Multi-linear Systems and Invariant Theory. in the Context of Computer Vision and Graphics. Class 4: Mutli-View 3D-from-2D. CS329 Stanford University
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1 Mult-lna Sytm and Invaant hoy n th Contxt of Comut Von and Gahc Cla 4: Mutl-Vw 3D-fom-D CS39 Stanfod Unvty Amnon Shahua Cla 4
2 Matal W Wll Cov oday Eola Gomty and Fundamntal Matx h lan+aallax modl and latv affn tuctu Why 3 vw? focal no Cla 4
3 Rmnd (fom cla ): [ I ;] P [ ; ] P x y P µ [ λ + n ; ]P P µ µ + λ + n Stand fo th famly of D octv tanfomaton btwn two fxd mag nducd by a lan n ac Cla 4 3
4 µ + Plan + Paallax P ( x, y,, µ ) [ I,] P [ ]P what do µ tand fo? what would w obtan aft lmnatng µ Cla 4 4
5 Cla 4 5 t K Z RK K + Rmnd (fom cla ): RK K, t K ) ( + K tn d R K P K ] ; [ P t R K ] [ Z Y X P
6 Cla 4 6 t K Z RK K + Z K n d + ) ( + K tn d R K ) ( Zd ZK n d + Z Y X K Z Rcall: ) ( ZK n d d Lt: Zd d + µ +
7 Not that, a dtmnd (ach) u to a cal. Lt Lt, µ B any fnc ont not ang fom µ + + µ b th homogahy w wll u Cla 4 7
8 µ Z d + µ µ µ P Z d d d µ P Z Z Rcall: µ d Zd Cla 4 8
9 Plan + Paallax µ + W hav ud 4 ac ont fo a ba: 3 fo th fnc lan fo th fnc ont (calng) Snc 4 ont dtmn an affn ba: d P d P Z µ Z Z d d µ calld latv affn tuctu Z Not: w nd 5 ont fo a octv ba. h 5 th ont th ft cama cnt. Cla 4 9
10 Not: A octv nvaant µ + ˆ ˆ µ µ ˆ µ + dˆ dˆ d d d dˆ P d Z dˆ P Z µ Z Z d d h nvaant ( octv dth ) ndndnt of both cama oton, thfo octv. 5 ba ont: 4 non-colana dfn two lan, and A 5 th ont fo calng. Cla 4
11 Not: An Affn Invaant What han whn cama cnt at nfnty? (aalll octon) µ Z d Z, Z Z d d d d P d P Z h nvaant ndndnt of both cama oton, and Affn. Z Cla 4
12 µ + Fundamntal Matx P ( x, y,, µ ) an [ ] ( ) ([ ] ) F Cla 4
13 Fundamntal Matx ([ ] ) F Dfn a blna matchng contant who coffcnt dnd only on th cama gomty (ha wa lmnatd) F do not dnd on th choc of th fnc lan [ ] λ [ ] [ ] ( + n ) Cla 4 3
14 Eol fom F Not: any homogahy matx ma btwn ol: c c Cla 4 4
15 Eol fom F F [ ] [ ] F [ ] Cla 4 5
16 Etmatng F fom matchng ont F,..., 8 Lna oluton F,..., 7 dt( F) N on-lna oluton dt( F) cubc n th lmnt of F, thu w hould xct 3 oluton. Cla 4 6
17 Cla 4 7 Etmatng F fom omogah F w-ymmtc (.. ovd 6 contant on F) + n F ] [ ] [ ) ( λ λ + n F ] [ ) ( ] [ λ λ F F homogahy matc a qud fo a oluton fo F
18 F Induc a omogahy F δ ] F [δ a homogahy matx nducd by th lan dfnd by th on of th mag ln δ and th cama cnt Cla 4 8
19 Poctv Rcontucton. Solv fo F va th ytm F (8 ont o 7 ont). Solv fo va th ytm F 3. Slct an abtay vcto δ δ 4. [ I ] and [ δ ] ] F a a a of cama matc. [ δ ] F + µ Cla 4 9
20 focal Gomty h th fundamntal matc comltly dcb th tfocal gomty (a long a th th cama cnt a not collna) F F 3 Lw: F F3 3 3 Each contant non-lna n th nt of th fundamntal matc (bcau th ol a th ctv null ac) Cla 4 3
21 focal Gomty 3 F 3 3 F3 3 F3 3 fundamntal matc ovd aamt. Subtact 3 contant, hu w hav that th tfocal gomty dtmnd by 8 aamt. h contnt wth th taght-fowad countng: 3x 5 8 (3 cama matc ovd 33 aamt, mnu th octv ba) Cla 4
22 What Go Wong wth 3 vw? contant ach, thu w hav -65 aamt Cla 4
23 What Go Wong wth 3 vw? t α 3 t + t hu, to nt t3 w nd only aamt (ntad of 3). t t t3 8-6 aamt a ndd to nt th tfocal gomty n th ca. but th aw fundamntal matc can account fo only 5! Cla 4 3
