Some Image Processing Algorithms on a RAP with Wider Bus Networks

Transcription

1 Some Image Pocessng Algothms on a RAP wth Wde Bus Netwoks Shung-Shng Lee, Sh-Jnn Hong, Hong-Ren Tsa and Yu-Hua Lee Natonal Tawan Insttute of Technology Depatment of Electcal Engneeng 4, Secton 4, Kee-Lung Road, Tape, Tawan, R. O. C. hong@pde.ee.ntt.edu.tw Febuay 9, 996 Abstact Based on the econfguable aay of pocessos wth wde bus netwoks [8], we popose a sees of algothms fo mage pocessng. Conventonally, only one bus s connected between two pocessos but n ths machne t has a set of buses. Such a chaactestc nceases the computaton powe of ths machne geatly. Based on the base-m numbe system, we fst ntoduce some basc opeaton algothms. Then thee elated applcatons ae deved n constant tme; one s the hstogam of an mage, anothe s the mage segmentaton and the othe s the mage labelng. Key Wods Image pocessng, paallel algothms, entopy, pefx sum, hstogam, Segmentaton, econfguable aay of pocessos, bus netwok, labelng. Intoducton Owng to the fact that the speed of the electon s lmted, the computaton powe of a sngle pocesso cannot be sgnfcantly nceased. Instead of desgnng supe pocessos, many eseaches have focused the attentons to the paallel pocessng systems. As we know, the computaton powe of a paallel pocessng system cannot be lnealy nceased n popoton to the numbe of pocessos nstalled to the system. It fully depends on the algothm desgned and the system achtectue poposed. The mesh-connected compute (abbevated to MCC) s one of these famous paallel pocessng systems [6]. Even though the achtectue of the MCC s smple and egula, ts achtectue s fxed at unnng tme and t s not adeuate fo global communcatons. Reseaches ovecame such dawbacks by euppng t wth a econfguable bus system. Seveal econfguable paallel pocessng systems have been poposed [6, 9,, 4, 5,,,, 4]. Thee ae pocesso aays wth a econfguable bus system (abbevated to PARBS) [, 4], econfguable meshes [4, 5], polymophc tous netwoks [9], bus automaton [], econfguable aay of pocessos [6], econfguable netwoks [], and polymophc pocesso aays []. Although the system bus of any econfguable paallel pocessng systems s econfguable at un tme thee s stll a dawback to these models. That s, the bandwdth of the buses between pocessos depends on the

2 logathmc ode of the numbe of the pocessos to be nstalled n the system. Conseuently, ths s no good fo those computatons that need wde communcaton bandwdth. Conventonally, eseaches solved ths poblem by nstallng the system wth an exta numbe of pocessos. In fact, ths poblem can be also solved usng a wde bus system achtectue nstead of usng moe pocessos. L and Maesca [, ] showed that t would take % moe slcon aea pe pocesso befoe they could contol the local swtchng between buses at the nstucton level. Ths mples that t would be moe effcent to save slcon aea by nceasng the bus capacty athe than by nceasng the pocesso complexty. Based on such a concept, Lee et al. [8] have poposed a econfguable aay of pocessos wth wde bus netwoks to solve Hough tansfoms effcently. In ths pape we also use t to solve some othe nteestngmage poblems such as segmentaton and labelng. The hstogam of an mage povdes a useful nfomaton fo segmentaton and fo measung the textual popetes of an mage. By defnton, the hstogam of an mage s a one-dmensonal aay H[]; =;;;G, such that H[] =k f k s the numbe of pxels wth gey-level. In ths pape, we use Lee s [8] esult to solve ths poblem. Afte computng the hstogam of an mage, the segmentaton of an mage based on the hstogam entopy can be pogessed. The technue fo mage segmentaton based on the smlaty