From: AAAI Technical Report SS-94-05. Compilation copyright © 1994, AAAI (www.aaai.org). All rights reserved.
uniformlyscattered throughoutk-space. Thus,unlike
other techniquesof constrained
imaging,
FRMR/
WiliTedprior knowledge
to control the image
acquisitionprocessas wenas the imagereconstruction
procedure.
Using a pattern recognition technique to
constrain the acquisition and reconstruction
of MRimages
YneCaoand DavidN. Levin
Departmentof Radiology,University of Chicago
Chicago, IL 60637
In an earlier repon2, FRMRIwasapplied to
simulatedand experimentalphantomimageswith small
matrices(16 x 32). In the following,wedescribethe
appficationof the methodto 64 x 64 imagesof the
humanbrain. This required the use of a numerical
optimizationtechnique(simulatedannealing)to find the
best subset of phase-encoded
signals to be measured.
Section2 is a brief discussionof the theoreticalbasis of
FRMRI,whichwasdescribedin detail in ref. 2. The
application of FRMRIto humanbrain imagesis
discussedin Section3. Theimplicationsof these
results are describedin Section4.
ABSTRACT
Atraining set of MRimagesof normaland
abnormalheadswasusedto derive a completeset of
orthonormalbasis functionswhichconvergedto headlike imagesmorerapidly than Fourierbasis functions.
Thenewimagerepresentationwasused to reconstruct
MRimagesof other headsfroma relatively small
numberof phase-encodedsignal measurements.The
trainingimages
alsodetermined
exactlyw.h/.ch.t .t .t .t~encodedsignals should be measuredto nnmmme
nnage
reconstructionerror. Thesesignals werenon-uniformly
scattered throughoutk.space. Experiments
showedthat
headimagesreconstructedwith the newmethodhad less
truncationartifacts than conventionalFourierimages,
reconstructedfromthe samenumber
of signals.
1.
2.
METHODOLOGY
The FRMRIalgorithmcan be divided into
twophases. Thefirst phaseconsists of preprocessing
or training computationswhichare performedjust once
for a givenclass of images(e.g. the class of Tlweightedaxial imagesthroughthe lateral ventricles of
the brain). Thesecalculations determine:(1) a formula
for reconstructingthe training imagesfroma reducedset
of PEsignals; (2) the indices of the phase-encodings
whichproducethe mostaccurate suchreconstruction.
Oncethe training phaseis completed
for a particular
class
ofimages,
"unknown"
objects
(objects notinthe
training set) can be imagedby followingthe steps.
First, the scanneracquiresa reducedset of PEsignals at
the locationsin k-space
specified
bythe training
computations.Usingformulasderivedin the training
phase,these measurements
are linearly transformedto
producean imageof the unknown
object.
INTRODUCTION
Conventionalmagneticresonanceimages
(MPA)
are reconstructedby Fourier uansformation
of
limited numberof phase-eneeded
if’E) and frequeneyencoded(FE)signals in the center ofk space. There
alwaysa tradeoff betweenimagingtime andS’l~rhl
resolution. If onewishesto reducedamacquisition
time, fewerPEsignals can be measured,andthe
resulting imageswill havemoretruncationartifacts and
worsespatial resolution. Manyconstrainedimage
reconstructionalgorithmsattemptedto improvethis
tradeoff by usingprior knowledge.
Mostof these
I did not attempt to optimizethe image
methods
acquisition;the measured
PEsignals werestill
uniformlyspacedin the center of k-space. Unmeasm’ed
high-ksignals wereestimatedfromthe measuredlow-k
data by using expficit mathematicalmodels,which
reflecteda particularauthor’sintuition aboutwhatthe
imagesshouldlook like.
2.1
Training images
Thealgorithmutilizes prior knowledge
in the
formof a set of "trainingimages",comprised
of a large
collection of high resolutionimagesof objects
resemblingthe"unknown"
objects
tobescanned. For
example,a large set of normalandabnormalbrain scans
wouldbe usedto train the algorithmto acquireand
reconstructbrain images.Thetraining imagesare
acquiredby conventionalMRscanning,i.e., by
measuringPEsignals, whichsamplerows in the kspacerepresentationof the image.Thesampledpoints
forma Cartesiangrid in the centerof k-spaceandhas
uniformdistribution in eachdirection. Thespacing
betweenpoints is determinedby the fields of view
(FOV)Df and Dpin the FEand the PEdirections,
respectively. Thenumberof sampledpoints in the FE
direction(Nf)arid in the PEdirection(Np)is determined
by the desiredresolutionalongeachaxis. Let gn(a)
representthe complex
data at the point with FEindexa
andPEindex n. Thevalues ofgn(a) for each fixed
valueof a (i.e. eachcolumn
in k-space)canbe regarded
Werecently developeda newapproachto
2, called Feature-Recognizing
constrainedMRimaging
MRIfb’R MRD.
