From: Martin Mares Date: Sat, 28 Nov 2009 21:37:43 +0000 (+0100) Subject: Prvni verze zapisku o FFT. X-Git-Url: http://mj.ucw.cz/gitweb/?a=commitdiff_plain;h=c3ea83834b9874fc87dcf228ad47ec0fe9d74bcf;p=ads2.git Prvni verze zapisku o FFT. --- diff --git a/9-fft/9-fft.tex b/9-fft/9-fft.tex new file mode 100644 index 0000000..b2e0639 --- /dev/null +++ b/9-fft/9-fft.tex @@ -0,0 +1,264 @@ +\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} $\equiv$ $(\forall k) +(kFFT($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 new file mode 100644 index 0000000..0769a79 --- /dev/null +++ b/9-fft/Makefile @@ -0,0 +1,3 @@ +P=9-fft + +include ../Makerules diff --git a/9-fft/img.eps b/9-fft/img.eps new file mode 100644 index 0000000..d7e20d4 --- 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pop pop}{v_plate +v_cpyl eq{pop 1 exch sub setgray pop pop}{v_plate v_cpmg eq{pop pop +1 exch sub setgray pop}{v_plate v_cpcy eq{pop pop pop 1 exch sub setgray}{1 +exch sub setgray pop pop pop}ifelse}ifelse}ifelse}ifelse}bd/setplatecolor{v_plate +v_cpky eq{1 exch sub 0 0 0 4 -1 roll setcmykcolor pop pop pop}{v_plate +v_cpyl eq{pop 1 exch sub 0 0 0 4 2 roll setcmykcolor pop pop}{v_plate +v_cpmg eq{pop pop 1 exch sub 0 0 0 4 1 roll setcmykcolor pop}{v_plate +v_cpcy eq{pop pop pop 1 exch sub 0 0 0 setcmykcolor}{1 exch sub 0 0 +0 4 -1 roll setcmykcolor pop pop pop}ifelse}ifelse}ifelse}ifelse}bd +/setcmykcolor where{pop}{/setcmykcolor{cmyk2rgb setrgbcolor}bd}ifelse +/setlogcmykcolor{v_gseps 1 eq{v_mono 1 eq{1 exch sub setgray pop pop +pop}{setcmykcolor}ifelse}{v_mono 1 eq{cmyk2rgb rgb2gray setgray}{setcmykcolor}ifelse}ifelse}bd +/setlogrgbcolor{v_gseps 1 eq{v_mono 1 eq{rgbtoplategray}{rgb2devcmyk +setplatecolor}ifelse}{v_mono 1 eq{rgb2gray setgray}{systemdict begin +setrgbcolor end}ifelse}ifelse}bd/setfillcolor{v_fct 0 eq{v_fc v_fm +v_fy v_fk setlogcmykcolor}{v_fr v_fg v_fb setlogrgbcolor}ifelse}bd +/setstrokecolor{v_sct 0 eq{v_sc v_sm v_sy v_sk setlogcmykcolor}{v_sr +v_sg v_sb setlogrgbcolor}ifelse}bd/setgfillcmyk{v_gseps 1 eq{v_mono +1 eq{cmyk2rgb rgb2plategray}{cmyk2rgb rgb2devcmyk setplatecolor}ifelse}{v_mono +1 eq{cmyk2rgb rgb2gray setgray}{setcmykcolor}ifelse}ifelse}bd/setgfillrgb{v_gseps +1 eq{v_mono 1 eq{rgb2plategray}{rgb2devcmyk setplatecolor}ifelse}{v_mono +1 eq{rgb2gray setgray}{systemdict begin setrgbcolor end}ifelse}ifelse}bd +/setgfillhsb{v_gseps 1 eq{v_mono 1 eq{systemdict begin sethsbcolor +currentrgbcolor end rgb2plategray}{systemdict begin sethsbcolor currentrgbcolor +end rgb2devcmyk setplatecolor}ifelse}{v_mono 1 eq{systemdict begin +sethsbcolor currentgray end setgray}{systemdict begin sethsbcolor end}ifelse}ifelse}bd +/Max{2 copy lt{exch}if pop}bd/Max3{2 copy lt{exch}if pop 2 copy lt{exch}if +pop}bd/Min{2 copy gt{exch}if pop}bd/Min3{2 copy gt{exch}if pop 2 copy +gt{exch}if pop}bd/clamp{3 1 roll Max 2 1 roll Min}bd/clamp01{0 Max +1 Min}bd/Pythag{dup mul exch dup mul add sqrt}bd/ssc{DeviceRGB setcolorspace +setcolor}bd/ssg{setgray}bd/p_render{}def/p_count 0 def/vis_flag true +def/DataString 3 string def/DataSrc{currentfile DataString readhexstring +pop}bd/DataStr1 1 string def/DataStr2 1 string def/DataStr3 1 string +def/DataSrc1{DataStr1}bd/DataSrc2{DataStr2}bd/DataSrc3{DataStr3}bd +/colorimage where{pop/ci{colorimage}bd}{/ci{pop pop/ci_datasrc exch +def matrix invertmatrix/ci_matrix exch def pop/ci_dataleft 0 def/ci_buf()def +/ci_dataofs 0 def 0 1 3 -1 roll 1 sub{/ci_y exch def dup 0 1 3 -1 +roll 1 sub{0 1 2{pop ci_dataleft 0 eq{ci_datasrc dup length/ci_dataleft +exch def/ci_buf exch def/ci_dataofs 0 def}if ci_buf ci_dataofs get +255 div/ci_dataofs ci_dataofs 1 add def/ci_dataleft ci_dataleft 1 sub +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 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