--- /dev/null
+\input lecnotes.tex
+\prednaska{8}{Fourierova transformace}{(K.Jakubec, M.Polák a G.Ocsovszky)}
+
+Násobení polynomù mù¾e mnohým pøipadat jako pomìrnì (algoritmicky) snadný problém. Asi ka¾dého hned napadne "hloupý" algoritmus - jednodu¹e vezmu koeficienty prvního polynomu a ka¾dým z nich pøenásobím v¹echny koeficienty toho druhého. Pokud øád prvního polynomu je $n$ a druhého $m$, tak èasová slo¾itost nám vyjde nìco jako $O(mn)$. To není a¾ tak ¹patné, v nejhor¹ím se dostaneme na $O(n^{2})$ (pokud $m = n$). Na první pohled se mù¾e zdát, ¾e rychleji to prostì nejde (pøeci musím v¾dy vynásobit "ka¾dej s ka¾dým"). Ve skuteènosti to ale rychleji fungovat mù¾e...ale k tomu je potøeba znát trochu tajemný algoritmus FFT neboli {\I Fast Fourier Transform}.
+
+
+\s{ Trochu algebry na zaèátek: }
+
+Libovolný polynom P øádu n mù¾eme mít reprezentován 2 rùznými zpùsoby:
+
+\itemize\ibull
+\: jeho koeficienty, èili èísly $a_{0}, a_{1}, \ldots ,a_{n}$
+\: jeho hodnotami v n + 1 rùzných bodech $x_{0}, x_{1}, \ldots , x_{n+1}$ èili èísly $P(x_{0} ), P(x_{1} ), \ldots ,P( x_{n+1} )$
+\:
+\endlist
+
+Pov¹imnìme si jedné skuteènosti - máme-li 2 polynomy $A$ a $B$ øádu $n$ a $x_{0}\ldots, x_{k}$ body, pak platí $C(x_{k}) = A(x_{k}) * B(x_{k}), k = 0,1,2 \ldots n+1.$ Toto èiní tento druhý zpùsob reprezentace polynomu velice atraktivním pro násobení. Problémem je, ¾e typicky máme polynom zadaný koeficienty a ne hodnotami v bodech. Tím pádem potøebujeme nìjaký hodnì rychlý algorimtus (= rychlej¹í ne¾ kvadratický, jinak bychom si nepomohli oproti hloupému algoritmu) na pøevod polynomu z jedné reprezentace do druhé a zase zpìt.
+
+Dále bychom si mìli uvìdomit, ¾e stupeò na¹eho výsledného polynomu C bude $\leq 2n+1$ (kde $n$ je stupnìm výchozích polynomù). To snad netøeba nijak vysvìtlovat, ka¾dý si to snadno ovìøí, jen dodáme, ¾e pokud chceme polynom C reprezentovat pomocí jeho hodnot v bodech, musíme vzít $2n + 2$ bodù. Tímto konèí malá algebraická vsuvka.
+
+\s{ Idea, jak by mìl algorimus pracovat: }
+\itemize\ibull
+\: vybereme 2n + 2 bodù $x_{0}, x_{1}, \ldots , x_{2n+2}$
+\: v tìchto bodech vyhodnotíme polynomy $A$ a $B$
+\: nyní ji¾ v lineárním èase získám polynom C (viz vý±e)
+\: invernznì pøevedu hodnoty polynomu C v 2n+2 bodech na jeho koeficienty
+\endlist
+
+Je asi vidìt, µe klíèové jsou kroky 2 a 4. Vybrání bodù jistì stihneneme pohodlnì v lineárním èas a vynásobení samotných hodnot té¾ (máme 2n+2 bodù, a $C(x_{k}) = A(x_{k}) * B(x_{k}), k = 0,1,2 \ldots 2n+2$, tak¾e na to nepotøebujeme více ne¾ $2n+2$ násobení).
+
+Celý trik spoèívá v chytrém vybrání onìch bodù, ve kterých budeme polynomy vyhodnocovat. Je na to potøeba vìdìt pár zajímavostí o komplexních èíslech, na stránce Matrina Mar¹e jsou k dispozici slajdy, zde to bude zapsáno o trochu struènìji.
+
+\s{ Vyhodnocení polynomu metodou rozdìl a panuj (algoritmus FFT): }
+Mìjme polynom P øádu $n$ a chceme jej vyhonotit v $n$ bodech. Vybereme si body tak, aby byly spárované, èili $\pm x_{0}, \pm x_{1} \ldots \pm x_{n/2-1} $. To nám výpoèet urychlí, proto¾e pak se druhé mocniny $x_{i}$ shodují s~druhými mocninami $-x_{i}$.
+
+Polynom P rozlo¾íme na dvì èásti, první obsahuje èleny se sudými koeficienty, druhá s lichými:
+
+$P(x) = p_{0}x^{0} + p_{2}x^{2} + \ldots + p_{n-2}x^{n-2} + p_{1}x^{1} + p_{3}x^{3} + \ldots + p_{n-1}x^{n-1}$
+
+$S(x^{2}) = p_{0}x^{0} + p_{2}x^{2} + ... + p_{n - 2}x^{n - 2}$
+$L(x^{2}) = p_{1}x^{1} + p_{3}x^{3} + ... + p_{n - 1}x^{n - 1}$
+
+Tak¾e obecnì $P(x) = S(x^{2}) + xL(x^{2})$ a $P(-x) = S(x^{2}) - xL(x^{2})$
+Jinak øeèeno, vyhodnocování P(x) v $n$ bodech se nám smrskne na vyhodnocení $S(x)$ a $L(x)$ (oba mají polovièní stupeò ne¾ P(x)) v $n/2$ bodech (proto¾e $(x_{i})^{2} = (-x_{i})^{2}$).
