Prvni verze zapisku o FFT.

diff --git a/9-fft/9-fft.tex b/9-fft/9-fft.tex
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+\input lecnotes.tex
+\def\imply{\Rightarrow}
+\prednaska{8}{Fourierova transformace}{ \vbox{\hbox{(K.Jakubec, M.Polák
+       a~G.Ocsovszky,}\hbox{\ V.Tùma, M.Kozák)}}}
+
+Násobení polynomù mù¾e mnohým pøipadat jako pomìrnì (algoritmicky) snadný
+problém. Asi ka¾dého hned napadne \uv{hloupý} algoritmus -- vezmeme
+koeficienty prvního polynomu a~vynásobíme ka¾dý se v¹emi koeficienty druhého
+polynomu a~pøíslu¹nì u~toho seèteme i~exponenty (stejnì jako to dìláme, kdy¾
+násobíme polynomy na~papíøe). Pokud stupeò prvního polynomu je $n$ a~druhého
+$m$, strávíme tím èas $\Omega(mn)$. Pro $m=n$ je to kvadraticky pomalé.
+Na~první pohled se mù¾e zdát, ¾e rychleji to prostì nejde (pøeci musíme
+v¾dy vynásobit \uv{ka¾dý s~ka¾dým}). Ve skuteènosti to ale rychleji fungovat
+mù¾e, ale k~tomu je potøeba znát trochu tajemný algoritmus FFT neboli {\I Fast
+Fourier Transform}.
+
+
+\ss{Trochu algebry na~zaèátek:}
+\>Libovolný polynom $P$ stupnì $n$ lze reprezentovat dvìma rùznými zpùsoby:
+
+\itemize\ibull
+\:svými koeficienty, èili èísly $p_{0}, p_{1}, \ldots ,p_{n}$, nebo
+\:svými hodnotami v~$n$ rùzných bodech $x_{0}, x_{1}, \ldots , x_{n}$, èili
+èísly $P(x_{0}),$ $P(x_{1}),$ $\ldots , P(x_{n})$.
+\endlist
+
+\>Dostateènost $n+1$ hodnot pro urèení druhým zpùsobem lze dokázat
+následnovnì: polynom stupnì $n$ má maximálnì $n$ koøenù (indukcí, je-li
+$k$ koøenem $P$, pak lze $P$ napsat jako $(x-k)Q$ kde $Q$ je polynom stupnì
+o~jedna men¹í, pøitom polynom stupnì 1 má jediný koøen); uvá¾íme-li
+dva rùzné polynomy $P$ a~$Q$ stupnì $n$ nabývající v~daných bodech stejných
+hodnot, tak $P-Q$ je polynom stupnì maximálnì $n$, ka¾dé
+z $x_{0}\ldots x_{n}$ je koøenem tohoto polynomu $\imply$ spor, polynom stupnì
+$n$ má $n+1$ koøenù $\imply$ $P-Q$ musí být nulový polynom $\imply$ $P=Q$.
+\medskip       
+
+\ss{Konvence:}
+Celé polynomy oznaèujeme velkými písmeny, jednotlivé èleny polynomù pøíslu¹nými
+malými písmeny (pø.: polynom $W$ stupnì $n$ má koeficienty $w_{0}, w_{1},
+w_{2},\ldots, w_{n}$).
+
+Pov¹imnìme si jedné skuteènosti -- máme-li dva polynomy $A$ a~$B$ stupnì $n$
+a~body $x_{0}, \ldots, x_{k}$, dále polynom $C=A \cdot B$ (stupnì $2n$), pak
+platí $C(x_{k}) = A(x_{k}) \cdot B(x_{k}), k = 0,1,2, \ldots, n.$ 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 (tj.
+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
+$\leq 2n$ (kde $n$ je stupeò výchozích polynomù). Pokud chceme polynom $C$
+reprezentovat pomocí jeho hodnot v~bodech, musíme tedy vzít alespoò $2n$
+bodù. Tímto konèí malá algebraická vsuvka.
+
+\s{Idea, jak by mìl algoritmus pracovat:}
+\algo
+\:Vybereme $2n$ bodù $x_{0}, x_{1}, \ldots , x_{2n-1}$.
+\:V tìchto bodech vyhodnotíme polynomy $A$ a~$B$.
+\:Nyní ji¾ v~lineárním èase získáme hodnoty polynomu $C$ v~tìchto bodech
+       (viz vý¹e).
+\:Pøevedeme hodnoty polynomu $C$ na~jeho koeficienty.
+\endalgo
+
+\>Je vidìt, ¾e klíèové jsou kroky 2 a~4. Vybrání bodù jistì stihneme pohodlnì
+v~lineárním èase a~vynásobení samotných hodnot té¾ (máme $2n$ bodù a~$C(x_{k})
+= A(x_{k}) \cdot B(x_{k}), k = 0,1,2, \ldots , 2n-1$, tak¾e na~to nepotøebujeme
+více ne¾ $2n$ 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~webové stránce pøedná¹ky jsou k dispozici slajdy, zde to bude zapsáno
+o~trochu struènìji.
+
+\ss{  Vyhodnocení polynomu metodou Rozdìl a~panuj (algoritmus FFT):}
+Mìjme polynom $P$ øádu $n$ a~chtìjme jej vyhodnotit 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 $x_{j}$
+shodují s~druhými mocninami $-x_{j}$.
