with Recon gurable Optical Buses
Chin-Hsiung Wuy Shi-Jinn Horngy Horng-RenTsaiz Jinn-FuLiny Tsrong-Lay Linx
yDepartment of ElectricalEngineering, National TaiwanUniversityof Science and Technology,
Taipei,Taiwan, R. O.C.
E-mail:[email protected]
zDepartment of Information Management, Chinese College of Commerce,
Taichung,Taiwan, R. O.C.
xSKuang Wu Institue of Technologyand Commerce, Taipei,Taiwan, R.O. C.
Abstract
Computingthemomentsofatwo-dimensional(2-D)
image involves a signi cant amount of
multiplication-s and additions in adire ctmethod. In this paper,we
use the suÆx sums to c ompute the 2-D moments
in-steadof using a dire ctmethod. This method c an
re-ducethe number of multiplications tremendously. By
integratingthe advantages of both optical transmission
and electronicc omputation, the 2-D moments c an b e
c omputedinconstanttimeona2-Darrayswithre c
on- gurableopticalbuses (AROB). This result achieves
optimal spe e d-up.
1 Introduction
Moments are byfar the most popular descriptors
for image regions and boundary segments. Many
im-portant geometric features of an image can be
de-termined from its moments. Hu [6] proposed a set
of moment invarian tsbased on the 10 loworder
mo-ments. These moment invarian tsare simple
func-tions of moments and independent of scaling,
trans-lation and rotation. Tocompute the moments of
a two-dimensional(2-D) image involvesa
signi can-t amount of multiplications and additions in adirect
method. Previously,somefastalgorithms for
comput-ingmomentshadbeenproposedusingvariousmethods
[2,3,5,7,12,13,14,15,16,17,18]. Sincethemoments
ofagraylevelimagearemorewidelyusedinmany
ap-plications, suchas texture analysis, wefocus on the
Thisworkwassupportedbythe NationalScienceCouncil
underthecon tractno.NSC-89-2213-E011-007.
moment computation of graylevelimages in this
pa-per.
Given an N N image, some parallel algorithms
wereproposed for computing moments of the image.
Reeves [13,14]presented twoparallel algorithms for
this problemusing theDirect algorithm onmesh
con-nectedcomputers(MCCs). Chen[2]presentedparallel
algorithms for computing the 16 loworder moments
basedonaso-calledcascade-partial-sumtechniqueon
MCCs. Chen's algorithmsruninO(N)timeona
lin-eararrayofsizeN andruninO(logN)timeona2-D
arraysof size N N, respectively. Recently,Cheng
et al. [3] proposed twospecial VLSI architectures for
computingthe2-Dmomentsoforder(p;q). Forthe
1-Dstructure,ittookO(max(p;q)N)timeusingN+1
processors. Forthe 2-Dstructure, it tookO(N) time
usingN(N+1)processors.
Themain contributionofthis paperisin designing
anoptimalspeed-upalgorithm forcomputingthe2-D
momentsona2-DAROB.We rstderivethe
relation-ships betweensuÆx sumsand moments. Then using
these relationships,anoptimalspeed-up algorithmfor
moment computation is developedon a 2-D AROB.
WhenthedomainoftheimageisO(logN)-bitinteger,
for a constant c, c 1, the proposed algorithm can
be run in O(1) time using N N 1+
1
c
processors. To
the best of our kno wledge,there wereno
algorithm-s before which could solvethis problem in constant
time and/orprovidesuchaperformance ona2-D
ar-rayarchitecture. Clearly,thisresultisbetterthanthe
previouslyknownalgorithmsproposedintheliterature
[2,3,13,14].
The rest of this paper is organizedas follows. We
giveabriefintroductiontotheAROBmodelinSection
be used in the parallel algorithm. Section 4 dev
elop-s ouralgorithm forcomputing2-D moments. Finally,
someconcludingremarks are includedin the last
sec-tion.
