EECS 442 Computer vision. Multiple view geometry Affine structure from Motion
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1 EECS 442 Computer visio Multiple view geometry Affie structure from Motio - Affie structure from motio problem - Algebraic methods - Factorizatio methods Readig: [HZ] Chapters: 6,4,8 [FP] Chapter: 2 Some slides of this lectures are courtesy of prof. J. Poce, prof FF Li, prof S. Lazebik & prof. M. Hebert
2 Applicatios Courtesy of Oford Visual Geometry Group
3 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 =,,
4 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
5 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
6 Fiite cameras p q r O Q R K R M TX P
7 Questio: R T??
8 Fiite cameras p q r R Q P O X T K R T R K M 3 3 Caoical perspective projectio matri Affie homography (i 3D) Affie Homography (i 2D) y s K o y o
9 Projective & Affie cameras X T K R T R K M y s K o y o Projective case Affie case
10 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)
11 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
12 Trasformatio i 2D Affiities: y H y t A y' ' a
13 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 Projective case Affie case Parallel projectio matri (poits at ifiity are mapped as poits at ifiity) Magificatio (scalig term)
14 Affie cameras X T K R y K 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
15 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
16 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 24 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
17 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
18 . Derivig the fudametal matri F A p P v p u Homogeeous system Dim=? 44
19 Derivig the fudametal matri F A where The Affie Fudametal Matri! Are the epipolar lies parallel or covergig?
20
21 Affie Epipolar Geometry
22 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 f Computed by least square ad by eforcig f = SVD
23 Estimatig projectio matrices from F A p P p
24 Affie ambiguity Affie p M P M Q - Q P A A
25 Estimatig projectio matrices from F A p P p
26 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 See HZ page 348
27 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
28 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 A X X X A b A X b A X A factorizatio method - Ceterig the data X k ik i ^
29 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 A X X X A b A X b A X A factorizatio method - Ceterig the data
30 Ceterig: subtract the cetroid of the image poits k k j i k i k i i j i k ik ij ij X X A b A X b A X j i ij X A 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
31 A factorizatio method - Ceterig the data X A ij i X j
32 Let s create a 2m data (measuremet) matri: m m m D cameras (2 m ) poits ( ) A factorizatio method - factorizatio
33 Let s create a 2m data (measuremet) matri: m m m m X X X A A A D 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
34 Factorizig the measuremet matri Source: M. Hebert
35 Factorizig the measuremet matri Sigular value decompositio of D: Source: M. Hebert
36 Factorizig the measuremet matri Sigular value decompositio of D: Sice rak (D)=3, there are oly 3 o-zero sigular values Source: M. Hebert
37 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
38 Factorizig the measuremet matri Obtaiig a factorizatio from SVD: M = motio S = structure D D
39 Affie ambiguity 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!)
40 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
41 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.
42 Net lecture Multiple view geometry Perspective structure from Motio
EECS 442 Computer vision. Multiple view geometry Affine structure from Motion
EECS 442 Computer visio Multiple view geometry Affie structure from Motio - Affie structure from motio problem - Algebraic methods - Factorizatio methods Readig: [HZ] Chapters: 6,4,8 [FP] Chapter: 2 Some
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