+
+Pøíklad:
+$3 + 4x + 6x^{2} + 2x^{3} + x^{4} + 10x^{5} = (3 + 6x^{2} + x^{4}) + x(4 + 2x^{2} + 10x^{4}):$
+
+
+Teï nám ov¹em vyvstane problém s oním párováním - druhá mocina pøece nemù¾e být záporná a tím pádem v u¾ v druhém levlu rekurze body spárované nebudou. Z tohoto dùvodu musíme pou¾ít komplexní èísla - tam druhé mocniny záporné býti mohou. Jako $x_{0} \ldots x_{n-1} $ si zvolíme n-tou komplexní odmocninu z 1. Máme n n-tých odmocnin z 1, rovnomìrnì rozesetých po jednotkové kru¾nici, búno $n=2^{k}, k \in N$ (jinak viz slajdy Martina Mare¹e). Jednotlivé odmociny vypadají takto: $1, \omega, \omega^{2} \ldots \omega^{n - 1} $, kde $\omega = e^{2 \pi i/ n}$. 2 poznámky:
+\itemize\ibull
+\: n-té odmocniny z 1 jsou spárované, èili $\omega^{j} = -\omega^{n/2 + j} $
+\: umocníme-li v¹echny na druhou, vznikne nám n/2 n-pùltých odmocnin z 1, které jsou i nadále spárované
+\endlist
+
+\s{Tak a teï koneènì ten slavný algoritmus: }
+FFT(P, $ \omega$)
+Vstup: $p_{0} \ldots p_{n-1}$ to jest vektor koeficientù polynomu P a $\omega$ co¾ je n-tá odmocina 1
+Výstup: Hodnoty polynomu v bodech $1, \omega, \omega^{2} \ldots \omega^{n - 1}$ èili èísla $P(1), P(\omega), P(\omega^{2}), \ldots P(\omega^{n - 1})$
+\algo
+\: pokud n = 1, vra» $P_{0}$ a konec
+\: jinak rozdìl P na sudé a liché koeficienty a zarekurzi se FFT(S, $\omega^{2}$) a FFT(L, $\omega^{2}$)
+\: pro $j = 0 \ldots n - 1$ spoèítej: $P(\omega^{j}) = S(\omega^{2j}) + \omega^{j} * L(\omega^{2j})$
+
+\endalgo
+
+
+\s{ Èasová slo¾itost:}
+ $T(n)=2T({n \over 2} ) + O(n) \Rightarrow$ slouitost $O(n)$ ... stejnì jako MergeSort.
+
+
+
+
+Máme tedy algoritmus, který "pøevede" koeficienty polynomu na hodnoty tohoto polynomu v námi zadaných bodech. Ale potøebujeme také algoritmus, který doká¾e reprezentaci polynomu pomocí hodnot pøevést zpìt na koeficienty polynomu. Tedy nìjaký inverzní algoitmus.
+Definuje me si DFT algoritmus, která vyu¾ívá maticovou reprezentaci a s jeho¾ pomocí získáme hledaný algoritmus.
+
+\s{Definice:}
+
+ {\sc Diskretní Fourierova transformace} $(DFT)$
+je funkce $f: { C ^n} -> { C ^n}$ , ze $y=f(x) \equiv \forall j \ y_{j} = \sum \limits ^{n-1}_{k=0} x_{k} . \omega ^{k} $
+
+\s{Poznámka:}
+
+Vezmeme polynom, který má $x_{kj}$ jako koeficienty a vyhodnotíme ho v bodì $\omega ^{j} [y_{j} = x(\omega^{j})] => {f}$ je linearní $=>$ mù¾eme napsat $f(x) = \Omega_{x} ,\ \Omega _{jk} =\omega ^{jk}$, kde $\Omega$ je matice
+
+
+\s{Jak najít inverzní matici?} Víme ¾e $\Omega =\Omega ^{T}$ protoue $\omega ^{jk} = \omega ^{kj}$ .
+
+\s{Jak vypadají øádky této matice?} Vyu¾ijeme následující lemma, které si ale napøed doká¾eme :)
+
+\s{Lemma:}
+
+\proof souèin
+$$\Omega _{j} \Omega _{k} = \sum \limits ^{n-1}_{l=0} \Omega _{jl} \overline{\Omega _{kl}} = \sum \limits _{l} \omega ^{jl} \overline{\omega ^{kl}} = \sum \limits _{l} \omega ^{jl} \omega ^{-kl} = \sum \limits _{l} \omega ^{(j-k)l } = \sum \limits ^{n-1}_{l} (\omega^{j-k}) ^{l} $$
+Proto¾e $ \overline{\omega^{kl}} = \overline{\omega} ^{kl} = {({1 \over \omega} )}^{kl} = \omega ^{-kl}$.