+
+Polynom $P$ rozlo¾íme na~dvì èásti, první obsahuje èleny se sudými exponenty,
+druhá s~lichými:
+$$P(x) = (p_{0}x^{0} + p_{2}x^{2} + \ldots + p_{n-2}x^{n-2}) + (p_{1}x^{1} +
+       p_{3}x^{3} + \ldots + p_{n-1}x^{n-1})$$
+\>se zavedením znaèení:
+$$P_s(x^{2}) = p_{0}x^{0} + p_{2}x^{2} + \ldots + p_{n - 2}x^{n - 2}$$
+$$P_l(x^{2}) = p_{1}x^{1} + p_{3}x^{3} + \ldots + p_{n - 1}x^{n - 1}$$
+
+\>Dohromady $P(x) = P_s(x^{2}) + xP_l(x^{2})$ a~$P(-x) = P_s(x^{2}) -
+xP_l(x^{2})$. Jinak øeèeno, vyhodnocování $P$ v~$n$ bodech se nám smrskne
+na~vyhodnocení $P_s(x)$ a~$P_l(x)$ (oba jsou polynomy stupnì $n/2$ 
+a~vyhodnocujeme je nyní v~$x^{2}$) v~$n/2$ bodech (proto¾e $(x_{i})^{2} =
+(-x_{i})^{2}$).
+
+\s{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 u¾ v~druhé úrovni 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 mocniny
+$n$-té primitvní odmocniny z jedné (oznaèíme si ji jako $\omega$). Pøipomeòme:
+$\omega$ je $n$-tá odmocnina z $1$ je {\it primitivní} $\equiv$ $(\forall k)
+(k<n)\ \omega^k \neq 1$ a~$\omega^n = 1$. Máme $n$ navzájem rùzných odmocnin
+z~jednièky, rovnomìrnì rozesetých po jednotkové kru¾nici, BÚNO $n=2^{k}, k \in
+N$ (jinak viz slajdy Martina Mare¹e). Jednotlivé $x$ vypadají takto: $1,
+\omega, \omega^{2}, \ldots , \omega^{n - 1} $, kde $\omega = e^{2 \pi i/ n}$.
+
+\s{Pìt poznámek:}
+\itemize\ibull
+\:Pro $k\neq j,\ 0\leq k<j<n$ je $\omega^k \neq \omega^j$, nebo» $\omega^j /
+\omega^k = \omega^{j-k}\neq 1$, proto¾e $j-k < n$ a~$\omega$ je $n$-tá
+primitivní (a~tedy $\omega^0, \ldots, \omega^{n-1}$ jsou rùzné).
+\:$\overline{\omega} = \omega^{-1}$, nebo» $\omega^{-1} = \omega^{-2\pi i/n}$
+a~tedy èísla $\omega,\omega^{-1}$ svírají vùèi reálné ose opaèný úhel a~jsou
+       komplexnì sdru¾ená.
+\:mocniny $\omega$ jsou spárované, èili $\omega^{j} = -\omega^{n/2 + j}$
+       (staèí si uvìdomit, ¾e $\omega^{n/2} = -1$ pro $n$ sudé)
+\:umocníme-li v¹echny $\omega^{0},\ldots, \omega^{n-1}$ na~druhou, vzniknou nám
+       odmnocniny z~$1$ které budou i~nadále spárované: $\omega^n$ je toti¾
+       $1$ a~exponenty lze tedy poèítat$\mod n$, $n$ je sudé $\imply$
+       dostaneme pùvodní posloupnost ze které zmizí liché mocniny $\omega$, ale $n$
+       je mocnina dvojky a~tedy i~$n/2$ je sudé (pro $n=2$ ne, ale tam je to
+       triviální), tak¾e sudé mocniny jsou spárované navzájem..
+\:$\omega^2$ je $n/2$-tá primitivní odmocnina z 1 -- je toti¾ $(\omega^{-2\pi
+       i/n})^2 = \omega^{-2\pi i/(n/2)}$ a~$n$ je mocnina dvojky.
+\endlist
+
+\ss{Celý algoritmus bude vypadat takto:}
+\>FFT($P$, $ \omega$)
+
+\>{\sl Vstup:} $p_{0}, \ldots , p_{n-1}$, koeficienty polynomu $P$, a~$\omega$,
+$n$-tá primitivní odmocina z jedné.
+
+\>{\sl 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 $n = 1$, vrátíme $p_{0}$ a~skonèíme.
+\:Jinak rozdìlíme $P$ na~sudé a~liché koeficienty a~rekurzivnì zavoláme
+       FFT($P_s$, $\omega^{2}$) a~FFT($P_l$, $\omega^{2}$) -- $P_l$ i~$P_s$
+       jsou stupnì max. $n/2-1$ a~$\omega^2$ je $n/2$-tá primitivní odmocnina.
+\:Pro $j = 0, \ldots , n/2 - 1$ spoèítáme:
+
+\:\qquad $P(\omega^{j}) = P_s(\omega^{2j}) + \omega^{j}\cdot P_l(\omega^{2j})$.
+
+\:\qquad $P(\omega^{j+n/2})=P_s(\omega^{2j})-\omega^{j}\cdot P_l(\omega^{2j})$.
+
+\endalgo
+
+
+\s{Èasová slo¾itost:}
+\>$T(n)=2T(n/2) + \Theta(n) \Rightarrow$ slo¾itost $\Theta(n \log n)$, jako
+MergeSort.
+
+
+Máme tedy algoritmus, který pøevede koeficienty polynomu na~hodnoty tohoto
+polynomu v~rùzných bodech. Potøebujeme ale také algoritmus, který doká¾e
+reprezentaci polynomu pomocí hodnot pøevést zpìt na~koeficienty polynomu.
+K~tomu nám pomù¾e podívat se na~ná¹ algoritmus trochu obecnìji.
