Lecture 9: Epipolar Geometry
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1 Lecture 9: Epipolar Geometry Professor Fei Fei Li Stanford Vision Lab 1
2 What we will learn today? Why is stereo useful? Epipolar constraints Essential and fundamental matrix Estimating F (Problem Set 2 (Q2)) Rectification Reading: [HZ] Chapters: 4, 9, 11 [FP] Chapters: 10 2
3 Recovering structure from a single view Pinhole perspective projection P p O w Calibration rig Scene C Camera K Why is it so difficult? Intrinsic ambiguity of the mapping from 3D to image (2D) 3
4 Recovering structure from a single view Pinhole perspective projection Courtesy slide S. Lazebnik Intrinsic ambiguity of the mapping from 3D to image (2D) 4
5 Two eyes help! 5
6 Two eyes help!? K =known x 2 x 1 R, T O 2 O 1 This is called triangulation K =known 6
7 Triangulation 2 2 Find X that minimizes d x, P X ) d ( x, P X ) ( X x 1 x 2 O 1 O 2 7
8 Stereo view geometry Correspondence: Given a point in one image, how can I find the corresponding point x in another one? Camera geometry: Given corresponding points in two images, find camera matrices, position and pose. Scene geometry: Find coordinates of 3D point from its projection into 2 or multiple images. This lecture (#9) 8
9 Stereo view geometry Next lecture (#10) Correspondence: Given a point in one image, how can I find the corresponding point x in another one? Camera geometry: Given corresponding points in two images, find camera matrices, position and pose. Scene geometry: Find coordinates of 3D point from its projection into 2 or multiple images. 9
10 What we will learn today? Why is stereo useful? Epipolar constraints Essential and fundamental matrix Estimating F Rectification Reading: [HZ] Chapters: 4, 9, 11 [FP] Chapters: 10 10
11 Epipolar geometry P p 1 p 2 e 1 e 2 Epipolar Plane Epipoles e 1, e 2 Baseline Epipolar Lines O 1 O 2 = intersections of baseline with image planes = projections of the other camera center = vanishing points of camera motion direction 11
12 Example: Converging image planes 12
13 Example: Parallel image planes P e 2 p 1 p 2 e 2 O 1 O 2 Baseline intersects the image plane at infinity Epipoles are at infinity Epipolar lines are parallel to x axis 13
14 Example: Parallel image planes 14
15 Example: Forward translation e 2 O 2 e 1 O 2 The epipoles have same position in both images Epipole called FOE (focus of expansion) 15
16 Epipolar Constraint Two views of the same object Suppose I know the camera positions and camera matrices Given a point on left image, how can I find the corresponding point on right image? 16
17 Epipolar Constraint Potential matches for p have to lie on the corresponding epipolar line l. Potential matches for p have to lie on the corresponding epipolar line l. 17
18 Epipolar Constraint P p M P u v 1 p p p M P u v 1 O R, T O M KI 0 M' KR T 18
19 Epipolar Constraint P p p O R, T O M K I 0 K 1 and K 2 are known (calibrated cameras) M' K R T M I 0 M ' R T 19
20 Epipolar Constraint P p p O R, T O T ( R p) Perpendicular to epipolar plane p T T (R p ) 0 20
21 21 Cross product as matrix multiplication b a b a ] [ z y x x y x z y z b b b a a a a a a skew symmetric matrix
22 Triangulation P p p O R, T O p T T (R p ) 0 p T T R p 0 (Longuet Higgins, 1981) E = essential matrix 22
23 Triangulation P p 1 l 1 l 2 e 1 e 2 p 2 O 1 O 2 E p 2 is the epipolar line associated with p 2 (l 1 = E p 2 ) E T p 1 is the epipolar line associated with p 1 (l 2 = E T p 1 ) E is singular (rank two) E e 2 = 0 and E T e 1 = 0 E is 3x3 matrix; 5 DOF 23
24 Triangulation P p p O O P M P u p M KI 0 v unknown 24
25 The image part with relationship ID rid8 was not found in the file. Triangulation p K 1 p P p p p T O T R p 0 (K 1 p) T O 1 T R K p 0 p T K T 1 T R K p 0 p T F p 0 25
26 Triangulation P p p O p T F p 0 O F = Fundamental Matrix (Faugeras and Luong, 1992) 26
27 Triangulation P p 1 p 2 e 1 e 2 O 1 O 2 F p 2 is the epipolar line associated with p 2 (l 1 = F p 2 ) F T p 1 is the epipolar line associated with p 1 (l 2 = F T p 1 ) F is singular (rank two) F e 2 = 0 and F T e 1 = 0 F is 3x3 matrix; 7 DOF 27
28 Why is F useful? p l = F T p Suppose F is known No additional information about the scene and camera is given Given a point on left image, how can I find the corresponding point on right image? 28
29 Why is F useful? F captures information about the epipolar geometry of 2 views + camera parameters MORE IMPORTANTLY: F gives constraints on how the scene changes under view point transformation (without reconstructing the scene!) Powerful tool in: 3D reconstruction Multi view object/scene matching 29
30 What we will learn today? Why is stereo useful? Epipolar constraints Essential and fundamental matrix Estimating F (Problem Set 2 (Q2)) Rectification Reading: [HZ] Chapters: 4, 9, 11 [FP] Chapters: 10 30
31 Estimating F The Eight Point Algorithm (Longuet Higgins, 1981) (Hartley, 1995) P p p O O u u P p P p v v p T F p
32 Estimating F p T F p 0 Let s take 8 corresponding points 32
33 Estimating F 33
34 Estimating F W Homogeneous system Rank 8 If N>8 W f 0 A non zero solution exists (unique) Lsq. solution by SVD! Fˆ f f 1 34
35 Estimating F p T Fˆ p 0 ^ The estimated Fmay have full rank (det(f) 0) (F should have rank=2 instead) ^ Find F that minimizes F Fˆ 0 Frobenius norm (*) Subject to det(f)=0 SVD (again!) can be used to solve this problem (*) Sqrt root of the sum pf squares of all entries 35
