THE PROJECTIVE GEOMETRY APPLIED TO PLANE RECTIFICATION

Transcription

1 THE PROJECTIVE GEOMETRY APPLIED TO PLANE RECTIFICATION Arias Pérez, B. () Gozález Auilera, D. (2) Gómez Laoz, J. (2) Sáez Martí, N. (3) () Departameto de Ieiería Miera, Uiversidad de Leó, Spai. (2) Dpto. Ieiería Cartoráfia y del Terreo. Uiversidad de Salamaa, Spai. dauilera@usal.es, fotod@usal.es (3) Faultad de CC Ararias y Ambietales, Uiversidad de Salamaa, Spai. ilda@usal ABSTRACT: I te preset artile te subjet of te retifiatio of diital imaes is approaed, from two poits of view. First (pramati), it solves te retifiatio for te partiular missio of oblique terrestrial potorammetry, ad oe demostrates tat wit a oly taki of te objet te retifiatio of fudametal plaes of te same oe is obtaied. I a seod poit of view (teoretial), oe udertakes te study ad aalysis of te eometri trasformatios of a plae to aoter plae (imae to imae), is appliatio i te retifiatio subjet, ad te deompositio of trasformatios i te plae eeri i simple trasformatios. INTRODUCTION Of form itrodutory to te retifiatio proess, i te first plae desribes te basi eometry of a oblique potoram, wi a be redued to te followi basi oepts: Foal (f) ad Priipal Poit (P), like ow parameters of te eometry of te amera. Vaisi Poits (A, B, C), like parameters of te positio of te amera respet to te objet. If te objet is redued to tree basi diretios, orrespodi to teir loal system of oordiates, tree vaisi poits eist, oe for ea basi diretio. Fiure.- Basi eometry of a sile imae Bot vaisi poits orrespodi to X ad Y-ais, B ad A respetively, are o te orizo lie (). Te vaisi poit orrespodi to Z-ais, C, is o te priipal lie or lie of maimum slope tat passes trou te priipal poit. Te orizo ad te priipal lie are perpediular to ea oter, i te plae of te potoram, wi arees wit te plae formed by te vaisi poits, A, B, C. Te priipal poit a be obtaied by itersetio of te tree eits of te triale formed by te vaisi poits, sie e is ortoetre, as it a be see i te fiure.

2 TRANSFORMATION OF THE VANISHING POINTS Te plae is defied as parallel to te plae defied by aes XZ of te system of te objet ad tat passes trou poit A (vaisi poit i te diretio Y). Of tis form tis retifiatio proess osists of te projetio of te plae o. Proess of retifiatio. Plae XZ We bei te projetio of te plae o te plae, ad sie it looks for to failitate te projetio eometrially, tis passes to use artesia systems tat tey ave i ommo te strait itersetio of tese plaes. Te strait CB is te strait itersetio of te plae of te potoram,, wit te plae parallel to te impliit plae i aes XZ of te objet, ad tat passes trou S. Te strait itersetio of te plaes ad is impliit i bot artesias systems tat are defied: Te system of referee XY, belos to te plae, plae of te potoram. Its orii i A, te ais X is defied passes trou tis orii ad P, wereas te ais Y is parallel to te strait lie formed by te oter two vaisi poits, C ad B. Te system of referee X2Y2, belos to te plae, plae of te retified imae. Its orii i A is defied, ad te ais Y2 is oiidet wit te ais Y, ad terefore it ompares to te strait CB. Terefore, tis proess of retifiatio is divided i tree differetiated steps, ad i ea oe of tem itermediate or auiliary systems are take:.- Trasferri from P to A, ad rotatio, betwee oplaar systems 2.- Projetio of plaes itself. 3.- Rotatio betwee oplaar systems. Te fiure sows 4 toeter imaes to appreiate te obtaied results. Upper left: oriial imae wit lies ad vaisi poits, triale, eits ad priipal poit. Upper rit: Rotated ad trasferred imae, result of step. Lower left: imae result of te projetio of te plae of te potoram o a flat parallel to plae XZ of te objet. Lower rit: Fial imae witout ow tur.