+
+
+\s{Definice:}
+\>{\I Diskretní Fourierova transformace} $(DFT)$
+je zobrazení $f: { {\bb C} ^n} \rightarrow { {\bb C} ^n}$, kde $$y=f(x) \equiv
+\forall j \ y_{j} = \sum \limits ^{n-1}_{k=0} x_{k} \cdot \omega ^{jk}$$
+(pøestavme si ji jako funkci vyhodnocující polynom s~koeficienty $x$ v~bodech
+$\omega^j$). Takovéto zobrazení je lineární a~tedy popsatelné maticí $\Omega$
+s~prvky $\Omega_{jk} = \omega^{jk}$
+
+
+\s{Jak najít inverzní matici?} Znaème $\overline{\Omega}$ matici, její¾ prvky
+jsou komplexnì sdru¾ené odpovídajícím prvkùm $\Omega$, a vyu¾ijeme následující
+lemma:
+
+\ss{Lemma:}$\quad \Omega\cdot \overline{\Omega} = n\cdot E$
+
+\proof $$ (\Omega\cdot \overline{\Omega})_{jk} = \sum_{l=0}^{n-1} \omega^{jl}
+\cdot \overline{\omega^{lk}} = \sum \omega^{jl} \cdot \overline{\omega}^{lk} =
+\sum \omega^{jl} \cdot \omega^{-lk} = \sum \omega^{l(j-k)}$$
+\itemize\ibull
+\:Pokud $j=k$, pak $ \sum \limits ^{n-1}_{l=0} (\omega ^{0}) ^{l} = n$.
+
+\:Pokud $j\neq k$, pou¾ijeme vzoreèek pro souèet geometrické posloupnosti, kde
+$a_{1}=1$ a~$q=\omega ^{(j-k) }$ a~dostaneme ${{\omega^{(j-k)n} -1} \over
+{\omega^{(j-k)} -1}} ={1-1 \over r- 1} = {0 \over \neq 0} = 0$, kde $r$
+je èíslo rùzné od jednièky (nebo» $\omega$ je $n$-tá primitivní odmoncina).
+\endlist
+\qed
+
+
+\>Na¹li jsme inverzi:
+$\Omega({1 \over n} \overline{\Omega}) = {1 \over n}\Omega \cdot
+\overline{\Omega} = E$, \quad $\Omega^{-1}_{jk} = {1 \over n}\overline{\omega^
+{jk}} = {1 \over n}\omega^{-jk} = {1 \over n} {(\omega^{-1})}^{jk}$, \quad
+(pøipomínáme, $\omega^{-1}$ je $\overline{\omega}$, $\omega$ je $n$-tá
+primitivní odmocnina z jednièky).
+
+Ná¹ algoritmus poèítá tedy i~inverzní transformaci, pouze místo $\omega_n$
+pou¾ijeme komplexnì zdru¾ené $\overline{\omega_n}$ a~matici vynásobíme $(1/n)$.
+Co¾ je skvìlé -- staèí znát pouze jeden algoritmus u~kterého staèí v~jednom
+pøípadì pou¾ít transformovanou matici a~vydìlit $n$.
+
+\s{Výsledek:} Pro $n= 2^k$ lze DFT na~${\bb C}^n$ spoèítat v~èase $\Theta(n
+\log n)$ a~DFT$^{-1}$ takté¾.
+
+\s{Dùsledek:}
+Polynomy stupnì $n$ lze násobit v~èase $\Theta(n \log n)$:
+$\Theta(n \log n)$ pro vyhodnocení, $\Theta(n)$ pro vynásobení a~$\Theta(n
+\log n)$ pro pøevedení zpìt.
+
+\s{Pou¾ití:}
+
+\itemize\ibull
+
+\:Zpracování signálu -- rozklad na~siny a~cosiny o~rùzných frekvencích
+$\Rightarrow$ spektrální rozklad.
+\:komprese dat -- napøíklad formát JPEG.
+\:Násobení dlouhých èísel v~èase $\Theta(n \log n)$.
+\endlist
+
+\s{Hardwarová implementace FFT}
+
+\figure{img.eps}{Pøíklad prùbìhu algoritmu na~vstupu velikosti 8}{3in}
+
+
+\>Obrázek ukazuje zapojení kombinaèního obvodu pro DFT pro vstup velikosti~8.
+Hladin bude v¾dy $\log_2 n$, tj. v~na¹em pøípadì $\log_2 8 = 3$ hladiny.
+
+\>Podívejme se na~pravou èást obrázku, tedy výstup celého obvodu. Èerná koleèka
+pøedstavují podobvody, rovnice vedle nich operaci, kterou provádìjí. Hodnoty
+$y_j$ znaèí hodnotu polynomu $P$ v~bodì $\omega^j$ kde $\omega^j$ je $j-tá$
+mocnina primitivní $n$-té odmocniny z jednièky. K jejímu spoètení ale
+potøebujeme znát hodnoty $s_k$ a~$l_k$ kde $k$ je z intervalu $[0, {n/2} -1]$
+a~$s_k$ a~$l_k$ jsou hodnoty polynomu stupnì ${n/2}$ v~bodì $\omega^{2k}$.
+V polynomu $s$ jsou sudé koeficienty a~v polynomu $l$ liché koeficienty
+polynomu $P$. Vidíme, ¾e se jedná pøesnì o~ná¹ rekurzivní algoritmus pro
+poèítání FFT a~tímto zpùsobem postavíme celou sí».
+\>Tímto obvodem jsme tedy získali nerekurzivní algoritmus pro poèítání FFT.
+V¹imìme si poøadí vstupních hodnot (koeficientù). Èísla jsou v~binárním tvaru
+0--7 pøeètená pozpátku. Pro pøedstavu jaké koeficienty polynomu $P$ se objevují
+v~rùzných hladinách, na~obrázku jsou naznaèena jejich èísla spolu s~pøíslu¹nými
+mocninami primitivní $n$-té odmocniny z jednièky.
+
+\s{Z~toho:}
+
+\itemize\ibull
+\:Kombinaèní obvod pro DFT
+s~$\Theta(\log n)$ hladinami
+a $\Theta(n)$ hradly na~hladinì.
+\:Nerekurzivní algoritmus (postupujeme zleva) v~èase $\Theta(n \log n)$.
+
+\endlist
+\bigskip
+
+\>Nakonec dodejme, ¾e poèítat lze nejen nad tìlesem komplexních èísel, ale
+v podstatì nad ka¾dým komutativním tìlesem s~$n$-tou primitivní odmocninou
+(pøíkladem jsou tìlesa øádu $p = 2^k + 1$, $p$ prvoèíslo) èi dokonce
+komutativním okruhem s~multiplikativními inversemi pro øád vektoru
+a~primitivní odmocninu.
