R

febb5a3d-420d-43e1-aaad-e4a16119559c 146 3 5 decimal

&[DATE] &[TIME] - &[FILENAME]

Rotate an RGB image

To lower interaction in this Smath document, each component has been extracted. Convenient for your visit to this gorgeous Smath demo.

Part 1: Extract [R,G,B] components from an image matrix

r M rows : c M cols 3 / : 12 mat red M 1 r 1 c submatrix : green M 1 r c 1 + (2 c * submatrix : blue M 1 r 2 c * 1 + (M cols submatrix : 41 sys

R

R 1241169677563715100000000000000000000000006183351688410011713415116618219921623324625325425425425425425425425425425425425425525512411799806243246000000000000000000000000000513294561779110712213915516918419921723224525125425425525425425425425425425425412411910484694934142000000000000000000000000000029213449657993108122138153166181195210225237247252254254255254254254254124121107897456422480000000000000000000000000000001310223449637789103116130144158171184198211225236246250254254254124122110947965483417500000000000000000000000000000000139162841546780911041161291421551671801912052172292401241231139984725643301430000000000000000000000000000000000114101728395163758698109120132144156166179124124116105897866514027134000000000000000000000000000000000000001271320314051617383931041141241241191099784746249392815510000000000000000000000000000000000000000022511162232405012412412111310391827261504131199200000000000000000000000000000000000000000000002231241241241161099988807263524436261683100000000000000000000000000000000000000000000001241241241211141079989827567585043352719115200000000000000000000000000000000000000000000124124124124119113108100928680746759524540332619127421000000000000000000000000000000000000000124124124124123119115109105989286817771666054484338332721161175321000000000000000000000000000000000124124124124124124121118114110106102979288838076716863585450464137332924201714118543211000000000000000000000001241241241241241241241241221211181161131101071041019793918885827976737068646057545249454239363331272422201816141211987554432211000124124124124124124124124124124124124123122121120119117116115113112110108106104103101999794939190888684828079777674727068666562605755535250494745444241393837124124124124124124124124124124124124124124124124124124124124124124123123123123122122122121121121121121120120120119119119119118117117115115114113112112112111110109108108107106106105105104103102124124124124124124124124124124124124124124124123123123122122122121120119118117116115114113112111110109108107106105104103103102100999796949392908987868684848281807978777675124124124124124124124124124123123121120117116114111109106104102999693918987858279767472706866625957555250484543413937353230272523212018171614131110101241241241241241241241221191161131091051029793908682797672696561575451474339363329242119161412975443211000000000000000124124124124124122118114110105100948984797571666055514641363227211713964321000000000000000000000000000001241241241241211161111069891858075696256494338322519148531000000000000000000000000000000000000012412412412311711010394878074675851443730221484200000000000000000000000000000000000000000012412412411911210493857769605143352617941000000000000000000000000000000000000000000000124124123115106958576685646382817820000000000000000000000000000000000000000000000001241241201111008778685544352312300000000000000000000000000000000000000000000124914191241241181079381715745342091000000000000000000000000000000000000000124101421324151617282124124115102877562473520810000000000000000000000000000000000001391424344657688089101112123135146124123111978269523924910000000000000000000000000000000001391728405365788810111212513715016117318519720912412210992776145301320000000000000000000000000000000161326385265789010411613014415616918219520822223324324925312412010687725338194000000000000000000000000000000614284257718599111127142155169182197211225238246251254254255254254124118101826546291000000000000000000000000000003112439557185100114130146160175190206222235246251254254255254254254254254254124117987959402030000000000000000000000000051631486581961111281451601761912082252402492542542542542542542542542542542542542543364 mat :

