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Brno University of Technology Vysoké Učení Technické V Brně BRNO UNIVERSITY OF TECHNOLOGY VYSOKÉ UČENÍ TECHNICKÉ V BRNĚ FACULTY OF MECHANICAL ENGINEERING FAKULTA STROJNÍHO INŽENÝRSTVÍ INSTITUTE OF MATHEMATICS ÚSTAV MATEMATIKY 3D SCENE RECONSTRUCTION USING CLIFFORD ALGEBRAS 3D REKONSTRUKCE SCÉNY POMOCÍ CLIFFORDOVÝCH ALGEBER MASTER'S THESIS DIPLOMOVÁ PRÁCE AUTHOR Bc. Jan Hrubý AUTOR PRÁCE SUPERVISOR doc. Mgr. Jaroslav Hrdina, Ph.D. VEDOUCÍ PRÁCE BRNO 2017 Zadání diplomové práce Ústav: Ústav matematiky Student: Bc. Jan Hrubý Studijní program: Aplikované vědy v inženýrství Studijní obor: Matematické inženýrství Vedoucí práce: doc. Mgr. Jaroslav Hrdina, Ph.D. Akademický rok: 2017/18 Ředitel ústavu Vám v souladu se zákonem č.111/1998 o vysokých školách a se Studijním a zkušebním řádem VUT v Brně určuje následující téma diplomové práce: 3D rekonstrukce scény pomocí Cliffordových algeber Stručná charakteristika problematiky úkolu: 3D rekonstrukce scény souvisí z Epipolární geometrii, tedy rekonstrukce pomocí projekcí pořízených ze dvou míst. Problematiku je možné řešit objektově orientovaným přístupem v různých Cliffordových algebrách (CGA, PGA, ...) Cíle diplomové práce: Nastudování algebraického základu teorie Clifforodových algeber. Nastudování potřebných geometrických modelů a to převážně konformní a projektivní geometrické algebry. Aplikace pro řešení konkrétních problémů binokulárního vidění. Seznam doporučené literatury: PERWASS, Ch. Geometric algebra with applications in engineering. Berlin: Springer, 2009. ISBN 35- 408-9067-X. DORST, L., D.H.F. FONTIJNE a S. MANN. Geometric algebra for computer science: an object- oriented approach to geometry. Amsterdam: Elsevier, 2007. Morgan Kaufmann series in computer graphics. ISBN 978-0-12-369465-2. ELL, T. A. Quaternion fourier transforms for signal and image processing. London: Wiley-ISTE, 2014. ISBN 978-184-8214-781. Fakulta strojního inženýrství, Vysoké učení technické v Brně / Technická 2896/2 / 616 69 / Brno Termín odevzdání diplomové práce je stanoven časovým plánem akademického roku 2017/18 V Brně, dne L. S. prof. RNDr. Josef Šlapal, CSc. doc. Ing. Jaroslav Katolický, Ph.D. ředitel ústavu děkan fakulty Fakulta strojního inženýrství, Vysoké učení technické v Brně / Technická 2896/2 / 616 69 / Brno Abstract This master thesis has as its goal to introduce to reader still quite new and unknown part of mathematics, the geometric algebra. First the basic definitions are presented and then the properties of general geometric algebra are studied. Large part of the text is dedicated to the Conformal geometric algebra which is currently one of the most frequently studied and applied geometric algebra. Its algebraic and geometric properties are described, particularly its ability do represent certain geometric object as vectors. We are also allowed to compute their intersections and conformal transformations. Next part of the text is devoted to applications of Conformal geometric algebra. First in description of a kinematics of a robotic arm and then in binocular vision. Abstrakt Tato diplomov´apr´acem´aza c´ılsezn´amitˇcten´aˇrese st´alejeˇstˇerelativnˇenovou a nezn´amou oblast´ımatematiky, s geometrickou