R

febb5a3d-420d-43e1-aaad-e4a16119559c 139 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 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 45255 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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Xv7/krdiXYJ9tTcT3ghNPJCIQ3wzo1+s/EqYYCI9DMAP4g9S8xH8o1Bg+8r7Qjg3ybMLVS8C5CWgnmGqheMJ4MRP8rMhfPPU1oBod0wpyfwe826Gkt9n/ByxMZQ2lFOoDoZOK/59hmsMFePsRNQmXofSBK1VDCPK1w8LpCZ3//oXQviTbQDR7rhSkjlNtrozYaHbDG8SnAp5Dbf01ZV+kahfZ5WzEt7HkDpBDWoJH8c/wTqfwfIaz9CoJXy/CLX/CmGOOfrbf709INqBgVIaa5YbM+FGpxmuMSJeo2cXienH7DmPC+Y8urkAnMXwHrKOw0kwD+E/R0b9mRDeZBn3Yi3MLhfh1Fe2D0Q7YFCwHuuU12fCeq8ZVo6RYxgdrzEkSiCl+27vR4C5zIBbYcnWjTkGtYSfZ3T9bJhtnQinjjoP78wOHhRNMGdmQo+h1L6HkTFN4gXOBW68UP4PF8w5VHORSfwGz8vrAZWgltnweRZmn2J753YwoWhdwDw/w4hqhi4TzPr7Q3iD1b5wSiCBh0wBsQQs6uFq73j4XPhdgPyZR9+WHVwoWgcw/wSYxWZcv3fxBZ7tLjM9nUM1KkYw/4vfH77Es8zfedRt28GGognmbwFzBTCuxljDt94LHBYg51m3CaXOoPnKJk+7O7WDD0VrA+ZbgLkEGKAUH6KKcuk4S/uJ2XD/LgLR7g4o2jpgngLMBcA8wG/9l2YBtLtANCa7u8QaI6H4+tOsXB8NLH5Z4O4NEO3uUUqyNRRz8qlQ/MFePZ2E8P/UXnO+UZQ3swAAAABJRU5ErkJggg== image 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Gray

1 time t0 - min 8.9

Rotate image 4 quadrants

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

M gray :

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