IMAGE RECONSTRUCTION FROM TRUNCATED PROJECTIONS :
A LINEAR PREDICTION APPROACH
N. Srinivasa*
*
V. Krishnan'
K.R. Ramakrishnan
Dept. of Elect. Engg., Indian Institute
x
K. Rajgopal
*
of Science, Bangalore, INDIA.
"School of Automation, Indian Institute of Science, Bangalore, INDIA.
'Dept. of Elec. Engg., University of Lowell,Lowell,
Abstract
Unambiguous
reconstruction
is
not
possible
with
truncated
projections.
In order to r e d u c e
t haem b i g u i titynhree c o n s t r u c t eidm a g es ,e v e r a l
a simple
'completion'
a u t h o rhsa vseu g g e s t etdh a t
involving
extrapolation
of t threu n c a t pe dr o j e c t ion
sufficient.
is
In this
paper
method
a based
Prediction
approach
proposed
is
for
on a Linear
o b t a i n i n gt h em i s s i n gp a r ti nt h ep r o j e c t i o n .R e c o n s t ruction
then
is carried
out
using
the
Convolution
111. Simulationresults
Backprojection(CBP)method
s h o w i n g t h e e f f e c t i v e n e s s of this method in extracting
significantly more information from truncated projections are presented.
Introduction
Image
reconstruction
from
projection
relies
onthe
fact t h a t h eo b j e c ft u n c t i o ni sr e p r e s e n t e d
c o m p l e t e l y by a set of lineintegrals so t h a t a knowledge of the projections is sufficient to unambiguously
d e t e r m i ntehset r u c t u r e
of t hoeb j e c t .
In medical
imaging,
projections
physically
correspond
the
to
l i n ea at tre n u a t i o n
of X-rays,
time-of-flight
the
of aunl t r a s o n ipcu l soetrhteo t arla d iaoc t i v i t y
isotope.
in aenl e m e n t av lo l u m e
of a distributed
O t h ea rp p l i c a t i oanr e awsh e ri m
e a g ea sr e c o n s t r u c tfer dop m
r o j e c t i oaN
nr sue c l eMa ra g n e t i c
Resonance,
electron
microscopy,
radio
astronomy,
etc.,.
In practice,therearemanysituationswherein
is t r u n c a t e (da l s roe f e r r e d
the
available
projection
to as 'limited
field
of v i e wa' n d' r e s t r i c t e dr e g i o n
F oirn s t a n c et ,hper o j e c t i o nasrnee c e scan') [ 2 ] .
s s a r i l tyr u n c a t e d
if the
object
being
examined
has
d i m e n s i o nl as r g et hr at nhf ei e l d
of view of t h e
as in t h e case of G a m m ac a m e r a s
imagingsystem,
nuclear
in medicine.
In 'clinical
situations
where
t h ed i s e a s e da r e ai sa l r e a d yl o c a l i z e df r o m
a previous
a followupscan
of t h eR e g i o n
Of Interest
studyand
a r e s t r i c t e rde g i o snc a inds o n e
(ROI)
is
required,
to minimize
the
radiation
dosage.
Here
order
in
as t h e
projection
data
intentionally
is
truncated
to radiation.
In
external
region
not
issubjected
electronmicroscopy,it
is knownthatinnegatively
of certain
biological
objects,
stained
preparations
the
overall
distribution
of stain
does
not
preserve
t h sey m m e t r i eostfh oe b j e c te, s p e c i a l l yt h oe u t e r
p a r t s of theprojections,thusresultingintruncated
cases, reconstruction
has
projections.
In atlhl e s e
Mass., USA.
to b e a t t e m p t e d w i t h t h e a v a i l a b l e t r u n c a t e d p r o j e c t ions.
Algorithms
based
Radon
the
on inversion
as the
CBP
when
directly
used
with
formula
such
truncated
projections
does
not
properly
reconstruct
of view.
t hoeb j e cf ut n c t i oonu t s i dtehfei e l d
a d d i t i o na ,r t e f a c et sx t e nidn t ohfei e lovdfi e w
[3,4,5]. Thus
indirect
methods
are being
devised
so t h a t a f a i r reer c o n s t r u c t i ocnaboneb t a i n e d
f rom truncated projections.
In
as Algebraic ReconstructI t e r a t i v em e t h o d ss u c h
(ART) and
Simultaneous
Iterative
ion
Technique
Reconstruction
Technique
(SIRT)
have
been
investigated
for
reconstruction
from
truncated
projections
[6].
major
A dis-advantage
of t haipsp r o aicsh
of c o m p u t e r
t hrai ett q u i r ecso n s i d e r a b laem o u n t
time.
