ressive - math.ncku.edu.tw
Transcript of ressive - math.ncku.edu.tw
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.
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.
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