2 The Computation Model
Thearraywitharecon gurableopticalbussystemis
de nedasanarrayofprocessorsconnectedtoa
recon- gurable optical bus system whose con guration can
bedynamicallychangedbysettingupthelocal
switch-esofeachprocessor,andmessagescanbetransmitted
concurrentlyonabusinapipelinedfashion. Two
relat-edmodelsusingrecon gurableopticalbuseshavebeen
proposed,namelythearraywithrecon gurableoptical
buses (AROB) [10]and linear arraywith a
recon g-urablepipelined bussystem(LARPBS) [8,9]. Dueto
unidirectionalsignalpropagationandpredictabledelay
of thesignalperunit length, theopticalbuses enable
synchronizedconcurrentaccess inapipelinedfashion.
TheAROBmodelisapowerfulcomputationmodel
whichincorporatessomeoftheadvantagesandc
harac-teristics ofrecon gurablemeshes [1] and mesheswith
optical buses[10]. A lineararraywith pipelined
opti-calbuses(LAPPB)[4]ofsizeN containsN processors
connectedtotheopticalbuswithtwocouplers.Oneis
usedtowritedataontheupper(transmitting)segment
ofthebusandtheotherisusedtoreadthedatafrom
thelower(receiving)segmentofthebus. A1-DAROB
extends thecapabilities of the LAPPBby permitting
eachprocessortoconnecttothebusthroughapairof
switc hes.Eachprocessorwithalocalmemoryis
identi- edbyauniqueindexdenotedasP
i0 ,0i
0
<N,and
eachswitch canbesettocrossorstraightbythelocal
processor. As for the properties of the optical buses,
the messagepropagatesunidirectionallyfrom rightto
left on the upper segment and from left to righton
the lower segment. Each processoruses a set of
con-trolregisterstostoreinformationneededtocontrolthe
transmissionandreceptionofmessagesbythat
proces-sor. AnexampleforalinearAROB ofsize5isshown
in Figure 1(a). Twointeresting switc h con gurations
derivablefromaprocessorofanLAROBarealsoshown
in Figure1(b).
A 2-DAROB ofsize MN containsMN
pro-cessorsarrangedina2-Dgrid. Eachprocessoris
iden-ti ed byaunique2-tupleindex (i
1 ; i 0 ),0i 1 <M, 0i 0
<N. Theprocessorwithindex(i
1 ; i 0 )is denot-ed byP i1;i0
. Each processorhas four I/Oports,
de-notedby S
j ,+S
j
,0j<2,tobeconnectedwitha
recon gurableopticalbussystem. Theinterconnection
amongthefour portsofaprocessorcanbecon gured
during the execution of algorithms. Formore details
- P 0 P 1 P 2 P4 (a) P 3 Æ unitdelay y u u i d Æ B B M B B M B B M B B M B B M Æ Æ straight cross (b) transmittingsegment receivingsegment d d d u u u u u u u u i i i i Æ Æ Æ switch } endprocessor leaderprocessor H H 6 = message selection }reference Z Z }
Figure 1: (a) An LAROB of size 5. (b) The switch
states.
ontheAROB,see[10].
3 Basic Operations
ForBooleanoperation,therewasaresultderivedby
PavelandAkl [11].
Lemma1 [11 ]GivenN Bo oleandataache witheither
0 or 1, the binary pre xsums of these N bits c an b e
c omputedinO(1) timeona1-DN AROB.
Next, wepropose the eÆcient algorithms for
com-puting integersuÆx sums. Given N integersa
i , 0
a
i
<N,0i<N,thesuÆxsumsoftheseN integers
isde ned as rps j = N 1 X i=j a i ; 0j<N: (1)
Using the radix-! system technique,rps
j
of Eq. (1)
can be computed more exibly. Since a
i
< N and
0 i <N, eachdigit has a valueranging from 0to
! 1 and a !-ary representation m
3 m 2 m 1 m 0 is equal to m 0 ! 0 +m 1 ! 1 +m 2 ! 2 +m 3 ! 3 in normal
!-aryrepresentation. Themaximumofrps
j
isatmost
N(N 1). Withthisapproach,a
i andrps j ofEq. (1) areequivalentto a i = 1 X k =0 m i;k ! k ; (2) rps j = 1 X l=0 n j;l ! l ; (3) where =blog ! Nc+1,0i<N, =blog ! N(N 1)c+1,and0m i;k ,n j;l <!.
