Basic projective geometry

Transcription

1 Bsc projectve geoetr Augusto Srt Stefno ubro Mrco Mrcon Fbo Antoncc Dprtento d Eettronc Inforzone e Bongegner Potecnco d Mno

2 Overvew Bsc projectve geoetr (P nd P ) rnsfortons n P Affne rectfcton n P Metrc rectfcton n P rnsfortons n P Affne rectfcton n P Metrc rectfcton n P

3 Introducton nd otvtons Projectvtes often rep objs t nfnt to fnte objs e.g. vnshng pt s vew of pt of ntersecton of pre nes (t nfnt) Geoetr of projectons s the stud of projectve nvrnts (whch nvove so objects t nfnt) In order to de wth objs t nfnt we need hoogeneous coordntes As sde effect n nonner retonshps becoe ner

4 Projectve geoetr nd rnsfortons n P Augusto Srt Stefno ubro Mrco Mrcon Fbo Antoncc Dprtento d Eettronc Inforzone e Bongegner Potecnco d Mno 4

5 5 Lnes nd ponts n hoogeneous for Pont coordntes () R oogeneous representton of nes b c he correspondence btw vectors (bc) nd nes s not one to one ( k) ( kb) ( kc) k ths defnes equvence csses of vectors ced hoogeneous vectors ( ) ( ) k kb kc b c k oogeneous representton of ponts A pont () es on the ne (bc) ff () hs s st true for (k k) therefore the equvence csses of vectors of the for (k k) re ced hoogeneous vectors

6 6 Intersecton of two nes Intersecton of two nes (bc) nd ( b c ) In fct fro the equtes We see tht Snce es on both nd t corresponds to ther ntersecton! Lne jonng two ponts nd ( ) ( ) Dut!

7 7 Dut prncpe For n theore of D projectve geoetr there ests correspondng du theore whch cn be derved b swppng the roes of ponts nd nes n the orgn theore

8 8 Ide ponts nd the ne t nfnt Intersecton of the two pre nes : : c b c b ( ) b c c b b b c c bc bc c b c b k j Pts whose thrd coord s zero re ced de pts he set of de pts ( ) s ced ne t nfnt wo pre nes ntersect wth the ne t nfnt t the se pont (drecton of the ne)

9 9 he ne t nfnt () oogeneous representton of the ne t nfnt: [ ] Ever de pont es on : [ b ] [ ] b b [ b ]

10 A ode for the projectve pne () Ponts nd nes re represented b rs nd pnes respectve through the orgn n R Y Y z z Z Z

11 A ode for the projectve pne() Lnes on the () pne (de pne) re de pts Y [ b ] [ b ] [ ] Z

12 Concs nd degree eq. n the pne b c d e f where / / In tr for (usng hoogeneuos coordntes): C C b / d / b / c e / d e / / f

13 Specfng conc Concs re copete specfed b set of fve pts Gven fve pts of the for ( ) we hve f e d c b ( ) ( ) f e d c b c c f e d c b In fct conc s the nu spce of ths 56 tr > we need fve pts n gener poston

14 ngent nes to concs: du concs A ne tngent to conc C t pont s gven b C A ne tngent to conc C stsfes n equton of the for C C s pont conc.e. n equton on pts A ne conc s n equton on nes (those tht re tngent to the conc) C Du conc (djont tr) ( ) C det C C C he epresson s vd when C s nvertbe (fu rnk) 4

15 Degenerte concs If C s not fu rnk then the conc s ced degenerte Rnk : two nes Rnk : repeted ne E.g. two nes nd ke degerte rnk- conc C s pont on ne (.e. ) es on the conc C: C ( )( ) ( )( ) he se hppens to pont on ne. he tr C s setrc nd hs rnk he nu vector s (ntersecton between nd ) Degenerte rnk- ne concs ncude two ponts nd It s de of the nes tht pss through ether of the pts. hs degenerte ne conc s the du of the two consdered pts. Note: becuse of the non nvertbt for deg. concs we hve: But n n cse s true tht: ( C ) C C C det ( C)I 5

16 Projectve trnsfortons Geoetr s the stud of propertes tht re nvrnt under groups of trnsforton (Ken) In the cse of D projectve geoetr we stud propertes of the projectve pne P tht re nvrnt under groups of trnsforton ced projectvtes Projectvt: nvertbe ppng fro P to P tht ps nes to nes A ppng h: P P s projectvt ff R det : h() P NB: projectvt s n nvertbe ner tr trnsforton n hoogeneous coordntes! Other nes: coneton hoogrph projectve tf 6