+
+\itemize\ibull
+\: Pokud $j\neq k$, pou¾ijeme vzoreèek pro souèet geometrické posloupnosti, kde $a_{1}=1$ a $q=\omega ^{(j-k) }$ a dostaneme ${{\omega^{(j-k)n} -1} \over {\omega^{(j-k)} -1}} ={1-1 \over @ -1} = {0 \over \neq 0} = 0$
+
+\: Pokud $j=k \sum \limits ^{n-1}_{l=0} (\omega ^{0}) ^{l} = n$.
+\endlist
+\qed
+
+a nyní slibované a u¾ i dokázané lemma
+
+\s{Lemma:} $\Omega _{j} . \Omega _{k} =$
+
+\itemize\ibull
+\: $0$ pro $j\neq k$
+
+\: $1$ pro $j=k$
+\endlist
+
+\s{Dùsledek:}$$\Omega*\overline{\Omega} = nE $$
+
+
+\>Jedná se o skalární souèin (jako pøedtím, èile prvek na pozici $ij$ je $0$ nebo $n$) $=>\Omega^{-1} = {1 \over n} \overline{\Omega}$
+
+
+\>na¹li jsme inverzi
+
+$\Omega({1 \over n} \overline{\Omega}) = {1 \over n}\Omega*\overline{\Omega} = E$
+
+$\Omega^{-1}_{jk} = {1 \over n}\overline{\omega^{jk}} = {1 \over n}\omega^{-jk} = {1 \over n} {(\omega^{-1})}^{jk} $
+
+kde $\omega^{-1}$ je $\overline{\omega}$
+
+
+\>Ná¹ algoritmus poèítá tedy i inverzní transformaci, akorát místo $\omega_n$ pou¾ijeme $\overline{\omega_n}$ a vydìlíme $n$. Co¾ je skvìlé - staèí znát pouze jeden algoritmus u kterého staèí v jednom pøípadì po¾ít jinou matici a vydìlit $n$!
+
+\s{Vìta:} pro $n= 2^k$ lze $DFT$ na $C^n$ spoèítat v èase $O(n \log n)$ a $DFT^{-1}$ takté¾
+
+\s{Dùsledek:}
+
+Polynomy stupnì $n$ lze násobit v èase $O(n \log n)$.
+
+$O(n \log n)$ pro vyhodnocení, $O(n)$ pro vynásobení a $O(n \log n)$ pro pøevedení zpìt
+
+\s{Pou¾ití:}
+
+\itemize\ibull
+
+\: zpracování signálu - rozklad na $\sin$ a $\cos$ o rùzných frekvencích $=>$ spektrální rozklad
+\: JPEG
+\: násobení dlouhých èísel v èase $O(n \log n)$
+\endlist
+
+
+\figure{img.eps}{Pøíklad prùbìhu algoritmu na vstupu velikosti 8}{3in}
+
+
+Je to schéma zapojení kombinaèního obvodu (tzv. "motýlek")
+
+\s{Z toho:}
+
+\itemize\ibull
+\: kombinaèní obvod pro DFT
+ s $O(\log n)$ hladinami
+ a $O(n)$ hradly na hladinì
+\: nerekurzivní algoritmus (postupujeme z leva) v èase $O(n \log n)$
+
+èísla vstupu jsou èísla v binárním tvaru 0-7 pøeètená pozpátku
+\endlist
+
+
+\bye
--- /dev/null
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+def}for setrgbcolor dup ci_y 3 -1 roll 1 add ci_y 1 add 4 copy 5 1
+roll 4 2 roll 5 -1 roll 1 1 4{pop ci_matrix transform 8 2 roll}for
+m l l l closepath fill}for}for pop}bd}ifelse/rci{/rciBuf 4 index 3
+index mul 7 add 8 div floor cvi string def{currentfile rciBuf readhexstring
+pop}bind false 3 ci}bd/cbsl{2 eq/cbslL2 xd 5 index/cbslw xd translate
+scale 8 [ 3 index 0 0 5 index 0 0 ] cbslL2{/DataStr1 cbslw string def
+currentfile/ASCII85Decode filter/RunLengthDecode filter DataStr1 readstring
+pop pop/DataStr2 cbslw string def currentfile/ASCII85Decode filter
+/RunLengthDecode filter DataStr2 readstring pop pop/DataStr3 cbslw
+string def currentfile/ASCII85Decode filter/RunLengthDecode filter
+DataStr3 readstring pop pop{DataStr1}bind{DataStr2}bind{DataStr3}bind
+true}{/DataSrc load false}ifelse 3 ci}bd/gbsl{2 eq/gbslL2 xd 5 index
+/gbslw xd translate scale 8 [ 3 index 0 0 5 index 0 0 ] gbslL2{/DataStr1
+gbslw string def currentfile/ASCII85Decode filter/RunLengthDecode filter
+DataStr1 readstring pop pop{DataStr1}bind}{/DataStr1 gbslw string def
+currentfile DataSrc1 readhexstring pop pop{DataStr1}bind}ifelse image}bd
+/cmi{/cmiBuf 4 index 3 index mul 7 add 8 div floor cvi string def{currentfile
+cmiBuf readhexstring pop}bind image}bd/cpal{4 mul string/cPalette exch
+def currentfile cPalette readhexstring pop}bd/cpci{/cpci_datasrc exch
+def matrix invertmatrix/cpci_matrix exch def/cpci_bpp exch def cpci_init
+0 1 3 -1 roll 1 sub{/cpci_y exch def dup cpci_bpp 4 eq{cpci_sampsleft
+1 eq{/cpci_sampsleft 0 def}if}if 0 1 3 -1 roll 1 sub{cpci_nextcol dup