of bghtness of mage objects s thesholdng. To choose a good theshold value of a gay-level mage, the mage can be segmented nto two pats, the object egons and the backgound. Based on the entopy of the hstogam, many authos [7, 9, ] have used Shannon s concept fo thesholdng an mage. Pun [9, ] and Kapu et al. [7] defned the entopy of an mage by assumng that an mage was entely epesented by ts gay level hstogam only. They then used ths entopy to choose a theshold value fo the mage. Based on ths concept, the seuental algothms fo fndng a theshold value of an mage take O(G ) tme. Hee G s the total gey levels of an mage. Recently, based on the entopy of the hstogam, Cnue et al. [] poposed some algothms fo mage thesholdng n O(log G) tme on a pyamdal machne wth a GG base pocessos. We pesent a paallel algothm fo the same poblem on the econfguable aay of pocessos wth wde bus netwoks n constant tme usng G +=c pocessos. Fo segmentaton, we fst tansfom the gaylevel mage nto a bnay mage. Then, we popose an mage labelng algothm fo ths bnay mage. The mage labelng poblem s to assgn a unue numbe to each connected component (object) of an mage. Kao et al. [5] and Alnuwe [] have poposed constant tme algothms fo the mage labelng poblem on econfguale paallel pocessng systems. Unde the RAPWBN model, we also can solve ths poblem n constant tme. Afte labelng an mage, each labeled object can be ecognzed o undestood by calculatng ts chaactestcs. The est of ths pape s oganzed as follows. The econfguable aay of pocessos wth wde bus netwoks and some basc opeatons ae descbed n Sectons and. Secton 4 ntoduces hstogam algothm. Secton 5 shows a segmentaton algothm based on the hstogam entopy of an mage. An mage labelng algothm s pesented n Secton 6. Fnally, some concludng emaks ae also ncluded n the last secton. The computaton model and Notatons A -dmensonal (-D) econfguable aay of pocessos wth wde bus netwoks (abbevated to RAPWBN) [8] of sze N contans N pocessos to be embedded wth a econfguable bus system. Each pocesso s dentfed by a unue ndex denoted as P j,whee j < N. The econfguable bus system conssts of an M-ow and N-

3 column netwok aay and the bandwdth of each bus of each netwok s assumed to be O(N =c )-bt, whee N s the numbe of pocessos and c s a constant fo c. Usually, assume O(N =c )=m. The M-ow and N-column bus netwoks have MN pots denoted by S ;j ; +S ;j and each pot has m-bt bus connecton swtches denoted by S ;j (k); +S ;j (k), fo <M; j<n and k<m. The th ow bus s constucted by connectng the j th -column s pot swtch +S ;j to the (j + ) th -column s pot swtch S ;j+,fo <Mand j<n. Each pocesso P j also has a column bus wth M pots, denoted as ]S ;j and each pot has m-bt bus connecton swtches denoted as ]S ;j (k), fo <Mand k<m. The m-bt column bus of a pocesso can be connected to any ow bus by settng the pot connecton swtches ]S ;j (k) to S ;j (k) and/o +S ;j (k) fo <M; j<nand k<m. Assume M = N = 4andm =, Fgue shows a lnea RAPWBN. Fo a unt of tme, assume each pocesso can ethe pefom one athmetc o logc opeaton, o access a local memoy wod, o set the local swtches wth the same connecton confguaton on the same column bus, o communcate wth othes by boadcastng data on a bus. It allows multple pocessos to boadcast data on the dffeent buses o to boadcast the same data on the same bus smultaneously at a tme unt, f thee s no collson. Any confguaton of the bus system can be deved by popely establshng the local connecton among the data bus of each pot wthn each pocesso. We use the notatons as used n [8]. Fo example, n a lnea RAPWBN, f the local connecton of a pocesso s f S ;j (k); +S ;j ((k + ) mod m); = ; ; 4; and k < mg then the m-bt data ae otated one bt afte passng though these thee swtches at the j th column. Fgue shows some nteestng swtch confguatons devable fom a pocesso of a RAPWBN. Fo smplcty, nstead of usng f S ;j (k); +S ;j (k);]s ;j (k); k<mg notaton to connect each bt one by one of the th ow and j th column bus netwok, we use f S ;j ; +S ;j ;]S ;j g notaton. A RAPWBN s opeated n a sngle nstucton steam, multple data steams (SIMD) model. Usually, the bus bandwdth s not unlmted between pocessos. We assume the bus bandwdth s bounded by m-bt so that an m-bt data can be tansfeed between pocessos n constant tme, whee m s an ntege. The I/O loadng (down load and up load) tme s fully dependent on how complex the I/O nteface between pocessos and pepheals wll be. It s dffcult to estmate accuately how much I/O tme should be ncluded to the tme complexty of an algothm. Theefoe, the tme complexty of an algothm s assumed to be the sum of the maxmal computaton tme among all pocessos and the communcaton tme among all pocessos. Ths assumpton was also used by many eseaches [9,, 4, 5, 7,,, 4]. Basc Opeatons Let B be a bnay seuence of sze N, wheeb = fb ;b ; :::; b N gand b s o, N. The pefx sum fo the bt one of a bnay seuence, ps j,sdefnedas ps j = jx = b ; () whee b = ; b j =, and j N. Fo example, assume B = f; ; ; ; ; ; ; g. Then ps = ; ps =;ps =;ps 5 =4;ps 6 =5; and ps 7 = 6. Fom E. (), the maxmum pefx sum of ps j s at most N. The pefx sum poblem has been solved n RAPWBN by Lee et al. [8], we nclude t as follows. Lemma [8] Gven a bnay seuence of length N, the pefx sum fo the bt one of t can be computed n O() tme on a lnea N RAPWBN each bus wth N =c -bt bandwdth, whee c s a constant and c.

4 Let psum j be the pefx sum of N s O(log N )- bt nteges and t s defned as psum j = jx = A ; () whee A <Nand j<n. Based on the pefx sum of a bnay seuence, Olau et al. [8] poposed an O() tme algothm fo ths poblem on econfguable meshes usng N N pocessos. Kao et al. [4] also solved ths poblem on a lnea RAP usng N +=c pocessos wth N =c -bt bus wdth, whee c s a constant and c. Kao also extended ths esult to eal numbes. Assume a eal numbe s epesented by a nomalzed floatngpont epesentaton whch conssts of two pats, mantssa and exponent. The esult fo the ntege numbe can be extended to the eal numbe by followngsteps. Fst, fnd the maxmum exponent pat fom these N eal numbes. Next, adjust all eal numbes wth the maxmum exponent pat. Then, compute the pefx sum of these N mantssa. Fnally, nomalze these N pefx sums. We lst Kao s esult as follows. Lemma [4] Gven N s O(log N )-bt nomalzed eal numbes, the pefx sum of these N eal numbes can be computed n O() tme on a lnea N +=c RAP wth N =c -bt bus wdth, whee c s a constant and c. Let A ;A ;;A N and A N be N s log N-bt unsgned nteges, whee A N and N. The maxmum (mnmum) opeaton of these N numbes s to detemne f thee s a numbe whch s not less (geate) than the othes. Based on the base-m numbe system and the pune-and-seach technue, the algothm fo fndng the maxmum of N s log N-bt unsgned nteges was poposed by Kao et al. []. Assume all numbes ae dstnct and each A s epesented by the base- numbe system, whee A = RX j= b ;j j ; () fo R = blog N c +, b ;j f;g, A N ; N and j R. Instead of usng the base- numbe system to epesent A, A can be epesented by the base-m numbe system as follows. A = TX k= a ;k m k ; (4) fo T = blog m N c +, N ; k T and a ;k m. Fom above euaton, each A s epesented by T dgts and each dgt s bounded wthn the nteval [;m ]. The maxmum (mnmum) numbe of these N unsgned nteges can be found usng the