Lecture 6 Stereo Systems Multi-view geometry
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1 Lectre 6 Stereo Systems Mlti-view geometry Professor Silvio Savarese Comptational Vision and Geometry Lab Silvio Savarese Lectre 6-29-Jan-18
2 Lectre 6 Stereo Systems Mlti-view geometry Stereo systems Rectification Correspondence problem Mlti-view geometry The SFM problem Affine SFM Reading: [AZ] Chapter: 9 Epip. Geom. and the Fndam. Matrix Transf. [AZ] Chapter: 18 N view comptational methods [FP] Chapters: 7 Stereopsis [FP] Chapters: 8 Strctre from Motion Silvio Savarese Lectre 6-29-Jan-18
3 Epipolar geometry P p p e e O 1 O 2 Epipolar Plane Epipoles e, e Baseline Epipolar Lines = intersections of baseline with image planes = projections of the other camera center
4 Epipolar Constraint P p p O 1 O 2 p T E p = 0 E = Essential Matrix (Longet-Higgins, 1981) E = [ T ] R
5 Essential matrix [ ] R T E = " $ E = $ $ $ 0 T z T y T z 0 T x T y T x 0 % ' ' R ' '
6 Epipolar Constraint P p p O 1 O 2 p T F p = 0 F F = Fndamental Matrix (Fageras and Long, 1992) [ ] -1 = K - T T R K
7 Parallel image planes P v e p p e y z x O 1 O 2! p $! Epipolar lines are horizontal p = p Epipoles go to infinity v p' = v-coordinates are eqal " 1 % " p' p v 1 $ %
8 Parallel image planes P v e p p e y z x O 1 O 2 Hint : K E=? 1 =K 2 = known R = I T = (T, 0, 0) x parallel to O 1 O 2
9 Essential matrix for parallel images [ ] R T E = " $ E = $ $ $ 0 T z T y T z 0 T x T y T x 0 % " ' $ ' R = $ ' $ ' T 0 T 0 % ' ' ' T = [ T 0 0 ] R = I
10 Parallel image planes P v v e l p p l e y z O 1 What are the directions of epipolar lines? x l = E p' " $ = $ $ T 0 T 0 %" ' $ ' $ ' $ ' v' 1 % " ' $ ' = $ ' $ 0 T T v' % ' ' ' horizontal!
11 Parallel image planes P v v e p p e y z O 1 x How are p and p related? p T E p' = 0
12 Parallel image planes P v v e p p e y z O 1 x How are p and p related? é0 0 0 ùæ ö æ 0 ö ( v ê ) T úç ç = Þ v ( v ) T Tv Tv ê - ú = Þ - = Þ = ê 0 T 0 úç 1 ç Tv ë ûè ø è ø v = v" T pep
13 Parallel image planes P v v e p p e y z O 1 x Rectification: making two images parallel Why it is sefl? Epipolar constraint v = v New views can be synthesized by linear interpolation
14 Rectification: making two images parallel H Cortesy figre S. Lazebnik
15 Application: view morphing S. M. Seitz and C. R. Dyer, Proc. SIGGRAPH 96, 1996, 21-30
16 Rectification P H 1 I 1 I s î 1 v I 2 î S v C 1 v î 2 H 2 C s C 2
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18
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20 From its reflection!
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22 Deep view morphing D. Ji, J. Kwon, M. McFarland, S. Savarese, CVPR 2017
23 Deep view morphing D. Ji, J. Kwon, M. McFarland, S. Savarese, CVPR 2017
24 Deep view morphing D. Ji, J. Kwon, M. McFarland, S. Savarese, CVPR 2017
25 Why are parallel images sefl? Makes trianglation easy Makes the correspondence problem easier
26 Point trianglation P v v e p p z e y z! p = " p p v 1 O $ % x! p' = " p' p v 1 $ % B O disparity = p p' B f z [Eq. 1] Disparity is inversely proportional to depth z!
27 Compting depth P (0, p v, 1) z f p p' O B O disparity = p p' B f [Eq. 1] z Disparity is inversely proportional to depth z!
28 Disparity maps p p' B f z p p [Eq. 1] Stereo pair Disparity map / depth map
29 Why are parallel images sefl? Makes trianglation easy Makes the correspondence problem easier
30 Correspondence problem P p belongs to l = F p p belongs to l = F T p p p O O Given a point in 3D, discover corresponding observations in left and right images [also called binoclar fsion problem]