+cpci_y 3 -1 roll 1 add cpci_y 1 add 4 copy 5 1 roll 4 2 roll 5 -1 roll
+1 1 4{pop cpci_matrix transform 8 2 roll}for m l l l closepath fill}for}for
+pop}bd/cpci_init{/cpci_sampsleft 0 def}bd/cpci_buf 1 string def/cpci_nextcol{cpci_bpp
+1 eq{cpci_sampsleft 0 eq{currentfile cpci_buf readhexstring pop pop
+/cpci_sampsleft 8 def}if cpci_buf dup 0 get dup 1 and setgray -1 bitshift
+1 exch put/cpci_sampsleft cpci_sampsleft 1 sub def}{cpci_bpp 4 eq{cpci_sampsleft
+0 eq{currentfile cpci_buf readhexstring pop pop/cpci_sampsleft 2 def}if
+cpci_buf 0 get dup 15 and exch -4 bitshift cpci_buf 0 3 -1 roll put
+/cpci_sampsleft cpci_sampsleft 1 sub def}{currentfile cpci_buf readhexstring
+pop 0 get}ifelse 4 mul dup 2 add cPalette exch get 255 div exch dup
+1 add cPalette exch get 255 div exch cPalette exch get 255 div setrgbcolor}ifelse}bd
+/setup1asciiproc{[ currentfile mystring/readhexstring cvx/pop cvx
+] cvx bind}bd/setup1binaryproc{[ currentfile mystring/readstring cvx
+/pop cvx ] cvx bind}bd level2{save/dontloadlevel1 xd}if/iw 0 def/ih
+0 def/im_save 0 def/setupimageproc 0 def/polarity 0 def/smoothflag
+0 def/mystring 0 def/bpc 0 def/beginimage{/im_save save def dup 0 eq{pop
+/setup1binaryproc}{1 eq{/setup1asciiproc}{(error, can't use level2 data acquisition procs for level1)print
+flush}ifelse}ifelse/setupimageproc exch ld/polarity xd/smoothflag xd
+/imat xd/mystring exch string def/bpc xd/ih xd/iw xd}bd/endimage{im_save
+restore}bd/1bitbwcopyimage{1 setgray 0 0 moveto 0 1 rlineto 1 0 rlineto
+0 -1 rlineto closepath fill 0 setgray iw ih polarity imat setupimageproc
+imagemask}bd/1bitcopyimage{setrgbcolor 0 0 moveto 0 1 rlineto 1 0 rlineto
+0 -1 rlineto closepath fill setrgbcolor iw ih polarity imat setupimageproc
+imagemask}bd/1bitmaskimage{setrgbcolor iw ih polarity [iw 0 0 ih 0
+0] setupimageproc imagemask}bd level2{dontloadlevel1 restore}if level2
+not{save/dontloadlevel2 xd}if/setup2asciiproc{currentfile/ASCII85Decode
+filter/RunLengthDecode filter}bd/setup2binaryproc{currentfile/RunLengthDecode
+filter}bd/myimagedict 9 dict dup begin/ImageType 1 def/MultipleDataSource
+false def end def/im_save 0 def/setupimageproc 0 def/polarity 0 def
+/smoothflag 0 def/mystring 0 def/bpc 0 def/ih 0 def/iw 0 def/beginimage{
+/im_save save def dup 2 eq{pop/setup2binaryproc}{dup 3 eq{pop/setup2asciiproc}{0
+eq{/setup1binaryproc}{/setup1asciiproc}ifelse}ifelse}ifelse/setupimageproc
+exch ld{[ 1 0 ]}{[ 0 1 ]}ifelse/polarity xd/smoothflag xd/imat xd/mystring
+exch string def/bpc xd/ih xd/iw xd}bd/endimage{im_save restore}bd/1bitbwcopyimage{1
+ssg 0 0 moveto 0 1 rlineto 1 0 rlineto 0 -1 rlineto closepath fill
+0 ssg myimagedict dup begin/Width iw def/Height ih def/Decode polarity
+def/ImageMatrix imat def/DataSource setupimageproc def/BitsPerComponent
+1 def/Interpolate smoothflag def end imagemask}bd/1bitcopyimage{ssc
+0 0 moveto 0 1 rlineto 1 0 rlineto 0 -1 rlineto closepath fill ssc
+myimagedict dup begin/Width iw def/Height ih def/Decode polarity def
+/ImageMatrix imat def/DataSource setupimageproc def/BitsPerComponent
+1 def/Interpolate smoothflag def end imagemask}bd/1bitmaskimage{ssc
+myimagedict dup begin/Width iw def/Height ih def/Decode polarity def
+/ImageMatrix imat def/DataSource setupimageproc def/BitsPerComponent
+1 def/Interpolate smoothflag def end imagemask}bd level2 not{dontloadlevel2
+restore}if
+level2{save/dontloadlevel1 xd}if/startnoload{{/noload save def}if}bd
+/endnoload{{noload restore}if}bd/testsystemdict{where{systemdict eq{true}{false}ifelse}{false}ifelse}bd
+/ncolors 1 def/colorimage where{pop true}{false}ifelse{/ncolors 0
+statusdict begin/processcolors where{pop pop processcolors}{/deviceinfo
