Affine Structure from Motion
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1 Affie Structure from Motio EECS Fall 24! Foudatios of Computer Visio!! Istructor: Jaso Corso (jjcorso)! web.eecs.umich.edu/~jjcorso/t/598f4!! Readigs: FP 8.2! Date: /5/4!! Materials o these slides have come from may sources i additio to myself (primarily Silvio Savarese to whom I am ultimately grateful); idividual slides referece specific sources.!
2 Pla 2 What is affie SFM?! Algebraic Methods from Two Views! Factorizatio!
3 Applicatio Courtesy of Oford Visual Geometry Group!
4 Structure from motio problem! X j j M m M mj 2j M 2 Give m images of fied 3D poits! ij M i X j, i,, m, j,,
5 Structure from motio problem! X j j M m M mj 2j M 2 From the m correspodeces ij, estimate:! m projectio matrices M i 3D poits X j motio! structure!
6 Affie structure from motio! (simpler problem)! Image! World! Image! From the m correspodeces ij, estimate:! m projectio matrices M i (affie cameras) 3D poits X j
7 p! q! r! O! Q! R! M K[ R T]X P!
8 Questio:! R T??
9 p! q! r! O! Q! R! K[ R T]X Caoical perspective projectio matri! P! M K33 Affie! Homography! (i 2D)! R T Affie homography! (i 3D)! K α s α y y o o
10 Projective & Affie cameras! [ ]X T K R T R K M y s K o y o α α Projective case! Affie case!
11 Weak perspective projectio! Whe the relative scee depth is small compared to its distace from the camera! ' m y' my Scalig fuctio of the distace (magificatio)!
12 Orthographic (affie) projectio! Whe the camera is at a (roughly costat) distace from the scee! ' y' y Distace from ceter of projectio! to image plae is ifiite!
13 Trasformatio i 2D y H y t A y' ' a
14 Projective & Affie cameras! [ ]X T K R y α s α o y o K T R K M T R K M y s K o y o α α Projective case! Affie case! Parallel projectio matri! (poits at ifiity are mapped as poits at ifiity)! Magificatio (scalig term)!
15 Affie cameras! [ ]X T K R y K α α T R K M b A 4affie] [4 3affie] 3 [ b a a a b a a a M X b AX Euc M b b Z Y X a a a a a a y [Homogeeous]! [o-homogeeous! image coordiates]! [ ] b A M M Euc ; P M Euc
16 Affie cameras! p! P! p! M camera matri! To recap:! from ow o we defie M as the camera matri for the affie case! p u v AP + b M P ; M [ A b]
17 The Affie Structure-from-Motio Problem! Give m images of fied poits P j (X i ) we ca write! N of cameras! N of poits Problem: estimate the m 2 4 matrices M i ad! the positios P j from the m correspodeces p ij.! How may equatios ad how may ukow?! 2m equatios i 8m+3 ukows! Two approaches:! - Algebraic approach (affie epipolar geometry; estimate F; cameras; poits)! - Factorizatio method!
18 Algebraic aalysis (2-view case)! - Derive the fudametal matri F A for the affie case! - Compute F A! - Use F A to estimate projectio matrices! - Use projectio matrices to estimate 3D poits!
19 . Derivig the fudametal matri F A! p! P! v! p! u! Homogeeous system! Dim?!44! Affie Epipolar Costrait!
20 Derivig the fudametal matri F A! where! The Affie Fudametal Matri!! Are the epipolar lies parallel or covergig?!
21
22 Affie Epipolar Geometry!
23 Estimatig F A! From correspodeces, we obtai a liear system o! the ukow alpha, beta, etc! Measuremets: u, u, v, v! v u v u v u v u ʹ ʹ ʹ ʹ f!!!!! Computed by least square ad by eforcig f! SVD!
24 Estimatig projectio matrices from F A! p! P! p!
25 Affie ambiguity Affie p M P ( M Q - )( Q P) A A
26 2. Estimatig projectio matrices! from epipolar costraits! If M i ad P i are solutios,! the M i ad P i are also solutios,! where! ad! Q is a affie! trasformatio.! Proof:!
27 3. Estimatig projectio matrices from F A! p! P! p!
28 Estimatig projectio matrices from F A! Choose Q such that! ~ M ' Where a,b,c,d ca be epressed as fuctio of the parameters of F A!
29 4. Estimatig the structure from F A! A! b! A! b! Ca be solved by least square agai!
30 3. Estimatig projectio matrices from epipolar costraits! Choose Q such that! A ~ ~ b [ ] T A ~ ʹ a ~ bʹ b [ d] T c Caoical affie cameras! Fuctio of the parameters of F!
