

the

sobjects

arerea

l-valu

edand

satisfy

Xm

in>

8σ

√2lo

gN

and

s≤

c0np

2lo

gN

.

Then

the

Lasso

estimate

Xwith

γ=

2 √2lo

gN

has

the

sam

esu

pport

as

Xwith

pro

bability

at

least

1−

2δ−

ρn(n−

1)π2

√np−

1

N−

2n2p(p

−1)e −

N(n

p−1) 2

−2N−1((2

πlo

gN

) −1/2

+sN

−1)−O

(N−2lo

g2)).

The

superreso

lutio

neff

ectca

noccu

rwhen

the

num

ber

pofra

n-

dom

pro

bes

islarg

e.Consid

er,fo

rexa

mple,

the

case

of

n=

1and

hen

ceth

eapertu

reA

isessen

tially

zero.

Sin

cea≤

1,

the

conditio

n

K√

2+

2K

2

√p

≤a0

log

N

and

s≤

c0p

2lo

gN

implies

that

the

Lasso

with

γ=

2 √2lo

gN

recovers

exactly

the

support

of

sobjects.

Num

eric

alre

sults

with

RI

05

1015

2025

300

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9 1

R

andom illum

inationsM

R

The

Lasso

perfo

rmance

com

pariso

nbetw

eenRIwith

n=

11,p

=6

and

MR

with

n=

11.

The

verticalaxis

isfo

rth

esu

ccesspro

bability

and

the

horizo

nta

laxis

isfo

rth

enum

ber

of

objects.

The

success

pro

bability

isestim

ated

from

1000

indep

enden

ttria

ls.

1015

2025

3035

4045

0

100

200

300

400

500

600

700

R

andom illum

inations−exactR

andom illum

ination−paraxialM

R−exact

MR−paraxial

The

num

bers

ofreco

verable

(by

the

Lasso

)objects

forRIwith

p=

(n+

1)/2

and

MR

as

nvaries.

The

curves

indica

tea

quadra

ticbeh

avio

rpred

ictedby

the

theo

ry.T

he

diff

erence

betw

eenreco

verieswith

the

exact

and

para

xialGreen

functio

ns

isneg

ligib

lein

both

the

RIand

MR

set-ups.

0100

200300

400500

6000

100

200

300

400Subspace Pursuit

Exact G

reen functionParaxial G

reen function

0100

200300

400500

6000 5 10 15 20

OST

Exact G

reen functionParaxial G

reen function

The

num

ber

ofreco

verable

objects

inth

eunder-reso

lvedca

seas

afu

nctio

nof

the

num

ber

of

senso

rsn

=1,2

,3,4

,5,6,8,10,12,15,

20,24,25,30,40,50,60,75,100,120,150,200,300,600

with

np=

600

fixed

.

MUSIC

alg

orith

m

•D

efine

the

data

matrix

Y=

(Yk,l )∈

Cn×

mas

Yk,l ∼

A(s

k ,dl ),

k=

1,...,n

,l=

1,...,m

where

we

keepopen

the

optio

nof

norm

alizin

gY

inord

erto

simplify

the

set-up.

The

data

matrix

isrela

tedto

the

object

matrix

X=

dia

g(ξ

j )∈

Cs×

s,j=

1,...s

by

the

mea

surem

ent

matrices

Φand

Ψas

Y=

ΦX

Ψ∗

where

Φand

Ψare,

respectively,

Φk,j

=1√n

e −iω

sk ·r

j∈C

n×s

Ψl,j

=1√n

e −iω

dl ·r

j∈C

m×

s.

•T

he

standard

version

ofM

USIC

alg

orith

mdea

lswith

the

case

of

n=

mand

sk=

dk ,k

=1,...,n

as

stated

inth

efo

llowin

gresu

lt.

Pro

positio

n2

(Kirsch

02,08)

Let

sk=

dk ,k

∈N

be

aco

unt-

able

setofdirectio

ns

such

thatany

analytic

functio

non

the

unit

sphere

thatva

nish

esin

sk ,∀

k∈

Nva

nish

esid

entica

lly.Let

K⊂

R3

be

aco

mpact

subset

conta

inin

gS.