+where{pop deviceinfo/Colors known{pop{deviceinfo/Colors get}}if}if}ifelse
+end def ncolors 0 ne{/colorimage testsystemdict/setcolortransfer testsystemdict
+/currentcolortransfer testsystemdict/currentcmykcolor testsystemdict
+and and and not{/ncolors 0 def}if}if}if ncolors dup 1 ne exch dup 3
+ne exch 4 ne and and{/ncolors 0 def}if ncolors 1 eq dup dup not startnoload{
+/expandbw{expandfactor mul round cvi bwclut exch get 255 div}bd/doclutimage{bwclut
+colorclut pop/bwclut xd bpc dup 8 eq{pop 255}{4 eq{15}{3}ifelse}ifelse
+/expandfactor xd [/expandbw load/exec load dup currenttransfer exch
+] cvx bind settransfer iw ih bpc imat setupimageproc image}bd}if not
+endnoload ncolors dup 3 eq exch 4 eq or dup dup not startnoload{/nullproc{{}}def
+/concatutil{/exec load 7 -1 roll/exec load}bd/defsubclut{1 add getinterval
+def}bd/spconcattransfer{/Dclut exch def/Cclut exch def/Bclut exch def
+/Aclut exch def/ncompute exch ld currentcolortransfer [{Aclut ncompute}concatutil
+] cvx [{Bclut ncompute}concatutil ] cvx [{Cclut ncompute}concatutil
+] cvx [{Dclut ncompute}concatutil ] cvx setcolortransfer}bd/setuprgbcluts{
+/bit3x rgbclut length 3 sub def/bit1x bit3x 3 idiv def/rclut rgbclut
+def/gclut rclut 1 bit3x defsubclut/bclut rclut 2 bit3x defsubclut}bd}if
+not endnoload ncolors 3 eq dup dup not startnoload{/3compute{exch bit3x
+mul round cvi get 255 div}bd/doclutimage{/rgbclut xd pop setuprgbcluts
+/3compute rclut gclut bclut dup spconcattransfer iw ih bpc imat [
+setupimageproc/exec load/dup load dup ] cvx nullproc nullproc true
+3 colorimage}bd}if not endnoload ncolors 4 eq dup dup not startnoload{
+/stuffclut{cmykindex 3 -1 roll put}bd/ftoint{1 exch sub 255 mul round
+cvi}bd/4compute{exch bit4x mul round cvi get 255 div}bd/computecmykclut{setuprgbcluts
+/bit4x rgbclut length 3 idiv 4 mul 4 sub def/cmykclut bit4x 4 add
+string def/cclut cmykclut def/mclut cclut 1 bit4x defsubclut/yclut
+cclut 2 bit4x defsubclut/kclut cclut 3 bit4x defsubclut/cmykindex 0
+def 0 1 bit1x{dup/cmykindex exch bit1x exch sub 4 mul def 3 mul dup
+rclut exch get 255 div exch dup gclut exch get 255 div exch bclut exch
+get 255 div setrgbcolor currentcmykcolor ftoint kclut stuffclut ftoint
+yclut stuffclut ftoint mclut stuffclut ftoint cclut stuffclut}for}bd
+/doclutimage{/rgbclut xd pop invalidcolortable?{computecmykclut}if
+/4compute cclut mclut yclut kclut spconcattransfer iw ih bpc imat
+[ setupimageproc/exec load/dup load dup dup ] cvx nullproc nullproc
+nullproc true 4 colorimage}bd}if not endnoload ncolors 0 eq dup dup
+not startnoload{/lookupandstore{3 mul 3 getinterval putinterval exch
+3 add exch 3 copy}bd/8lookup/lookupandstore ld/4lookup{/byte 1 index
+def -4 bitshift lookupandstore byte 15 and lookupandstore}bd/2lookup{
+/byte 1 index def -6 bitshift lookupandstore byte -4 bitshift 3 and
+lookupandstore byte -2 bitshift 3 and lookupandstore byte 3 and lookupandstore}bd
+/colorexpand{mystringexp 0 rgbclut 3 copy 7 -1 roll/mylookup load
+forall pop pop pop pop pop}bd/createexpandstr{/mystringexp exch mystring
+length mul string def}bd/doclutimage{/rgbclut xd pop/mylookup bpc 8
+eq{3 createexpandstr/8lookup}{bpc 4 eq{6 createexpandstr/4lookup}{12
+createexpandstr/2lookup}ifelse}ifelse ld iw ih bpc imat [ setupimageproc
+/exec load/colorexpand load/exec load] cvx false 3 colorimage}bd}if
+not endnoload/colorimage where{pop true}{false}ifelse dup{/do24image{iw
+ih 8 imat setupimageproc false 3 colorimage}bd}if dup dup startnoload
+not{/rgbtogray{/str xd/len str length def/smlen len 3 idiv def/rstr
+str def/gstr str 1 len 1 sub getinterval def/bstr str 2 len 2 sub getinterval
+def str dup 0 1 smlen 1 sub{dup 3 mul rstr 1 index get .3 mul gstr