pune-and-seach technue; fo each dgt a ;k of A,fa ;k s geate (less) than a j;k then A j s puned. Ths pocess epeats fom the most sgnfcant dgt to the least sgnfcant dgt. We lst Kao s esult as follows. Lemma [] The maxmum (mnmum) of N s log N-bt unsgned nteges can be computed n O() tme on a lnea N RAP each bus wth N =c -bt bandwdth, whee c s a constant and c. The above two lemmas ae also tue n the RAP- WBN model, as we can use a RAPWBN wth oneow and N-column bus netwoks to smulate the RAP. 4 Hstogam The computaton of the hstogam s a basc opeaton n mage pocessng and compute vson. It can be used fo segmentaton and measung the textual popetes of an mage. The hstogam computaton has been studed extensvely by seveal eseaches usng vaous computaton models [8,,, 8]. In ths pape, we use the algothm poposed by Lee et al. [8] fo the hstogam computaton. Fo the sake of completeness, we nclude the algothm as follows. Assume the ange of the gaylevel of each pxel s bounded by [;G ],whee

5 G s any constant. The hstogam of an mage wth n n pxels can be epesented by a one dmenson aay H[k], k = ; ;:::;G, such that H[k] =lf l s the numbe of pxels wth gaylevel k. Followng Lemma, the numbe of each H[] can be computed by the th ow bus and ts coespondng allocated pocessos. Afte dentfyng the last pocesso that holds gay-level, we stoe the hstogam of the mage pxels each wth gay-level back to pocesso. The detaled hstogam algothm s descbed as follows. Assume the RAPWBN conssts of n PE s and each bus has m-bt bandwdth. Intally, each pxel of the n n mage wth gay-level g x;y ; x; y < n and g x;y < G,sstoednthegl(j) local vaable of pocesso P j,wheej = x n + y, by ow majo ode. Fnally the hstogam aay H[j]; j =;;:::;G, of an n n mage s stoed n the h(j) local vaable of pocesso P j fo all j<g. Algothm HISTOGRAM Input : gl Output: h. // Set the local connecton of each ow bus. // Each pocesso P j sets ts local connecton f S ;j ; +S ;j g, whee < G; j < n. Then, fo each pocesso P j ; j < n, establsh the local connecton f S ;j (k); +S ;j ((k + ) mod m); #S ;j ; k<mg,f = gl(j) fo <G.. // Compute the hstogam. // Use the th ow bus to count up the numbe of pxels h() that has gey-level gl(j) =fo each pocesso P j, j<n ; <G. By Lemma, at the th ow bus, the pocesso whose allocated mage pxel has gey-level wll be contbuted to the computaton only. Hence, the hstogam can be computed by Lemma fo each ow bus smultaneously.. // Identfy the last pocesso that has the geylevel fo the th ow bus. // Set the local connecton f S ;j ; +S ;j g fo each pocesso P j ; j < n, whee <G,f 6= gl(j); set the local connecton f+s ;j ; #S ;j g, othewse. Then the pocesso P n boadcasts a sgnal * (o any sgnal) by the bt of pot +S ;n (.e. +S ;n ()) though the establshed bus. The pocesso whch eceves the sgnal * on the th ow bus s the pocesso that has the gey-level. By Step, the esult h() s stoed n ths pocesso. 4. // Stoe h() back to pocesso P.// Thepocesso, whch s stoed the h(), copes t back to pocesso P though the th ow bus. We also have the followng theoem. Theoem [8] Fo an nn mage, the hstogam can be computed n constant tme on a lnea n RAPWBN each bus wth (n ) =c -bt, whee c s a constant and c. 5 Segmentaton based on the hstogam entopy The most mpotant thng egadng segmentaton s to choose a gay-level as a theshold whch dvdes then gay-level mage nto a b-level mage. Let G L = ; ; ;G bethesetofgaylevels and F = [f(x; y)]nn be an mage of sze n = N, wheef(x; y) s the gay-level at (x; y) and f (x; y) G L.LetN be the numbe of pxels each wth the gay-level. Then, GX = N = N: (5) Pun [9, ] and Kapu et al. [7] defned the entopy of an mage (hstogam) as H = GX = p log p ; (6)