3 Fiure 2.- Proess of Retifiatio. Plae XZ Retifiatio of plae YZ Wit te previous steps oe as bee able to projet te oriial imae o a plae parallel to te impliit plae i aes XZ of te system assoiated to te objet. Te resampli of te plae orrespodi to plae YZ also a be divided i tree steps similar to te orrespodi oes to plae XZ:.- Trasferri from P to B, ad rotatio, betwee oplaar systems 2.- Projetio of plaes itself. 3.- Rotatio betwee oplaar systems. Strait lie AC a be iterpreted like te itersetio betwee te plae of te potoram wit te plae defied by te poits SAC, parallel plae to te impliit plae i aes YZ of te objet. Sie te plae o wi it is tried to projet is also parallel to tis plae of te objet ad passes trou poit B (vaisi poit i X), te ais Y defies as a parallel strait lie to te strait itersetio betwee te plaes p ad te oe tat otais to poits SAC. Tis strait itersetio allows to iterpret te tree steps eometrially, ad allows te deompositio of te projetio i te tree desribed pases. Te Fiure 3 sows te 4 obtaied toeter imaes to appreiate te result. Upper left: oriial imae wit lies ad vaisi poits, triale, eits ad priipal poit. Te plae to retify is plae YZ of te objet, i olour orae i te imae. Upper rit: Rotated ad trasferred imae, wit wi te eit orrespodi to poit B is ompares to -ais of te imae. It is te imae result of step. Lower left: imae result of te projetio of te plae of te potoram o a flat parallel to plae YZ of te objet. As it a be observed te effet of te ollows iterferes to a reat etet i te result. Lower rit: Fial imae witout ow tur, result of step 3. If it is observed wit torouess it a distiuis te olour poits orae orrespodi to retified plae YZ.

4 Fiure 3.- Retifiatio of Plae YZ Iverse retifiatio As it a be observed i te imaes obtaied by meas of te proess of previous retifiatio, tey eist ollow i te imaes beause te projetio is disreet. Fiure 4.- Iverse retifiatio However, te reviewed proess of retifiatio is te ituitive oe, te oe tat omes from te idea, ad diret retifiatio a be deomiated. Aoter added disadvatae is related to te size (ad sale) of te resulti imae, sie e is impossible a priori to determie is size. It is ertai tat it is possible to be alulated of previous form to te wide resampli ad te i oe of te retified imae, but a be imposed before resample eiter te sale or te size. Terefore, a solutio to tese problems passes to make resampli of iverse form, i.e., to ross piels of te resulti imae oe by oe, to look for its orrespodi piels i te oriial imae, to take or to iterpolate its olour ad to plae tem i te imae destiy. Of tis form, te iverse retifiatio before resolves te reviewed problems. O te oe ad, we rossi piels of te imae destiy, ollows will ot take plae, sie always tere is piel orrespodi. Ad o te oter ad, it is possible to be otrolled te size of te retified imae, ad terefore, te relatio (sale) wit respet to te oriial imae. I additio, e is eeri o te part of diverse autors to reommed te use of iterpolatios to still avoid diverse effets i tis proess of iverse retifiatio (to stair stepped...), as it is eplaied more aead.