+2 index get .59 mul add bstr 3 -1 roll get .11 mul add round cvi put
+dup}for pop 0 smlen getinterval}bd/do24image{iw ih 8 imat [ setupimageproc
+/exec load/rgbtogray load/exec load ] cvx bind image}bd}if endnoload
+/doimage{iw ih 8 imat setupimageproc image}bd level2{dontloadlevel1
+restore}if level2 not{save/dontloadlevel2 xd}if/myappcolorspace/DeviceRGB
+def/rgbclut 0 def/doclutimage{/rgbclut xd pop bpc dup 8 eq{pop 255}{4
+eq{15}{3}ifelse}ifelse/hival xd [/Indexed myappcolorspace hival rgbclut]
+setcolorspace myimagedict dup begin/Width iw def/Height ih def/Decode
+[0 hival] def/ImageMatrix imat def/DataSource setupimageproc def/BitsPerComponent
+bpc def/Interpolate smoothflag def end image}bd/do24image{myappcolorspace
+setcolorspace myimagedict dup begin/Width iw def/Height ih def/Decode
+[0 1 0 1 0 1] def/ImageMatrix imat def/DataSource setupimageproc def
+/BitsPerComponent 8 def/Interpolate smoothflag def end image}bd level2
+not{dontloadlevel2 restore}if
+/NumSteps{dtransform matrix defaultmatrix idtransform Pythag currentscreen
+pop pop 72 exch div div}bd/FindMinSteps{v_ft 4 eq{urx startX sub abs
+llx startX sub abs Max ury startY sub abs lly startY sub abs Max Pythag
+2 3.14159265 mul mul 0}{v_ft 2 eq{endY startY sub endX startX sub Pythag
+endY2 startY sub endX2 startX sub Pythag gt{endY startY sub endX startX
+sub}{endY2 startY sub endX2 startX sub}ifelse}{endY startY sub endX
+startX sub}ifelse}ifelse NumSteps}bd/cxe{/v_cxe exch def}bd/cxm{pop
+/v_cxm exch def}bd/cxmt{pop pop}bd/cxt{pop}bd/S_eoclip{currentflat{{eoclip}stopped{dup
+currentflat exch sub 20 gt{([Error: PathTooComplex; OffendingCommand: eoclip]\n)print
+flush exit}{currentflat 2 add setflat}ifelse}{exit}ifelse}loop setflat}bd
+/S_clip{currentflat{{clip}stopped{dup currentflat exch sub 20 gt{([Error: PathTooComplex; OffendingCommand: clip]\n)print
+flush exit}{currentflat 2 add setflat}ifelse}{exit}ifelse}loop setflat}bd
+/S_eofill{currentflat{{eofill}stopped{dup currentflat exch sub 20
+gt{([Error: PathTooComplex; OffendingCommand: eofill]\n)print flush
+exit}{currentflat 2 add setflat}ifelse}{exit}ifelse}loop setflat}bd
+/gpbbx{pathbbox/ury exch def/urx exch def/lly exch def/llx exch def}bd
+/lineargfill{initgfill{false initgfx/distance endX startX sub endY
+startY sub Pythag def/incD distance Steps div def endY startY sub endX
+startX sub atan newpath llx lly urx ury Bx startX startY translate
+rotate gpbbx eGy ssg newpath llx lly urx ury Bx S_eofill sGy ssg newpath
+llx lly 0 ury Bx S_eofill/fillX 0 def 0 1 Steps 1 sub{stepgfx newpath
+fillX lly fillX incD add dup/fillX exch def ury Bx S_eofill pop}for}if}bd
+/radialgfill{initgfill{false initgfx/distance endX startX sub endY
+startY sub Pythag def/incD distance Steps div def eGy ssg newpath llx
+lly urx ury Bx S_eofill/distance 0 def 0 1 Steps 1 sub{stepgfx newpath
+startX startY distance 0 360 arc closepath distance incD add dup/distance
+exch def 0 rmoveto startX startY distance 0 360 arc closepath S_eofill
+pop}for}if}bd/ellipticgfill{initgfill{true initgfx sGy ssg newpath
+llx lly urx ury Bx S_eofill [ endX startX sub endY startY sub endX2
+startX sub endY2 startY sub startX startY ] concat Steps 1 sub -1 0{stepgfx
+/i exch def 0 0 moveto 0 0 i Steps div 0 360 arc fill}for}if}bd/conicalgfill{initgfill{urx
+startX sub abs llx startX sub abs Max ury startY sub abs lly startY
+sub abs Max Pythag startY endY sub startX endX sub Pythag div/radius
+exch def true initgfx [ endX startX sub endY startY sub startY endY
+sub endX startX sub startX startY ] concat/slice 180 Steps div def
+/startangle 0 def Steps -1 0{pop stepgfx 0 0 moveto 0 0 radius startangle
+neg dup slice add arc fill 0 0 moveto 0 0 radius startangle dup slice