6 whee p = N =N. We use the concept as used by Kapu et al. [7] to defne the entopy of the object (black) and the backgound (whte) as follows. The entopy of the black poton of an mage s H B (s) = sx = p P s log p P s ; (7) and the entopy of the whte poton of the mage s H W (s) = GX =s+ p p log ; (8) P s P s whee s s a theshold and P s = P s = p. The total entopy of the pattoned mage s H T (s) = H B (s) +H W (s): (9) Then, fo all s, s G, thesthat maxmzes the H T (s) s the theshold value of the mage. Instead of usng a lnea RAPWBN fo the computatons, we can confgue t nto a two dmensonal RAPWBN though the followng mappng technue. Fst, eset (dsconnect) all swtches. Second, pocesso P sets ts local connecton f+s ; ;]S ; g,fmod n = ; sets ts local connecton f S ; ;]S ; g,fmod n = n ; sets ts local connecton f S ; ; +S ; ;]S ; g,othewse. Thd, each pocesso P, mod n = j sets ts local connecton f S j+;; +S j+;;]s j+;g. Fo the pupose of dffeentaton, the swtches n bus- of all pocessos ae called ow-swtch and those n othe buses ae called column-swtch. Afte the econfguaton, the swtch settng s shown n Fgue (a) fo n = ; by ths way, the onedmensonal aay of pocessos can be vewed as a two dmensonal aay of pocessos, as shown n Fgue (b). Assume each pocesso has thee vaables to keep ts ndces, ; x and y, whee x = n,y = mod n; s used to epesent the ndex of a pocesso n a one-dmensonal aay of pocessos and x and y aeusedtoepesent the ndex of the same pocesso n a coespondng two-dmensonal aay of pocessos. The mappng between and x and y ae one-to-one and onto. Fo calculatng the H B (s); H W (s)and H T (s), we aange the pocessos to compute these entopes as shown n Fgue 4. That s, H B (s) s computed n the lowe tangula pocessos and H W (s) s computed n the uppe tangula pocessos. Assume I s a 4 4mageandI = f(; ; ; ); (; ; ; ); (; ; ; ); (; ; ; )g. A snapshot of algothm SEGMENTATION s shown n Fgue 5. In algothm SEGMENTA- TION, we assume the numbe of pocessos s n + c and the numbe of gay levels G s n. Algothm SEGMENTATION Input : The gay-level gl of each pxel. Output: Abnaymage.. // Compute the hstogam. // By algothm HISTOGRAM, we can compute the hstogam n a lnea RAPWBN. Assume the fnal esults H();H();;H(G )and H(G ) ae stoed n h(; );h(;);, h(;g )and h(;g )local vaables of pocesso P ; ;P ; ;, P ;G and P ;G, espectvely.. // Reconfgue the achtectue. // Map the one-dmensonal aay of pocessos of sze n + c to the two-dmensonal aay of pocessos of sze n n + c though the mappng technue. See Fgue.. // Calculate the pobablty p of each gaylevel. // Each pocesso P ;y ; y<g calculates the pobablty p (;y)= h(;y) N,whee p (;y)sthe local vaable of pocesso P ;y. 4. // Compute the pefx sum of the pobablty p. // Accodng to Lemma, the pefx sum of p (;y); y G can be computed

7 on pocessos P ; ; n. The sum of each pefx s stoed n the local vaables p s (;y)of pocesso P ;y ; y G. 5. // Boadcast the p s.// Each pocesso P ;y ; y<g sends ts p s to pocesso P y;y. Then, pocesso P y;y ; y G, boadcasts p s to pocesso P y;k ; k G. 6. // Boadcast the p.// Pocesso P ;y boadcasts p (;y)to pocesso P x;y ; x; y G. 7. // Compute the ndvdual entopy. // Pocesso P x;y ; x; y G computes the entopy p(x;y) p(x;y) p s(x;y) log p s(x;y) and stoes t n ts local vaable hb(x; y), fx y; computes the entopy (x;y) p ( p s(x;y)) log p (x;y) ( p s(x;y)) and stoes t n ts local vaable hw(x; y), othewse. 