From: Martin Mares Date: Tue, 18 Dec 2007 08:57:31 +0000 (+0100) Subject: Nulta verze kapitoly o FFT. X-Git-Url: http://mj.ucw.cz/gitweb/?a=commitdiff_plain;h=303a617310903c1543eb1415c96b79d1d96e4157;p=ads2.git Nulta verze kapitoly o FFT. --- diff --git a/8-fft/8-fft.tex b/8-fft/8-fft.tex new file mode 100644 index 0000000..2b9da46 --- /dev/null +++ b/8-fft/8-fft.tex @@ -0,0 +1,166 @@ +\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 diff --git a/8-fft/Makefile b/8-fft/Makefile new file mode 100644 index 0000000..f7c7ba1 --- /dev/null +++ b/8-fft/Makefile @@ -0,0 +1,3 @@ +P=8-fft + +include ../Makerules diff --git a/8-fft/img.eps b/8-fft/img.eps new file mode 100644 index 0000000..5a4b207 --- /dev/null +++ b/8-fft/img.eps @@ -0,0 +1,518 @@ +%!PS-Adobe-2.0 EPSF-1.2 +%%Creator: Xara X +%%For: (Unregistered user) (Unregistered company) +%%Title: (Priklad-prubehu-velikosti-8.xar) +%%CreationDate: (09/12/07) (07:34 PM) +%%BoundingBox: 6 4 265 132 +%%HiResBoundingBox: 6.135 4.944 264.089 131.873 +%%AWColourTable +%%+h (Red) 0.0 100.0 100.0 +%%+h (Orange-Red) 15.0 100.0 100.0 +%%+h (Orange) 30.0 100.0 100.0 +%%+h (Orange-Yellow) 45.0 100.0 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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 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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 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