36 Example 36 Data courtesy of R. Mohr and B. Boufama.
37 Example Mean errors: 10.0pixel 9.1pixel 37
38 Normalization Is the accuracy in estimating F function of the ref. system in the image plane? E.g. under similarity transformation (T = scale + translation): q i T i p i q i T i p i Does the accuracy in estimating F change if a transformation T is applied? 38
39 Normalization The accuracy in estimating F does change if a transformation T is applied There exists a T for which accuracy is maximized Why? 39
40 Normalization W f 0, Lsq solution by SVD f 1 F SVD enforces Rank(W)=8 Recall the structure of W: Highly un balance (not well conditioned) Values of W must have similar magnitude More details HZ pag
41 Normalization IDEA: Transform image coordinate system (T = translation + scaling) such that: Origin = centroid of image points Mean square distance of the data points from origin is 2 pixels q i T i p i q i T i p i (normalization) 41
42 The Normalized Eight Point Algorithm 0. Compute T i and T i 1. Normalize coordinates: q i T i p i q i T i p i 2. Use the eight point algorithm to compute F q from the points q and q. i i 3. Enforce the rank 2 constraint. 4. De normalize F q : F T T F T q F q q T F q det( Fq q )
43 Example Without transformation Mean errors: 10.0pixel 9.1pixel With transformation Mean errors: 1.0pixel 0.9pixel 43
44 What we will learn today? Why is stereo useful? Epipolar constraints Essential and fundamental matrix Estimating F Rectification Reading: [HZ] Chapters: 4, 9, 11 [FP] Chapters: 10 44
45 Rectification P p p O O Make two camera images parallel Correspondence problem becomes easier 45
46 Rectification P v v e 2 p p e 2 y u u K O z x O K Parallel epipolar lines Epipoles at infinity v = v Let s see why. 46
47 Rectification P v v e 2 p p e 2 y u u K O z x O K K 1 =K 2 = known x parallel to O 1 O 2 E [ t ] R 47
48 48 b a b a ] [ z y x x y x z y z b b b a a a a a a Cross product as matrix multiplication
49 Rectification P v v e 2 p p e 2 y u u K O z x K 1 =K 2 = known x parallel to O 1 O 2 E [ t ] R 0 0 T 0 T 0 O K v = v? 49
50 50 Rectification v T Tv Tv T v u v u T T v u O O P e 2 p p e 2 K K x y z u v u v 0 E p p T v = v
51 Rectification p p H O O GOAL of rectification : Estimate a perspective transformation H that makes images parallel Impose v =v This leaves degrees of freedom for determining H If not appropriate H is chosen, severe projective distortions on image take place We impose a number of restriction while computing H [HZ] Chapters: 11 (sec ) 51
52 Rectification H Courtesy figure S. Lazebnik 52
53 Application: view morphing S. M. Seitz and C. R. Dyer, Proc. SIGGRAPH 96, 1996,
54 Application: view morphing If rectification is not applied, the morphing procedure does not generate geometrically correct interpolations 54
55 Application: view morphing 55
56 Application: view morphing 56
57 Application: view morphing 57
58 Application: view morphing 58
59 The Fundamental Matrix Song 59
60 What we have learned today? Why is stereo useful? Epipolar constraints Essential and fundamental matrix Estimating F (Problem Set 2 (Q2)) Rectification Reading: [HZ] Chapters: 4, 9, 11 [FP] Chapters: 10 60
61 Supplementary materials 61
62 Making image planes parallel P K I 0 P P P K R T P p p K e e K O R,T O 0. Compute epipoles e K R T T [e 1 e 2 1] T e K T 62
63 Making image planes parallel P p p e e O O 1. Map e to the x axis at location [1,0,1] T (normalization) e [e 1 T 1 e2 1] 0 1 T H R T 1 H H 63
64 Making image planes parallel P p p e 2. Send epipole to infinity: O O H e T T Minimizes the distortion in a neighborhood (approximates id. mapping) 64
65 Making image planes parallel P H = H 2 H 1 e p p e O O 3. Define: H = H 2 H 1 4. Align epipolar lines 65
66 Projective transformation of a line (in 2D) H A v t b l H T l 66
67 Making image planes parallel P H = H 2 H 1 e p l l p e 4. Align epipolar lines O O 3. Define: H = H 2 H 1 H T l H T l [HZ] Chapters: 11 (sec ) These are called matched pair of transformation 67
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