8. // Calculate the total entopy H T n each ow. // Compute the total entopy H T (x) fo each x; x G, by the pefx sum on the entopy stoed n each ow usng Lemma. Let H T (x) be stoed n the local vaable ht(x; G ) of pocesso P x;g.thenset ht(x; G ) to ht(x; G ) n pocesso P x;g. 9. // Fnd the maxmum entopy among all H T (x); x G. // a. Pocesso P x;g boadcasts ht(x; G ) to pocesso P x;k ; k G and P x;k stoes t n local vaable ht (x; k). b. Pocesso P x;x boadcasts ht (x; x) to pocesso P k;x ; k G and P k;x stoes t n local vaable ht (k; x). c. Pocesso P x;y ; x; y G, compaes ht (x; y) and ht (x; y) and sets ts flag(x; y) =, f ht (x; y) ht (x; y); sets flag(x; y) =, othewse. d. Usng Lemma, we can compute the pefx sum on the flag stoed n each ow. Pocesso P x;g has the total sum of flag stoed n each ow and f the total sum of flag s eual to G then ths pocesso boadcasts ts ndex x (the theshold value) to all pocessos.. // Obtan the bnay mage. // Pocesso P x;y ; x; y n, compaes ts gay-level value gl(x; y) wth the theshold value and sets ts gay-level to be one, f gl(x; y) s geate than o eual to the theshold value; sets ts gay-level to be zeo, othewse. Theoem Let the sze of an mage be n n and the numbe of gay levels be n. Based on the hstogam entopy of the mage, the RAPWBN of sze n + c can coectly compute the segmentaton n constant tme. Poof: The numbe of pxels wth the same gaylevel, N, s computed n Step. Then, the pobablty of each gay-level, p, s calculated n Step. The pefx sum of the pobablty p, p s, s computed n Step 4. The ndvdual entopy of Es.(7) and (8) s calculated n Step 7. Snce H B (s) s computed wthn the ange to s and H W (s) s computed wthn the ange s + tog, the total entopy of E.(9) fo each s can be computed by the s th ow n Step 8. Then, the theshold value s found n Step 9. Fnally, the bnay mage s obtaned by Step. The tme complexty s analyzed as follows. By Lemma, Step and Step 8 each takes O() tme usng N + c pocessos fo each ow. Othe steps each also takes O() tme. Hence, the total tme complexty s O(). Q:E:D: If the lnea RAPWBN wth n pocessos s confgued to n c c+ n c+ c+ two dmensonal aay, we can compute the segmentaton wth gay-level G n c c+. Ths leads to the followng coollay.

8 Coollay Let the sze of an mage be n n and the numbe of gay levels be n c c+. Based on the hstogam entopy of the mage, the RAPWBN of sze n can coectly compute the segmentaton n constant tme. 6 Image labelng Afte segmentaton, we get a bnay mage whch has only black and whte pxels. Let the black pxels become a goup, f they ae neghbos (by founeghbo defnton). Then, we gve each goup a unue label. The labelng algothm s descbed as followng. Algothm LABELLING Input : Abnaymage. Output: A bnay mage wth ts black pxels labelled.. // Vaable ntalzaton. // Pocesso P x;y ; x; y n local vaable flag(x; y) to zeo. setsthe. // Constuct each goup. // Pocesso P x;y wth a black pxel checks the gay-level of ts fou neghbos and then beaks the connecton wth the pocesso whch has the whte pxel by settng ethe ow-swtch o column-swtch dsconnected.. // Fnd the epesentatve of each goup. // In each goup, we can fnd the pocesso wth the maxmum ndex by usng Lemma. Then, the flag of ths pocesso (the epesentatve of the coespondng goup) s set to be "". 4. // Fnd the label of each goup. // Compute the pefx sum on the flag stoed n each pocesso. Followng Lemma, each pocesso wth flag = obtans a unue numbe (.e. label) to epesent ts coespondng goup. 5. // Boadcast the label by the epesentatve pocesso. // Reconstuct each goup as stated n Step. Then, the pocesso who s the epesentatve of ts coespondng goup boadcasts the label to all membes of ts goup. Theoem Fo an n n bnay mage, the mage labellng poblem can be solved n constant tme on a lnea n RAPWBN each bus wth (n ) =c -bt, whee c s a constant and c. Poof: The coectness