Then

there

existsn0

such

thatfo

rany

n≥

n0

the

follo

win

gch

aracteriza

tion

hold

sfo

revery

r∈K:

r∈S

ifand

only

ifφr≡

1√n(e −

iωs1 ·r,e −

iωs2 ·r,···

,e −iω

sn ·r)

T∈

Ran(Φ

).

Moreo

ver,th

era

nges

of

Φand

Yco

incid

e.

Rem

ark

1As

aco

nseq

uen

ce,r∈

Sif

and

only

ifP

φr

=0

where

Pis

the

orth

ogonalpro

jection

onto

the

null

space

of

Y∗

(Fred

holm

altern

ative).

And

the

loca

tions

ofth

esca

tterersca

nbe

iden

tified

by

the

singularities

ofth

eim

agin

gfu

nctio

n

J(r)

=1

|Pφr | 2

.

•T

heore

m11

Suppose

δs+

1<

1and‖E‖2

=ε.

The

thresh

old

ing

rule

then

the

object

support

Sca

nbe

iden

tified

by

the

thresh

old

ing

rule

r∈K

:J

ε(r)≥

2

(

1−

δs+

1 (1+

δs )

2+

δs −

δs+

1

)−2

under

the

follo

win

gbound

on

the

noise-to

-scatterer

ratio

(NSR)

ε

ξmin

<

√√√√(1

+δs )

2ξ2m

ax

ξ2m

in+

(1−

δs )

2∆−

(1+

δs )

ξmax

ξmin

where∆

=m

in

ν∗

((1

+δs )

2

(1−

δs )

2ξ2m

ax

ξ2m

in

)

,1

5 √2

(

1−

δs+

1 (1+

δs )

2+

δs −

δs+

1

)

ν∗ (x)

=−2x−

1+

√(2

x+

1)2+

16

16

and

ξmax /

ξmin

isth

edyn

am

icra

nge

ofsca

tterers.

MUSIC

simula

tions

1015

2025

300 50

100

150

200

250

300

BPM

USIC

10

15

20

25

30

5

10

15

20

25

30

A=

10

BP

MU

SIC

Com

pariso

nof

MUSIC

and

BP

perfo

rmances,

with

both

usin

gth

ewhole

data

matrix:

the

num

ber

sof

recovera

ble

scatterers

versus

the

num

ber

ofsen

sors

nwith

A=

100

(left),th

ewell-reso

lvedca

se,and

A=

10

(right),

the

under-reso

lvedca

se.In

the

well-reso

lvedca

se,BP

delivers

am

uch

better

(quadra

tic-in-n

)perfo

rmance

than

MUSIC

;in

the

under-reso

lvedca

se,M

USIC

outp

erform

sBP

whose

perfo

rmance

tendsto

be

unsta

ble

inth

isreg

ime.

The

num

bers

ofre-

covera

ble

scatterers

by

BP

areca

lcula

tedbased

on

successfu

lreco

v-ery

ofatlea

st90

outof100

indep

enden

trea

lizatio

nsoftra

nsceivers

and

scatterers

while

the

success

rate

ofM

USIC

is100%

.

1015

2025

300 5 10 15 20 25 30

A=100

BPM

USIC

150160

170180

190200

60 80

100

120

140

160

180

200A=100

BPM

USIC

Com

pariso

nof

MUSIC

and

BP

perfo

rmances

with

BP

emplo

ying

only

single

colu

mn

ofth

edata

matrix:

the

num

ber

sofreco

verable

scatterers

versus

the

num

ber

of

senso

rsn

with

A=

100

for

n∈

[10,3

0]

(left)and

n∈

[150,2

00]

(right).

Both

BP

curves

show

aro

ughly

linear

beh

avio

rwith

slope

lessth

an

that

of

the

MUSIC

curves.

0.50.6

0.70.8

0.91

1.11.2

1.30.75

0.8

0.85

0.9

0.95 1n=10,s=9

0.20.22

0.240.26

0.280.3

0.320.34

0.75

0.8

0.85

0.9

0.95 1n=100,s=9

15.516

16.517

17.518

0.84

0.86

0.88

0.9

0.92

0.94

0.96

0.98 1n=100,s=99

Success

pro

bability

ofth

eM

USIC

reconstru

ction

versus

apertu

refo

rn

=10,s

=9

(left),n

=100,s

=9

(mid

dle)

and

n=

100,s

=99

(right).