In this scheme,a set of training
imagesreflected our prior knowledge
about the objects
to be imaged.Anautomaticpattern recognition
techniquewasusedto examinethe space of all possible
imagesandto t’mdthat subspaee,whichbest described
the training set. Since the imagesof subsequently
scannedobjects werepresumed
to lie in this subspace,
they couldbe reconstructedfroma subset of the usual
array of PEsignals. Furthermore,the training images
automaticallydeterminedwhichsignals should be
measured
in orderto reduceimagereconstructionerror.
In mostcases, the optimal measurements
werenon215
as a complexcolumnvector in an Np-dimensional
imagespace. Since the Ixaining imagesare similar to
oneanother,the vectorscorresponding
to different
training imagesare usuallyclusteredin the imagespace.
2.3
Let<gn(a)>
betherawdA~array
ofthe
average
training
image,
anddefine
fn(a)
tobethe
diffe~nce
benueen
therawdata
ofa specific
image
and
this
average
training
image:
fn(a)
= gn(a)
- <&nfa)>
TheFoudcx
Iransformation
offn(a)
isa complex
function
l(x,y),
which
represents
thedifference
between
the averagetraining imageandthe imagecorresponding
to gnfa ):
~’z+tq’u
l(x,y)ffi
. t ~’.~".fs(a)e
(I)
Dt.~-~p
.=~...s;.= .17"..up
Theaim of the FRMRIalgorithmis to recover the
functionl(x,y) froma subset of the N/) PEsignals.
relxesentedby gnfa) orfnfa). Thedesiredimage
isthen
produced
by adding
theaverage
waining
image
tol(x,y).
2.2
FR basis functions
Theimagespaceis spanned
bymany
complete
sets
oforthononnal
unit
vectorswhichate "rotated"
withrespect
totheaxes
corresponding
totheFomier
imagecomponentsfn(a).
Theimage’s
components
(vn(a))
suc
h an
alt
ernative
rdinate
coo tem
sysate
related to its Fourier components
by an Npx Np
unitary
Iransformation,Unn’:
f.(a)=
~’~Um,(a)vs(a)
(2)
a’=I...N#
Theadoptionof such a "rotated" coordinatesystemis
equivalentto representing
the imagewith a completeset
of orthonormalfunctions Van(X,y):
l(x,Y)
= ..~..s¢!
a=~...sc
v.(a)Vas(x.Y)
O)
whe~
V,n(x,y)
Image reconstruction
If wewantto reconstruct
animagefroma
subset of PEsignals with indices [L) (L < Np),it is
necessaryto truncate the FRexpansionofeach column
of k-spacedata sOthat it has L or fewerterms.In othe~
words,Eq(2) andEqO) mustbe truncatedafter the firs
M(a)terms whereM(a)< L. Then, the L measured
components
Offn(a)Canbeused
tO CalC))latOthe values
ofthefirst
M(a)FRcomponents,
vn(a
in
).Asshown
reference
2,these
~ components
canbeusedto
estimate the Np-Lunmeasured
PEsignals:
fh(a)= [Q(aXR(a)+R(a))’lR(a)+]hlfk(a)
k e{C}
whereRkm(a)is an L x M(a)matrix with elements
Rkm(a)= Ukm(a)for m = l...M(a) and k e {L)
whereQign(a)is the (Np-L)x M(a)matrix: Qhm(a)
Uhra(a)form= l..a~lfa) andh ~.tL). Finally,
the
imagecanbereconslructedby conventionalFourier
transformation(Eq (1)) of the L measuredPEsignals
andthe Np-LestimatedPEsignals in Eq(6).