G

G 000000052543648410412414316418320322224525425425425425425425425425425425425425425525425425425425425425425425425425425425425425124222620919317716114312711195786145271700000001133151698910612614416418220122023925225425425425425425425425425425425425425425425425425425425425425425425425525425425425425425124323021419918316915213612210591750000000032037567291107125141161177196213231247254254254254254254254254254254254254254254254254254254254254254254254254254255254254254254254253247239226210197181168153138000000000825405873911061231391551721882052212372502542542542542542542542542542542542542542542542542552542542542542542542542542542542542542542542542542532482402292152020000000001122741587288103117133148163178193209224238250254254254254254254254254254254254254254254254254254255254254254254254254254254254254254255254254254254254253252000000000011327405568829711012413715116517919320822123524725325425425425425425425425425425425425425425425425425425425525425425425425425425425425425425425425525425400000000000111243649627488101113125138150163176189201214228239249253254254254254254254254254254254254254254254254254254254254254255254254254254254254254254254254000000000000171930415365768810011012213414415716817919120321422723624625025425425425425425425425425425425425425425425425425425425425425425425525425425425425400000000000000311223241526373839310411412413514415616517618619620621722823624524925225425425425425425425425425425425425425425425425425425425425425425425400000000000000013111928364756657383921021101201271371461561641741831932012102192292352432482502532542542542542542542542542542542542542542542542542540000000000000000002513202835445260687584919910711512313013814615516216917818619420120821722523223724324724925225425425425425425425425425425400000000000000000000024815202734404854606874818894100108114121127133140146155161167173179188195201207213219227231236241244246249251000000000000000000000000013581317222733384450556165717683889499103109115121126132136141146154160164170174179185193198000000000000000000000000000000012346913172023262934394348535760646871768287919499102106109115121000000000000000000000000000000000000000000124456689121418212325273031333638410000000000000000000000000000000000000000000000000000000000000001000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000011334456677991100000000000000000000000000000000000013445791216182124272932353944495256586265687175800000000000000000000000000000134610151822263136404751566064697480869094991031071131201241281331371411461531580000000000000000000000024612162229354047535965717784909610110711312012613113614314915716216717317818619419820420921422022623100000000000000000001381521293643505765728087931011081151231301361441521601661721801891962032092162242302352412442472502522542542542542540000000000000000028162533415058677584921011091181261341431521601681771851952022102192282342402462492522542542542542542542542542542542542542542540000000000000003112130394960697888991071181261371461561651761851952042142242332402472502532542542542542542542542542542542542542542542542542542542542540000000000000192131425364748697107118129140150161173184194205216228236246250253254254254254254254254254254254254254254254254254254254254254254254254255254000000000000316283852647588101112124136148160172184196209221233244250254254254254254254254254254254254254254254254254254254254254254255254254254254254254254254000000000005193145597185991111251381511641771922052182312442502542542542542542542542542542542542542542542542542542542542552542542542542542542542542542542542542540000000000520334863779210612113615016417819320922323724925425425425425425425425425425425425425425425425425425425425425425425425425425425425425425425425425425425425400000000031933506581971121291431601751912072222382502542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542532502422340000000011430486582991151321491671832002172352492542542542542542542542542542542542542542542542552542542542542542542542542542542542542542542542542532482392272121991851720000000072643628099116134152171188208225244253254254254254254254254254254254254254254254255254254254254254254254254254255254254254254254249241227211198182168153137123107000000021937587697114134153174193211232249254254254254254254254254254254254254254255254254254254254254254254254255254254254254254247236219203188172156139124108937659443364 mat :

B

B 25525425425425425425425425425425425425425425425425425425425424322520418416414512510585675032153000000000000000000000000000000255254254254254254254254254254254254254254254254254254254254254249231214194176156138119100826649331750000000000000000000000000000255254254254254254254254254254254254254254254254254254254254254254253241225207189171154136119101846852372392000000000000000000000000025525525425425425425425425425425425425425425425425425425425425425425425425023822220518917215614012410791756147331982000000000000000000000025525525425425425425425425425425425425425425425425425425425425425425425425425424923922420819417816314813311710288736047342110500000000000000000002552552542542542542542542542542542542542542542542542542542542542542542542542542542542522432312162031891741601471331191059277665342301994100000000000000255255254254254254254254254254254254254254254254254254254254254254254254254254254254254254253249241230216204191178166153140128115102917767564433231484000000000025525525425425425425425425425425425425425425425425425425425425425425425425425425425425425425425425425325124423522521220119017816715614313212210998887566564536271811731000255255254254254254254254254254254254254254254254254254254254254254254254254254254254254254254254254254254254254252251244237229218208199188176166158146135126116104968675675848393224162552552552542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542522512452392332262152061981891781691621531421331251171069789822552552552542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542522502492432372322272212102031961891801711641572552552552552542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542532522512492482442382342302552552552552552542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542552552552552552552542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542552552552552552552552552542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542552552552552552552552552552552552552552552542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552552542542542542542542542542542542542542542542542542542542542542542542552552552552552552552552552552552552552552552552552552552552552542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542552552552552552552552552552552552542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542552552552552552552552542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542552552552552552542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542552552552552542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542522512502552552552552542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542532522512492452392342302262182092031972552552552542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542532522512472402352302212112031961881771691621541431351281212552552552542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542532522482412352262152061971881771671591481381301201091009183726456472552552542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542542532512442372262152051941821711621501381281181069687746656453729201395225525525425425425425425425425425425425425425425425425425425425425425425425425425425425425425425425324623822621420319017716615414113011810594837060494030201273000000025525525425425425425425425425425425425425425425425425425425425425425425425425425425425425324623622320919618317015714313011710391766654433121116100000000000025525525425425425425425425425425425425425425425425425425425425425425425425425425425124122721319818316915413912511196826956443120940000000000000000025525525425425425425425425425425425425425425425425425425425425425425425425425023822320719117516014412811297826754402614610000000000000000000025525525425425425425425425425425425425425425425425425425425425425425425224022320618917215413812110488715743281350000000000000000000000002552542542542542542542542542542542542542542542542542542542542542542452282091911721551371191018367513519710000000000000000000000000025525425425425425425425425425425425425425425425425425425425425223821819918016014112110283664932164000000000000000000000000000003364 mat :