algebrou. Nejdˇr´ıve jsou uvedeny z´akladn´ıdefinice a pot´ejsou studov´any vlastnosti obecn´egeometrick´ealgebry. Dalˇs´ıvelk´aˇc´asttextu se vˇenuje Konformn´ıgeometrick´ealgebˇre,kter´aje v souˇcasnostijedna z nejv´ıcezkouman´ych a aplikovan´ych geometrick´ych algeber. Jsou pops´any jej´ıalgebraick´ea geometrick´evlast- nosti, konkr´etnˇeschopnost reprezentovat urˇcit´egeometrick´eobjekty jako vektory. Takt´eˇz umoˇzˇnujepoˇc´ıtat jejich pr˚unikya konformn´ıtransformace. Dalˇs´ıˇc´asttextu je zamˇeˇrena na aplikace Konformn´ıgeometrick´ealgebry, nejdˇr´ıve k popisu kinematiky robotick´eruky a pot´ev binokul´arn´ımvidˇeni. key words geometric algebra, Conformal geometric algebra, rotation, translation, robotic arm, fot- ward kinematics, inverse kinematics, binocular vision kl´ıˇcov´aslova geometrick´aalgebra, Konformn´ıgeometrick´aalgebra, rotace, translace, robotick´aruka, dopˇredn´akinematika, inverzn´ıkinematika, binokul´arn´ıvidˇen´ı HRUBY,´ J. 3D rekonstrukce sc´enypomoc´ıCliffordov´ychalgeber. Brno: Vysok´euˇcen´ı technick´ev Brnˇe,Fakulta strojn´ıhoinˇzen´yrstv´ı,2018. 62 s. Vedouc´ıdiplomov´epr´ace doc. Mgr. Jaroslav Hrdina, Ph.D. Rozˇs´ıˇren´yabstrakt V posledn´ıch letech roste z´ajemmatematik˚ui inˇzen´yr˚uo nov´yprostˇredekumoˇzˇnuj´ıc´ı geometrick´ev´ypoˇcty, o tzv. geometrick´ealgebry. Jedn´ase o zobecnˇen´ı pojm˚uˇc´ıslo a vektor do jednoho pojmu multivektor, kter´yumoˇzˇnujepopis mnoha geometrick´ych ob- jekt˚ua snadnou pr´acis nimi. Pˇredkl´adan´adiplomov´apr´acepˇredstavuje ´uvodn´ı text do dan´eproblematiky a m˚uˇzeb´ytpovaˇzov´anaza ´uvodn´ı studijn´ı text do studia geo- metrick´ych algeber. Pojem geometrick´aalgebra pˇredstavuje speci´aln´ıpˇr´ıpadCliffordovy algebry nad re´aln´ymiˇc´ısly. Existuje velk´emnoˇzstv´ıgeometrick´ych algeber liˇs´ıc´ıse al- gebraick´ymia geometrick´ymivlastnosti. Nˇekter´egeometrick´ealgebry byli jiˇzznaˇcnˇe matematicky prozkoum´any a jejich vlastnosti byly dostateˇcnˇeprok´az´any a ocenˇeny, jin´e geometrick´ealgebry na sv˚ujobjev a uzn´an´ıteprve ˇcekaj´ı. Prvky geometrick´ych algeber jsou zobecnˇen´aˇc´ısla.Kupˇr´ıkladukomplexn´ıˇc´ıslavzniknou pˇrid´an´ımkomplexn´ıjednotky k re´aln´ymˇc´ısl˚um. Pokud m´ısto jedn´ejednotky, rovnaj´ıc´ı se v druh´emocninˇeminus jedn´e,pˇrid´amek re´aln´ymˇc´ısl˚umtyto jednotky tˇria vhodnˇeurˇc´ımevztahy mezi nimi, dostaneme algebru kvaternion˚u,nach´azej´ıc´ıznaˇcn´euplatnˇen´ınapˇr.pro popis rotac´ınebo v anal´yzeobrazu. Dalˇs´ızobecnˇen´ıkvaternion˚ujsou oktoniony, kter´emaj´ısedm jednotek rovnaj´ıc´ıse v druh´emocninˇeminus jedniˇcce. Komplexn´ıˇc´ısla,kvaterniony a oktoniony pˇredstavuj´ıpouze nˇekolik pˇr´ıklad˚ugeometrick´ych algeber. Dalˇs´ıdostaneme pˇrid´av´an´ım jednotek, kter´ese v druh´ych mocnin´ach nerovnaj´ıpouze minus jedn´e,ale tak´eplus jedn´e a eventu´alnˇetak´enule. Konkr´etn´ıgeometrickou algebru tedy obdrˇz´ımedefinov´an´ımtˇr´ı ˇc´ısel urˇcuj´ıc´ı poˇcetjednotek rovnaj´ıc´ı se v druh´emocninˇejedn´e,minus jedn´ea nule. Zmˇenoutˇechto ˇc´ısel dost´av´amer˚uzn´egeometrick´ealgebry s odliˇsn´ymigeometrick´ymi vlastnostmi. Nˇekter´etakto z´ıskan´egeometrick´ealgebry nach´azej´ı v posledn´ıch letech ˇsirok´euplatnˇeni v mnoha teoretick´ych i aplikovan´ych discipl´ın´ach a jejich studiem se zab´yv´ast´alerostouc´ıkomunita