2.4 Optimal PE signals to be measured
In principle, the imagecan be reconstructed
from any subset of L PEsignals. However,some
choices
ofthese
measured
signals
mayleadto
amplification of envrs dueto measurement
noise and
series truncation. Theoptimal
PEsignals
to be
measured
are those which
lead to least error in the FR
reconsgucfion
of the training images.If e is the root
meansquarednoise in the PEsignal measurements,
th,
averageerror in the FRreconstructionof the training
imagesis:
< ~r> =e2,,~...t¢
NiLe2 + Tr[(Q(a)÷Q(a)XR
41
(a)*R(a))
t
I. e#, ~ Y Un’ n(a)dq’~’
(4)
+ ~.~...s, Tr[W(a)T(a)W(a)+]
Sinceall of these alternativebasissets arc
complete,eachonecan be usedto representan arbitrary
imageexactly as well as the Fourier expansionin Eq
(1). However,
if these representations
arcmmcaled
at
anypoint, there is onespecial basis set whichprovides
the mostaccuraterepresentationof the training images.
Asdemonstrated
in reference2, the optimalbasis set
can be derivedfromTnn’(a),the covariancematrixof the
training images:
(5)
TM.(a)
=l ~. f~(ad)fs(aJS*
CO
whe~
W~(a)
= -[Q(a)(R(
a)÷R(
a)) 4
w~.(a)
=&h.
for h 4 (Ll, t e tLl, h’ ~(L} (8)
Ourtaskis to find the indices {L) of the PEsignal
measurements
whichminimize<E~,>.Notice that the
matrices Qand R in Eq (7) dependon the mmcation
points M(a)andthe indices (L). order to avoid to
examinethe hugesearch spacedefinedby the
simultaneousvariation of the M(a)andthe indices (L)
wehaveused the followingitemtive procedureto look
for optimalvaluesof M(a)and(L}. First, weesfimat,
the values of M(a)whichled to an acceptableamount
averagemmcation
error in the representationof the
wainingimagesby M(a)FRbasis functions. Next.
simulatedannealingwasusedto identify the optimal
indices[L), i.e. the optimalk-spacelocationsfor the
measuredPEsignals used to calculate those M(a)
J j.,~..a
wherefn(aj)is the Fourier representationof the/h
training image.Theunitary transformationto the
optimalcoordinatesystem(i.e. Unn’(a)in (2)) is
matrix whichtransformsTnnfa) into diagonalform
with descendingeigenvalues.In other words,the
optimalimagebasis functionsare the principal
components
of the columnvectors of the training
2.
images
216
After the optimalset of measuredPEsignals
wasfoundin this way,weexamined
the eigenvaluesof
the matrix R(a)+R(a),whichmust be inverted to
performthe FRreconstruction.If a smalleigenvalue
wasfound, M(a)waslowereduntil all eigenvalueswere
acceptablylarge, therebyavoidingerror amplification
duringimage
reconstruction.
components.
Finally M(a ) wasadjusted,as needed,
minimize
ill-conditioningof the calculationof the M(a)
FRcomponentsfrom the optimally.located L PE
measurements.
Asstated above,the M(a)wereinitially chosen
to lead to an acceptablelevel of truncationerror. As
shown
in reference2, the meansquarederror in the
truncatedFRrepresentationof the training imagesis
givenby the sumof the last (Np.M)eigenvaluesof
Tnn’(a). Themeansquaredmagnitudeof the training
imagesis givenby the sumof all of these eigenvalues.
Therefore,the relative size of the meansquared
truncationerror is givenby the ratio of these two
quantities:
2.5
Imaging unknownobjects
After
thetraining procedureis completed,
"unknown"
imagescan be quicklyreconstructed. First,
the operator choosesthe numberL of PEsignals to be
acquired;for example,this mightbe determinedby the
time available for scarming.Next, the scannermeasures
the optimalPEsignals with indices (L} determined
duringthe training phase.Imagereconstructionis
performedby "back-projecting"these measuredFourier
components
onto the M(a)-ffnnensionalreconstruction
subspace,spannedby the In’st M(a)FRbasis functions.
<Era(a)2>-i,.M(a~l..A’p
gi(a)/,-.~..N, k(a)
where),i(a) is the I th eigenvalueof Tnn’(a).TheM(a)
wereinitially chosento lead to an acceptablevalueof
this relativetruncationerror.