Functions/parameters ........ to construct _° rot, λ background

gray R G + B + 3 / :

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 gray

gray Min gray Max 21 mat 8616721 mat

m n q m n m n - 22 mat :

θ f θ cos θ sin - θ sin θ cos 22 mat :

gray rows gray cols 12 mat 336412 mat

Absolute value of a vector v Abs j 1 v rows range a j el v j el abs : for a 21 line :

t0 1 time :

M n Round i 1 M rows range j 1 M cols range m i j el M i j el n round : for for m 21 line :

Part 2: Bilinear program to rotate the [R,G,B]

This rotation program is courtesy from Paul W. [Mathsoft Collab], adapted Smath R G B deg λ rotate m R rows : n R cols : 12 mat θ deg π * 180 / : A θ f : bounds m n q A * : j 12 range uv j el bounds j col Abs max 0 round : for u uv 1 el : v uv 2 el : 12 mat N m n u v 22 mat 1 + 2 / : mc N 11 el : nc N 12 el : uc N 21 el : vc N 22 el : 22 mat prepare pixel canvas wht λ 11113 mat transpose * : RGB u 3 v * matrix : x 1 u range y 1 v range s0 A x y 21 mat uc vc 21 mat - (* eval mc nc 21 mat + : s s0 Floor s0 Floor 1 + augment : w s 2 col s0 - s0 s 1 col - augment : sum 00013 mat transpose : p 12 range q 12 range i s 1 p el : j s 2 q el : 12 mat a R i j el G i j el B i j el 31 mat : a wht : try sum sum w 1 p el w 2 q el * a * + eval : 41 line for for k 13 range RGB x y v k 1 - (* + el sum k el mwround : for 61 line for for RGB 131 line eval :

could not vectorize for "absolute" |,| ?

sum k el 0 round sum k el mwround 12 mat

mwround("matrix") - (X) rounds the elements of X to the nearest integers.

Part 3: Utilities to recover image

Might work for earlier than
"try, on error" i 0 > (i m ≤ (& (j 0 > (& (j n ≤ (& a R i j el G i j el B i j el 31 mat : a wht : if

rgb R G B 45235 rotate :

To make it work in older SMath
(without try()) a R i j el G i j el B i j el 31 mat : a wht : try i 0 > (i m ≤ (& j 0 > (& j n ≤ (& a R i j el G i j el B i j el 31 mat : a wht : if ≡

`

rgb rgb :

R G B ° rotation λ background rotate

M i extract j i : m M rows : n M cols 3 / : 12 mat M 1 m 1 j 1 - (n * + j n * submatrix 31 line :

R rgb 1 extract : G rgb 2 extract : B rgb 3 extract : 13 mat

Gray R G + B + (3 / :

Paul W. wrtote: "Note that the original is of low resolution and that the bilinear interpolation of the rotation program has a small smoothing effect".

image R G B 13 mat :

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

Result from the rotation

rgb 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 image 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Gray

1 time t0 - min 7.4

Rotate image 4 quadrants

M C:\SmathFile\Venturi 50.png READBMP :

M gray :

Unlike Mathcad: Smath 6179 does not support the "submatrix" reverse cols order.Here is the Smath "submatrix mirror".

Explode mirror Matrix M mirror ct 1 : j M cols ct range mir 1 ct el M j col : ct ct 1 + : 21 line for mir Unest N v mir 1 el : j 2 mir cols range v v mir j el augment : for v 71 line :

Mirror M mirror :

M Min M Max 21 mat 8616721 mat

Inverse 91 M - :

M rotate_90 M mirror transpose :

M rotate90 M transpose :

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M rotate_90 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Inverse 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167 M -

Davide: 10 April, 20017 M mirror M 1 M rows range M cols 1 range el :

N M mirror :