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1 Applcaton o paralll rson two-dmnsonal ast Forr transorm alorthm analo o th Cool-Tk alorthm or dtal ma procssn o satllt data Mkhal osko * Valr Ttatchko Mkhal Lapchk Marna Ralna and Tatana Yamskkh Sbran Fdral Unrst Insttt o Spac and Inormaton Tchnolo Sobodn prospct 79 Krasnoarsk 664 Rssan Fdraton Omsk Stat Pdaocal Unrst abrzhnaa Tkhachskoo 4 Omsk Rssan Fdraton Abstract. In modrn sstms o rmot snsn two-dmnsonal ast Forr transorm FFT has bn wdl sd or dtal procssn o satllt mas and sbsqnt ma ltrn. Ths artcl prods a paralll rson two-dmnsonal ast Forr transorm alorthm analo o th Cool-Tk alorthm and ts mplmntaton or procssn th satllt ma o Krasnoarsk and ts sbrban aras. Introdcton At prsnt Earth rmot snsn s closl rlatd to dtal ma procssn as th arospac mas that ar rch n dtal ar commonl rprsntd n dtal orm o a rastr tp. Adjstmnt o contrast and brhtnss spatal ltrn Forr transorm and sbsqnt rqnc ltrn ar otn sd to mpro ma qalt []. Th tradtonall appld alorthm or comptn two-dmnsonal ast Forr transorm FFT s th sqntal applcaton o a on-dmnsonal FFT rst or all rows thn or all colmns. Th artcl dscrbs a rson two-dmnsonal ast Forr transorm alorthm analo o th Cool-Tk alorthm wth th rdcd nmbr o compl opratons compard to th tradtonall sd alorthm. A arant o th alorthm paralllzaton to acclrat calclatons s shown n th artcl. Th s o th alorthms to prorm ma procssn on dtal mas s consdrd. Th ltrn procdr s mplmntd n sral stas: Data radn and swappn Paralll comptaton o th Cool-Tk alorthm Fltrn th obtand Forr transorm Rrs FFT Calclaton Ima acqston * Corrspondn athor: mnosko@and.r Th Athors pblshd b EDP Scncs. Ths s an opn accss artcl dstrbtd ndr th trms o th Crat Commons Attrbton Lcns 4.

2 Dtald dscrpton o ths stas can b ond n []. A two-dmnsonal analo o th Cool-Tk alorthm s consdrd n [] ts mltdmnsonal arant n [3] and a mor dtald dscrpton o th alorthm s prodd n [4]. In ths papr w prsnt a paralll two-dmnsonal rson o th Cool-Tk alorthm. Frst w consdr two dmnsonal analo o th Cool-Tk alorthm or comptn th ast Forr transorm. Th Cool-Tk alorthm Gn a ncton o two-dmnsonal prodc snal wth als n th compl spac whr = s - s s a post ntr. Grascal ma that s s pls n hht and wdth can b takn as an ampl o sch a ncton. Th al o th ncton s a compl nmbr whch ral componnt s qalnt to th brhtnss als o th corrspondn pl wth coordnats n th ran -55 and th manar componnt o th nmbr s zro. Thn th Forr transorm F o ths snal can b rprsntd as ollows: F whr = s = -. W conrt ormla as ollows: th coordnats and ar dcomposd nto n and odd parts. Thn th ntal sm can b ddd nto or qatons: F Each o th sms obtand n s a FFT snal wth th lmnts or s s n or odd coordnats o th sorc snal Thn n th nt stp w dd th coordnats and nto two qal-szd sbsts and + + rspctl = -. Thn or th actors and rom th scond sbst w t: 3.

