IMAGE
RESTORATION
USING
HOS
AND
THE
RADON
TRANSFORM
Elisa
Say ol*,
Ch ysos omos
L.
Nikias*,
and
Toni
Gad**
Signal
and
Image P ocessing Ins i u e
Uni e si y o Sou he n Cali o nia
Los
Angeles
CA
90089-2564
el:
(213) 7404654,
ax:
(213) 7404651
email:
[email p o ec ed]
*Depa men o Signal
~beo y
and
~ommunica im.
Uni e si a Polibica
de
Ca alunya
08080
Ba celona,
Spain
*
i
El3400
Apdo.
3O002
ABSTRACT^
We p opose he use
o
highe -o de s a is ics
(HOS)
o
s udy he p oblem
o
image es o a ion. We conside
images deg aded by linea
o
ze o phase blu ing poin
sp ead i ne ions (PSF) and addi i e Gaussian noise.
The
complexi y associa ed wi h he combina ion
o
wo-
dimensional signal p ocessing and highe -o de
s a is ics is educed by means
o
he
Radon
T ans o m
The
p ojec ion a each angle is an one-dimensional
signal ha can
be
p ocessed by any exis ing 1-D highe -
o de s a is ics-based me hod. We apply wo me hods
ha ha e p o en o a ain good one-dimensional signal
econs uc ion, especially in he p esence
o
noise.
A e he ideal p ojec ions ha e been es ima ed, he
In e se Radon T ans o m gi es he es o ed image.
Simula ion esul s a e p o ided.
1
INTRODUCTION
Highe -O de s a is ics ha e been success ully applied
be o e
U,
he p oblem o
2-D
signal econs uc ion. In
[l]
he phase o he bispec um is used o econs uc
images
deg aded by
a
ji e y channel wi h addi i e one
in e e ence. In
[2]
he bispec um is used
o
es ima e a
andomly ansla ing and o a ing objec om
a
sequence
o noisy images. Ano he applica ion is desc ibed in
[3]
whe e in a ian Highe -O de Spec a ea u es
o
p ojec ions
a e
used
o objec classi ica ion.
The Bispec um
B lwl,w2)
o a one-dimensional,
de e minis ic. disc e e- ime signal
in),
is
de ined
as
he
Fou ie T ans o m o i s iple co ela ion unc ion and
is
gi en by
[41 151
whe e
F(w)
is
he Fou ie T ans o m
o
in).
The
phase
o he Bispec um is he e o e
his
wo k
was
suppo ed
by
he
O ice
o
Na al Resea ch unde con ac
N0oO14-92-J-1034
and he Spanish Minis y
o
Educa ion and Science.
dBpwpw2)
=
NWl)
+
W2)
-
Nq+w2)Q)
whe e
#(w)=.CF(w).
The Bispec a, and HOS in
gene al, a e insensi i e
o
addi i e Gaussian
noise
and
p ese e he Fou ie phase o he signal up
o
a
linea
phase ac o .
Ou
goal is
o
exploi such p ope ies o
he "a ion o
2-D
blu ed
and
noisy images.
We ha e de eloped echniques ha conside
wo
HOS-based me hods ha ha e p o en
o
a ain
good
1-D
signal econs uc ion. The m one is he Bispec um
I e a i e Recons uc ion Algo i hm
@IRA)
desc ibed
in
[6]
ha was u ilized o signal econs uc ion om he
phase o he bispec um. Al hough i was applied o
es o a ion o images, he p ocessing
was
made
on
a
line
by line
basis.
When
he
image
is
a ec ed by
a
ocus blu
he dis o ing il e is
2-D
and he econs uc ion o
lines
does no lead o good
esul s.
This me hod canno be
ex ended o he wo dimensional
case
since
i
is
based
on
he ceps al coe icien s ha
can
no
be
de ined in
2-D.