17 7 Projectve trnsfortons A projectvt s copete specfed b hoogeneous tr (8 d.o.f.) Epe : centr projecton ppng Mps ponts of onto ponts of If coordnte fres re Eucden (rectner) t s ced perspectvt (6 d.o.f.) Epe of ppcton: reovng perspectve dstorton h h h h h h h h h

18 8 Projectve trnsfortons Epe : unnce trnsfer btw ges through pnr surfce Epe : two ges wth the se cer center (cer pnnng/zoong) Epe 4: projectng shdow

19 9 rnsforton of nes nd concs rnsforton rnsfored nes In fct the foowng ncdence retons hods ( ) ( ) rnsfored concs C C ( ) ( C C C )( ) C rnsfored du concs C C C C ( ) ( )( C ) C

20 A herrch of trnsfortons ncresng No. of nvrnts Projectve trnsfortons for group PL() ced Projectve Lner group PL() s the hoogeneous verson (sced versons of the se tr re dentfed) of the gener ner group GL() here re spec csses of trnsfortons tht correspond to subgroups of PL() Isoetres Eucden group Spec Eucden group SE() Srtes Affntes Projectvtes Incresng No. of d.o.f.

21 he nuber of nvrnts he nuber of functon ndependent nvrnts s not ser thn the No. of degrees of freedo of the confgurton ess the nuber of degrees of freedo of the trnsforton

22 Isoetres rnsfortons tht preserve Eucden dstnce R s rotton tr (R RI) nd t s trnston -vector When det R- the soetr perfors roto-trnston wth refecton (not subgroup) When det R the Eucden group speczes n the Spec Eucden group SE() nd odes rgd otons Invrnts Length Anges Are Pres of nes E R t

23 Srtes Isoetres wth sotropc scng S sr t Invrnts Anges Pres of nes Rtos of two engths (scng fctor cnces out) Rtos of res (squred scng fctor cnces out)

24 4 Affntes Non-sngur ner trnsforton foowed b trnston A t A det A f decoposton through SVD AUDV where U nd V re orthogon trces A UDV R Invrnts ( )( UV VDV ) ( θ ){ R( φ) DR( φ) } D Pres Rto of engths of pre ne segents Rto of res λ λ

25 5 Projectvtes A Non-sngur ner trnsforton Invrnts Cross-rto of four coner ponts det P P v v t A ( ) det Cross j j j

26 Geoetrc propertes Group Mtr Dstorton Invrnts Projectve 8 d.o.f h h h h h h h h h Concurrenc conert order of contct: ntersecton ( pt contct) tngenc ( pt of contct); cross rto (rto of rto of engths) Affne 6 d.o.f. t t Pres rto of res rto of engths on coner or pre nes ner cobntons of vectors he ne t nfnt Srt 4 d.o.f. sr sr sr sr t t Rto of engths nge. he crcur ponts IJ Eucden r r t d.o.f. Length re r r t 6

27 7 Decoposton of projectve tf Strtng fro Eucden fre gener projectve trnsforton cn be decoposed s foows K upper-trngur tr wth det(k) v. he go s to retreve etrc nforton on fro the observton of We w show tht the best tht we cn do s to obtn etrc reconstructon of up to srt tf S t R K v I S A P s v

28 8 Decoposton of projectve tf We w foow the reverse pth fro to : P ( dof) oves the vnshng pts t nfnt A ( dof) ffects the ffne propertes but does not ffect the ne t nfnt S (4 dof) s gener srt tf tht does not ffect the ffne or projectve propertes upper trngur tr K v t R K A v t A v I K t R P A S s v v v s P A S P A S

29 A prctc epe () he orgn ge of pnr surfce (the budng fçde) wth perspectve dstorton. (b) he sntheszed ge wth front orthogon vew of the fçde. See rte-zssern 9

30 Dstorton rsng under centr projecton Iges of ted foor. () he trnsforton wth respect to the orgn s srt. A squre te s ged s squre (b) An ffne trnsforton s dded (crces becoe epses orthogon nes on the re foor becoe non-orthogon). (c) A further projectve trnsforton s dded (pre nes on the re foor becoe convergent). See rte-zssern