of ths algothm can be vefed by Lemma and Lemma. By Lemma, t dentfes the epesentatve of each goup. By Lemma, each goup s labelled unuely. The tme complexty can be easly vefed to be O(). Q:E:D: 7 Concluson The system bus bandwdth detemnes the capacty of data communcaton between pocessos. Accodng to the esults as shown n [, ], we know that the slcon aea used by the swtchng contol mechansm s fa less than that used by the pocesso. Instead of nceasng the numbe of pocessos, we extend the numbe of buses to ncease the powe of a paallel pocessng system. Such a stategy of utlzng the econfguable aay of pocessos wth wde bus netwoks not only has the advantage of savng slcon but also nceases the system powe enomously. In ths pape, to demonstate the powe of the RAPWBN, thee elated mage poblems such as hstogam, segmentaton and labellng ae poposed fo ths machne. All poblems can be solved n constant tme. The RAPWBN s desgned to be sutable fo mage pocessng. We beleve that lots of mage elated poblems can be solved n constant tme n the nea futue.

9 Refeences [] H. M. Alnuwe, "Fast algothms fo mage labelng on econfguable netwok of pocessos," Intenatonal Paallel Pocessng Symposum, pp , 99. [] L. Cnue, S. Levald and A. Rosenfeld, "Fast pyamdal algothms fo mage thesholdng," Patten Recognton, vol. 8, no. 6, pp. 9 96, 995. [] T. W. Kao and S. J. Hong, "Paallel computng two neaest neghbo poblems on a RAP", Seventeenth Annual Compute Scence Confeence, pp. 5 44, Jan [4] T. W. Kao and S. J. Hong, "Computng lst ankng on a RAP wth wde bus netwoks", Poceedng of the 994 Intenatonal Confeence on Paallel and Dstbuted Systems, pp. 8, Dec [5] T. W. Kao and S. J. Hong, "Effcent paallel algothms fo mage pocessng on CRAP", Techncal Repot, Depatment of Electcal Engneeng, Natonal Tawan Insttute of Technology, 99. [6] T.W.Kao,S.J.HongandH.R.Tsa,"Computng connected components and some elated applcatons on a RAP," Poc. Int. Conf. Paallel Poc., vol., pp , 99. [7] J. N. Kapu, P. K. Sahoo and A. K. C. Wong, "A new method fo gay level pctue thesholdng usng the entopy of the hstogam, " Compute Gaphcs, Vson and Image Pocessng, vol. 9, pp. 7 85, 985. [8] S. S. Lee, S. J. Hong, T. W. Kao and H. R. Tsa, "Optmal computng Hough tansfom on a econfguable aay of pocessos wth wde bus netwok," to appea n Patten Recognton, 995. [9] H. L and M. Maesca, "Polymophc-tous netwok," IEEE Tansactons on Computes, vol. 8, no. 9, pp. 45 5, Sep [] H. L and M. Maesca, "Polymophc-tous achtectue fo compute vson," IEEE Tansactons on Patten Analyss and Machne Intellgence, vol., no., pp. 4, Ma [] W. M. Ln and V. K. P. Kuma, "Effcent hstogammng on hypecube SIMD machnes," Compute Vson, Gaphcs, and Image Pocessng, vol. 49, pp. 4-, 99. [] M. Maesca, "Polymophc pocesso aays," IEEE Tans. Paa. and Dst. Sys., 4, pp , 99. [] M. Maesca and H. L, "Connecton autonomy n SIMD computes: a VLSI mplementaton," Jounal of Paallel and Dstbuted Computng, vol. 7, no., pp., 989. [4] R. Mlle, V. K. P. Kuma, D. Ress and Q. F. Stout, "Meshes wth econfguable buses," Poceedngs of the MIT Confeence on Advanced Reseach n VLSI, pp. 6 78, Ma. 988 [5] R. Mlle, V. K. P. Kuma, D. Ress and Q. F. Stout, "Data movement opeatons and applcatons on econfguable VLSI aays," Poceedngs of the IntenatonalConfeence on Paallel Pocessng, vol., pp. 5 8, Aug [6] D. Nassm and S. Sahn, "Data boadcastng n SIMD computes, " IEEE Tans. Comput., vol. C-, pp. -7, Feb. 98. [7] S. Olau, J. L. Schwng and J. Zhang, "Fundamental data movement algothms fo econfguable mesh," Poc. -th Annual Intenatonal Phoenx Confeence on Computes and Communcatons, Scottsdale, Azona, pp , Apl, 99.