Several
authors
have
shown
that
reasonable
a c c u r aitm
e a gw
e siltehs saer t e f a cct os usl dt i l l
CBP method of reconstruction
b eo b t a i n e df r o mt h e
[3,4,51.
u s'icnogm p l e tter du 'n c apt er od j e c t i o n s
The
completion
of projection
in
[3] is a c h i e v e d
byestimatingthemissingdatabyfittingpolynomial
f u n c t i o ntso
a few
end
points
on
either
side
of
thetruncatedprojection.
. In [ 5 ] , t h ec r o s ss e c t i o n a l
outline of theobjectfunctionisobtainedbyoptical
is approxm e t h o d sa n dt h em i s s i n gp r o j e c t i o nd a t a
i m a t e d as the p r o d u c t of the path length and a
meanvalue of t h eo b j e c tf u n c t i o nw h i c hi se s t i m a t e d
from
the
available
data.
This
paper
presents
a
method for c o m p l e t ittnhrgue n c a tperdo j e c t i o n
by enabling
using
Linear
Prediction
theory,
there
to
algorithms
based
Radon
on inversion
formula
e x t r a csti g n i f i c a n t l m
y o rien f o r m a t i o fnr o m
incomplete projections.
Projection completion by Linear Prediction
Linearpredictormodelshavebeendeveloped
for many
signals
for
different
applications
such
as prediction or forecasting, control, data compression
and spectral e s t i m a t i o nO. n e
o i the most widely
a truncated
investigated
model
for
extrapolating
a s p e c t r adle n s i tfyu n c t i o n
signal
or
interpolating
is the
Autoregressive
(AR) model.
Fitting
an
AR
model to t ha ve a i l a bol end ei m e n s i o ndaalitsa
equivalent to the
Maximum
Entropy
Method
[71.
M a x i m uemn t r o p yr o c e s s i ne gn s u rtehsfaetw e s t
possibleassumptionsaremadeabouttheunmeasured
for modelling
the
d a tTathm
.ihiseso t i v a t i o n
as an o u t p u t of a linearpredictor.
p r o j e c t i o nd a t a
34.3.1
ICASSP 86, TOKYO
CH2243-4/86/0000-1733
$1.00
0 1986 IEEE
1733
SimulationResults
L e t g(x,y) r e p r e s e n t a t w od i m e n s i o n a ol b j e c t
a c i r c l e of radius
function
which
zero
is outside
T hRe a d otnr a n s f o r m
[ R g(]8 , tit)sh e
b (Fig. 1).
:RB (Fig.1).
lineintegral of t h e f u n c t i o n a l o n g t h e l i n e
w h e r ed si st h e l e m e n t a dl i s t a n c eo nt h el i n eA B
representedbytheequation
x cos O+ y sin0 = t
Theprojection
[Rg] (0 ,t)is
a set of l i n ei n t e g r a l s
to t h e
evaluated
along
parallel
lines
perpendicular
0 withthex-axis
as
vectorwhichmakesanangle
showninFig.
1.
.
Consider a d a t m
a atrix
P w h e r ea ne l e m e n t
p(i,j)
represents
a line
integral
[Rg]
(i,j)
and
each
a c o m p l e tper o j e c t i o n
at aann g l e
r o rwe p r e s e n t s
as missing
A
i.
truncatedprojectionmanifestsitself
c o l u mtinhdnseam
t aa t r iFxi. g .
2 d e p i ctths e
of v i e ws i t u a t i o nw h e r teh R
e O(Ir < a
limited
field
0
'
s 0 < 179".
inFig. I ) is projected from all directions
E a c h of t haev a i l a b lter u n c a t epdr o j e c t i o n ,
a row
as a A
nR
process
i tnhde a tm
a a t r i xim
,s o d e l l e d
N.
p(i,j)
expressed
is
approximately
as
order
of
a linear combination of the 'past' values.
A . .
where
p(l,l)
.
IS
an
approximation
of p(i,j)
and
aN(k)
of o r d e r N. T he r r oor r
a r teh A
e Rc o e f f i c i e n t s
r e s i d u able t w etehaec R
t u a.vl.a l u. e .
of
p(i,j)
and
the
predicted value p(1,~) IS glven by
A .
e(j) = p(i,j) - p ( l , ~ )=
N
1
a#)
p(i,j-k)
k=O
whereaN(0)
= 1.