31 Recover of ffne nd etrc propertes Chrcterzton of projectvtes beond srtes 4 dof ore thn the 4 of srtes Such d.o.f. cqure phsc nterpretton through he ne t nfnt ( dof) he two crcur ponts on ( dof) Projectve dstorton be reoved once the ge of s specfed Affne dstorton be reoved once the ge of the crcur ponts s specfed

32 he ne t nfnt Under projectve trnsforton de pts be pped to fnte pts nd vce-vers therefore the ne t nfnt be pped to fnte ne nd vce-vers he ne t nfnt rens t nfnt ff the trnsforton s n ffnt he ne t nfnt s nvrnt under n ffne trnsforton s whoe not just pont-wse! A pont on the ne t nfnt s gener trnsfored nto nother pont on the ne t nfnt t A A t A - A

33 Recover of ffne propertes fro Suppose tht under projectve trnsforton the ne t nfnt s pped nto fnte ne We reove projectve dstorton nd eve the ffne dstorton on

34 4 Recover of ffne propertes fro p ( ) p p p

35 Affne rectfcton v vnshng pt See rte-zssern 5

36 6 Recover of ffne propertes fro We obtned n ffne reconstructon: Now we hve to upgrde to etrc reconstructon s s K t R K t R

37 7 Crcur ponts here re two (nd on two) pts on tht re nvrnt under srt tfs: the crcur (or bsoute) pts I nd J wth cnonc coordntes Pr of cope conjugte de pts J I ( ) ( ) I I sn cos sn cos cos sn sn cos se s s t s s t s s s θ θ θ θ θ θ θ θ θ

38 8 Crcur ponts he ne coes fro the fct tht ever crce ntersects t the crcur ponts he generc crce b c dz ez fz c b dz ez fz ntersects (z) n the soutons of the eq. whch re I() nd J(-)

39 9 he du to the crcur ponts he du of the crcur ponts s degenerte rnk- ne conc of the for he du conc s nvrnt under projectve trnsforton ff the trnsforton s srt ( ) ( ) C JI IJ C C C C S S S Unke the ne t nfnt does not becoe re nd vsbe conc under projectve tf! herefore we w hve to use ths nforton n pct for (through esureents constrnts) C

40 4 Epe of esureent constrnt: nges Anges n Eucden geoetr where ( ) nd ( ) re the nors to the two nes nd Such nges cn be epressed n tr for s ( )( ) C C C I I I cos θ ( ) ( ) ( )( ) cos θ

41 4 Epe of esureent constrnt: nges Notce tht ths nge esureent s nvrnt under generc projectve trnsforton In prtcur f θ/ we hve esureent constrnt of the for rrespectve of the pped hoogrph We wnt to use constrnts of ths sort to deterne hoogrph tht puts the crcur ponts bck to ther cnonc poston hs hoogrph s our etrc rectfcton C C C C C C C C C C C C

42 4 Metrc rectfcton he du conc C ncorportes we need If the pt tf s then C C therefore decoposng nto chn of the for we obtn projectve (v) nd ffne (K) coponents re deterned drect fro the ge C of C but the srt coponent rens undeterned ( ) ( ) ( )( )( ) ( ) ( ) P A A P P A S S A P S A P S A P KK v v KK KK C C C C t R K v I S A P s v

43 4 Metrc rectfcton v orthogon nes After copenston of the projectve dstorton: [ ] [ ] v v C KK K s n upper-trngur b tr wth det(k) therefore C hs degrees of freedo We need constrnts!

44 44 Metrc rectfcton v orthogon nes Orthogon ne pr n the word pne One constrnt: We need t est two ne prs ( ) ( ) s KK C ( ) s s s s s s s s k k k k k k k k k k det ; s KK K K

45 45 Metrc rectfcton v orthogon nes Wth Choesk decoposton: hen one cn obtn etrc reconstructon: We cnnot do better thn KK Cho( ) K s s s t R K t R K t R

46 46 ( ) ) ) ( ) ( ) ( ( c KK v v KK KK Metrc rectfcton v orthogon nes We cn so skp the step of ffne rectfcton usng ore nge constrnts Fro the orgn perspectve ge we hve where c(bcdef) re the eeents of the tr We cn stck fve orthogont constrnts to bud 56 tr w be the nu spce of ths tr C C C