As rps j = N 1 i=j 1 k =0 m i;k ! k = P 1 k =0 P N 1 i=j m i;k ! k , let d j;k be the sum of N j coeÆcientsm i;k ,0i<N,whichisde ned as d j;k = N 1 X i=j m i;k ; (4) where0k< . Thenrps j
canbealsoformulatedas
rps j = 1 X k =0 d j; k ! k ; (5) where0d j;k (! 1)(N j). Let cy j;0 = 0and d j; u = 0, u < . The
rela-tionshipbetweenEqs.(3)and(5)isdescribedbyEqs.
(6)-(10). sum j;t =cy j; t +d j; t ; 0t< ; (6) cy j;t+1 =sum j;t div !; 0t< ; (7) n j; t =sum j;t mod !; 0t< ; (8) where sum j; t
isthesumat thetthdigitposition and
cy
j;t
isthecarrytothetthdigitposition. Hence,n
j;t
is thecoeÆcientofrps
j
under theradix-! system.
S-incecy j;0 =0,thency j;1 (! 1)(N j)=!N j, thency j;2 [(! 1)(N j)=!+(! 1)(N j)]=! N j (N j)=! 2 ,thency j;t+1 N j (N j)=! t+1 .
Therefore, the carry to the (t+1) th
digit position is
not greater than N j, wehavecy
j; t+1 N j, 0t< . Alsocy j;t+1 isnolargerthancy j+1;t+1 +1 becauseof m j;t <!. Letq i;t+1 beabinarysequence
representing the binary quotient sequence of cy
j;t+1 . Then,cy j; t+1 canbeexpressedas cy j;t+1 = N 1 X i=j q i;t+1 : (9)
FromEq.(9), weobtain
cy j;t+1 =cy j+1;t+1 +q j;t+1 : (10) That is, q j;t+1 = cy j; t+1 cy j+1;t+1 . Since rps j
N(N 1),thenumberofdigitsrepresentingrps
j under
radix-!isnotgreaterthan ,where =blog
! N(N
1)c+1. Therefore, insteadof computingEq. (1), we
rst compute the coeÆcient m
i;k for eacha i . Then eachn j; t
canbecomputed byEqs. (4), (6)-(10).
Fi-nally,rps
j
canbecomputedbyEq. (3). Forthesakeof
readability,letP 0
i;j
,0i<N,0j<!,denotethe
2-Dlogicalprocessorcorrespondingtothe1-Dphysical
processorP
i!+j
. ThesuÆxsumsofN integerseachof
size (logN)-bit can be computed in a constant time
ona1-D!N AROB.Thedetailed algorithm(SSA) is
describedin the following. Initially,a
i
is allocatedto
thelocal variablea(i;0)ofprocessorP 0
i;0
,0i<N.
Finally,thesuÆxsumsofeacha
i
isstoredtothelocal
variablerps(i;0)ofprocessorP 0 i;0 ,0i<N. ProcedureSSA(a; rps) 1: Processor P 0 i;0
, 0 i < N, copies a(i;0) (i.e.,
a i )toprocessorP 0 i;j ,0j<!. 2: ProcessorP 0 i;0 , 0i<N,setsq(i;0)[0]=0. 3: for k=0 to do Steps3.1-3.6 3.1: ProcessorP 0 i;j , 0i<N, 0j<!,
com-putesm(i; j)[k]froma(i;j)byusingEq. (2).
3.2: ProcessorP 0
i;! j 1
, 0i <N, 0j <!,
setsb(i;! j 1)=1ifj<m(i; ! j 1)[k];
b(i; ! j 1)=0,otherwise. 3.3: Processor P 0 i;0 , 0 i < N, sets b(i; 0) = q(i;0)[k].
3.4: ApplyLemma1tocomputethebinarysuÆx
sums on b(i; j) and store the result to the
localvariablesum(i; 0)[k]ofprocessorP 0 i;0 . 3.5: ProcessorP 0 i;0
computescy(i; 0)[k+1](i.e.,
cy
i;k +1
in Eq. (7)) and n(i; 0)[k] from
sum(i; 0)[k] byusing Eqs. (7) and (8),
re-spectively.