Addi ionally,
BIRA
does no pe o m well
when
he
Z-
ans o m o he signal has ze os
on
he uni ci cle,
unless
an
exponen ial window
o
he o m
an
mo es
he
ze os ou wa ds o inwa ds. An adequa e pa ame e
U
is
usually di icul o ind. Fo una ely,
as
we will
see,
we
can o e come bo h p oblems by econs uc ing om
p ojec ions. The second app oach
we
s udy is
he
Weigh
Slice
me hod
(WS)
[7]
ha eco e s he signal o in em
e en in he case o ze os
on
he uni ci cle. Howe e ,
al hough possible, i s ex ension
o
2-D
will
inc ease
eno mously i s analy ical complexi y. We p esen
an
algo i hm using he WS me hod o e he p ojec ions
o
he image.
In Sec ion
2
we desc ibe he imaging sys em ha
we
assume. We e iew he p ojec ion heo em ha allows
he es o a ion o images om hei p ojec ions. In
Sec ion
3
we de elop wo di e en me hods o he
es o a ion ei he om he
phase
o he Bispec um
o
om he phase
o
he Fou ie T ans o m.
In
Sec ion
4
some examples
a e
gi en. Finally, Sec ion
5
is de o ed
o conclusions and ema ks.
0-7803-1238493 $3.00
0
1993
IEEE
76
..
.
.
I_-.
2
PRELIMINARY DEFINITIONS
To add ess he p oblem o eco e ing images om
deg aded e sions o i ,
we
assume a linea imaging
sys em wi h space in a ian poin sp ead unc ion and
addi i e Gaussian noise. Thus. o he con inuos model
(3)
whe e g(x,y) is he obse ed image, h(x,y) he poin
sp ead unc ion, (sy) he o iginal image and n(x,y) he
addi i e Gaussian noise.
The 2-D signal eco e y p oblem can be uniquely
decomposed in o many l-D signal econs uc ion
p oblems. The Radon ans o m o a 2-D unc ion, lx,y),
deno ed pe(s), is de ined
as
i s line in eg al along a line
inclined a
an
angle
8
om he y-axis and a a dis ance
s
om he o igin.
A
undamen al ela ionship, he
P ojec ion Theo em, ela es he l-D Fou ie ans o m o
he p ojec ion
pds),
wi h he cen al slice, a angle
e,
o
he 2-D Fou ie ans o m o he objec l y)
gp&)
=
hp&)
* ps(S)+Fl,S+G@ )
=
H(e, F(6@
(4)
whe e
gpds).
hpe(s),
pds)
a e he p ojec ions o g(x,y),
h(x,y), (x,y) a angle
8,
and
G(e, ),
H(e. ),
F(8, )
a e
he Fou ie ans o ms in pola coo dina es o g(x,y),
h(x,y), and lx,y), espec i ely.
We conside ha he Op ical T ans e Func ion
(OF),
he Fou ie ans o m o he PSF, has linea o ze o
phase.
As
we
see
om
(4).
when he phase o he
OTF
is
ze o, he phase o
he
l-D Fou ie ans o m o each
p ojec ion G(O, ), co esponds o he phase o he
Fou ie ans o m o he p ojec ion o he o iginal image,
In
mos applica ions, blu deg ada ions due o a numbe
o
sou ces
a e
conside ed o ha e linea o ze o phase. Fo
example in
[8]
models o eal ou -o - ocus images a e
in es iga ed. I is shown ha he phase is ei he ze o o
linea . Fu he mo e, a unique disc e e ep esen a ion
esul s in a PSF suppo ha co e s he minimum
numbe o pixels and whe e he
OTF
has ze o phase.
I is well known ha he disc e iza ion o he in e se
o mula o he Radon T ans o m is no easily de ined.
Finding accu a e and as algo i hms is a cu en opic o
esea ch. On he o he hand, i he PSF suppo is no an
in ege numbe , in e ms o numbe o pixels, some
dis o ion occu s. Despi e hese p oblems, es o a ion
om p ojec ions is s ill easible.
F(e.5).