10 [8] S. Olau, J. L. Schwng and J. Zhang, "Fast compute vson algothms fo econfguable mesh," Image and Vson Computng, vol., no. 9, pp. 6-66, Nov. 99. [9] T. Pun, "A new method fo gay-level pctue thesholdngusng the entopy of the hstogam," Sgnal Pocessng, vol., pp. - 7, 98. [] T. Pun, "Entopc thesholdng: a new appoach," Sgnal Pocessng, vol., pp. - 9, 98. [] J. Rothsten, "Bus automata, bans, and mental models," IEEE Tansactons on Systems, Man, and Cybenetcs, vol. 8, no. 4, pp. 5 5, Ap. 988.? - j #S ; > S ; +S ; P P P P h h h h h h h h h h h h h h h h Fgue : A lnea RAPWBN of sze 4 wth 4 4 bus netwoks, each bus netwok wth -bt bus wdth. [] A Schuste, "Dynamc econfgung netwoks fo paallel computes: algothms and complexty bounds," Doctoal Dssetaton, Dept. of Compute Scence, Hebew Unvesty, Isael, 99. [] B. F. Wang, G. H. Chen and F. C. Ln, "Constant tme sotng on a pocesso aay wth a econfguable bus system," Infomaton Pocessng Lettes, vol. 4, no. 4, pp. 87 9, Ap. 99. [4] B. F. Wang, G. H. Chen and H. L, "Confguatonal computaton: a new computaton method on pocesso aays wth econfguable bus system," Poceedngs of the Intenatonal Confeence on Paallel Pocessng, pp. III-4 III-49, Aug. 99. f S ;j ; +S ;j ; #S ;j g f #S ;j (k); S ;j ((k + ) mod 4); +S ;j ((k + ) mod 4); k g hhh h f S ;j ; +S ;j g f #S ;j ;+S ;j g f S ;j (); +S ;j (( + ) mod 4); g f g Fgue : Some local swtch confguatons of a RAPWBN.

11 column- column- column- P P P P P 4 P 5 P 6 P 7 P 8 P 9 P P P P P 4 P 5 s = bus- ow- ow- ow- ow- bus-column- bus- bus- bus-4 (a) The swtch settng fo a two-dmensonal aay. s = s = s = Calculate each tem of H W (s). 6 P@ ; P ; P ; P ; P ; P ; P ; ; P ; P ; P P@ ;? calculate each tem of H B(s). Fgue 4: The aangementof the H T (s) calculaton on pocessos.? x - y P P P P P ; P ; P ; P ; P 4 P 5 P 6 P 7 P ; P ; P ; P ; P 8 P 9 P P P ; P ; P ; P ; P P P 4 P 5 P ; P ; P ; P ; (b) The emulated two-dmensonal aay of pocessos. Fgue : The mappng between a one-dmensonalaay of pocessos and a two-dmensonal aay of pocessos.