+
+\bye

diff --git a/9-fft/Makefile b/9-fft/Makefile
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+P=9-fft
+
+include ../Makerules

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+pop 1 exch sub 0 0 0 4 -1 roll}{/ucurve where{pop 3{1.0 exch sub 3\r
+1 roll}repeat 3 copy 2 copy gt{exch}if pop 2 copy gt{exch}if pop dup\r
+ucurve readcurve exch bcurve readcurve clamp01 3{5 1 roll dup 3 1 roll\r
+sub clamp01}repeat 5 1 roll pop 4 1 roll ycurve readcurve 4 1 roll\r
+mcurve readcurve 4 1 roll ccurve readcurve 4 1 roll}{rgb2cmyk}ifelse}ifelse}def\r
+/rgb2keyG{3 copy dup 3 1 roll eq 3 1 roll eq and{pop pop}{Max3}ifelse\r
+1 exch sub bcurve readcurve clamp01}bd/rgb2key{Max3 1 exch sub bcurve\r
+readcurve clamp01}bd/rgb2cyanG{3 copy dup 3 1 roll eq 3 1 roll eq and{pop\r
+pop pop 0}{rgb2cyan}ifelse}bd/rgb2cyan{3 copy Max3 1 exch sub ucurve\r
+readcurve 4 1 roll pop pop 1 exch sub exch sub ccurve readcurve clamp01}bd\r
+/rgb2magentaG{3 copy dup 3 1 roll eq 3 1 roll eq and{pop pop pop 0}{rgb2magenta}ifelse}bd\r
+/rgb2magenta{3 copy Max3 1 exch sub ucurve readcurve 4 1 roll pop\r
+1 exch sub 3 1 roll pop sub mcurve readcurve clamp01}bd/rgb2yellowG{3\r
+copy dup 3 1 roll eq 3 1 roll eq and{pop pop pop 0}{rgb2yellow}ifelse}bd\r
+/rgb2yellow{3 copy Max3 1 exch sub ucurve readcurve 4 1 roll 1 exch\r
+sub 4 1 roll pop pop sub ycurve readcurve clamp01}bd/rgb2plategray{v_keyg\r
+0 eq v_plate v_cpky eq{{rgb2key}{rgb2keyG}ifelse}{v_plate v_cpyl eq{{rgb2yellow}{rgb2yellowG}ifelse}{v_plate\r
+v_cpmg eq{{rgb2magenta}{rgb2magentaG}ifelse}{v_plate v_cpcy eq{{rgb2cyan}{rgb2cyanG}ifelse}{{rgb2key}{rgb2keyG}ifelse}ifelse}ifelse}ifelse}ifelse\r
+1 exch sub setgray}bd/dc{0 def}bd/aca{/v_cpnone 0 def/v_cpcy 1 def\r
+/v_cpyl 2 def/v_cpmg 3 def/v_cpky 4 def/v_gseps 0 def/v_keyg 0 def\r
+/v_plate v_cpnone def/v_mono 0 def/v_wr dc/v_fc dc/v_fm dc/v_fy dc\r
+/v_fk dc/v_fg dc/v_fr dc/v_fg dc/v_fb dc/v_sc dc/v_sm dc/v_sy dc/v_sk\r
+dc/v_sg dc/v_sr dc/v_sg dc/v_sb dc/v_sct 0 def/v_fct 0 def/v_ft 0 def\r
+/v_cxe 0 def/v_cxm 0 def/v_sa -1 def/v_ea -1 def/sR dc/sG dc/sB dc\r
+/mR dc/mG dc/mB dc/eR dc/eG dc/eB dc/sC dc/sM dc/sY dc/sK dc/eC dc\r
+/eM dc/eY dc/eK dc/sH dc/sS dc/sV dc/eH dc/eS dc/eV dc/sGy dc/eGy\r
+dc/mGy dc/ci_datasrc dc/ci_matrix dc/ci_dataleft dc/ci_buf dc/ci_dataofs\r
+dc/ci_y dc/rciBuf dc/cbslw dc/cmiBuf dc/cPalette dc/cpci_datasrc dc\r
+/cpci_matrix dc/cpci_bpp dc/cpci_y dc/cpci_sampsleft dc/cpci_nextcol\r
+dc/cpci_buf dc/startX dc/startY dc/endX dc/endY dc/endX2 dc/endY2 dc\r
+/fillX dc/urx dc/ury dc/llx dc/lly dc/incD dc/distance dc/slice dc\r
+/startangle dc/Steps dc/incH dc/incS dc/incV dc/incR dc/incG dc/incB\r
+dc/incGy dc 0.25 setlinewidth [] 0 setdash 0 setlinejoin 0 setlinecap}bd\r
+aca/setplategray{v_plate v_cpky eq{1 exch sub setgray pop pop pop}{v_plate\r
+v_cpyl eq{pop 1 exch sub setgray pop pop}{v_plate v_cpmg eq{pop pop\r
+1 exch sub setgray pop}{v_plate v_cpcy eq{pop pop pop 1 exch sub setgray}{1\r
+exch sub setgray pop pop pop}ifelse}ifelse}ifelse}ifelse}bd/setplatecolor{v_plate\r
+v_cpky eq{1 exch sub 0 0 0 4 -1 roll setcmykcolor pop pop pop}{v_plate\r
+v_cpyl eq{pop 1 exch sub 0 0 0 4 2 roll setcmykcolor pop pop}{v_plate\r
+v_cpmg eq{pop pop 1 exch sub 0 0 0 4 1 roll setcmykcolor pop}{v_plate\r
+v_cpcy eq{pop pop pop 1 exch sub 0 0 0 setcmykcolor}{1 exch sub 0 0\r
+0 4 -1 roll setcmykcolor pop pop pop}ifelse}ifelse}ifelse}ifelse}bd\r
+/setcmykcolor where{pop}{/setcmykcolor{cmyk2rgb setrgbcolor}bd}ifelse\r
+/setlogcmykcolor{v_gseps 1 eq{v_mono 1 eq{1 exch sub setgray pop pop\r
+pop}{setcmykcolor}ifelse}{v_mono 1 eq{cmyk2rgb rgb2gray setgray}{setcmykcolor}ifelse}ifelse}bd\r
+/setlogrgbcolor{v_gseps 1 eq{v_mono 1 eq{rgbtoplategray}{rgb2devcmyk\r
+setplatecolor}ifelse}{v_mono 1 eq{rgb2gray setgray}{systemdict begin\r
+setrgbcolor end}ifelse}ifelse}bd/setfillcolor{v_fct 0 eq{v_fc v_fm\r
+v_fy v_fk setlogcmykcolor}{v_fr v_fg v_fb setlogrgbcolor}ifelse}bd\r
+/setstrokecolor{v_sct 0 eq{v_sc v_sm v_sy v_sk setlogcmykcolor}{v_sr\r