+add dup/startangle exch def arc fill}for}if}bd/initgfill{gpbbx{S_eoclip}{S_clip}ifelse
+startX endX eq startY endY eq and sGy eGy eq or dup{sGy ssg newpath
+llx lly urx ury Bx S_eofill}if not}bd/initgfx{/flag exch def flag{sGy
+eGy/sGy exch def/eGy exch def}if eGy sGy sub abs 256 mul FindMinSteps
+Min 256 Min ceiling 1 Max/Steps exch def eGy sGy sub Steps div/incGy
+exch def sGy/mGy exch def}bd/stepgfx{mGy ssg mGy incGy add/mGy exch
+def}bd/linearfill{initfill{false initfx endX startX sub endY startY
+sub Pythag/distance exch def/incD distance Steps div def endY startY
+sub endX startX sub atan newpath llx lly urx ury Bx startX startY translate
+rotate gpbbx v_cxe 0 eq{eR eG eB setgfillrgb}{eH eS eV setgfillhsb}ifelse
+newpath llx lly urx ury Bx S_eofill v_cxe 0 eq{sR sG sB setgfillrgb}{sH
+sS sV setgfillhsb}ifelse newpath llx lly 0 ury Bx S_eofill/fillX 0
+def 0 1 Steps 1 sub{stepfx newpath fillX lly fillX incD add dup/fillX
+exch def ury Bx S_eofill pop}for}if}bd/radialfill{initfill{false initfx
+endX startX sub endY startY sub Pythag/distance exch def/incD distance
+Steps div def v_cxe 0 eq{eR eG eB setgfillrgb}{eH eS eV setgfillhsb}ifelse
+newpath llx lly urx ury Bx S_eofill/distance 0 def 0 1 Steps 1 sub{stepfx
+newpath startX startY distance 0 360 arc closepath distance incD add
+dup/distance exch def 0 rmoveto startX startY distance 0 360 arc closepath
+S_eofill pop}for}if}bind def/ellipticalfill{initfill{true initfx v_cxe
+0 eq{sR sG sB setgfillrgb}{sH sS sV setgfillhsb}ifelse newpath llx
+lly urx ury Bx S_eofill [ endX startX sub endY startY sub endX2 startX
+sub endY2 startY sub startX startY ] concat Steps 1 sub -1 0{stepfx
+/i exch def 0 0 moveto 0 0 i Steps div 0 360 arc fill}for v_cxe 0
+eq{sR sG sB eR eG eB/sB exch def/sG exch def/sR exch def/eB exch def
+/eG exch def/eR exch def}if}if}bd/conicalfill{initfill{urx startX
+sub abs llx startX sub abs Max ury startY sub abs lly startY sub abs
+Max Pythag startY endY sub startX endX sub Pythag div/radius exch def
+true initfx [ endX startX sub endY startY sub startY endY sub endX
+startX sub startX startY ] concat/slice 180 Steps div def/startangle
+0 def Steps -1 0{pop stepfx 0 0 moveto 0 0 radius startangle neg dup
+slice add arc fill 0 0 moveto 0 0 radius startangle dup slice add dup
+/startangle exch def arc fill}for}if}bd/initfill{gpbbx{S_eoclip}{S_clip}ifelse
+startX endX eq startY endY eq and v_cxe 2 ne sR eR eq sG eG eq and
+sB eB eq and and or dup{sR sG sB setgfillrgb newpath llx lly urx ury
+Bx S_eofill}if not}bd/initfx{/flag exch def v_cxe 0 eq{flag{sR sG sB
+eR eG eB/sB exch def/sG exch def/sR exch def/eB exch def/eG exch def
+/eR exch def}if eR sR sub abs 256 mul eG sG sub abs 256 mul Max eB
+sB sub abs 256 mul Max FindMinSteps Min 256 Min ceiling 1 Max/Steps
+exch def/incR eR sR sub Steps div def/incG eG sG sub Steps div def
+/incB eB sB sub Steps div def sR/mR exch def sG/mG exch def sB/mB
+exch def}{sR sG sB rgb2hsb/sV exch def/sS exch def/sH exch def eR eG
+eB rgb2hsb/eV exch def/eS exch def/eH exch def eH sH sub abs v_cxe
+1 eq{dup 0.5 gt{1 exch sub}if}{dup 0.5 lt{1 exch sub}if}ifelse 256
+mul eS sS sub abs 256 mul Max eV sV sub abs 256 mul Max FindMinSteps
+Min 256 Min ceiling 1 Max/Steps exch def v_cxe 1 eq{/incH eH sH sub
+dup abs 0.5 gt{dup 0 ge{1 sub}{1 add}ifelse}if Steps div def/incS eS
+sS sub Steps div def/incV eV sV sub Steps div def}{/incH eH sH sub
+dup abs 0.5 le{dup 0 ge{1 sub}{1 add}ifelse}if Steps div def/incS eS
+sS sub Steps div def/incV eV sV sub Steps div def}ifelse flag{/sH eH
+/eH sH def def/sS eS/eS sS def def/sV eV/eV sV def def/incH incH neg
+def/incS incS neg def/incV incV neg def}if}ifelse}bd/stepfx{v_cxe 0