Givenan initial choicefor M(a),the nextstep
wasto find the indices [LJ of measuredPEsignals
whichminimized
the error in the FRreconstructionof
the training images;namely,Eq(7). This involved
lookingat manypossible choicesof {L); for example,
for Np- 64andL ffi 32there are 64!/32!(64-32)I
1018choicesfor these indices. Weusedsimulated
3 since it is an efficient meansof finding
annealing
global or nearly global minimain such hugesearch
spaces.In our case, the state of the systemto be
annealedwasdefinedby the indices(L.h andthe
system’senergywasgivenby the teconslructionerror in
Eq(7). After the initial state of the systemwaschosen
randomly,a proposedperturbation("move")was
stochasticallygenerated;i.e. someindices in (L) were
deleted, and an equal numberof indices were~ to the
set. TheenergychangeASassociated with the
proposedmovewascalculated. If dE < 0, the move
wasacceptedanddefinedthe next state of the system;
i.e. the systemalwayswent"downlfill". If AE> 0, the
con’esponding
Boltzmann
factor
e"AE/T
wascalculated,
where
T wastheuser-controlled
"tempemtme"
ofthe
system.
Themovewasaccepted
withhigh(low)
probability
iftheBoltzmann
factor
wasclose
toI (0);
otherwise,
themovewasrejected.
Inother
words,
the
systemwaslikely to makelarge "uphill" movesat high
T, but usually madeonly small uphill movesat low T.
After a large series of moveswasmade,the temperature
parameterT waslowered,anotherseries of moveswas
generated,T wasloweredagain, etc. If the initial
temperatureis high enough,enoughmovesare madeto
approachthermodynamic
equilibriumat each
temperature,andT is loweredslowlyenough,this
procedureis highly likely to movethe systemto a
3. In other words,the
global or nearly global minimum
systemwas"frozen"or annealedin the "ground"state
(or nearlygroundstate).
3.
3.1
APPLICATION TO IMAGES OF THE
HUMAN HEAD
Training images
Weapplied the FRalgorithmto 64 x 64
imagesof the humanhead. First, Tl-weightedaxial
imagesof 75 normaladult volunteersand patients
withoutlarge lesions werecollectedby a 1.5 T scanner.
A spin echopulse sequencewasused with parameters:
TR/rE= 500/16 msec, slice thickness = 6 mm,FOV=
24 cm,and2 excitations. If necessary,the imageswere
manuallyIranslated a fewpixels in the FEand PE
directionsso that the headwasretrospectivelycentered
in the FOVno matter howthe patient waspositionedin
the scanner.Next, the k-spacedata werenormalized
to
unit magnitudeat k=0in order to compensate
for
variations in receiver gain fromscanto scan. Finally,
the averagetraining imagewassubtractedfromeach
image.
3.2
FR basis functions
Thetraining imageswereusedto calculate and
diagonalizethe covariancematrixTnn’(a)(E,q (5)). This
proceduredeterminedthe FRbasis functionsVan(X,y)
given by Eq (4). Thesecompriseda completeset
orthonormal
functionswhichwere"tailored" to
convergerapidly to head-likeimages.Thefirst FR
functionsdepict features whichare common
in the
training images,suchas the scalp, diploicfat, brain
surface, andventricles. Higherorderbasis functions
describefeaturesrarely foundin the trainingset; for
example,they are distributedin regionswhichare filled
with air in mostimages.
3.3
Optimal PE signals
to be measured
We"Irained"the algorithmto reconstruct
imagesfrom8, 16, 24, 32, or 40 PEsignals. For each
217
slightly less aocmatethan the FRimagesreconstructed
from the samenumberof signals with TR= 500 ms.
This was becauseimagesacquired with TR= 300 ms
andI000msare expectedto be "farther" fromthe
cluster of training images(TR= 500 ms).
Nevertheless,the FRreconstructions for TR= 300 ms
and1000msstill havesignificantly less truncation
artifact than the comparable
FTreconstructions.This
suggeststhat the FRalgorithmderivedfroma single
Tl-weighted
training set can be usedto reconstruct
imagesacquiredwith a widevariety of Tl-weighted
pulse sequences.TheFRalgorithmwasalso used to
reconmuct
a training subject, after the data were
modifiedto containa simulated"lesion" unlike any
lesion in the training set. It is apparentthat the FR
algorithmreconstructedthe lesion accuratelywhenit
waslocatedinside the head. This is becausethe
IruncatedFRseries is fairly completeover the head
regionwheremosttraining imageshavea variety of
structural
detail.~.
of these choicesof L, the first step wasto estimate
M(a),the numberof FRbasis functions to be usedin
the reconstructionof the ath columnin k-space. As
describedin Section2, this wasdoneby requiringthat
the averagemnr, afion ezror <Erel(a)2>
be less than a
certain upperlimit: e.g. 0.01 for L=32.This criterion
usually led to valuesof M(a)whichwerelowerfor
central columnsin k-spacethan for peripheralcolumns.