+v_sg v_sb setlogrgbcolor}ifelse}bd/setgfillcmyk{v_gseps 1 eq{v_mono\r
+1 eq{cmyk2rgb rgb2plategray}{cmyk2rgb rgb2devcmyk setplatecolor}ifelse}{v_mono\r
+1 eq{cmyk2rgb rgb2gray setgray}{setcmykcolor}ifelse}ifelse}bd/setgfillrgb{v_gseps\r
+1 eq{v_mono 1 eq{rgb2plategray}{rgb2devcmyk setplatecolor}ifelse}{v_mono\r
+1 eq{rgb2gray setgray}{systemdict begin setrgbcolor end}ifelse}ifelse}bd\r
+/setgfillhsb{v_gseps 1 eq{v_mono 1 eq{systemdict begin sethsbcolor\r
+currentrgbcolor end rgb2plategray}{systemdict begin sethsbcolor currentrgbcolor\r
+end rgb2devcmyk setplatecolor}ifelse}{v_mono 1 eq{systemdict begin\r
+sethsbcolor currentgray end setgray}{systemdict begin sethsbcolor end}ifelse}ifelse}bd\r
+/Max{2 copy lt{exch}if pop}bd/Max3{2 copy lt{exch}if pop 2 copy lt{exch}if\r
+pop}bd/Min{2 copy gt{exch}if pop}bd/Min3{2 copy gt{exch}if pop 2 copy\r
+gt{exch}if pop}bd/clamp{3 1 roll Max 2 1 roll Min}bd/clamp01{0 Max\r
+1 Min}bd/Pythag{dup mul exch dup mul add sqrt}bd/ssc{DeviceRGB setcolorspace\r
+setcolor}bd/ssg{setgray}bd/p_render{}def/p_count 0 def/vis_flag true\r
+def/DataString 3 string def/DataSrc{currentfile DataString readhexstring\r
+pop}bd/DataStr1 1 string def/DataStr2 1 string def/DataStr3 1 string\r
+def/DataSrc1{DataStr1}bd/DataSrc2{DataStr2}bd/DataSrc3{DataStr3}bd\r
+/colorimage where{pop/ci{colorimage}bd}{/ci{pop pop/ci_datasrc exch\r
+def matrix invertmatrix/ci_matrix exch def pop/ci_dataleft 0 def/ci_buf()def\r
+/ci_dataofs 0 def 0 1 3 -1 roll 1 sub{/ci_y exch def dup 0 1 3 -1\r
+roll 1 sub{0 1 2{pop ci_dataleft 0 eq{ci_datasrc dup length/ci_dataleft\r
+exch def/ci_buf exch def/ci_dataofs 0 def}if ci_buf ci_dataofs get\r
+255 div/ci_dataofs ci_dataofs 1 add def/ci_dataleft ci_dataleft 1 sub\r
+def}for setrgbcolor dup ci_y 3 -1 roll 1 add ci_y 1 add 4 copy 5 1\r
+roll 4 2 roll 5 -1 roll 1 1 4{pop ci_matrix transform 8 2 roll}for\r
+m l l l closepath fill}for}for pop}bd}ifelse/rci{/rciBuf 4 index 3\r
+index mul 7 add 8 div floor cvi string def{currentfile rciBuf readhexstring\r
+pop}bind false 3 ci}bd/cbsl{2 eq/cbslL2 xd 5 index/cbslw xd translate\r
+scale 8 [ 3 index 0 0 5 index 0 0 ] cbslL2{/DataStr1 cbslw string def\r
+currentfile/ASCII85Decode filter/RunLengthDecode filter DataStr1 readstring\r
+pop pop/DataStr2 cbslw string def currentfile/ASCII85Decode filter\r
+/RunLengthDecode filter DataStr2 readstring pop pop/DataStr3 cbslw\r
+string def currentfile/ASCII85Decode filter/RunLengthDecode filter\r
+DataStr3 readstring pop pop{DataStr1}bind{DataStr2}bind{DataStr3}bind\r
+true}{/DataSrc load false}ifelse 3 ci}bd/gbsl{2 eq/gbslL2 xd 5 index\r
+/gbslw xd translate scale 8 [ 3 index 0 0 5 index 0 0 ] gbslL2{/DataStr1\r
+gbslw string def currentfile/ASCII85Decode filter/RunLengthDecode filter\r
+DataStr1 readstring pop pop{DataStr1}bind}{/DataStr1 gbslw string def\r
+currentfile DataSrc1 readhexstring pop pop{DataStr1}bind}ifelse image}bd\r
+/cmi{/cmiBuf 4 index 3 index mul 7 add 8 div floor cvi string def{currentfile\r
+cmiBuf readhexstring pop}bind image}bd/cpal{4 mul string/cPalette exch\r
+def currentfile cPalette readhexstring pop}bd/cpci{/cpci_datasrc exch\r
+def matrix invertmatrix/cpci_matrix exch def/cpci_bpp exch def cpci_init\r
+0 1 3 -1 roll 1 sub{/cpci_y exch def dup cpci_bpp 4 eq{cpci_sampsleft\r
+1 eq{/cpci_sampsleft 0 def}if}if 0 1 3 -1 roll 1 sub{cpci_nextcol dup\r
+cpci_y 3 -1 roll 1 add cpci_y 1 add 4 copy 5 1 roll 4 2 roll 5 -1 roll\r
+1 1 4{pop cpci_matrix transform 8 2 roll}for m l l l closepath fill}for}for\r
+pop}bd/cpci_init{/cpci_sampsleft 0 def}bd/cpci_buf 1 string def/cpci_nextcol{cpci_bpp\r
+1 eq{cpci_sampsleft 0 eq{currentfile cpci_buf readhexstring pop pop\r
+/cpci_sampsleft 8 def}if cpci_buf dup 0 get dup 1 and setgray -1 bitshift\r
+1 exch put/cpci_sampsleft cpci_sampsleft 1 sub def}{cpci_bpp 4 eq{cpci_sampsleft\r
+0 eq{currentfile cpci_buf readhexstring pop pop/cpci_sampsleft 2 def}if\r
+cpci_buf 0 get dup 15 and exch -4 bitshift cpci_buf 0 3 -1 roll put\r
