Computing a complete histogram of an image in Log(n) steps and minimum expected memory requirements using hypercubes

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1 Computng complete hstogrm of n mge n Log(n) steps nd mnmum expected memory requrements usng hypercubes TAREK M. SOBH School of Engneerng, Unversty of Brdgeport, Connectcut, USA. Abstrct Ths work frst revews n lredy-developed, exstng determnstc prllel lgorthm [] to compute the complete hstogrm of n mge n optml number of steps ( log n ) on hypercube rchtecture nd utlzng memory spce on the order of O x log x) where x s the number of gry levels n the mge, t ech processng element. The pper then ntroduces our mprovement to ths lgorthm's memory requrements by ntroducng the concept of rndomzton nto the lgorthm. (. Introducton The frst lgorthm [] to be revewed n ths pper s concerned wth the tsk of computng the complete hstogrm of n gry-level vlues n log n steps. The lgorthm s descrbed for hypercubes nd computes the complete hstogrm n log n tme ndependent of the rnge of gry level vlues. The computton of the complete hstogrm of n such vlues tkes plce n seres of log n steps; fter whch, the hstogrm for vlue cn be found n the lowest-ddressed processor whose ddress ends n. The lgorthm mkes use of the ssocton of suffxes of dt vlues of ncresng wdth wth suffxes of processor ddresses. We shll begn by defnng the hstogrm of n mge nd the uses of the hstogrm n dfferent mge processng pplctons, then we shll defne the SIMD hypercube multprocessor nd descrbe ts nterconnectons. The lgorthm n [] wll be revewed fter tht. Fnlly, we present n mprovement to ths lgorthm s memory requrements v the usge of rndomzton.. The gry level hstogrm. One of the smplest nd most useful tools n dgtl mge processng s the gry level hstogrm. The gry level hstogrm s functon showng, for ech gry level, the number of pxels n the mge tht hve tht gry level. The bscss s the gry level nd

2 the ordnte s the frequency of occurrence (number of pxels). Whle the hstogrm of ny mge contns consderble nformton, certn types of mges re completely specfed by ther hstogrms. When n mge s condensed nto hstogrm, ll sptl nformton s dscrded. There re mny uses for the gry level hstogrm. One mportnt use s n dgtzng prmeters, whch s due to the fct tht the hstogrm ndctes whether or not n mge s properly scled wthn the vlble gry level. Another mportnt use s n boundry threshold selecton, s contour lnes provde n effectve wy to estblsh the boundry of smple object wthn n mge. The contours my be, for exmple, the 'dp' between two peks n the hstogrm n the cse of lght re wthn drk re or vce vers. The re nd the ntegrted optcl densty of smple object cn be computed from ts mge hstogrm too. 3. The SIMD hypercube mcroprocessor. k k A hypercube of dmenson k hs nodes nd k edges. A hypercube of dmenson s nlogous to squre, the hypercubes of dmenson d 3 cn be recursvely defned s obtned from two hypercubes of dmenson ( d ) ech, by connectng correspondng nodes of the two hypercubes. Tht s, two cells shre drect connecton f nd only f ther correspondng hypercube vertces re connected by hypercube edge. Furthermore, we cn see tht two cells wll shre drect connecton f nd only f ther ddresses dffer n exctly one bt poston (.e., one n ech dmenson). In the SIMD multprocessor model (sngle-nstructon, multple dt), whch s our model, ll processng elements execute sequence of nstructons, sent from one controller. 4. The Algorthm. Ths secton explns nd revews the lgorthm ntroduced n [], so tht the reder of ths rtcle cn follow lter the development of our own contrbuton, whch s nmely, rndomzng the lgorthm. The hstogrm whch s to be computed wll be represented s set of ordered prs, ech pr wll contn n ndex (represented n bnry; for exmple, m, m,..., 3,,, wll represent n ndex of length m btes, whch mples tht the m mxmum number of gry levels llowed s ), nd count, whch s the hstogrm vlue of the correspondng ndex (tht s the number of pxels whch hve the ndex vlue s ts gry level). 4.. Intl confgurton. Intlly ech processng element (PE) n the hypercube nclude one nd only one pr, whch s n fct one pxel vlue, whch mens tht the count component of the pr hve the vlue "" throughout the hypercube. Ths my be consdered s f ech PE of the