3.6: Processor P 0
i;0
, 1 i < N, computes
q(i; 0)[k+1]:=cy(i; 0)[k+1] cy(i+1;0)[k+
1],wherecy(N; 0)[k+1]=0.
4: //ComputeEq. (3)toobtainthesuÆxsums. //
P 0
i;0
computes the nal rps(i;0) from n(i; 0)[k]
byusingEq.(3).
EndfSSAg
Lemma2 GivenN integerseachofsizeO(logN)-bit,
the suÆxsumsofthese N integerscanebcomputedin
O( )timeon a1-D!N AROBfor !2.
Proof: The correctness of procedure SSA directly
followsfrom Lemma 1 and Eqs. (1)-(10). Step 3.6
is used to implement Eqs. (9) and (10). The time
complexity isanalyzed asfollows. Steps1and 2each
takeO(1) time. Each iteration of Step 3 takesO(1)
time, and the number of iterations will be repeated
times. Hence,thetotaltimecomplexityofStep3is
O( ). Finally,Step4takesO( )time. Hence,thetotal
timecomplexityoftheproposedalgorithmisO( ). 2
Forsimplicity,assume!=N 1
c
,wherecisaconstant
andc1. Then =blog
!
N(N 1)c+1=b2cc+1is
bit,thesuÆxsumsoftheseN integerscanebcomputed
in O(1) timeon a 1-D N 1+
1
c
AROB for some xedc
andc1.
Procedure SSA can be used to compute the suÆx
sumsofN integers,each rangingfrom 0toN k
. Suc h
anintegercanberepresentedasasequenceof(atmost)
k radixN digits. Thus, theproblem canbe reduced
torepeatprocedureSSAforeverydigitintheradixN
representation, separately. ByCorollary1, this takes
O(k)timeona1-DN 1+
1
c
AROB.Hence,wehavethe
followingcorollary.
Corollary 2 GivenN integerseachofsizeO(logN k
)-bit,thesuÆxsumsoftheseN integerscanebcomputed
in O(k) timeon a1-D N 1+
1
c
AROB for some xedc
andc1.
4 Computing 2-D Moments
Fora 2-D image A = a(x;y), 1 x;y N, the
momentoforder(p+q)is de nedas:
m pq = N X x=1 N X y=1 x p y q a(x;y); (11)
where a(x;y) is an integerrepresenting the intensity
function (graylevelorbinaryvalue)at pixel(x;y).
Forthe sakeof simplicity,we rst consider a 1-D
digitalimagea(x);themomentofitwith orderpis
m p = N X x=1 x p a(x): (12)
Nowwede nethe highorder suÆxsumfunctions S
i ,
0ip+1asfollows. Initially ,thezero-ordersuÆx
sumsisde nedas
S
0
(j)=a(j); 1jN; (13)
andtheone-ordersuÆxsumsS
1 arede ned as S 1 (j) = N X n=j a(n)= N X n=j S 0 (n)=a(j)+ N X n=j+1 a(n) = S 0 (j)+S 1 (j+1); 1jN: (14) Similarly, S i
, 2 i < p+1, can be de ned and
recursiv elycomputedas
S i (j) = N X n 1 =j N X n2=n1 N X ni=ni 1 a(n i )= N X n 1 =j S i 1 (n 1 ) = S i 1 (j)+ N X n1=j+1 S i 1 (n 1 ) = S i 1 (j)+S i (j+1); 1jN; (15) i
initions, note that S
i
can be obtained byrecursiv ely
computingthesuÆxsumsoverthesuÆxsumsS
i 1 .
Next,wewill usethesuÆxsumsfunctions to
com-putemomentm
p
,0p3.Letb
p
(x),1xN
rep-resentthe partialmomentof orderpand b
p
(1)=m
p .