47 Epe 47

48 48 Mesureents on Pnes 4 Approch: unwrp then esure 4

49 49 estton b pont correspondences () Gven set of ponts n P nd set of correspondng ponts (kewse n P ) estte the trnsforton tht ps ech n. he equton nvoves hoogeneous vectors: eft nd rght sdes dffer b sce fctor. he use of the cross product s ore usefu. ( ) Ah h A A A A h h h h A h h h h h h h h h h h h h h 4 ). ( re necessr for esttng Four ponts ; On two equtons re ndependent. _row_a t second_row_a frst_row_a. row of the row vector tht represents s ; w w hrd w w w w w -th DL (Drect Lner rnsforton) ethod

50 5 estton b pont correspondences () If ore thn 4 ponts re used (Ah s over-deterned) nd f there re errors n the evuton of the pont coordntes no ect souton for Ah ests (ecudng the zero souton). he nu of Ah shoud be serched (wth the condton for epe h ). It corresponds to fnd the nu of Ah / h. he prevous condton corresponds to fnd the (unt) egenvector of A A wth est egenvue. It corresponds so to fnd the (unt) sngur vector correspondng to the sest sngur vue of A. A sr pproch cn be used esttng b usng ne correspondences ( ).

51 5 estton b pont correspondences () When the estton probe s over-deterned the souton s gven b the nzton (for the consdered ponts) the so ced gebrc dstnce: It s so possbe to defne nd use n Eucden dstnce: When the two dstnces re dentc. he nzton of the Eucden dstnce consderng 4 or ore ponts eds to sove equtons tht re qudrtc n the eeents ( ) ( ) ( ) ( ) ( ) ˆ ; ˆ ˆ ˆ ˆ ˆ g g w w w w w w d d h ε ˆ w w ( ) ( ) ( ) ˆ / ˆ / ˆ / ˆ / ˆ w w w w d

52 5 Sur Affne rectfcton Deterne the ge of the ne t nfnt Drect esureent on the ge pne Deterne the hoogrph tht reps the ne t nfnt to ts cnonc poston Cosed-for coputton Metrc rectfcton Deterne the ge of the conc t nfnt Epot orthogont constrnts nd sove ner sste Deterne the hoogrph tht reps the conc t nfnt to ts cnonc poston Choesk fctorzton estton b usng DL ethod

53 Projectve geoetr nd rnsfortons n P 5

54 54 Ponts nd projectve trnsfortons A pont (YZ) R s represented n hoogeneous coordntes s 4-vector ( 4 ) P If 4 then / 4 Y / 4 Z / 4 (YZ) P If 4 then s pont t nfnt A projectve trnsforton ctng on P s nonsngur ner tf on hoogeneous 4-vectors (44 tr 5 d.o.f.)

55 Pnes A pne n -spce be wrtten s Y Z 4 Scbe preters: d.o.f. More gener: whch ens tht the pt ( 4 ) P es on the pne the frst three coponents of correspond to the pne nor 55

56 56 Jon nd ncdence retons hree pts defne pne Ech one of the pts es on the pne pne nu spce of rnk- 4 tr Cosed-for epresson for the pne defned s the jon of three pts hree pnes defne pt Projectve trnsforton ( ) ( ) ( ) ~ ~ ~ ~ ~ ~ ~ ~ ~ ~

57 57 Preterzed ponts on pne Ponts on the pne be wrtten s M he 4 tr M generte the rnk- nu spce of.e. M nd the -vector P preterzes pts on M s not unque If (bcd) P nd then M cn be wrtten s M b c [ p I ] p d

58 58 Preterzed ponts on pne [ ] [ ] ( ) [ ] [ ] [ ] [ ] [ ] ; d d c c b b d c b d c b d c b d c b p p p p p p d c b p p p d c b d c b M p I p M M herefore spns the ponts of the consdered pne

59 Qudrcs nd du qudrcs A qudrc s surfce n P defned b the equton Q Q s setrc 44 tr 9 d.o.f. (4 ndependent eeents n setrc tr for sce) 9 pts n gener poston defne qudrc he eet btw pne nd qudrc Q s conc C A coordnte sste for the pne be defned b the copeent spce to s M Q C pt trnsforton: M QM C M QM Q Q ; Q Q 59

60 he herrch of trnsfortons nvrnts Projectve trnsfortons for group ced Projectve ner group here re spec csses of trnsfortons tht correspond to subgroups Isoetres Eucden group Spec Eucden group SE() Srtes Affntes Projectvtes d.o.f. 6