31 Correspondence problem P p p O O When images are rectified, this problem is mch easier!
32 Correspondence problem A Cooperative Model (Marr and Poggio, 1976) Correlation Methods (1970--) Mlti-Scale Edge Matching (Marr, Poggio and Grimson, ) [FP] Chapters: 7
33 Correlation Methods (1970--) p Where is p' is? image 1 Image 2! p = " v 1 $ %! p' = " ' v 1 $ %
34 Correlation Methods (1970--) p Where is p' is? image 1 Image 2 100! p = " v 1 $ %! p' = " ' v 1 $ %
35 Correlation Methods (1970--) p Where is p' is? image 1 Image What s the problem with this?
36 Window-based correlation v W p image 1 image 2 Pick p a window W arond Bild vector w p = (, v )
37 Window-based correlation W v image 1 image 2 ' = + d = 1 Example: W is a 3x3 window in red w is a 9x1 vector w = [100, 100, 100, 90, 100, 20, 150, 150, 145] T Pick p a window W arond p = (, v ) Bild vector w Slide the window W along v = v in image 2 and compte w () for each Compte the dot prodct w T w () for each and retain the max vale
38 Window-based correlation W v image 1 image 2 ' = + d = 1 Example: W is a 3x3 window in red w is a 9x1 vector w = [100, 100, 100, 90, 100, 20, 150, 150, 145] T What s the problem with this?
39 Changes of brightness/exposre Changes in the mean and the variance of intensity vales in corresponding windows!
40 Normalized cross-correlation W v image 1 image 2 ' = + d = +1 Find that maximizes: (w w) T ( w "() w ") (w w) ( w "() w ") [Eq. 2] w = mean vale within W w'() = located at bar in image 1 mean vale within W located at in image 2
41 Example Image 1 Image 2 v scanline NCC Credit slide S. Lazebnik
42 Effect of the window s size Window size = 3 Window size = 20 Smaller window - More detail - More noise Larger window - Smoother disparity maps - Less prone to noise Credit slide S. Lazebnik
43 Isses Fore shortening effect O O Occlsions O O
44 Isses To redce the effect of foreshortening and occlsions, it is desirable to have small B / z ratio! However, when B/z is small, small errors in measrements imply large error in estimating depth Small B / z ratio Large B / z ratio O O O O e e
45 Isses Homogeneos regions mismatch
46 Repetitive patterns Isses
47 Correspondence problem is difficlt! - Occlsions - Fore shortening - Baseline trade-off - Homogeneos regions - Repetitive patterns Apply non-local constraints to help enforce the correspondences
48 Non-local constraints Uniqeness For any point in one image, there shold be at most one matching point in the other image Ordering Corresponding points shold be in the same order in both views Smoothness Disparity is typically a smooth fnction of x (except in occlding bondaries)
49 Lectre 6 Stereo Systems Mlti-view geometry Stereo systems Rectification Correspondence problem Mlti-view geometry The SFM problem Affine SFM Silvio Savarese Lectre 5-29-Jan-18
50 Strctre from motion problem Cortesy of Oxford Visal Geometry Grop
51 Strctre from motion problem X j x 1j M m M 1 x mj x 2j M 2 Given m images of n fixed 3D points x ij = M i X j, i = 1,, m, j = 1,, n
52 Strctre from motion problem X j x 1j M m M 1 x mj x 2j M 2 From the mxn observations x ij, estimate: m projection matrices M i n 3D points X j motion strctre
53 Affine strctre from motion (simpler problem) World point X j Image 1 x ij Image i From the mxn observations x ij, estimate: m projection matrices M i (affine cameras) n 3D points X j
54 Perspective x = M X! = " m 1 m 2 m 3 $! X = % " m 1 X m 2 X m 3 X $ %! M = A b " v 1 $ = % ém ê ê m êë m ù ú ú úû x E = ( m 1 X m 3 X, m 2 X m 3 X )T Affine x = M X! = " x E = (m 1 X, m 2 X) T magnification m 1 m 2 m 3 $ X % =! " A b! = " $ X =! " A b [Eq. 3] m 1 X m 2 X 1 $ %! % % $ % % " x y z 1! M = " m 1 m 2 m 3 = AX E + b $ $ %! = A 2 x3 b 2 x1 " 0 1x3 1! = " X E = m 1 m ! " x y z $ % $ % $ %
55 Affine cameras Image 1 x 1j X j x i j Image i For the affine case (in Eclidean space) x ij = A i X j + b i [Eq. 4] 2x1 2x3 3x1 2x1
56 The Affine Strctre-from-Motion Problem Given m images of n fixed points X j we can write x ij = A i X j + b i for i = 1,,m and j = 1,,n N. of cameras N. of points Problem: estimate m matrices A i, m matrices b i and the n positions X j from the m n observations x ij. How many eqations and how many nknown? 2m n eqations in 8m + 3n - 9 nknowns
57 The Affine Strctre-from-Motion Problem Two approaches: - Algebraic approach (affine epipolar geometry; estimate F; cameras; points) - Factorization method
58 Next lectre Mltiple view geometry: Affine and Perspective strctre from Motion
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