24 What El Go Wong: Rocton F F3 3 Gvn, and th aw F-mat on can dctly dtmn th oton of th matchng ont h fal whn th 3 cama cnt a collna bcau all th ln of ght a colana thu th only on ola ln! F 3 3 F 3 Cla 4 4
25 h focal Contant [ I ]P [ A ]P [ B ]P x y x y Cla 4 5
26 h focal Contant [ A ]P [ ] A P [ A ] P [ B ]P [ ] B P [ B ] P [ I ]P ( x ) P ( y ) P Cla 4 6
27 Cla P B B A A y x Evy 4x4 mno mut vanh! of tho nvolv all 3 vw, thy a aangd n 3 gou Dndng on whch vw th fnc vw. h focal Contant
28 Cla 4 8 B B A A y x h fnc vw Choo ow fom h Choo ow fom h W hould xct to hav 4 matchng contant ),, ( f h focal Contant
29 Exandng th dtmnant: h focal Contant A + µ A + µ, B + µ B + µ, lmnat µ A B ( )( A ) ( )( B,, ) Cla 4 9
30 h focal Contant [ A ] P What gong on gomtcally: ( A, ) a lan C y C P x 4 lan ntct at P! Cla 4 3 C
31 h focal no ( )( A ) ( )( B ) Nw ndx notaton: -mag, -mag, -mag 3 A + µ a + µ a ont n mag a ln n mag a ont n mag Cla 4 3
32 l h focal no l, a th two ln concdnt wth,.. l m m m, a th two ln concdnt wth,.. l a + µ l m b + µ m Elmnat µ l m m l ( )( b ) ( )( a ) Cla 4 3
33 Cla 4 33 h focal no ) )( ( ) )( ( l m m l a b Raang tm: ) ( m l a b h tfocal tno : a b,, m l
34 h focal no l m l x y m x y h fou tlnat : x 3 - x x 33 + x 3 - y 3 - y x 33 + x 3 - x 3 - x y 33 + y 3 - y 3 - y y 33 + x 3 - Cla 4 34
35 Cla 4 35 h focal no β α + δ γ + ) )( ( + + δ γ β α A tlnaty a contacton wth a ont-ln-ln wh th ln a concdnt wth th ctv matchng ont.
36 Slc of th focal no Now that w hav an xlct fom of th tno, what can w do wth t?? h ult mut b a contavaant vcto (a ont). h ont concdnt wth fo all ln concdnt wth h ont octon quaton (wll wo whn cama cnt a collna a wll). Not: octon obl aft obvng 7 matchng ont, (bcau on nd 7 matchng tlt to olv fo th tno). h n contat to octon ung aw fundamntal matc Whch qu 8 matchng ont (n od to olv fo th F-mat). Cla 4 36
37 Slc of th focal no 3 Cla 4 37
38 Slc of th focal no? h ult mut b a ln. q Ln octon quaton O 3 matchng ln a ncay fo O q olvng fo th tno (comad to 7 matchng ont) O Cla 4 38
39 Slc of th focal no δ? h ult mut b a matx. δ δ th octon quaton δ a homogahy matx 3 δ δ a famly of homogahy matc (fom to ) nducd by th famly of lan concdant wth th 3 d cama cnt. Cla 4 39
40 Slc of th focal no δ th homogahy matx fom to 3 nducd by th lan dfnd by th mag ln δ and th cond cama cnt. δ? δ h ult a ont on th ola ln of δ on mag 3 th octon quaton 3 F 3 δ Cla 4 4 δ
41 Slc of th focal no δ G G I a ont on th ola ln δ F 3 an( G) (bcau t ma th dual lan onto collna ont) F 3 δ null ( G) F null( G ) F 3δ δ 3 δ Cla 4 4
42 Cla Paamt fo th focal no a b ) ( ) ( n a n b + + n n + a 4 aamt ( ) mnu fo global cal mnu fo calng, to b unt vcto mnu 3 fo ttng n uch that B ha a vanhng column n 8 ndndnt aamt W hould xct to fnd 9 non-lna contant among th 7 nt of th tno (admblty contant).
43 8 Paamt fo th focal no What han whn th 3 cama cnt a collna? (w aw that aw F-mat account fo 5 aamt). A B 3 B A h ovd two addtonal (non-lna) contant, thu 8-6. Cla 4 43
44 Itm not Covd n Cla Dgnat confguaton (Lna Ln Comlx, Quatc Cuv) h ouc of th 9 admblty contant (com fom th homogahy lc). Concatnaton of tfocal tno along a qunc Quadfocal tno (and t laton to th homogahy tno) Cla 4 44
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