3 hypercube contns hstogrm consstng of sngle pr, whch s obtned by prng the gry level vlue (ndex) n the cell wth the count "". The gol my be consdered then to 'combne' ll those hstogrms n dfferent cells to form the complete hstogrm of the mge, whch s to be dstrbuted n some resonble wy throughout the hypercube n order to be retreved esly. 4.. The bsc de behnd the lgorthm n []. The bsc de behnd the lgorthm s very smple, t s n fct the de of 'combnng' ll the vlues dstrbuted throughout hypercube nto sngle processng element n log n steps, where n s the number of cells wthn the hypercube multprocessor. Combnng the set of vlues n the hypercube s very smple nd stndrd procedure. It cn be descrbed usng smple lgorthm consstng of loop tht s to be performed k tmes, where k s equl to the quntty log n. Durng terton number j the vlue tht s stored n the processor k,...,,, j,..,,,, s to be sent to the processor k,...,,, j,..,,,, whch s done n exctly one tme step, s ths s hypercube edge. Then, combned wth the vlue stored n the ltter processor, t cn be seen tht fter k steps the 'combnton' of ll the elements tht were orgnlly dstrbuted throughout the hypercube wll be found n sngle processor nmely the processor k,k,k 3,...,,, 4.3. The problems rsng when usng the bsc lgorthm. In [], t cn be redly seen tht the prevous sort of lgorthm cn be redly ppled to lrge clss of problems, nmely the clss of problems where the mount of storge tht s requred to store the combned vlue fter the current terton does not ncrese or ncrese sgnfcntly, but does so wth slow rte s more vlues re combned. If ths ws not true, the mount of storge requred would be n fct exponentl. One such problem my be the ddton problem. For the problem of combnng hstogrms, ths bsc lgorthm wll not be sutble, becuse the output of the operton of combnng two hstogrms my be twce s lrge s ether of the orgnl hstogrms. The consequence 3

4 of ths wll be the exponentl growth n the storge requred for the hstogrm n gven processng element The descrpton of the lgorthm The mn de behnd the lgorthm tht the uthors developed n [] s to try to llevte the problem wth the bsc lgorthm of the exponentl growth n memory requrement t the processng elements by llowng the hstogrm nformton to remn dstrbuted to certn degree whle stll ggregtng t n useful wy n seres of log n steps. Ths s to be performed usng the followng lgorthmc steps, presented n []: The lgorthm s stll loop wth k tertons wth k = log n. At ech terton j of the loop durng the frst m steps, the followng s to be performed. ( n = number of PE's;.e., number of pxels, m = log (number of gry levels)) All the prs wth ndex (n bnry): m m,...,,, j,...,,, whch re n processng element n the hypercube whose ddress s: k,...,,, j,...,,, re sent to the PE n the hypercube whose ddress s:,...,,, j,..,,, At the sme tme, n complementry fshon, ll prs wth the ndex: m, m,..., j +,, j,...,, whch re n processng element n the hypercube whose ddress s:,...,,, j,...,,, re sent to the PE n the hypercube whose ddress s: k,...,,, j,...,,, 4

5 At the very frst terton of the loop, every pr n the hypercube whose ndex ends wth zero, whch s n PE whose ddress, s:,,...,, wll be send to the PE, k,...,, lso ny pr wth n ndex endng n one, whch s n PE whose ddress s: wll be send to the PE,,...,, Durng the second terton, every pr n the hypercube whose ndex ends wth,, whch s n PE whose ddress s:, k,...,, k k,...,,,, wll be send to the PE, k,...,,,, lso ny pr wth n ndex endng n,, whch s n PE whose ddress s: k,...,,,, wll be send to the PE, k,...,,, It cn be redly notced tht some use s beng mde of the ssocton of suffxes of the ndces nd suffxes of the PE ddresses. After the vlues re sent, whenever two prs wth the sme ndex (.e., two pxels wth the sme gry level) re collected n the sme PE, they re to be combned to form one pr wth the sme ndex nd wth the count vlue equl to the sum of count vlues of ech ndvdul pr, thus formng the hstogrm. At frst glnce, t mght seem tht the problem of the exponentl growth n memory requrements t ech PE stll exsts, due to fcts:. The number of bts requred for the count ncreses by one ech tme combne s performed.. The possblty tht mny prs wth ndces whch re the sme n the lst severl bts my be ntlly locted n processng elements such tht they ll hppen fterwrds to grvtte to sngle cell, whch mples tht the possblty for exponentl growth n the number of prs stored n prtculr processng element stll exsts. However, when one tkes closer look t the operton of ths lgorthm, the stuton wll turn to be much better thn t frst ppered to be. Regrdng the frst concern - the ncrese of the number of bts needed for the count by one fter ech terton - t cn be seen tht there s no need to store fter terton j the lst j + 5