+eq{mR mG mB setgfillrgb mR incR add/mR exch def mG incG add/mG exch
+def mB incB add/mB exch def}{sH sS sV setgfillhsb sH incH add dup 0
+le{1 add}{dup 1 ge{1 sub}if}ifelse/sH exch def sS incS add/sS exch
+def sV incV add/sV exch def}ifelse}bd
+/ar{}bd/arr{pop pop pop pop pop pop pop}bd/ae{pop pop pop pop pop
+pop}bd/aoa{pop}bd/apl{pop}bd/apc{}bd/aof{pop pop}bd/aafs{pop pop pop}bd
+/O{pop}bd/R{pop}bd/axop{pop pop pop pop}bd/g{/v_ft 0 def/v_fc 0 def
+/v_fm 0 def/v_fy 0 def 1 exch sub/v_fk exch def/v_fct 0 def}bd/G{
+/v_sc 0 def/v_sm 0 def/v_sy 0 def 1 exch sub/v_sk exch def/v_sct 0
+def}bd/k{/v_fk exch def/v_fy exch def/v_fm exch def/v_fc exch def/v_ft
+0 def/v_fct 0 def}bd/K{/v_sk exch def/v_sy exch def/v_sm exch def/v_sc
+exch def/v_sct 0 def}bd/Xa{/v_fb exch def/v_fg exch def/v_fr exch def
+/v_ft 0 def/v_fct 1 def}bd/XA{/v_sb exch def/v_sg exch def/v_sr exch
+def/v_sct 1 def}bd/a_tc{exch pop 1 exch sub dup 3 -1 roll exch div
+exch dup 4 -1 roll exch div 3 1 roll dup 5 -1 roll exch div 4 1 roll
+5 -1 roll exch div 4 1 roll}bd/x{a_tc k}bd/X{a_tc K}bd/Xx{0 eq{x}{pop
+pop Xa}ifelse}bd/XX{0 eq{X}{pop pop XA}ifelse}bd/awr{/v_wr exch def}bd
+/w{setlinewidth}bd/j{setlinejoin}bd/J{setlinecap}bd/d{setdash}bd/asc{pop
+pop J}bd/aec{pop pop J}bd/csah{pop pop pop}bd/ceah{pop pop pop}bd/cst{pop
+pop}bd/cdp{pop}bd/m{moveto}bd/l{lineto}bd/c{curveto}bd/Bx{4 copy 5
+1 roll 4 2 roll 5 -1 roll m l l l closepath}bd/Cp{Bx clip newpath}bd
+/a_fp{gsave v_wr 0 ne v_ft 0 eq{setfillcolor{eofill}{fill}ifelse}{v_ft
+1 eq{linearfill}{v_ft 2 eq{ellipticalfill}{v_ft 3 eq{radialfill}{v_ft
+4 eq{conicalfill}{v_ft 8 eq{lineargfill}{v_ft 9 eq{ellipticgfill}{v_ft
+10 eq{radialgfill}{conicalgfill}ifelse}ifelse}ifelse}ifelse}ifelse}ifelse}ifelse}ifelse
+grestore}bd/*u{/p_count p_count 1 add def}bd/*U{/p_count p_count 1
+sub def p_count 0 eq{vis_flag{p_render}{newpath}ifelse}if}bd/B{/p_render{a_fp
+setstrokecolor stroke}bd p_count 0 eq{vis_flag{p_render}{newpath}ifelse}if}bd
+/b{closepath B}bd/F{/p_render{a_fp newpath}bd p_count 0 eq{vis_flag{p_render}{newpath}ifelse}if}bd
+/f{closepath F}bd/S{/p_render{setstrokecolor stroke}bd p_count 0 eq{vis_flag{p_render}{newpath}ifelse}if}bd
+/s{closepath S}bd/H{/p_render{newpath}bd p_count 0 eq{vis_flag{p_render}{newpath}ifelse}if}bd
+/h{closepath H}bd/N{H}bd/n{h}bd/cag{dup 7 add/v_ft exch def dup 2
+eq{pop/endY2 exch def/endX2 exch def}{7 eq{pop pop/v_ft 8 def}if}ifelse
+/endY exch def/endX exch def/startY exch def/startX exch def/eGy exch
+def/sGy exch def}bd/caz{dup/v_ft exch def dup 2 eq{pop/endY2 exch def
+/endX2 exch def}{7 eq{pop pop/v_ft 1 def}if}ifelse/endY exch def/endX
+exch def/startY exch def/startX exch def/eB exch def/eG exch def/eR
+exch def/sB exch def/sG exch def/sR exch def}bd/cax{dup/v_ft exch def
+dup 2 eq{pop/endY2 exch def/endX2 exch def}{7 eq{pop pop/v_ft 1 def}if}ifelse
+/endY exch def/endX exch def/startY exch def/startX exch def pop pop
+8 3 roll pop pop/sB exch def/sG exch def/sR exch def/eB exch def/eG
+exch def/eR exch def}bd/axm{/endY exch def/endX exch def/startY exch
+def/startX exch def}bd/alyr{pop pop 1 eq{true}{false}ifelse/vis_flag
+exch def pop pop}bd
+/t{moveto show}bd/ts{moveto false charpath S}bd/tf{moveto true charpath
+F}bd/tb{3 copy moveto true charpath F moveto false charpath S}bd/selectfont
+where{pop}{/selectfont{dup type/integertype eq{exch findfont exch scalefont
+setfont}{exch findfont exch makefont setfont}ifelse}bd}ifelse/sf{selectfont}bd
+/u{}bd/U{}bd/anu{pop}bd
+end
+%%EndResource
+%%EndProlog
+%%BeginSetup
+save XaraStudio1Dict begin
+%%EndSetup
+0 cxe
+2 0 cxm
+1 awr
+0.250 w
+2 j
+[ ] 0 d
+0 J
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projects / ads2.git / commitdiff