5 Partial retifiatios I te followi imae te imae of eample wit plae XZ is observed to retify i olour orae. Also te oter elemets a be observed (lies of flit, eits...) ad te imae of te rit we a observe te result by meas of te iverse proedure. Fiure 5.- Partial retifiatios As te effet of ollows a be verified is orreted, but it follows witout bei solved te questio of te sale. I additio, aoter ioveiet produt of te size is added, as mu i dimesio as i memory of te retified imae. Te retified plae, i olour orae, is orret, everteless, retifies wit te same projetio te oter basi plae of te objet. Terefore, te problem of te sale like te relative oe to te retifiatio limits beomes eessary to solve so mu. Te Fiure 6 is a eample of diret retifiatio (rit), ad te same eample wit te partial retifiatio of a sile faade (left). Saled retifiatios Fiure 6.- Partial retifiatios It is possible to apply a fator of sale, or so tat te retified imae is ot reater of determied eit/widt of imae, or so tat te imaes of two plaes wit a ommo ede ave te same sale, or to establis a ertai sale to te imae retified wit views to its prited eit. For it, oe well-kow te wised sale, is due to apply i te followi passaes of te retifiatio: I all te data, i.e., as mu i te oordiates of te vaisi poits like i tose of te priipal poit. I te oordiates of te poits tat delimit te retifiatio limit. I te oordiates of te retified imae, at te ed of te alulatios, but i iverse sese to as it is applied i te oter data.

6 Fiure 7.- Saled retifiatios Metods of Iterpolatio Te iterpolatio of te imaes as a reat importae i te proesses of retifiatio of imaes. Te two followi imaes display te result for te same plae, i te imae of te left is ot applied o type of iterpolatio, wereas i te imae of te rit te biliear iterpolatio is applied. PROJECTIVE GEOMETRY Fiure 8.- Metods of Iterpolatio I tis setio a itrodutio to Projetive Geometry beomes ad te projetive trasformatios of a plae study. By meas of tese trasformatios it is possible to model te eometri projetio tat it passes to im to a plae tat is see by a perspetive amera from a poit of view. Wit tese trasformatios some properties oserve as te oliearity (air lies are see like air lies) wereas oter properties ot (te parallel lies are eerally ot see like parallel lies). I a elemetary level eometry is te study of poits ad lies, ad teir relatios.

7 Trouout istory eometry as bee oeived iitially like a et eometri disiplie, i wi te lies ad poits study witout osideri a system of oordiates. Later, by meas of te itrodutio of a system of Cartesia oordiates it is maaed to alebraizar to eometry. Tis way, te eometri oraizatios a be desribed like oordiates ad alebrai oraizatios. By meas of te alebrai relatios a appropriate matematial represetatio is obtaied to implemet aloritms ad to proram omputaioales metods. I some ases eometry maaes to better visualize a ive problem, i oters alebra a represet it ad solve it more easily. Trasformatios i te plae I eeral terms, if a omoeous vetor of tree dimesios is ad ive by (, 2, 3)T tat represets a poit i a plae, te oordiates of tis poit i te plae are defied as (; ad) = (/3, 2/3): X 3 Y 2 3 were X ad Y reeive te ame of omoeous oordiates. Projetive eometry 2D is te study of te properties of te projetive plae P2 tat are ivariat uder a roup of trasformatios kow like proyetivity. A proyetivity is a reversible trasformatio ive by : P 2 P 2 of way so tat a air lie is trasformed like a air lie. Te proyetivity tis defied like: were H is a o-siular matri 33: (m) = m = Hm Trasformatio Matri H Ivariats Eulidea R T t Let betwee poits Similarity sr T t Ales betwee strait lies, reaso betwee two distaes Affie A T t Parallel lies, reaso betwee two areas Geeral Cross ratio

8 Affie Trasformatio Te matri a bot be disturbed i te rotatio ad sale fators, by meas of te Deompositio i Siular Values, SVD. T V D U A were: V is te matri of rotatio orrespodi to te ale. D is a first diaoal wose elemets are te sale fators. U it is a rotatio matri, tat stops te ase 2D, implies a tur i te turs of or, wit wi te impliit fators of sale i matri D iterae. Projetive Trasformatio Te projetive trasformatio is te eeralizatio of te liear trasformatios R 2 R 2, ito tat te parallel lies are ot trasformed eessarily like su. It is possible to be epressed of te form: y V t A T 3 2 were V T is te differee betwee te projetive trasformatio ad te affie. I tis oe tese values tey are ull. For tat reaso, a projetive trasformatio i simpler oters a be trasformed. Deompositio of te Projetive Trasformatio Te rotatios of Gives a be used to passes of te Projetive Trasformatio to Affie: y f d e b a z y f d e b a Gives Rotatios f d e b a s s =os, s=si. Also it is possible to be aed of te Projetive Trasformatio to Affie taks to te lie i te ifiite. If obtaied vaisi poits i te fiure of te projetive trasformatio are uited bot obtais a strait lie, deomiated by several autors like te lie at ifiite. I te ase of te ompatible trasformatio, strait appiess is i te ifiite, sie by defiitio te strait parallel bars are ut i te ifiite. Terefore, a way to obtai a ompatible struture from a projetive oe osists of taki tat lie to te ifiite.