vˇedc˚ua inˇzen´yr˚u,kon´ase mnoho konferenc´ıa je st´ale v´ıcezˇrejm´e,ˇzegeometrick´ealgebry nab´ızej´ıefektivn´ızp˚usoby jednak k inovativn´ımˇreˇsen´ı klasick´ych probl´em˚u,jednak k ˇreˇsen´ıprobl´em˚unov´ych. Tato diplomov´apr´aceprezen- tuje jednu z nejv´ıcev souˇcasnostipouˇz´ıvan´ych geometrick´yhalgeber, tzv. Konformn´ı geometrickou algebru, umoˇzˇnuj´ıc´ısnadnou reprezentaci line´arn´ıch a sf´erick´ych geomet- rick´ych objekt˚ua efektivn´ızach´azen´ıs nimi. V jednotliv´ych kapitol´ach probereme r˚uzn´e aspekty Konformn´ıgeometrick´ealgebry, od z´akladn´ıch algebraick´ych definic, pˇrespopis transformac´ı geometrick´ych objekt˚uaˇzpo aplikace v poˇc´ıtaˇcov´emvidˇen´ı. Prvn´ıˇc´ast prvn´ıkapitoly se zamˇeˇrujena z´akladn´ıalgebraick´edefinice a vˇety. Je definov´anatzv. algebra nad polem, coˇzje z´akladn´ıstavebn´ık´amencel´ehokonceptu geometrick´ych al- geber. Jedn´ase o koneˇcnˇedimenzion´aln´ıvektorov´yprostor s dodateˇcnoubiline´arn´ıop- erac´ı,tj. operac´ıline´arn´ıv obou promˇenn´ych. Reˇcenoobr´acenˇe,pokudˇ definujeme na vektorov´emprostoru operaci pˇriˇrazuj´ıc´ıdvˇemavektor˚umnov´yvektor, kter´aje line´arn´ı v obou promˇenn´ych, z´ıskan´astruktura se naz´yv´aalgebra nad polem, kde pole je je stejn´e pole, nad kter´ymje definovan´ypˇr´ısluˇsn´yvektorov´yprostor. D´aleje uvedena slavn´aFu- biniova vˇetaˇr´ıkaj´ıc´ı,ˇzeexistuj´ıpouze tˇriasociativn´ıalgebry nad polem re´aln´ych ˇc´ısel, na kter´ych m˚uˇzeme zav´estoperaci dˇelen´ı- re´aln´aˇc´ısla,komplexn´ıˇc´ıslaa kvaterniony. Pˇripomeˇnme,ˇzeoktoniony takovou strukturu netvoˇr´ı,protoˇzejejich n´asoben´ınen´ıaso- ciativn´ı. N´asledujenˇekolik pˇr´ıklad˚uzn´am´ych struktur, kter´etvoˇr´ıalgebru nad polem, napˇr.tˇr´ıdimenzion´aln´ıEuklidovsk´yvektorov´yprostor s vektorov´ymsouˇcinem,polynomy, ˇctvercov´ematice apod. Druh´aˇc´astprvn´ıkapitoly je zamˇeˇrena na pˇresnoudefinic´ıgeomet- rick´ealgebry a popis jejich vlastnost´ı. Nejdˇr´ıve je definov´anvhodn´yvektorov´yprostor. Vezmeme koneˇcnˇedimenzion´aln´ı Euklidovsk´yvektorov´yprostor a definujeme n´asoben´ı mezi jeho b´azov´ymivektory takov´ymzp˚usobem, ˇzeje antikomutativn´ıa b´azov´evektory se v druh´ych mocnin´ach rovnaj´ıminus jedn´e,jedn´enebo nule. R˚uzn´ekombinace n´asobk˚u b´azov´ych vektor˚upotom tvoˇr´ıprvky b´azegeometrick´ealgebry. Takto z´ıskan´ageometrick´a algebra pˇredstavuje vektorov´yprostor vyˇsˇs´ıdimenze, neˇzbyl p˚uvodn´ıvektorov´yprostor. Kupˇr´ıkladudvouprvkov´amnoˇzinab´azov´ych vektor˚uvygeneruje ˇctyˇrdimenzion´aln´ıgeo- metrickou algebru, tˇr´ıprvkov´amnoˇzinab´azov´ych vektor˚uvygeneruje osmidimenzio´aln´ı geometrickou algebru apod. Prvky geometrick´ealgebry nazveme multivektory a defin- ujeme mezi nimi operaci n´asoben´ı. V dalˇs´ım textu n´asledujed˚ukaz bilinearity pr´avˇe definovan´ehon´asoben´ıa geometrick´aalgebra je tedy definov´anakorektnˇe. D´alepopisu- jeme nˇekolik d˚uleˇzit´ych pojm˚upouˇz´ıvan´ych v geometrick´ych algebr´ach a jejich pˇr´ıklady. Prvn´ıkapitola pokraˇcujedefinicemi dalˇs´ıch dvou velmi d˚uleˇzit´ych operac´ımezi multivek- tory, tzv. vnitˇrn´ıma vnˇejˇs´ımproduktem. Tyto operace naleznou uplatnˇenipozdˇejiv t´eto pr´acipˇripopisu geometrick´ych objekt˚ua pˇri
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