+/cpci_sampsleft cpci_sampsleft 1 sub def}{currentfile cpci_buf readhexstring\r
+pop 0 get}ifelse 4 mul dup 2 add cPalette exch get 255 div exch dup\r
+1 add cPalette exch get 255 div exch cPalette exch get 255 div setrgbcolor}ifelse}bd\r
+/setup1asciiproc{[ currentfile mystring/readhexstring cvx/pop cvx\r
+] cvx bind}bd/setup1binaryproc{[ currentfile mystring/readstring cvx\r
+/pop cvx ] cvx bind}bd level2{save/dontloadlevel1 xd}if/iw 0 def/ih\r
+0 def/im_save 0 def/setupimageproc 0 def/polarity 0 def/smoothflag\r
+0 def/mystring 0 def/bpc 0 def/beginimage{/im_save save def dup 0 eq{pop\r
+/setup1binaryproc}{1 eq{/setup1asciiproc}{(error, can't use level2 data acquisition procs for level1)print\r
+flush}ifelse}ifelse/setupimageproc exch ld/polarity xd/smoothflag xd\r
+/imat xd/mystring exch string def/bpc xd/ih xd/iw xd}bd/endimage{im_save\r
+restore}bd/1bitbwcopyimage{1 setgray 0 0 moveto 0 1 rlineto 1 0 rlineto\r
+0 -1 rlineto closepath fill 0 setgray iw ih polarity imat setupimageproc\r
+imagemask}bd/1bitcopyimage{setrgbcolor 0 0 moveto 0 1 rlineto 1 0 rlineto\r
+0 -1 rlineto closepath fill setrgbcolor iw ih polarity imat setupimageproc\r
+imagemask}bd/1bitmaskimage{setrgbcolor iw ih polarity [iw 0 0 ih 0\r
+0] setupimageproc imagemask}bd level2{dontloadlevel1 restore}if level2\r
+not{save/dontloadlevel2 xd}if/setup2asciiproc{currentfile/ASCII85Decode\r
+filter/RunLengthDecode filter}bd/setup2binaryproc{currentfile/RunLengthDecode\r
+filter}bd/myimagedict 9 dict dup begin/ImageType 1 def/MultipleDataSource\r
+false def end def/im_save 0 def/setupimageproc 0 def/polarity 0 def\r
+/smoothflag 0 def/mystring 0 def/bpc 0 def/ih 0 def/iw 0 def/beginimage{\r
+/im_save save def dup 2 eq{pop/setup2binaryproc}{dup 3 eq{pop/setup2asciiproc}{0\r
+eq{/setup1binaryproc}{/setup1asciiproc}ifelse}ifelse}ifelse/setupimageproc\r
+exch ld{[ 1 0 ]}{[ 0 1 ]}ifelse/polarity xd/smoothflag xd/imat xd/mystring\r
+exch string def/bpc xd/ih xd/iw xd}bd/endimage{im_save restore}bd/1bitbwcopyimage{1\r
+ssg 0 0 moveto 0 1 rlineto 1 0 rlineto 0 -1 rlineto closepath fill\r
+0 ssg myimagedict dup begin/Width iw def/Height ih def/Decode polarity\r
+def/ImageMatrix imat def/DataSource setupimageproc def/BitsPerComponent\r
+1 def/Interpolate smoothflag def end imagemask}bd/1bitcopyimage{ssc\r
+0 0 moveto 0 1 rlineto 1 0 rlineto 0 -1 rlineto closepath fill ssc\r
+myimagedict dup begin/Width iw def/Height ih def/Decode polarity def\r
+/ImageMatrix imat def/DataSource setupimageproc def/BitsPerComponent\r
+1 def/Interpolate smoothflag def end imagemask}bd/1bitmaskimage{ssc\r
+myimagedict dup begin/Width iw def/Height ih def/Decode polarity def\r
+/ImageMatrix imat def/DataSource setupimageproc def/BitsPerComponent\r
+1 def/Interpolate smoothflag def end imagemask}bd level2 not{dontloadlevel2\r
+restore}if\r
+level2{save/dontloadlevel1 xd}if/startnoload{{/noload save def}if}bd\r
+/endnoload{{noload restore}if}bd/testsystemdict{where{systemdict eq{true}{false}ifelse}{false}ifelse}bd\r
+/ncolors 1 def/colorimage where{pop true}{false}ifelse{/ncolors 0\r
+statusdict begin/processcolors where{pop pop processcolors}{/deviceinfo\r
+where{pop deviceinfo/Colors known{pop{deviceinfo/Colors get}}if}if}ifelse\r
+end def ncolors 0 ne{/colorimage testsystemdict/setcolortransfer testsystemdict\r
+/currentcolortransfer testsystemdict/currentcmykcolor testsystemdict\r
+and and and not{/ncolors 0 def}if}if}if ncolors dup 1 ne exch dup 3\r
+ne exch 4 ne and and{/ncolors 0 def}if ncolors 1 eq dup dup not startnoload{\r
+/expandbw{expandfactor mul round cvi bwclut exch get 255 div}bd/doclutimage{bwclut\r
+colorclut pop/bwclut xd bpc dup 8 eq{pop 255}{4 eq{15}{3}ifelse}ifelse\r
+/expandfactor xd [/expandbw load/exec load dup currenttransfer exch\r
+] cvx bind settransfer iw ih bpc imat setupimageproc image}bd}if not\r
+endnoload ncolors dup 3 eq exch 4 eq or dup dup not startnoload{/nullproc{{}}def\r
+/concatutil{/exec load 7 -1 roll/exec load}bd/defsubclut{1 add getinterval\r