9 Fiure 9.- Lie at ifiite Te lie i te ifiite a be etrated projetively of te eometry of te trasformed fiure, i.e., a be epressed of te form: y = m +, ad te matries orrespodi to te projetive ad affie trasformatio are, respetively: f d e b a f d e b a of were it is possible to be etrated: + y + =. Terefore, parameters m, of te previous strait lie a be epressed based o terms, of te matri of projetive trasformatio: m Tus, it is possible to passes of te projetive struture of te fiure to a affie by marriae trou tese parameters, i.e., taki te lie tat uites te vaisi poits to te ifiite. For it te followi steps a be followed:. Te parameters of te projetive trasformatio alulate, so ad as it is eplaied at te beii of tis subjet. 2. It is applied to te projetive fiure (resampli te imae) te equatios tat allow to take tis lie to te ifiite. Aoter form, simpler, osists of alulati te equatio of te strait lie, i.e., te parameters m,; ad from tese, to alulate te terms, of te projetive trasformatio: m ad to take te lie to te ifiite te followi projetive trasformatio is applied: m witout foretti tat te oordiates tat are obtaied, (, y, z), are omoeous (projetive spae): y m z y ad to obtai te real oordiates te trasformatio is due to apply:

10 X z y Y z Tis matri epressio a be epressed i terms of te system assoiate te projetive oriial fiure (X,Y) ad i terms of te trasformed fiure ompatible (X,Y) of te followi way: X X X m Y ( ) Y Y X m Y ( ) Te followi imae presets/displays a imae model of a objet wit two fudametal plaes, (orae ad yellow). It is possible to be observed tat te lies tat i te previous imae esaped to te vaisi poits B ad C, ow esape to te ifiite. REFERENCES Braüer-Burardt, C., Voss, K., Mooular 3d-Reostrutio Of Buildis, Jea.De/P/D/Publi.Html. Gómez Laoz, J., Fotorametría Y Patrimoio, Cieia Y Teoloía De La Ieiería Geodésia Y Cartoráfia, Prorama De Dotorado 2-23, Esuela Politéia Superior De Ávila. Hartley R., Zisserma A., Multiple View Geometry I Computer Visio, 2, Cambride. Hemmleb, M., Wiedema, A., Diital Retifiatio Ad Geeratio Of Ortoimaes I Aritetural Potorammetry, Berli. Oreja Pedraza, V., Aputes De Fotorametría, Esuela Politéia Superior De Ávila. Truo E., Verri A., Itrodutory Teiques For 3-D Computer Visio, Ed. Pretie Hall, 998. Heuvel, F. Va De, Vaisi Poit Detetio For Aritetural Potorammetry, Iteratioal Arives Of Potorammetry Ad Remote Sesi, 32, Part 5, 998. Wiedema, A., Diital Ortoimaes I Aritetural Potorammetry Usi Diital Surfae Models, Iteratioal Arives Of Potorammetry Ad Remote Sesi, Xi, Part. B5, 996. Wiedema, A., Moré, J., Tau, R., Arimedes3d - A Iterated System For Te Geeratio Of Aritetural Ortoimaes, Iteratioal Arives for Potorammetry ad Remote Sesi, XXXVIII, 23, Atalya.