Lecture 10: Multi view geometry
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1 Lecture 10: Multi view geometry Professor Fei Fei Li Stanford Vision Lab 1
2 What we will learn today? Stereo vision Correspondence problem (Problem Set 2 (Q3)) Active stereo vision systems Structure from motion Reading: [HZ] Chapters: 4, 9, 11 [FP] Chapters: 10 2
3 What we will learn today? Stereo vision Correspondence problem (Problem Set 2 (Q3)) Active stereo vision systems Structure from motion Reading: [HZ] Chapters: 4, 9, 11 [FP] Chapters: 10 3
4 Two eyes help!? p 2 p 1 R, T O 2 O 1 4
5 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. Previous lecture (#9) 5
6 Stereo view geometry This 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. 6
7 Stereo view geometry Pinhole perspective projection P p p O 1 O 2 Subgoals: Solve the correspondence problem Use corresponding observations to triangulate (depth estimation) 7
8 Depth estimation Pinhole perspective projection P v v e 2 p p e 2 y u u K z O x O K Assume cameras are calibrated and rectified Assume correspondence problem is solved Goal: estimate depth (triangulate) 8
9 Depth estimation Pinhole perspective projection P p p z f f O Baseline B O p p' B z f = disparity Note: Disparity is inversely proportional to depth 9
10 Depth estimation Stereo pair Pinhole perspective projection p p' B z f x x Disparity map / depth map Disparity map with occlusions 10
11 Depth estimation Pinhole perspective projection P p p O 1 O 2 Subgoals: Solve the correspondence problem Use corresponding observations to triangulate 11
12 Correspondence problem Pinhole perspective projection P p p O 1 O 2 Given a point in 3D, discover corresponding observations in left and right images [also called binocular fusion problem] 12
13 Correspondence problem Pinhole perspective projection A Cooperative Model (Marr and Poggio, 1976) Correlation Methods (1970 ) Multi Scale Edge Matching (Marr, Poggio and Grimson, ) [FP] Chapters: 11 13
14 Correlation Methods Pinhole perspective projection p Pick up a window around p(u,v) 14
15 Correlation Methods Pinhole perspective projection Pick up a window around p(u,v) Build vector W Slide the window along v line in image 2 and compute w Keep sliding until w w is maximized. 15
16 Correlation Methods Pinhole perspective projection Normalized Correlation; minimize: (w w)(w w ) (w w)(w w ) 16
17 Correlation Methods Pinhole perspective projection Left Right scanline Norm. corr Credit slide S. Lazebnik 17
18 Correlation Methods Pinhole perspective projection Smaller window More detail More noise Larger window Smoother disparity maps Less prone to noise Window size = 3 Window size = 20 Credit slide S. Lazebnik 18
19 Issues with Correlation Methods Pinhole perspective projection Fore shortening effect O O Occlusions O O It is desirable to have small B/z ratio! 19
20 Issues with Correlation Methods Pinhole perspective projection small B/z ratio O O Small error in measurements implies large error in estimating depth 20
21 Issues with Correlation Methods Pinhole perspective projection Homogeneous regions Hard to match pixels in these regions 21
22 Issues with Correlation Methods Pinhole perspective projection Repetitive patterns 22
23 Results with window search Pinhole perspective projection Data Ground truth Window based matching Credit slide S. Lazebnik 23
24 Improving correspondence: Non local constraints Uniqueness For any point in one image, there should be at most one matching point in the other image 24
25 Improving correspondence: Non local constraints Uniqueness For any point in one image, there should be at most one matching point in the other image Ordering Corresponding points should be in the same order in both views Courtesy J. Ponce 25
![Dynamic Programming Pinhole perspective projection [Uses ordering constraint] (Baker and Binford, 1981) Courtesy J. Ponce Nodes = matched feature points (e.g., edge points).](/docs-images/89/99075258/images/26-0.jpg "Arcs = matched intervals along the epipolar lines. Arc cost = discrepancy between intervals.")