3
RESTORATION FROM THE PHASE OF
THE PROJECTIONS
I was shown in
[9]
ha a sequence o leng h
N
can
be
uniquely speci ied i he phase unc ion
is
known
a
(&I)
dis inc equencies in he in e al
O<w<z,
excep o a
cons an ac o . Likewise in
[6],
i is shown ha a
sequence is uniquely speci ied
i€
he phase o he
bispecuum is known a
N{N-l)n
dis inc equencies in
he egion
O<WI+W~<
z,
W~<WI,
wl 0,
excep o
a
cons an ac o and a linea shi . This
is
ue unde he
assump ions ha he Z- ans o m o he signal has no
ze os in ecip ocal pai s.
The e o e, i he p ojec ion o he blu ing il e
has
leng h
da
and he blu ed p ojec ion
has
leng h
Lb
he e
is
a
unique signal o leng h
Lbe
=
Le
-
(de
-1)
wi h he
same phase unc ion (o phase bispec um) han he
blu ed p ojec ion. This signal
is
no hing bu he
p ojec ion o he o iginal image.
We nex desc ibe wo HOS-based me hods ha
can
be
used o he econs uc ion o he p ojec ions. Fo mo e
de ails on he algo i hms
see
161
and
171
3.1 Recons uc ion
wi h
BIRA
As
we men ioned be o e, BIRA econs uc s
a
signal
om he phase o i s
bispecuum.
I employs he
ceps al
coe icien s ha o a signal pe(n) a e
de ined
as
L,
whe e aipa
bipe
a e
he ze os inside and ou side
he
uni
ci cle espec i ely, and
Cipe
a e he poles inside he uni
ci cle o he Z- ans o m o pe(n). The las e m is no
included o ini e sequences
(L2
=O).
The ceps al
coe icien s a e ela ed o
he
phase o he bispec um
as
and
o
he powe ceps um
as
whe e
Aipe
is a cons an . I s ela ion
o
he signal is
pe(n)
=
F1/eF/C P~m)l),
n=O,..,N-1
(9)
77
Wh
C
m)
=
P
e(
-l/mA
e(")
D0
op
m=O
l/mB
8-m)
md)
P
is he ceps um o he signal.
The algo i hm
s a s
om he ue sequence o he phase
o he bispec um o a leas
N(N-1)/2
samples and
an
ini ial guess o he powe ceps um. I con e ges
o
a
unique solu ion o he ceps al coe icien s and
consequen ly o he signal, excep o
a
cons an ac o
and a space shi .
An
addi ional p ocedu e is added
o
conml he space shi . The scale ac o is calcula ed
aking in o accoun ha he p ojec ion is a posi i e
unc ion and ha
all
he p ojec ions ha e he same a ea,
he olume o
he
image.
As
we
see
in
(5)
and
(6)
in
he case ha he e a e ze os
on
he
uni ci cle, he ceps al coe icien s a e in ini e
sequences. I s unca ion leads o inco ec solu ions.
Two di e en al e na i es a e p oposed when p ojec ions
wi h ze os on he uni ci cle
a e
de ec ed.
A)
When he numbe o such p ojec ions
is
low
:
In his case we es o e he image om he es o he
p ojec ions since he omission o ew p ojec ions does
no al e he es o a ion signi ican ly.
B)
When he numbe o such p ojec ions is high
:
We
y
o
iden i y he coe icien s o he blu ing il e
using he ollowing p ocedu e:
B1)
Recons uc he p ojec ions ha ha e nei he ze os
on he uni ci cle no ecip ocal pai s.
B2)
F om (4) we
see
ha i we know gpB(s) and pe(s)
we
can
ob ain
hpHs)
by decon olu ion. The sequence
is
he p ojec ion o he blu ing il e a
an
angle Band i s
coe icien s a e weigh ed sums o
he
coe icien s o he
blu ing il e . We build a sys em o equa ions o he
ollowing o m
xa=y,
(11)
whe e
X
is
he
ma ix
o he weigh s
o
he coe icien s a
he
speci ied angles,
a
is he coe icien ec o and
y
is
he ec o o he es ima ed blu ing il e p ojec ed a he
speci ied angles.
As
an
example suppose we ha e only
he p ojec ion a angle
B
=
0"and a squa e 3x3 PSF wi h
coe icien s named om
a
o
i.