The
AR
parameters
are
obtained
minimising
by
to
t hme e aotnor t as lq u a r eedr r owr i trhe s p e c t
e a c h of tphaer a m e t e V
r sa. r i oaulsg o r i t h m
a rse
[81.
Each
available t o o b t a itnhAecRo e f f i c i e n t s
projection
then
is completed
by e x t r a p o l a t i nt gh e
truncated
projection
forwards
and
backwards
using
of
t hdee r i v eldi n e aprr e d i c t oTr .hfei n i teex t e n t
the
object
which
may
be
known
apriori
or
can
be
to determine
determined
experimentally
used
is
t h ne u m b e r
of points t o be x t r a p o l a t e od eni t h e r
side.
If d a t a points in t he ex t r a p o l a t esde g m e n t
set t o zero.Thecompleted
a r ne e g a t i v et,h e ya r e
projection
thus
consists
of the
original
truncated
p r o j e cetaxihnotedrna p o l asteegdm e nTthse.
completion of t h e d a t a m a t r i x i s t h u s a c h i e v e d w i t h o u t
as t h eo b j e c ft u n c t i o n
makinganyassumptionssuch
or c o n s t a nftotrhaer eoau t s i dtehRe O I
izse r o
r ae a s u r e m e n t s
( a < r < b) nrioetrq u i r ae nseyx t m
o u t s i d e t h e ROI.
of 9 superimposed
T h e test phantomcomposed
(64x64)
used
in t h ec o m p u t esr i m u ellipses of size
3.
180 equispaced
lation
studies
shown
is Fig.
in
a t 91 values of t (corresprojections,eachsampled
ponding t o a parallelprojection)distributeduniformly
of view
(a=45)
encompassing
o vtehfruef li le l d
thewholephantomwerecomputed.Thereconstruct i t ht h i fsu l l
tion of t h pe h a n t o mw a cs a r r i e do uw
d a tuas i ntghCeBm
P e t h o(dF i g .
4).
The
Rama[l] and a Hamming
c h a n d r a n - Lakshminarayanfilter
window
was
used
for
filtering
and
smoothing
the
projection
data.,
This
operation
was
carried
out
tfihrne q u e ndcoym auisniFn Fag lTg o r i t h m s .
A weightedbackprojectionwasthenusedtoobtain
t hree c o n s t r u c t eidm a g e .
In o r d e r t o s i m u l a tteh e
m e a s u r e m e n t s of truncated
projections,
the
outer
t o rays
p a r t s of t hper o j e c t i odna tcao r r e s p o n d i n g
of i n t e r e s t
w h i c hd on o ti n t e r s e c tt h ec i r c u l a rr e g i o n
t o zero.
Three
cases of t h e ROI
( r > aw) e rsee t
w i t h a = 35, 30 and
25
are
considered.
Fig.
5a,
6a an7sdha o wt hrse c o n s t r u c t i oonb t a i n ebdy
using
the
CBP
method
directly
with
the
truncated
cases a = 3 5 a, = 3 0a n da = 2 5
p r o j e c t i o n sf o rt h et h r e e
r e s p e c t i v e l yT. heer r o rpsr e s e natreev i d e n t lcyo n of t ht er u n c a t epdr o j e c t i o n s
siderable.
Completion
as mentionedintheprevious
w e r et h e nc a r r i e do u t
section
assuming
elliptical
an
boundary.
A study
of t h e p a r t i a l a u t o c o r r e l a t i o n f u n c t i o n a n d t h e r e s i d u a l
for
various
projections
showed
that
an
AR
model
of order 5 was
sufficient
t o m o d et lhter u n c a t e d
[91
d a t a in t hper e s e ncta s eT.hBe u ragl g o r i t h m
whichisfound
to b ee f f i c i e n tf o rs h o r td a t al e n g t h s
was
used
t o o b t a ti hnAecRo e f f i c i e ndt si r e c t l y
from
the
data
values.
Fig.
5b,
6b
and
7b
shows
t h ee f f e c t i v e n e s s of theproposedmethod
of complecases
tion of t h et r u n c a t e dp r o j e c t i o n sf o rt h et h r e e
r e s p e c t i v e l y .F o rt h es a k e
of comparison,thereconstruction
obtained
by
using
the
positive
constraint
additive
ART
method
with
truncated
projection
f otrhfei r st twcoa s eas= 3a5nad= 3a0rseh o w n
in Fig. 8 a n d 9.