31 Estimatig projectio matrices from epipolar costraits! Choose Q such that! By re-eforcig the epipolar costrait, we ca compute a, b, c, d directly from the measuremets!
32 Remider: epipolar costrait! p! P! v! p! u! Homogeeous system!
33 Estimatig projectio matrices from epipolar costraits! Choose Q such that! ~! ~! A! b! Re-eforce the Epipolar costrait!
34 Estimatig projectio matrices from epipolar costraits! Choose Q such that! A! b!
35 Estimatig projectio matrices from epipolar costraits! Liear relatioship betwee measuremets ad ukow! Ukow: a, b, c, d! Measuremets: u, u, v, v! From at least 4 correspodeces, we ca solve this liear system! ad compute a, b, c, d (via least square)! The cameras ca be computed! How about the structure?!
36 4. Estimatig the structure from F A! A! b! Ca be solved by least square agai!
37 First recostructio. Mea reprojectio error:.6piel! Secod recostructio. Mea re-projectio error: 7.8piel!
38 A factorizatio method Tomasi & Kaade algorithm C. Tomasi ad T. Kaade. Shape ad motio from image streams uder orthography: A factorizatio method. IJCV, 9(2):37-54,! November 992.!! Ceterig the data! Factorizatio!
39 A factorizatio method - Ceterig the data Ceterig: subtract the cetroid of the image poits! ˆ ij ij A i X j k ik k X k i i j A X -! + b i i A Xˆ j k X k! ( A X + b ) i k i -! i ik
40 Ceterig: subtract the cetroid of the image poits! ( ) j i k k j i k i k i i j i k ik ij ij ˆ ˆ A X X X A b A X b A X + + A factorizatio method - Ceterig the data i j i ij b A X +
41 Ceterig: subtract the cetroid of the image poits! ( ) + + k k j i k i k i i j i k ik ij ij ˆ X X A b A X b A X j i ij X A ˆ A factorizatio method - Ceterig the data Assume that the origi of the world coordiate system is at the cetroid of the 3D poits!! After ceterig, each ormalized poit ij is related to the 3D poit X i by!
42 A factorizatio method - Ceterig the data!! X!! ˆ A ij i X j
43 m m m D ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ! "!! cameras (2 m )! poits ( )! A factorizatio method - factorizatio Let s create a 2m data (measuremet) matri:!
44 Let s create a 2m data (measuremet) matri:! [ ] m m m m X X X A A A D! "! #!! ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ cameras (2 m 3)! poits (3 )! The measuremet matri D M S has rak 3! (it s a product of a 2m3 matri ad 3 matri)! A factorizatio method - factorizatio (2 m )! M! S!
45 Factorizig the measuremet matri Source: M. Hebert
46 Factorizig the measuremet matri Sigular value decompositio of D:! Source: M. Hebert!
47 Factorizig the measuremet matri Sigular value decompositio of D:! Sice rak (D)3, there are oly 3 o-zero sigular values! Source: M. Hebert!
48 Factorizig the measuremet matri Obtaiig a factorizatio from SVD:! S structure! M Motio (cameras)! What is the issue here?! D has rak>3 because of - measuremet oise!!!!! - affie approimatio!
49 Factorizig the measuremet matri Obtaiig a factorizatio from SVD:! M motio! S structure! D! D!
50 Affie ambiguity M S C C - The decompositio is ot uique. We get the same D by usig ay 3 3 matri C ad applyig the trasformatios M MC, S C - S! We ca eforce some Euclidea costraits to resolve! this ambiguity (more o et lecture!)!
51 Algorithm summary. Give: m images ad features ij! 2. For each image i, ceter the feature coordiates! 3. Costruct a 2m measuremet matri D:! Colum j cotais the projectio of poit j i all views! Row i cotais oe coordiate of the projectios of all the poits i image i! 4. Factorize D:! Compute SVD: D U W V T! Create U 3 by takig the first 3 colums of U! Create V 3 by takig the first 3 colums of V! Create W 3 by takig the upper left 3 3 block of W! 5. Create the motio ad shape matrices:! M M U 3 ad S W 3 V 3 T (or U 3 W 3 ½ ad S W 3 ½ V 3 T )! 6. Elimiate affie ambiguity!
52 Recostructio results C. Tomasi ad T. Kaade. Shape ad motio from image streams uder orthography: A factorizatio method. IJCV, 9(2):37-54, November 992.
53 Net Lecture: Perspective SFM 53 Readigs: FP 8.3!
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