Then, he sys em o
equa ions will be
111000000
000111000
000000111
[
.I
a
b
C
d
c
8
h
i
.I
a+b+c
d+c+
In gene al, we can sol e he
sys em
o
equa ions
b
ind
he Coe icien ec o i
he
ank
o he ma ix is
equal
o g ea e han
Nc.
Using a
LS
app oxima ion he
solu ion will be
21
=
(XHX)-'
XH
y
(12)
B3)
Once we know he blu ing il e coe icien s we
can
ei he decon ol e he 2-D noise- ee image o he Id
noise ee p ojec ions.
3.2
Recons uc ion wi h he
WS
me hod
We
can
uniquely econs uc he p ojec ions om he
phase only o he Fou ie ans o m e en in
he
case
ze os lie on he
uni
ci cle [9]. Howe e , in
he
p esence
o noise he phase
is
deg aded. The e o e, we decompose
he econs uc ion o a blu ed noisy p ojec ion
in
wo
s eps. In he i s s ep, a me hod based on
HOS
educes
Gaussian noise p ese ing he phase unc ion. We choose
he WS me hod because i wo ks when hen?
a e
oo s
on
he uni ci cle and i
has
be e pe o mance han o he
me hods. In he second s ep we deblu he signal om i s
Ph.
S ep
1)
I is known ha o a causal and exponen ially
s able sys em wi h inpu assumed o
be
independen ,
iden ically dis ibu ed and non Gaussian, wi h skewness
p.
he ou pu bispec um
BX(wl,w2)
exis s and
is
gi en
by [31, [dl
BX(wpw$
=
B
H(w1)
WW~)
H*(wl +w2)
(13)
As
we
see,
inding he coe icien s o he il e
H(w)
is
equi alen
o
es ima e he sequence
F(w)
in (1).
The Weigh Slice WS) algo i hm
has
been p e iously
used o ob ain he pa ame e s o a non-minimum
FIR
sys em. The pa ame e s can
be
exp essed
as
a
linea
combina ion o cumulan slices. I was shown ha
i
he e exis s a se o weigh s ha gi es
a
causal slice,
hen he
FIR
sys em can be iden i ied. The sys em o
equa ions
is
exp essed
as
SW
=
bo (14)
whe e
S
is
he ma ix o cumulan s
s
1
C,is
he es ima ed
n h
o de cumulan o he ou pu
signal,
w
is
he
weigh ec o
w
=
(
w2, w3cj), w4(j,k),
...
)
(16)
and
bo
is
he coe icien ec o
bo
=
(0
...
0
I
...
b(q-I) b(9)).
(17)
The unknowns a e he ec o
w,
and he las q elemen s
The ma ix equa ion is sol ed in
wo
s eps
:
1) Compu a ion o he minimum no m weigh s ha gi e
a
causal
W-slice
o
bo.
su
wm
=
(0,
..
,
0,I)
wm
=
SU#
1
whe e
Su
is he
ma ix
o med om he uppe
q+l
ows
o
S,
and
Su#
deno es he pseudo in e se o
Sua
2)
Compu a ion o he coe icien s
as
bo=Swm=SSu#l
S ep
2)
The ec o
bo
is he sequence o he noise-
educed blu ed p ojec ion. Once
bo
has been calcula ed,
we
deblu he signal using a simila algo i hm han he
one desc ibed in
[9].
I is equi alen o BIRA, i
s a s
wi h he
Vue
sequence o he phase o he Fou ie
T ans o m and an ini ial guess o i s magni ude.
4
EXAMPLES
In he nex examples, he Radon T ans o m
is
calcula ed
om an in e pola ion
o
he Ca esian sampling g id o
he
2-D
FFT
o he image o a pola g id. The In e se
Radon
T ans o m
is
implemen ed in a e e se way.
Fig. la shows a 24x24 image. The econs uc ed image
a e
applying Radon T ans o m ollowed by he In e se
Radon ans o m is shown on Fig. Id. The dimensions o
he
2-D
FIT
and
2-D
IFFT
a e
64x64
o
a oid dis o ion
in he
high
equencies. Fig. lb p esen s a blu ed
e sion o his image when a
3x3
Gaussian il e wi h
pa ame e
a
=
0.4
is used. Fig.
le
shows he es o ed
image om he Fou ie phase o he p ojec ions. In Fig.