Inthefollowing,wewillshowthatb
p
(x)isarecurrence
function,anditcanbeexpressed asalinear
combina-tionofS i asde ned previously. Lemma3 Letb 0 (x) = P N i=x a(i), then m 0 = S 1 (1) and b 0 (x)=S 1 (x), 1x N,where m 0 is the zer o-ordermoment. Proof: Let b 0 (1) = P N i=1 a(i), b 0 (2) = P N i=2 a(i), ,b 0 (N)=a N . Thenb 0 (x)=S 1 (x)byEq. (14)and m 0 =b 0 (1)=S 1 (1). 2 Lemma4 Letb 1 (x)= P N i=x (i x+1)a(i),thenm 1 = S 2 (1) andb 1 (x)=S 2 (x), 1xN,where m 1 isthe one-ordermoment. Proof: b 1 (1)canberewrittenas b 1 (1) = N X i=1 ia(i)= N X i=2 (i 1)a(i)+ N X i=1 a(i) = N X i=3 (i 2)a(i)+ N X i=2 a(i)+ N X i=1 a(i) = = N X i=1 a(i)+ N X i=2 a(i)++ N X i=N a(i) = N X i=1 N X j=i a(j)= N X i=1 S 1 (i)=S 2 (1): (16)
FromEq. (16), observe that P N i=2 (i 1)a(i) = P N i=1 S 1 (i) P N i=1 a(i) = P N i=2 S 1 (i) = S 2 (2): Con-sequently, P N i=x (i x+1)a(i) =S 2 (x), 1 x N.
These imply that b
1 (x) = S 2 (x). Then, weconclude that m 1 = P N i=1 ia(i)=b 1 (1)=S 2 (1). 2 Lemma5 Letb 2 (x) = P N i=x (i x +1) 2 a(i), then m 2 = S 3 (1)+S 3 (2) and b 2 (x) = S 3 (x)+S 3 (x+1), 1xN,wherem 2
isthe two-ordermoment.
Proof: Based onLemma4, b
2 (1)canberewritten as b 2 (1) = N X i=1 i 2 a(i)= N X i=2 (i 1) 2 a(i)+ N X i=2 (i 1)a(i) + N X i=1 ia(i)
= N X i=1 ia(i)+2 N X i=2 (i 1)a(i)+2 N X i=3 (i 2)a(i) ++2 N X i=N (i N+1)a(i) = S 2 (1)+2S 2 (2)+2S 2 (3)++2S 2 (N) = N X i=1 S 2 (i)+ N X i=2 S 2 (i) = S 3 (1)+S 3 (2): (17)
From Eq. (17), observe that
P N i=2 (i 1) 2 a(i)= P N i=1 S 2 (i)+ P N i=2 S 2 (i) P N i=2 (i 1)a(i) P N i=1 ia(i)=S 3 (1)+S 3 (2) S 2 (2) S 2 (1)= S 3 (2)+S 3 (3): Consequently, P N i=x (i x+1) 2 a(i) = S 3 (x)+S 3
(x +1), 1 x N. These imply that
b 2 (x) = S 3 (x)+S 3
(x+1). Then, weconclude that
m 2 = P N i=1 i 2 a(i)=b 2 (1)=S 3 (1)+S 3 (2). 2 Lemma6 Letb 3 (x) = P N i=x (i x +1) 3 a(i), then m 3 = S 4 (1)+4S 4 (2)+S 4 (3) and b 3 (x) = S 4 (x)+ 4S 4 (x+1)+S 4 (x+2), 1 x N,where m 3 is the
thre e-ordermoment.
Proof: TheproofissimilartoLemma 5. 2
Fromthe abovearguments, it is easy to see that
b
p
(x) is a recurrence function with b
p (1) = m p and m p (N)=m p 1
(N)=a(N). Sosolvingtherecurrence
function, wecan nd the relationbetweenb
p and S
i .
Similarly, the higher order moments (p > 3) can be
computed using the suÆx sum functions. According
totheabovederivations, weconcludethatthep-order
momentm
p
canbeexpressedasalinearcombination
ofthesuÆxsumsS
i
,0ip+1.However,thenoise
of images ofhigher order momentsaremore sensitive
than that of lowerorder moments. In this paper, we
concernaboutthelowerordermomentsupto order3
sincetheselowerordermomentscan providesuÆcient
informationforpatternrecognition[2,6].