6 bts of ny ndex, snce these bts wll t tht tme be gven by the lst j + bts of the processng element ddress whch contns tht ndex. It s true tht the number of bts needed to store the count wll ncrese by one fter ech terton but t the sme tme the number of bts requred to store the ndex wll decrese by one. Thus, the totl number of bts requred to store pr wll remn constnt throughout the whole lgorthm. The vlue of ths constnt s smply equl to the number of bts requred to store the gry level plus one (.e.: ( m + )) The rgument bout the exponentl growth n the number of prs to be stored t ech processng element wll lso be found to be not exctly the cse, nd ths s dscussed by the uthors n detls n [] The forml lgorthm The followng s the fnl lgorthm n [], descrbed n pseudo-code lke lnguge. Defne n = number of nodes n hypercube (number of pxels). k = log n. x = number of gry levels. m = log x. For j = to ( m ) do (***) Send ll the prs wth ndex m m,...,,, j,...,,, whch re n processng element n the hypercube whose ddress s k,...,,, j,...,,, to the PE n the hypercube whose ddress s,...,,, j,...,,, In prllel, send ll prs wth the ndex m m,...,,, j,...,,, whch re n processng element n the hypercube whose ddress s 6

7 ,...,,, j,...,,, to the PE n the hypercube whose ddress s k,...,,, j,...,,, If two prs hve the sme ndex re collected n the sme PE then they re to be combned to form one pr wth the sme ndex nd wth the count vlue equl to the sum of count vlues of ech ndvdul pr ; end For ; If k = m then STOP else (###) For j = m to ( k ) Do Send the count vlue stored n the PE whose ddress s k,...,,, j,...,,, to the PE whose ddress s,...,,, j,...,,, sum both count vlues nd leve the combned hstogrm n the ltter PE. end For ; end If. The lgorthm ssumes tht the number of bts to represent the gry levels re never more thn the number of bts requred to represent the number of pxels n n mge, whch s logcl nd true ssumpton for nerly ll relstc stutons n computer vson pplctons Complexty nlyss of the determnstc lgorthm. It cn be seen tht wth regrds to the tme complexty of the lgorthm, tht the lgorthm runs exctly n log n loop tertons, the tme for ech loop terton my be consdered s one tme step (ncludng routng nd the combnng opertons performed wthn processng element), thus t s log n tme complexty process. Wth regrds to spce complexty, the mxmum spce requred t sngle PE wll be n O( ) spce c 7

8 per cell where c s the number of bts requred to store one of the hstogrm domn vlues. To be more specfc, the lgorthm requres (log x + ) log x spce per PE where x s the dscretzed number of gry levels. 5. Comments bout the determnstc lgorthm The presented lgorthm n [] descrbes n ntellgent pproch to solvng the problem of fndng the hstogrm of n mge usng populr prllel rchtecture n number of steps equl to the dmeter of the hypercube ( log n ), whch s the best complexty tht cn be hoped for. The lgorthm mkes use of the ssocton of dt vlues (gry level vlues) nd PE ddresses to keep growng collecton of vlues dstrbuted so tht ther growth s mngeble. The spce complexty s descrbed bove c s O ( ). It my be possble to reduce the spce by the use of some encodng technques durng the storge of hstogrms tht result whle tertng. However, the lgorthm would then hve to be modfed ccordngly nd ts behvor my become more complcted. The lgorthm works ncely for the cses when m k, s the forml descrpton of the lgorthm mply. 6. The Rndomzed verson of the lgorthm A rndomzed lgorthm s best descrbed s n lgorthm where some of the decsons re mde bsed upon the outcome of con flps. The de s to prove tht the lgorthm wll behve n certn mnner wth hgh probblty. In our problem, we re concerned wth provng tht rndomzng the lgorthm wll mke the memory requrements t ech processng element less thn those for the orgnl lgorthm wth hgh probblty. Typclly, we men probblty n for ny >. Generlly spekng, we cn dvde rndomzed lgorthms nto two clsses. The frst clss s one n whch the output s correct wth hgh probblty nd s gurnteed to use certn mount of resources. In the second one the lgorthm s gurnteed to produce correct output, usng certn mount of resource wth hgh probblty. The frst clss s clled "Monte Crlo" lgorthm, the second s clled "Ls Vegs" lgorthm, our new lgorthm s of the "Ls Vegs" type. 8