IC
he
image is blumd wi h he same il e and Gaussian
noise o
SNR=25
dbs is added. The es o ed image
is
shown in Fig.
I
whe e he Ws me hod
is
applied using
one slice only, he hi d o de cumulan
is
es ima ed
hm
50
ealiza ions. Ano he example
is
shown
in Fig.
2a
and
Fig.
24
whe e he o iginal image
is
64x64
and he
2-D
FFTs
a e
256x128.
Fig.
2b
is
a
blu ed e sion
o
his
image wi h a 5x5 Gaussian il e and
a
=
0.4.
Fig.
2c
is
he es o ed image om he Fou ie phase. F om his
image
we
see
ha
high equencies
ha e
been eco e ed.
Howe e , we obse e some dis o ion. This is due
o
ela i e shi ing among p ojec ions. Fig.
3
shows
a
ypical p ojec ion, i s blu ed e sion, he econs uc ed
signal and he di e ence be ween
he
o iginal and he
es o ed signal.
5
CONCLUSIONS AND REMARKS
In his pape ,
we
ha e
shown
ha
i is possible
o
es o e
a blu ed and noisy image om
HOS
o
i s
p ojec ions.
Two di e en app oaches we e gi en
o
econs uc
he
1-
D
signals om ei he he phase o he Fou ie T ans o m
o he phase o he bispec um. Al hough he complexi y
using
HOS
o e
2-D
signals is educed, he
compu a ional load is s ill
high.
Some o he applica ions
o
HOS
o e he p ojec ions
a e
unde in es iga ion.
REFERENCES
[l]
S.A.
Diana , M. R. Raghu ee . “Fas Algo i hms
o phase and magni ude econs uc ion om
bispec a”,
Op ical Enginee ing, May 1990.
[2]
B.M.Sadle , “Shi and Ro a ion In a ian Objec
Recons uc ion using Bispec um”,
Wo kshop
on
HOS
analysis, Vail, Colo ado, I989,pag
106
[3]
V. Chand an,
S.
Elga , “Posi ion. o a ion, and scale
in a ian ecogni ion o images using Highe -o de
spec a”,
P oc.
ICASSP ‘92,
S.
F.,pp -213-216.
[4]
C.
L. Nikias,
M.
R. Raghu ee ,”Bispec um
es ima ion
:
A Digi al Signal P ocessing
F amewo k“,
P oc.
o
IEEE,
V-7.5,
No.
7, July
87.
151
J.M.
Mendel, “Tu o ial on Highe -O de S a is ics
(Spec a) in signal p ocessing and sys em heo y:
Theo e ical esul s and some applica ions”,
P oc.
o
IEEE,
Vo179, No.
3,
No . 1991.
163
A.
Pe opulu,
C.
L.
Nikias, “Signal Recons uc ion
om he phase o he Bispec um”,
ZEEE
T ansac ions
on
Signal P ocessing,
Vol.
40,
No.
3,
pug
601.
[A
J.A.
Fonollosa,
J.
Vidal, “Sys em Iden i ica ion
using a linea combina ion o cumulan slices”,
submi ed
o
IEEE
T ansac ions
on
Signal
P ocessing.
181
M.I.
Sezan,
G.
Pa lo ic,
M.,
Tekalp, “On modeling
he ocus blu on Image es o a ion”,
P oc. ICASSP
‘92, Albuque que.NM,
pp
248.5-2488.
[9]
M.
H. Hayes,
J.
S.
Lim and A.
V.
Oppenheim.
“Signal Recons uc ion om
he
phase o
magni ude”,
IEEE
on
ASSSP,
Vol
28,
No.
6,
Dec
1980.
79
Fig.
la
....................................
..
Fig. lb Fig.
lc
ig. Id Fig.
le
Fig. I
Fig.
2a
Fig. 2b Fig.
2c
e)
RcmmWcIed
om
b
wFl
0
......
...
-50
.
02l4060~
Fig. 2d
80
Fig.3