26 Dynamic Programming Pinhole perspective projection [Uses ordering constraint] (Baker and Binford, 1981) Courtesy J. Ponce Nodes = matched feature points (e.g., edge points). Arcs = matched intervals along the epipolar lines. Arc cost = discrepancy between intervals. Find the minimum cost path going monotonically down and right from the top left corner of the graph to its bottom right corner. 26
27 Dynamic Programming (Baker and Binford, 1981) (Ohta and Kanade, 1985) 27
28 Improving correspondence: Non local constraints Uniqueness For any point in one image, there should be at most one matching point in the other image Ordering Corresponding points should be in the same order in both views Smoothness Disparity is typically a smooth function of x (except in occluding boundaries) 28
29 Stereo matching as energy minimization Pinhole perspective projection Y. Boykov, O. Veksler, and R. Zabih, Fast Approximate Energy Minimization via Graph Cuts, PAMI 01 I 1 I 2 D W 1 (i) W 2 (i+d(i)) D(i) E Edata ( I1, I2, D) Esmooth ( D) E W ( i) W ( i D( i 2 data 1 2 )) i Esmooth D( i) D( j) neighbors i, j Energy functions of this form can be minimized using graph cuts 29
30 Stereo matching as energy minimization Pinhole perspective projection Y. Boykov, O. Veksler, and R. Zabih, Fast Approximate Energy Minimization via Graph Cuts, PAMI 01 Ground truth Window based matching Energy minimization 30
31 Two frame stereo correspondence algorithms Click here 31
32 Stereo SDK vision software development kit A. Criminisi, A. Blake and D. Robertson 32
33 Foreground/background segmentation using stereo V. Kolmogorov, A. Criminisi, A. Blake, G. Cross and C. Rother. Bi layer segmentation of binocular stereo video CVPR
34 3D Urban Scene Modeling using a stereo system 3D Urban Scene Modeling Integrating Recognition and Reconstruction, N. Cornelis, B. Leibe, K. Cornelis, L. Van Gool, IJCV
35 What we will learn today? Stereo vision Correspondence problem Active stereo vision systems Structure from motion Reading: [HZ] Chapters: 4, 9, 11 [FP] Chapters: 10 35
36 Active stereo (point) P projector p Replace one of the two cameras by a projector Single camera Projector geometry calibrated What s the advantage of having the projector? O 2 Correspondence problem solved! 36
37 Active stereo (stripe) projector Projector and camera are parallel Correspondence problem solved! O 37
38 Active stereo (stripe) Digital Michelangelo Project Optical triangulation Project a single stripe of laser light Scan it across the surface of the object This is a very precise version of structured light scanning 38
39 Active stereo (stripe) Digital Michelangelo Project 39
40 Active stereo (shadows) J. Bouguet & P. Perona, 99 S. Savarese, J. Bouguet & Perona, 00 Light source O 1 camera,1 light source very cheap setup calibrated light source 40
41 Active stereo (shadows) J. Bouguet & P. Perona, 99 S. Savarese, J. Bouguet & Perona, 00 41
42 Active stereo (color coded stripes) L. Zhang, B. Curless, and S. M. Seitz 2002 S. Rusinkiewicz & Levoy 2002 projector Dense reconstruction Correspondence problem again Get around it by using color codes O 42
43 Active stereo (color coded stripes) Rapid shape acquisition: Projector + stereo cameras L. Zhang, B. Curless, and S. M. Seitz. Rapid Shape Acquisition Using Color Structured Light and Multi pass Dynamic Programming. 3DPVT
44 What we will learn today? Stereo vision Correspondence problem Active stereo vision systems Structure from motion Reading: [HZ] Chapters: 4, 9, 11 [FP] Chapters: 10 44
45 Multiple view geometry the structure from motion problem P j p 1j M m M 1 p mj p 2j M 2 Given m images of n fixed 3D points; p ij = M i P j, i = 1,, m, j = 1,, n 45
46 Multiple view geometry the structure from motion problem Courtesy of Oxford Visual Geometry Group 46
47 Multiple view geometry the structure from motion problem P j p 1j M m M 1 p mj p 2j From the mxn correspondences p ij, estimate: m projection matrices M i n 3D points P j motion structure M 2 47
48 Structure from motion ambiguity SFM can be solved up to a N degree of freedom ambiguity In the general case (nothing is known) the ambiguity is expressed by an arbitrary affine or projective transformation p M j i P j M K i i R i T i H P 1 j M j H p j M i P j -1 M H H P i j 48
49 Affine ambiguity 49
50 Prospective ambiguity 50
51 Structure from motion ambiguity A scale (similarity) ambiguity exists even for calibrated cameras For calibrated cameras, the similarity ambiguity is the only ambiguity A reconstruction up to scale is called metric [Longuet Higgins 81] 51
52 Multiple view geometry overview Recover camera and geometry up to ambiguity Use affine approximation if possible (affine SFM) Algebraic methods Factorization methods Metric upgrade to obtain solution up to scale (remove perspective or affine ambiguity) Self calibration Bundle adjustment (optimize solution across all observations) 52
53 Affine structure from motion a simpler problem Image p P p Image M = camera matrix From the mxn correspondences p ij, estimate: m projection matrices M i (affine cameras) n 3D points P j 53
