Projective geometry- 2D

Transcription

1 Projecive geomer- D Acknowledgemens Marc Pollefes: for allowing e use of is ecellen slides on is opic p:// Ricard Harle and Andrew Zisserman, "Muliple View Geomer in Compuer Vision"

2 Homogeneous coordinaes Homogeneous represenaion of lines a + b + c ( a,b,c) ( ka ) + ( kb) + kc, k ( a,b,c ) ~ k( a,b,c) equivalence class of vecors, an vecor is represenaive Se of all equivalence classes in R 3 (,,) forms P Homogeneous represenaion of poins (, ) on l ( a,b,c) if and onl if a + b + c (,, )( a,b,c) (,, ) l (,,) ~ k(,,), k e poin lies on e line l if and onl if ll ( ) Homogeneous coordinaes,, Inomogeneous coordinaes (, ) 3 bu onl DOF 4//4 Projecive Geomer D

3 Poins from lines and vice-versa Inersecions of lines e inersecion of wo lines l and l' is l l' Line joining wo poins e line roug wo poins and ' is l ' Eample 4//4 Projecive Geomer D 3

4 Ideal poins and e line a infini Inersecions of parallel lines l ( a, b, c) and l' ( a, b, c' ) l l' ( b, a,) Eample Ideal poins (,,) Line a infini l (,,) Noe a is se lies on a single line, P R l Noe a in P ere is no disincion beween ideal poins and oers 4//4 Projecive Geomer D 4

5 Summar e se of ideal poins lies on e line a infini, inersecs e line a infini in e ideal poin A line parallel o l also inersecs in e same ideal poin, irrespecive of e value of c. In inomogeneous noaion, is a vecor angen o e line. I is orogonal o (a, b) -- e line normal. us i represens e line direcion. As e line s direcion varies, e ideal poin varies over. --> line a infini can be oug of as e se of direcions of lines in e plane. 4//4 Projecive Geomer D 5

6 A model for e projecive plane Poins represened b ras roug origin Lines represened b planes roug origin plane represens line a infini eacl one line roug wo poins eacl one poin a inersecion of wo lines 4//4 Projecive Geomer D 6

7 Duali l l l l l' l ' Duali principle: o an eorem of -dimensional projecive geomer ere corresponds a dual eorem, wic ma be derived b inercanging e role of poins and lines in e original eorem 4//4 Projecive Geomer D 7

8 Conics Curve described b nd -degree equaion in e plane a + b + c + d + e + f or omogenized a a, 3 a + b + c + d3 + e3 + f3 3 or in mari form C wi C a b / d / b / c e / d / e / f 5DOF: { a : b : c : d : e : f } 4//4 Projecive Geomer D 8

9 4//4 Projecive Geomer D 9 Five poins define a conic For eac poin e conic passes roug f e d c b a i i i i i i or ( ),,,,, c f i i i i i i ( ) f e d c b a,,,,, c c sacking consrains ields

10 angen lines o conics e line l angen o C a poin on C is given b lc l C 4//4 Projecive Geomer D

11 Dual conics A line angen o e conic C saisfies l C * l In general (C full rank): * C C Dual conics line conics conic envelopes 4//4 Projecive Geomer D

12 Degenerae conics A conic is degenerae if mari C is no of full rank e.g. wo lines (rank ) C lm + ml e.g. repeaed line (rank ) m l C ll l Degenerae line conics: poins (rank ), double poin (rank) Noe a for degenerae conics ( * C ) * C 4//4 Projecive Geomer D

13 Projecive ransformaions Definiion: eorem: A projecivi is an inverible mapping from P o iself suc a ree poins,, 3 lie on e same line if and onl if ( ),( ),( 3 ) do. A mapping :P P is a projecivi if and onl if ere eis a non-singular 33 mari H suc a for an poin in P reprened b a vecor i is rue a ()H Definiion: Projecive ransformaion ' ' ' or ' H 8DOF projecivicollineaionprojecive ransformaionomograp 4//4 Projecive Geomer D 3

14 Mapping beween planes cenral projecion ma be epressed b H (applicaion of eorem) 4//4 Projecive Geomer D 4

15 Removing projecive disorion selec four poins in a plane wi know coordinaes ' ' ' ' ' + + ' ( ) + 3 ( ) + 3 ' + ' (linear in ij ) 3 33 ( consrains/poin, 8DOF 4 poins needed) Remark: no calibraion a all necessar, beer was o compue (see laer) 4//4 Projecive Geomer D 5

16 ransformaion of lines and conics For a poin ransformaion ' H ransformaion for lines l' H - ransformaion for conics l C ' H - CH - ransformaion for dual conics * C' * HC H 4//4 Projecive Geomer D 6

17 Disorions under cener projecion Similari: squares imaged as squares. Affine: parallel lines remain parallel; circles become ellipses. Projecive: Parallel lines converge. 4//4 Projecive Geomer D 7

18 Class I: Isomeries (isosame, mericmeasure) ' ' ε cosθ ε sinθ sinθ cosθ ε ± orienaion preserving: orienaion reversing: ε ε ' H E R 3DOF ( roaion, ranslaion) R R special cases: pure roaion, pure ranslaion Invarians: leng, angle, area 4//4 Projecive Geomer D 8 I

19 4//4 Projecive Geomer D 9 Class II: Similariies (isomer + scale) cos sin sin cos ' ' s s s s θ θ θ θ ' R H s S I R R also know as equi-form (sape preserving) meric srucure srucure up o similari (in lieraure) 4DOF ( scale, roaion, ranslaion) Invarians: raios of leng, angle, raios of areas, parallel lines

20 4//4 Projecive Geomer D Class III: Affine ransformaions ' ' a a a a ' A H A non-isoropic scaling! (DOF: scale raio and orienaion) 6DOF ( scale, roaion, ranslaion) Invarians: parallel lines, raios of parallel lengs, raios of areas ( ) ( ) ( ) φ φ θ DR R R A λ λ D

21 Class VI: Projecive ransformaions ' H P A v v v ( v v ), 8DOF ( scale, roaion, ranslaion, line a infini) Acion non-omogeneous over e plane Invarians: cross-raio of four poins on a line (raio of raio) 4//4 Projecive Geomer D

22 4//4 Projecive Geomer D Acion of affiniies and projeciviies on line a infini + v v v v A A v A A Line a infini becomes finie, allows o observe vanising poins, orizon. Line a infini sas a infini, bu poins move along line

23 4//4 Projecive Geomer D 3 Decomposiion of projecive ransformaions v v s P A S v v A I K R H H H H RK + v A s K de K upper-riangular, decomposiion unique (if cosen s>) H.5. cos 45 sin 45. sin 45 cos 45 o o o o H Eample:

24 Overview ransformaions Projecive 8dof Concurrenc, collineari, order of conac (inersecion, angenc, inflecion, ec.), cross raio Affine 6dof Similari 4dof Euclidean 3dof a a sr sr r r a a r r sr sr Parallellism, raio of areas, raio of lengs on parallel lines (e.g midpoins), linear combinaions of vecors (cenroids). e line a infini l Raios of lengs, angles. e circular poins I,J lengs, areas. 4//4 Projecive Geomer D 4