+def}bd/spconcattransfer{/Dclut exch def/Cclut exch def/Bclut exch def\r
+/Aclut exch def/ncompute exch ld currentcolortransfer [{Aclut ncompute}concatutil\r
+] cvx [{Bclut ncompute}concatutil ] cvx [{Cclut ncompute}concatutil\r
+] cvx [{Dclut ncompute}concatutil ] cvx setcolortransfer}bd/setuprgbcluts{\r
+/bit3x rgbclut length 3 sub def/bit1x bit3x 3 idiv def/rclut rgbclut\r
+def/gclut rclut 1 bit3x defsubclut/bclut rclut 2 bit3x defsubclut}bd}if\r
+not endnoload ncolors 3 eq dup dup not startnoload{/3compute{exch bit3x\r
+mul round cvi get 255 div}bd/doclutimage{/rgbclut xd pop setuprgbcluts\r
+/3compute rclut gclut bclut dup spconcattransfer iw ih bpc imat [\r
+setupimageproc/exec load/dup load dup ] cvx nullproc nullproc true\r
+3 colorimage}bd}if not endnoload ncolors 4 eq dup dup not startnoload{\r
+/stuffclut{cmykindex 3 -1 roll put}bd/ftoint{1 exch sub 255 mul round\r
+cvi}bd/4compute{exch bit4x mul round cvi get 255 div}bd/computecmykclut{setuprgbcluts\r
+/bit4x rgbclut length 3 idiv 4 mul 4 sub def/cmykclut bit4x 4 add\r
+string def/cclut cmykclut def/mclut cclut 1 bit4x defsubclut/yclut\r
+cclut 2 bit4x defsubclut/kclut cclut 3 bit4x defsubclut/cmykindex 0\r
+def 0 1 bit1x{dup/cmykindex exch bit1x exch sub 4 mul def 3 mul dup\r
+rclut exch get 255 div exch dup gclut exch get 255 div exch bclut exch\r
+get 255 div setrgbcolor currentcmykcolor ftoint kclut stuffclut ftoint\r
+yclut stuffclut ftoint mclut stuffclut ftoint cclut stuffclut}for}bd\r
+/doclutimage{/rgbclut xd pop invalidcolortable?{computecmykclut}if\r
+/4compute cclut mclut yclut kclut spconcattransfer iw ih bpc imat\r
+[ setupimageproc/exec load/dup load dup dup ] cvx nullproc nullproc\r
+nullproc true 4 colorimage}bd}if not endnoload ncolors 0 eq dup dup\r
+not startnoload{/lookupandstore{3 mul 3 getinterval putinterval exch\r
+3 add exch 3 copy}bd/8lookup/lookupandstore ld/4lookup{/byte 1 index\r
+def -4 bitshift lookupandstore byte 15 and lookupandstore}bd/2lookup{\r
+/byte 1 index def -6 bitshift lookupandstore byte -4 bitshift 3 and\r
+lookupandstore byte -2 bitshift 3 and lookupandstore byte 3 and lookupandstore}bd\r
+/colorexpand{mystringexp 0 rgbclut 3 copy 7 -1 roll/mylookup load\r
+forall pop pop pop pop pop}bd/createexpandstr{/mystringexp exch mystring\r
+length mul string def}bd/doclutimage{/rgbclut xd pop/mylookup bpc 8\r
+eq{3 createexpandstr/8lookup}{bpc 4 eq{6 createexpandstr/4lookup}{12\r
+createexpandstr/2lookup}ifelse}ifelse ld iw ih bpc imat [ setupimageproc\r
+/exec load/colorexpand load/exec load] cvx false 3 colorimage}bd}if\r
+not endnoload/colorimage where{pop true}{false}ifelse dup{/do24image{iw\r
+ih 8 imat setupimageproc false 3 colorimage}bd}if dup dup startnoload\r
+not{/rgbtogray{/str xd/len str length def/smlen len 3 idiv def/rstr\r
+str def/gstr str 1 len 1 sub getinterval def/bstr str 2 len 2 sub getinterval\r
+def str dup 0 1 smlen 1 sub{dup 3 mul rstr 1 index get .3 mul gstr\r
+2 index get .59 mul add bstr 3 -1 roll get .11 mul add round cvi put\r
+dup}for pop 0 smlen getinterval}bd/do24image{iw ih 8 imat [ setupimageproc\r
+/exec load/rgbtogray load/exec load ] cvx bind image}bd}if endnoload\r
+/doimage{iw ih 8 imat setupimageproc image}bd level2{dontloadlevel1\r
+restore}if level2 not{save/dontloadlevel2 xd}if/myappcolorspace/DeviceRGB\r
+def/rgbclut 0 def/doclutimage{/rgbclut xd pop bpc dup 8 eq{pop 255}{4\r
+eq{15}{3}ifelse}ifelse/hival xd [/Indexed myappcolorspace hival rgbclut]\r
+setcolorspace myimagedict dup begin/Width iw def/Height ih def/Decode\r
+[0 hival] def/ImageMatrix imat def/DataSource setupimageproc def/BitsPerComponent\r
+bpc def/Interpolate smoothflag def end image}bd/do24image{myappcolorspace\r
+setcolorspace myimagedict dup begin/Width iw def/Height ih def/Decode\r
+[0 1 0 1 0 1] def/ImageMatrix imat def/DataSource setupimageproc def\r
+/BitsPerComponent 8 def/Interpolate smoothflag def end image}bd level2\r
+not{dontloadlevel2 restore}if\r