9 7. The Rndomzton Scheme. The rndomzton scheme we descrbe s bsclly method to chnge the step mrked (***) n the old lgorthm to be rndomzed routng step. Ths s nsted of sendng ll the prs to the specfc processor,...,,, j,...,,, we send them rndomly n frst phse to the complement) k j processors ( c mens (), k,...,,, j,...,, (), k,..., c,, j,...,, ( k j ), k c,...,,, j,...,, ( k j ) c, k,...,,, j,...,, nd then, second phse, sends ll 'msplced' prs ;.e., prs wth zero n ther j th poston n PE's ( k j ) to the proper PE's ( ' ), k,..., c,, j,...,, ( ( k j )' ), k c,...,,, j,...,, ( ( k j)' ) k c, k,..., j +,, j,...,, Note tht ths scheme comples wth the de behnd the orgnl lgorthm regrdng tht the suffxes of the ndces of the prs s the sme s the suffxes of the processng elements ddresses, wth the dvntge tht the prs re dspersed wthn the hypercube more thn before. 9

10 8. Probblstc Anlyss of Memory Requrements. An nlyss of the memory loctons requred t ech PE s to follow. The nlyss s done for specfc processng element [ y ] t the frst step of the rndomzton procedure, the sme nlyss could be ppled to ll PE's wthn the hypercube wthout loss of generlty, the sme method could then be used for the followng tertons tll j = m. The fct tht the dstrbutons used to clculte the number of prs tht re expected to be t certn processor (n the probblty tree) re bnoml dstrbutons nd tht the sum of k j such dstrbutons s stll polynoml rndom vrble could be ppled to use Chernoff's lemm. Assume X s the rndom vrble of the number of prs (memory loctons needed), nd m s the expected vlue we clculted, then: E ( X ) = X Imples (snce X s polynoml) tht the Probblty ( X ( + epslon)m ) s m epslon e << n where > Whch stsfes tht the memory requrements re less thn m wth probblty n, for ny >, s descrbed before n the rndomzton crter. 9. Complexty nlyss of the rndomzed lgorthm It could be clerly seen tht the memory requrements hs decresed sgnfcntly when pplyng ths rndomzton scheme. The number of bts for every pr s stll equl to log x + t every tme durng the tertons. Due to the 'blnce' we dscussed n the probblstc nlyss, the number of prs wll lwys be close to constnt. Even f the number of prs ncresed to be proportonl to ( log x ), ths wll defntely be better thn x. One my rgue tht sendng the prs to more thn one processor for rndomzton purposes wll ncrese the tme complexty to be more thn O( log n ) steps, but ths s not true becuse the number of prs wll lwys be close to constnt s we proved. Thus, the totl memory requrements wll be O( C log x + (log n log x) ) bts. The second term s for the count ncrese by one for the loop (###) fter the ndex suffx s scnned for ll the PE's. The tme complexty remns O( log n ). As n exmple, for mges tht we used n the orgnl lgorthm the memory requrements were 44 bts for ech PE. For the rndomzed lgorthm, the memory requrements wll be equl to C ( log 56 + ) + ( log log 56)

11 I.E., C 9 + 8, whch, f C ws even equl to log x, wll be equl to 8 bts only for ech PE wth very hgh probblty nd the effect wll be more when the number of gry levels n n mge s more.. Summry nd Comments The presented rndomzed lgorthm descrbes method of sound mprovement wth regrds to memory requrements when compred to the determnstc lgorthm. The new memory requrements could be used t ech PE nd one cn be sure tht they wll suffce wth very hgh probblty. Rndomzed lgorthms provde lower bounds for resources lke tme nd memory thn those provded by determnstc lgorthms n mny cses, especlly n routng nd routng-relted problems. The tme complexty remns optml nd of the order of the dmeter of the hypercube. References: [] Bestul,T.,nd Dvs,S.L., "On Computng Complete Hstogrms of Imges n Log(n) steps usng hypercubes", IEEE Trnsctons on Pttern Anlyss nd Mchne Intellgence, vol., no., Feb [] Bllrd, D. H.,nd Brown, C. M., "Computer Vson", Prentce Hll, 98. [3] Hwng, K., nd Brggs, F. A., "Computer Archtecture nd Prllel Processng", McGrw Hll, 984. [4] Rjsekrn, S.,nd Tsntls, T.,"Optml Routng Algorthms For Mesh-Connected Processor Arrys", Algorthmc, Volume 8, Number, pp -38, Jnury 99. [5] Vlnt, L. G., "A Scheme For Fst Prllel Communcton", SIAM Journl on Computng, Volume, Number, pp , 98. [6] Rbn, M. O., "Probblstc Algorthms", n Algorthms nd Complexty, J. F. Trub, Ed., Acdemc Press, pp. -4, 976.