For example,for L = 32, M(a)was10, 27, and 30 for a
= 32, 16 and 1, respectively. Next, simulatedannealing
wasusedto determineoptimalor nearly-optimalindices
(L} of the PEsignals to be measured.Theenergyto be
minimizedwasgiven by Eq (7). Weused a value of
(the noise in each PEsignal), whichwouldproduce
signal-to-noiseratio of 7 in a conventionalFTimage.
These calculations took 1-15 CPUhours on a Convex
C-3supercompuer.
This training procedureis practical
since it only has to be doneoncefor eachclass of
images.
3.4
Image reconstruction
4.
Theaboveproceduredetermined
the indices(/.,)
of PEsignals whichled to a reasonablysmall average
error in the FRreconstructionof the training images.
TheFRreconsu’uctionformula,Eq (6), uses those
signals to reconstructthe ath columnof a k-spaceimage
as a sumof M(a)FRbasis functions. Notice that this
imagereconstructionformulainvolvesthe inverse of a
matrix R(a)+R(a).For a few k-space columnsthis
matrix hadsmall eigenvalues(e.g. conditionnumber
greater than200)whichcausedits inverseto be illconditioned.Therefore,wedecreasedM(a)for those
columns(i.e. usedfewerFRbasis functions)so that the
matrixinversewasbetter definedandthere wasless error
ampfificationin the reconstructionprocess.
DISCUSSION
ConventionalMRIhas spatially-uniform
resolutionbecauseit utilizes Fourierbasis functions
(sines andcosines)whichoscillate uniformlyacrossth~
FOV.Furthermore,the measuredPEsignals are
uniformlydistributed in the center of k-space.This
strategy for imageacquisitionandreconstructionis
appropriate
in situations in whichthere is no prior
informationabout the object being scanned. However,
the method
suffers fromthe fact that sharpedgesare
poorly representedby Fourier functions. Suchedges
generatetruncationartifacts whichcan obscureadjacent
structures. In contrast, FRMRIutilizes prior
informationin the formof training imagesto generate
"customized"set of basis functionswhichconvergeto
similar imagesmorerapidly than the generic Fourier
functions. Thesebasis functions maybe highly
localizeddistributions andcan containsharpedges. Th
FRimagereconstructionprocessis characterizedby
Spatial resolution whichvaries across the FOV,
Regionscontainingmuchstructm’al detail in the
training imagesare morehighly resolved and"come
into focus" earlier in the FRseries expansion.TheFt
expansionof an imageis best calculatedfroma set of
PEsignals whichsamplek-space in a non-uniform
fashion, also dictated by the training images.Thus,th
FRstrategy is to use prior informationto optimizetin
scanningprocessas well as the imagereconstruction
Oncethis training procedurewascompleted,
"unknown"
objects were imagedby measuringthe
optimalL PEsignals and substituting themin Eq(6).
The FRimagesof a typical "unknown"
headwere
reconstructedfrom8, 16, 24, 32 and40 PEsignals.
Themeasuredsignals wereacquiredwith the samepulse
sequenceusedto collect the wainingdata. Theresults
are comparedto conventionalFTreconstructionsfrom
the samenumberof PEsignals, distributed uniformly
in the center of k-space. TheFRimageshaveless
truncation artifact than the FTimages. Common
featuresin the training images(the scalp, diploic
marrow,brain surface, ventricles, andinterventricular
septum)"comeinto focus" even for highly mmcatedFR
reconstructions(e.g. L = 8, 16, 24). This happens
becausethese featuresare "built into" the first FRbasis
functionsandare addedinto the FRseries, complete
with highlyresolvededges, as soonas they are
"detected"in the measured
signals.
1. Z.P. Liang, F.E. Boada,R.T. Constable, E.M.
Haacke,P.C. Lauterbur, and M.R.Smith. Rev.
Mag.Res. in Med., VoL4, pp. 67-185, 1992.
2. Y. Cao and D~. Levin. Mag. Res. Med., Vol.
30, pp. 305-317,1993.
3. S. Kirkpatrick, C.D.Gelatt, and M.P. Vecchi.
Science, Vol. 220, pp. 671-680,1983.
TheFRalgorithmwasused to reconstruct
imagesof unknown
headsscannedwith different TRs:
such as TR=300and 1000ms. The FRreconstructions
from PEsignals with TR= 300 ms and 1000mswere
218