+/NumSteps{dtransform matrix defaultmatrix idtransform Pythag currentscreen\r
+pop pop 72 exch div div}bd/FindMinSteps{v_ft 4 eq{urx startX sub abs\r
+llx startX sub abs Max ury startY sub abs lly startY sub abs Max Pythag\r
+2 3.14159265 mul mul 0}{v_ft 2 eq{endY startY sub endX startX sub Pythag\r
+endY2 startY sub endX2 startX sub Pythag gt{endY startY sub endX startX\r
+sub}{endY2 startY sub endX2 startX sub}ifelse}{endY startY sub endX\r
+startX sub}ifelse}ifelse NumSteps}bd/cxe{/v_cxe exch def}bd/cxm{pop\r
+/v_cxm exch def}bd/cxmt{pop pop}bd/cxt{pop}bd/S_eoclip{currentflat{{eoclip}stopped{dup\r
+currentflat exch sub 20 gt{([Error: PathTooComplex; OffendingCommand: eoclip]\n)print\r
+flush exit}{currentflat 2 add setflat}ifelse}{exit}ifelse}loop setflat}bd\r
+/S_clip{currentflat{{clip}stopped{dup currentflat exch sub 20 gt{([Error: PathTooComplex; OffendingCommand: clip]\n)print\r
+flush exit}{currentflat 2 add setflat}ifelse}{exit}ifelse}loop setflat}bd\r
+/S_eofill{currentflat{{eofill}stopped{dup currentflat exch sub 20\r
+gt{([Error: PathTooComplex; OffendingCommand: eofill]\n)print flush\r
+exit}{currentflat 2 add setflat}ifelse}{exit}ifelse}loop setflat}bd\r
+/gpbbx{pathbbox/ury exch def/urx exch def/lly exch def/llx exch def}bd\r
+/lineargfill{initgfill{false initgfx/distance endX startX sub endY\r
+startY sub Pythag def/incD distance Steps div def endY startY sub endX\r
+startX sub atan newpath llx lly urx ury Bx startX startY translate\r
+rotate gpbbx eGy ssg newpath llx lly urx ury Bx S_eofill sGy ssg newpath\r
+llx lly 0 ury Bx S_eofill/fillX 0 def 0 1 Steps 1 sub{stepgfx newpath\r
+fillX lly fillX incD add dup/fillX exch def ury Bx S_eofill pop}for}if}bd\r
+/radialgfill{initgfill{false initgfx/distance endX startX sub endY\r
+startY sub Pythag def/incD distance Steps div def eGy ssg newpath llx\r
+lly urx ury Bx S_eofill/distance 0 def 0 1 Steps 1 sub{stepgfx newpath\r
+startX startY distance 0 360 arc closepath distance incD add dup/distance\r
+exch def 0 rmoveto startX startY distance 0 360 arc closepath S_eofill\r
+pop}for}if}bd/ellipticgfill{initgfill{true initgfx sGy ssg newpath\r
+llx lly urx ury Bx S_eofill [ endX startX sub endY startY sub endX2\r
+startX sub endY2 startY sub startX startY ] concat Steps 1 sub -1 0{stepgfx\r
+/i exch def 0 0 moveto 0 0 i Steps div 0 360 arc fill}for}if}bd/conicalgfill{initgfill{urx\r
+startX sub abs llx startX sub abs Max ury startY sub abs lly startY\r
+sub abs Max Pythag startY endY sub startX endX sub Pythag div/radius\r
+exch def true initgfx [ endX startX sub endY startY sub startY endY\r
+sub endX startX sub startX startY ] concat/slice 180 Steps div def\r
+/startangle 0 def Steps -1 0{pop stepgfx 0 0 moveto 0 0 radius startangle\r
+neg dup slice add arc fill 0 0 moveto 0 0 radius startangle dup slice\r
+add dup/startangle exch def arc fill}for}if}bd/initgfill{gpbbx{S_eoclip}{S_clip}ifelse\r
+startX endX eq startY endY eq and sGy eGy eq or dup{sGy ssg newpath\r
+llx lly urx ury Bx S_eofill}if not}bd/initgfx{/flag exch def flag{sGy\r
+eGy/sGy exch def/eGy exch def}if eGy sGy sub abs 256 mul FindMinSteps\r
+Min 256 Min ceiling 1 Max/Steps exch def eGy sGy sub Steps div/incGy\r
+exch def sGy/mGy exch def}bd/stepgfx{mGy ssg mGy incGy add/mGy exch\r
+def}bd/linearfill{initfill{false initfx endX startX sub endY startY\r
+sub Pythag/distance exch def/incD distance Steps div def endY startY\r
+sub endX startX sub atan newpath llx lly urx ury Bx startX startY translate\r
+rotate gpbbx v_cxe 0 eq{eR eG eB setgfillrgb}{eH eS eV setgfillhsb}ifelse\r
+newpath llx lly urx ury Bx S_eofill v_cxe 0 eq{sR sG sB setgfillrgb}{sH\r
+sS sV setgfillhsb}ifelse newpath llx lly 0 ury Bx S_eofill/fillX 0\r
+def 0 1 Steps 1 sub{stepfx newpath fillX lly fillX incD add dup/fillX\r
+exch def ury Bx S_eofill pop}for}if}bd/radialfill{initfill{false initfx\r
+endX startX sub endY startY sub Pythag/distance exch def/incD distance\r
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