54 Affine structure from motion a simpler problem Image p P p Image M = camera matrix p u v AP b M P 1 ; M A b 54
55 Affine structure from motion a simpler problem Given m images of n fixed points P j (=X i ) we can write N of cameras N of points Problem: estimate the m 24 matrices M i and the n positions P j from the mn correspondences p ij. How many equations and how many unknown? 2m n equations in 8m+3n unknowns Two approaches: Algebraic approach (affine epipolar geometry; estimate F; cameras; points) Factorization method 55
56 Factorization method C. Tomasi and T. Kanade. Shape and motion from image streams under orthography: A factorization method, Two steps: Centering the data Factorization P k Centering: subtract the centroid of the image points ^ p i p ik Assume that the origin of the world coordinate system is at the centroid of the 3D points pˆ ij p ij 1 n n k 1 p ik A i P j b i 1 n n k 1 A i P k b i A i P j 1 n n k 1 P k A i Pˆ j 56
57 Factorization method C. Tomasi and T. Kanade. Shape and motion from image streams under orthography: A factorization method, Two steps: Centering the data Factorization After centering, each normalized point p ij is related to the 3D point P i by pˆ ij A i P j 57
58 Factorization method D Let s create a 2m n data (measurement) matrix: pˆ pˆ pˆ m1 pˆ pˆ pˆ m2 pˆ pˆ pˆ 1n 2n mn cameras (2m ) points (n ) 58
59 59 Factorization method Let s create a 2m ndata (measurement) matrix: n m mn m m n n P P P p p p p p p p p p ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ A A A D cameras (2m 3) points (3 n ) The measurement matrix D = M S has rank 3 (it s a product of a 2mx3 matrix and 3xn matrix) (2m n) M S
60 Factorization method Source: M. Hebert 60
61 Factorization method Singular value decomposition of D: Source: M. Hebert 61
62 Factorization method Singular value decomposition of D: Since rank (D)=3, there are only 3 non zero singular values Source: M. Hebert 62
63 Factorization method Obtaining a factorization from SVD: Motion (cameras) structure What is the issue here? D has rank>3 because of measurement noise affine approximation Source: M. Hebert 63
64 Factorization method Obtaining a factorization from SVD: Motion (cameras) structure 64
65 Factorization method 1. Given: m images and n features p ij 2. For each image i, center the feature coordinates 3. Construct a 2m n measurement matrix D: Column j contains the projection of point j in all views Row i contains one coordinate of the projections of all the n points in image i 4. Factorize D: Compute SVD: D = U W V T Create U 3 by taking the first 3 columns of U Create V 3 by taking the first 3 columns of V Create W 3 by taking the upper left 3 3 block of W 5. Create the motion and shape matrices: M = M = U 3 and S = W 3 V 3 T (or U 3 W 3½ and S = W 3½ V 3T ) Source: M. Hebert 65
66 Multiple view geometry overview Recover camera and geometry up to ambiguity Use affine approximation if possible (affine SFM) Algebraic methods Factorization methods Metric upgrade to obtain solution up to scale (remove perspective or affine ambiguity) Self calibration Bundle adjustment (optimize solution across all observations) 66
67 Self calibration [HZ] Chapters: 18 Prior knowledge on cameras or scene can be used to add constraints and remove ambiguities Obtain metric reconstruction (up to scale) Condition N. Views Constant internal parameters 3 Aspect ratio and skew known Focal length and offset vary Aspect ratio and skew known Focal length and offset vary 4* 5* skew =0, all other parameters vary 8* 67
68 Multiple view geometry overview Recover camera and geometry up to ambiguity Use affine approximation if possible (affine SFM) Algebraic methods Factorization methods Metric upgrade to obtain solution up to scale (remove ambiguity) Self calibration Bundle adjustment (optimize solution across all observations) 68
69 Bundle adjustment Non linear method for refining structure and motion Minimizing re projection error It can be used before or after metric upgrade [HZ] Chapters: 18 p j? m E( M, P) n i1 j1 D 2 p, M P ij i j M 1 P j p 1j p 3j P 1 M 2 P j p 2j M 3 P j P 2 P 3 69
70 Bundle adjustment [HZ] Chapters: 18 Non linear method for refining structure and motion Minimizing re projection error It can be used before or after metric upgrade Advantages Handle large number of views Handle missing data m E( M, P) n i1 j1 D 2 p, M P ij i j Limitations Large minimization problem (parameters grow with number of views) Requires good initial condition 70
71 Photo synth Noah Snavely, Steven M. Seitz, Richard Szeliski, "Photo tourism: Exploring photo collections in 3D," ACM Transactions on Graphics (SIGGRAPH Proceedings),2006, Bundler is a structure from motion (SfM) system for unordered image collections (for instance, images from the Internet) written in C and C++. 71
72 Additional references D. Nistér, PhD thesis 01 See also 5 point algorithm M. Pollefeys et al 98 M. Brown and D. G. Lowe, 05 72
73 What we have learned today? Stereo vision Correspondence problem (Problem Set 2 (Q3)) Active stereo vision systems Structure from motion Reading: [HZ] Chapters: 4, 9, 11 [FP] Chapters: 10 73
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