Note

the

diff

erent

apertu

rera

nges

forth

eth

reeplo

ts.T

he

success

rate

isca

lcula

tedfro

m1000

trials.

Increa

sing

the

num

ber

of

transceivers

forth

esa

me

num

ber

ofsca

tterersred

uces

the

apertu

rereq

uired

for

the

sam

esu

ccessra

te.T

he

reductio

nof

apertu

reis

aboutth

reefo

lds(left

tom

iddle).

On

the

oth

erhand,hig

her

num

ber

of

scatterers

with

the

sam

enum

ber

of

transceivers

also

dem

ands

larger

apertu

refo

rth

esa

me

success

rate.

The

increa

sein

apertu

reis

about

7tim

es(m

iddle

torig

ht).

1011

1213

1415

0.75

0.8

0.85

0.9

0.95 1

050

100150

200250

300350

4000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

100101

102103

104105

1060.8

0.82

0.84

0.86

0.88

0.9

0.92

0.94

0.96

0.98 1

Success

pro

bability

ofM

USIC

versusth

enum

ber

oftra

nsceivers

with

A=

0.5

,s=

9(left),

A=

0.2

,s=

9(m

iddle)

and

A=

15,s

=99

(right).

The

pro

babilities

areca

lcula

tedfro

m1000

indep

enden

ttria

ls.

11.5

22.5

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9 1

A=100

!

00.05

0.1

0.15

0.2

0.25

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

A=10

!

Success

pro

bability

of

MUSIC

reconstru

ction

of

s=

10

scatterers

with

n=

100

transceivers

versusth

enoise

levelσ

inth

ewell-reso

lvedca

seA

=100

(left)and

the

under-reso

lvedca

seA

=10

(right).

The

success

rate

isca

lcula

tedfro

m1000

trials.

Note

the

diff

erentsca

lesof

σin

the

two

plo

ts.Noise

sensitivity

increa

sesdra

matica

llyin

the

under-reso

lvedca

se.

100120

140160

180200

2200.65

0.7

0.75

0.8

0.85

0.9

0.95 1A=100,s=10,!=1.5

100200

300400

500600

700800

9001000

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

Success

pro

bability

ofM

USIC

reconstru

ction

of

s=

10

scatterers

as

afu

nctio

nof

nwith

σ=

150%

inth

ewell-reso

lvedca

seA

=100

(left)and

σ=

5%

inth

eunder-reso

lvedca

seA

=10

(right).

The

success

rate

reach

esth

epla

teau

of85%

near

n=

1000

inth

eunder-

resolved

case.

The

success

rate

isca

lcula

tedfro

m1000

trials.

refe

rences

•A.F.:

Com

pressive

inverse

scatterin

gII.

Multi-sh

ot

SIS

Om

ea-

surem

ents

with

Born

scatterers

Inverse

Pro

blem

s26

(2010),

035009

•A.F.:

Com

pressive

inverse

scatterin

gI.

hig

h-freq

uen

cySIM

O/M

ISO

and

MIM

Om

easu

remen

tsIn

versePro

blem

s26

(2010),

035008

•A.

F.,

P.

Yan

and

T.

Stro

hm

er:Com

pressed

Rem

ote

Sen

sing

ofSparse

Object.

SIA

MJ.Im

agin

gSci.

Volu

me

3,Issu

e3,pp.

595-6

18

(2010).

•A.

F.:

Exa

ctLoca

lizatio

nand

Superreso

lutio

nw

ithnoisy

data

and

random

illum

inatio

narXiv:1008.3146

•A.F.:

The

MUSIC

alg

orith

mfo

rsp

arseobjects:

aco

mpressed

sensin

ganalysis

arXiv:1006.1678

Conclu

sions

•In

versesca

ttering

inth

efra

mew

ork

ofco

mpressed

sensin

g.

•Random

incid

ent

and

scatterin

gdirectio

ns

•Random

illum

inatio

n

•Superreso

lutio

n

THANK

YO

U!