Asafurtherre nementofthe2-Dmoment
compu-tation,thedoublesummation inEq. (11)canbesplit
intotwoseparatesummations: m pq = N X x=1 x p N X y=1 y q a(x;y)= N X x=1 x p m q;x ; (18) wherem q;x
istheq-orderrowmomentofrowx,andit
isde ned as: m q;x = N X y=1 y q a(x;y): (19)
thealgorithmforcomputing2-Dmomentsm
pq canbe
decomposedintotwophases.FirstcomputetheN 1-D
rowmomentsm
q;x
,1xN,foreachrowby
apply-ing Eq. (19)in parallel. Thentherowmomentsm
q;x
are used to compute the 2-D moments m
pq
. Instead
of computingEq. (19)directly, the1-Dmomentscan
becomputedby Lemmas3-6. Thus, m
q;x
, 0q3,
1x N, canbe computedbyusing thesuÆxsum
functions asshown inLemmas 3-6;m
pq
, 0p;q3,
can be also computed byusing the suÆx sums
func-tionsbysettingS 0 (j)tom q;j , 0j <N,initiallyin Eq. (13).
Initially,assume that the gray-levelimage A is
s-tored in the local variable a(i; j) of processor P
i;j ,
1 i; j N. Finally, the results are stored in the
local variablem
pq
(1;1) of processor P
1;1
. Following
the de nitions of moments, the suÆx sums, and the
relationshipsbetweenthem,thedetailedmoment
algo-rithm(MOMENT)islistedasfollows.
AlgorithmMOMENT;
1: Foreachrowx, 1 x N, apply Corollary 2
to compute the high order suÆx sum functions
S
i
, 1 i 4; byinitializing S
0
(x;y) = a(x;y),
1y N. Afterthat, theresultsS
i
, 1i4;
are stored in local variableS
i
(x;y) in processor
P
x;y .
2: Foreachrowx,1xN,computethe1-Drow
moments m
q;x
, 0q 3, accordingto Lemmas
3-6. After that, the 1-Drow momentsm
q;x , 0
q3,1xN,are storedinthelocal variable
m q (x;1)ofprocessorP x; 1 . 3: Route m q
(x;1), located on the rst column
pro-cessor P x; 1 , 1 x N, to the (q +1)th row processorP q+1;x .
4: Foreachrowx, 1 x N, apply Corollary 2
to compute the high order suÆx sums
function-s S 0 i , 1 i 4, byinitializing S 0 0 (x+1;y) = m q
(x+1;y), 0x 3,1y N. After that,
the resultsS 0
i
, 1 i 4, are stored in thelocal
variableS 0 i (x+1;y)inprocessorP x+1;y .
5: Computethe2-Dmomentsm
pq
,0p;q3,
ac-cording to Lemmas 3-6. After that, the2-D
mo-ments m
pq
, 0 p;q 3, are stored in the local
variablem
pq
(q+1;1)ofprocessorP
q+1;1 .
Theorem1 Given anNN gray levelimageA,the
2-D moments upto order3c an b ec omputed in O(1)
time on an N N 1+
1
c
AROB for a c onstant c and
followsfromEqs.(11)-(19),Lemmas3-6andCorollary
2. Steps 1 and 4 implement Eqs. (13)-(15). Step 2
implements Eq. (19). Step 5 implements Eq. (18).
Thetimecomplexityisanalyzedasfollows. Settingk=
4inCorollary2,Step1takesO(1)timeusingN N 1+
1
c
processors. Similarly, Setting k = 8 in Corollary 2,
Step4alsotakesO(1)time. Steps2,3and5eachtak e
O(1)time. Hence,thetimecomplexityisO(1). 2
5 Concluding Remarks
In this paper, we rst derivethe relationships
be-tweenthe suÆxsums and the moments. Then using
these relationships, wepropose an optimal algorithm
for2-D momentcomputation ona2-DAROB. When
thedomain oftheimage isO(logN)-bit integer,fora
constantc,c 1, theproposed algorithm canberun
in O(1)time using NN 1+
1
c
processors. This result
isbetterthananypreviouslyresultsderivedinthe
lit-erature[2,3,13,14].
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