3 W pl 3 nto th ormla to t: F F F F whr = -. Formla 4 dscrbs th two-dmnsonal ast Forr transorm battrl n analo o Cool-Tk alorthm. It s schmatcall dpctd n Fr. Wth ths bttrl w can splt th sorc snal and th Forr transorm wth lmnts nto or sb-snals ach wth th nmbr o lmnts s s [4]. Ths rdcs th nmbr o mltplcatons and addtons o compl nmbrs rqrd to compt FFT. 4 F.. Two-dmnsonal ast Forr transorm battrl n analo o Cool-Tk alorthm Dcomposton 4 can b appld to ach sb-snal sral tms ntl w t sb-snals o 4 lmnts whch FFTs ar bn comptd drctl. Thn or th snal o th lmnts th total nmbr o mltplcatons rqrd to compt compl nmbrs wll b 3 lo and th nmbr o addtons - [3]. 4 lo For comparson th standard mthod o comptn two-dmnsonal FFT b rows and colmns wll rqr lo mltplcatons and lo addtons. 3

4 Th paralll Cool-Tk alorthm To tst th rnnn tm o an alorthm or calclatn two-dmnsonal ast Forr transorm alorthm analo o th Cool-Tk alorthm a smlaton proram was wrttn n th C++ prorammn lana [4]. Two prncpal mthods o alorthm paralllzaton wr sd: OpnMP tartd toward s on shard mmor sstms and MPI or dstrbtd mmor sstms. Tstn was carrd ot on th clstr nod o th SFU sprcomptr wth IBM HS XM Xon Qad cor E GHz 64 Gb RAM [5]. Th rslt o tstn or a sstm wth dstrbtd mmor on a snl clstr nod s prsntd n Tabl. Th two-dmnsonal analo o th Cool-Tk alorthm procsss th snal wth s s th nmbr o sampls n s tratons. In th rst traton d to th prlmnar prmtaton o th lmnts th snal data s ddd nto qads o rlatd rtcall and horzontall lmnts spacd b on lmnt rom ach othr; n th scond traton nto qads o lmnts sparatd b two lmnts rom ach othr; n th thrd traton - b th lmnt; n th last s- traton b- s- lmnt. For sch mplmntaton o th alorthm n shard mmor sstms paralll data strctrs ar dsnd b splttn mltpl data n ach traton nto arras o ntrconnctd sts o th lmnts or ach sparat thrad. In a dstrbtd mmor sstms a smlar parttonn nto sts o rlatd data occrs at ach traton wth sbsqnt transr o data btwn th procsss or ndpndnt comptatons. Tabl. Th rslt o tstn a two-dmnsonal paralll analo o th Cool-Tk alorthm n a sstm wth dstrbtd mmor n sconds Sz 4*4 48*48 496*496 89*89 Th nmbr o D FFT or rows and D Cool-Tk FFT procsss colmns Th data ar prsntd raphcall n Fr whr th FFT alorthm b rows and colmns s dnotd b FFT RC and th analo o th Cool-Tk alorthm b FFT CT. 4

5 F.. Comparson o rnnn tm or aros alorthms Ima ltrn LandSat-8 ma o Krasnoarsk and ts srrondn aras datd Aprl 7 6 [6] whch s shown n Fr on th lt was sd as a tst snal. Th ornal ma rsolton s 88 * 87 pls t was conrtd to th narst powr o two: and thn scald or th powrs -5. On th rht sd o th r 3 th rslt rom hh-pass ltrn s shown. In ths cas th contors ar mor d: th rr ln th bondars o th rock aras. F. 3. Ornal ma - Th rslt rom hh-pass ltrn Th ornal ma s shown on th lt sd o th r 4. Th rslt rom low-pass ltrn s on th rht sd. In ths cas small sharp chans n montanos trran ar not so notcabl on th nral backrond that s small dtals ha bn rmod. 5

6 F. 4. Ornal Ima - Rslt rom low pass ltrn Conclson It s shown that th paralll analo o th Cool-Tk alorthm or comptn twodmnsonal FFT s ctd on ara 4 tms astr than th standard mthod or comptn two-dmnsonal ast Forr transorm b rows and colmns. Rrncs. R.C. Gonzalz and R.E. Woods Dtal Ima Procssn Scond Edton Parson Edcaton Inc 4. R. Blaht Fast alorthms or dtal snal procssn Addson-Wsl Pblshn Compan V.S. Ttatchko O.I. Ksl M.V. osko Calclatn th n-dmnsonal Fast Forr Transorm PRIA V.S. Ttatchko Two-dmnsonal ast Forr transorm Procdn o th Intrnatonal Form on Stratc Tchnolo IFOST Sprcomptr compl o SFU Accss mod: 6. LandSat-8 ma o Krasnoarsk and ts srrondn aras Accss mod: 6