61 6 f subgroups nd ther nvrnts Group Mtr Dstorton Invrnts Projectve 5 d.o.f Affne d.o.f. Srt 7 d.o.f. Eucden 6 d.o.f. A v A sr R t v t v t v t v Cross rto; ntersecton nd tngenc of surfces n contct; sgn of gussn curvture Retve dstnces ong drecton of pres; pres of pnes; voue rtos; centrods; the pne t nfnt Retve dstnces nges the bsoute conc Ω Voue bsoute dstnces

62 6 he pne t nfnt he pne t nfnt hs cnonc poston ( ) wo pnes re pre ff the ntersect on A ne s pre to nother ne or to pne f the pont of ntersecton s on he pne t nfnt s nvrnt under the projectve trnsforton ff s n ffnt s on gob fed not pont-wse s the on pne tht s nvrnt under n ffnt s fed under ffntes but sees (s oved b) projectve tf he d.o.f. of esure the projectve coponent of gener hoogrph wth respect to n ffnt Wht s eft re the 5- d.o.f. of ths gener tf copred to n ffnt

63 6 Recover of ffne propertes fro Fct: s fed under projectve trnsforton ff t s n ffnt Consequence: usng we reove the projectve dstorton nd eve on ffne dstorton Affne esureents be done on the ffne rectfed structure Note: f ( 4 ) 4 then sutbe trnsforton to p bck to () s ( ) 4 I Affne rectfcton In fct we hve: ( ) ( ) 4 4 I

64 64 he bsoute conc he bsoute conc Ω s pont conc on In etrc fre Ω s specfed b pr of equtons ( ) Ω : 4 On (wth 4 ) the defnton of Ω becoes ( ) ( ) I so tht Ω corresponds to conc C wth tr CI he conc Ω s geoetrc representton of the 5 ddton d.o.f. requred to specf etrc propertes n n ffne coordnte fre

65 65 he bsoute conc he bsoute conc s fed under the projectve tf ff s srt Proof: he bsoute conc es on therefore ts nvrnce pes the nvrnce of. As consequence the tf ust be ffne A A on the bsoute conc I s epected to be nvrnt under A : A IA ths pes tht A s orthogon ( sced rotton possb wth refecton). t v I A A I A A

66 he bsoute conc: propertes Ω s nvrnt under srtes s whoe not pont-wse A crces ntersect Ω n two pts he support of the crce s ntersects n ne nd ths ne ntersects Ω n two pts whch re the crcur pts of A spheres ntersect n Ω Metrc propertes: nge btw two nes wth drectons (-vectors) d nd d cosθ ( )( d Ω d d Ω d ) d Ω d d d : ponts (on ) n whch the two nes ntersects Ω : tr representton of the bsoute conc on orthogont dω d 66

67 67 he bsoute du qudrc Ω s conc on the pne t nfnt therefore t s defned b two eqs he du of the bsoute conc s degenerte du qudrc n D-spce ced the bsoute du qudrc Q he bs du qudrc conssts of the pnes tngent to Ω so tht Ω s the r of Q (r qudrc) 44 hoogeneous tr of rnk wth cnonc for Q I A pne (v k) s n the enveope defned b Q ff Q whch becoes vv. v s the ne n whch the pne (v k) eets the pne t nfnt nd s tngent to the bsoute du qudrc ff viv. herefore the enveope Q s de up of just those pnes tngent to the bsoute conc Just one equton nsted of two!

68 68 he bsoute du qudrc he bsoute conc nd the du bsoute qudrc n D spce Q Ω

69 69 he bsoute du qudrc: propertes he bsoute du qudrc s fed under the projectve tf ff s srt Q Q Q qudrc du v v A v Av AA t v A I v t A I k k Q Q Q qudrc du In prtcur ths s true when v nd Aorthogon tr ( s srt)

70 7 he bsoute du qudrc: propertes he pne t nfnt s the nu vector of the du bsoute qudrc If n cnonc for If not n cnonc for he nge btw two pnes nd s ( )( ) cos Q Q Q θ ( ) Q Q Q Q Q ( ) I Q

71 Net Bsc snge-vew geoetr Cer ode Coputton of the projecton tr Cer Cbrton wo-vew geoetr Eppor constrnts Fundent tr nd essent tr Retrevng the cer tr fro F Reconstructon Coputton of the fundent tr 7