Conclusion
Theproposedmethod
of c o m p l e t i n g t h e t r u n c a t e d
a significantlybetterreconstprojectionsresultsin
t o the
reconstruction
obtained
ruction
compar,ed
by
using
the
incompleted
ones.
Qualitatively
the
r e c o n s t r u c t e di m a g e sa r ev e r yg o o dw i t h i nt h ef i e l d
of view.
Also,
features
which
are
visual
ly
missing
of viewcanbeseen
in t h er e g i o no u t s i d et h ef i e l d
w h etnhter u n c a t epdr o j e c t i o nasrceo m p l e t e da,l thoughquantitativeerrorscanbeobserved.
A detailederroranalysisisbeingcarriedout.Anadvantage
of thismethod
is t h a ti td o e sn o tr e q u i r ea d d i t i o n a l
of view.
The
time
m e a s u r e m e n tosu t s i dtehfei e l d
t a k e nf o rt h ec o m p l e t i o n
of t h et r u n c a t e dp r o j e c t i o n s
is being
extended
to
insignificant.
is
The
method
i n c l u dtehoe t h etrw o
cases - t hhe o l l o w
a n tdh e
limited angle situations.
34. 3. 2
1734
ICASSP 86, TOKYO
Predictionessentiallyreliesonthe
fact t h a t
is a b l et oe x t r a c tt h ec o r r e l a t i o n
t h el i n e a rp r e d i c t o r
p r e s e n t in t h ed a t aT. h es i m p l ec o m p l e t i o nm e t h o d
of theprojectionindivipresentedheremodelseach
a one
dimensional
predictor
model.
dually
using
fact t hatalhptler o j e c t i o n s
This
overlooks
the
as t h e ya r ea l ld e r i v e d
arenotactuallyindependent
from
t hsea mcer o ssse c t i o n .
In o r d eteorx p l o i t
this fact, further
investigations
are
being
carried
out
using
two
dimensional
linear
predictor
models
and block adaptive methods.
References
[I]
G . N . R a m a c h a n d raA
ann.dV . L a k s h m i n a r a y a n ,
'Three
dimensional
reconstruction
from
radiographs
: Application of Convolution
andelectronmicrographs
instead of Fourier
Transform',
Proc.
Nat.
Acad.
Sci., vo1.68, pp2236-2240,1971.
[2]A.K.Louis
andF.Natterer,'Mathematicalproblems
of Computerized
Tomography',
Proc.
IEEE,
vo1.71,
~ ~ 3 7 9 - 3 8 91983.
,
[3] R.M.Lewitt
and
R.H.T.Bates,
'Image
reconstruction
from
projections',
Optik,
pp19-33;Part 111 : pp189-204,1978.
vol.
50, P a r t
I:
J.C.Gore
[4] and
%Leeman,
'The
reconstruction
ofobjectsfromincompleteprojections',Phys.Med.
Bioi., Vo1.25, pp129-136,1980.
[ 5 ] W . W a g n'eRre, c o n s t r u c t i of nr rosems t r i c t e d
- Newmeans
to r e d u c et h ep a t i e n t
r e g i o ns c a nd a t a
dose',IEEETrans.Nucl.
Sci., Vol. NS-26,pp2866-2869,
April1979.
[6] B.E.Oppenheim,
'More
accurate
algorithms
for
iterative
3-dimensional
reconstruction',
IEEE
Trans.
Nucl.Sci,,NS-21,pp72-77,1974.
[7] A . V a n D e n B'oAsl,t e r n a tiinvtee r p r e t a t i o n
maximum
entropy
spectral
analysis',
IEEE
Trans.
Inform.Theory, Vol.1T-17, pp493-494,1971.
of
[SI V.Krishnan,
'Adaptive
Estimation
Algorithms',
of Science,Bangalore,
LectureNotes,IndianInstitute
India,October1984.
Fig. 3
[9] J.P.Burg,
'nAeawn a l y sties c h n i q uf oet irm e
at theNATOAdvancedStudy
s e r i e sd a t a ' ,p r e s e n t e d
Inst.SignalProcessingwithemphasisonUnderwater
Acoustics,Enschede,Netherlands,1968.
Fig.
4
34. 3. 3
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Fig. 5a
Fig.
Fig.
6a
Fig.
6b
Fig.
7a
Fig,
Fig.
7b
Fig.
8
34.
1
5b
9
3. 4
ICASSP 86, TOKYO