1. Advanced Computer Vision Structure from Motion1 Chapter 7 S TRUCTURE FROM M OTION Presented by Prof. Chiou-Shann Fuh & Pradnya Borade 0988472377 r99922145@ntu.edu.tw

2. Structure from Motion2 Today’s Lecture Structure from Motion What is structure from motion? Triangulation and pose Two-frame methods

3. What Is Structure from Motion? 1.Study of visual perception. 2.Process of finding the three-dimensional structure of an object by analyzing local motion signals over time. 3.A method for creating 3D models from 2D pictures of an object. Structure from Motion3

4. Example Structure from Motion4 Picture 1 Picture 2

5. Example (cont). Structure from Motion5 3D model created from the two images

6. Example Structure from Motion6 Figure7.1: Structure from motion systems: Orthographic factorization

7. Example Structure from Motion7 Figure 7.2: line matching

8. Example Structure from Motion8 a b cd e Figure 7.3: (a-e) incremental structure from motion

9. Example Structure from Motion9 Figure 7.4: 3D reconstruction of Trafalgar Square

10. Example Structure from Motion10 Figure 7.5: 3D reconstruction of Great Wall of China.

11. Example Structure from Motion11 Figure 7.6: 3D reconstruction of the Old Town Square, Prague

12. 7.1Triangulation A problem of estimating a point’s 3D location when it is seen from multiple cameras is known as triangulation. It is a converse of pose estimation problem. Given projection matrices, 3D points can be computed from their measured image positions in two or more views. Structure from Motion12

13. Triangulation (cont). Find the 3D point p that lies closest to all of the 3D rays corresponding to the 2D matching feature locations {x j } observed by cameras {P j = K j [R j | t j ] } t j = -R j c j c j is the jth camera center. Structure from Motion13

14. Triangulation (cont). Structure from Motion14 Figure 7.7: 3D point triangulation by finding the points p that lies nearest to all of the optical rays

15. Triangulation (cont). The rays originate at c j in a direction The nearest point to p on this ray, which is denoted as q j, minimizes the distance. which has a minimum at Hence, Structure from Motion15

16. Triangulation (cont). Structure from Motion16

17. Triangulation (cont). Structure from Motion17

18. Triangulation (cont). The squared distance between p and q j is The optimal value for p, which lies closest to all of the rays, can be computed as a regular least square problem by summing over all the r j 2 and finding the optimal value of p, Structure from Motion18

19. Triangulation (cont). If we use homogeneous coordinates p=(X,Y,Z,W), the resulting set of equation is homogeneous and is solved as singular value decomposition (SVD). If we set W=1, we can use regular linear least square, but the resulting system may be singular or poorly coordinated (i.e. all of the viewing rays are parallel). Structure from Motion19

20. Triangulation (cont). For this reason; it is generally preferable to parameterized 3D points using homogeneous coordinates, especially if we know that there are likely to be points at generally varying distances from the cameras. Structure from Motion20

21. 7.2Two-Frame Structure from Motion In 3D reconstruction we have always assumed that either 3D points position or the 3D camera poses are known in advance. Structure from Motion21

22. Two-Frame Structure from Motion (cont). Structure from Motion22 Figure 7.8: Epipolar geometry: The vectors t=c 1 – c 0, p – c 0 and p-c 1 are co-planar and the basic epipolar constraint expressed in terms of the pixel measurement x 0 and x 1

23. Two-Frame Structure from Motion (cont). Figure shows a 3D point p being viewed from two cameras whose relative position can be encoded by a rotation R and a translation t. We do not know anything about the camera positions, without loss of generality. We can set the first camera at the origin c 0 =0 and at a canonical orientation R 0 =I Structure from Motion23

24. Two-Frame Structure from Motion (cont). The observed location of point p in the first image, is mapped into the second image by the transformation : the ray direction vectors. Structure from Motion24

25. Two-Frame Structure from Motion (cont). Structure from Motion25 Taking the cross product of both the sides with t in order to annihilate it on the right hand side yields Taking the dot product of both the sides with yields

26. Two-Frame Structure from Motion (cont). The right hand side is triple product with two identical entries We therefore arrive at the basic epipolar constraint : essential matrix Structure from Motion26

27. Two-Frame Structure from Motion (cont). The essential matrix E maps a point in image 0 into a line in image 1 since All such lines must pass through the second epipole e 1, which is therefore defined as the left singular vector of E with 0 singular value, or, equivalently the projection of the vector t into image 1. Structure from Motion27

28. Two-Frame Structure from Motion (cont). The transpose of these relationships gives us the epipolar line in the first image as and e 0 as the zero value right singular vector E. Structure from Motion28

29. Two-Frame Structure from Motion (cont). Given the relationship How can we use it to recover the camera motion encoded in the essential matrix E? If we have n corresponding measurements {(x i0,x i1 )}, we can form N homogeneous equations in the elements of E= {e 00 …..e 22 } Structure from Motion29

30. Two-Frame Structure from Motion (cont). : element-wise multiplication and summation of matrix elements z i and f: the vector forms of the and E matrices. Given N>8 such equation, we can compute an estimate for the entire E using a Singular Value Decomposition (SVD). Structure from Motion30

31. Two-Frame Structure from Motion (cont). In the presence of noisy measurement, how close is this estimate to being statistically optimal? In the matrix, some entries are product of image measurement such as x i0 y i1 and others are direct image measurements (even identity). Structure from Motion31

32. Two-Frame Structure from Motion (cont). If the measurements have noise, the terms that are product of measurement have their noise amplified by the other element in the product, which lead to poor scaling. In order to deal with this, a suggestion is that the point coordinate should be translated and scaled so that their centroid lies at the original variance is unity; i.e. Structure from Motion32

33. Two-Frame Structure from Motion (cont). such that Structure from Motion33 and n= number of points. Once the essential matrix has been computed from the transformed coordinates; the original essential matrix E can be recovered as

34. Two-Frame Structure from Motion (cont). When the essential matrix has been recovered, the direction of the translation vector t can be estimated. The absolute distance between two cameras can never be recovered from pure image measurement alone. Ground control points in Photogrammetry: knowledge about absolute camera, point positions or distances. Required to establish the final scale, position and orientation. Structure from Motion34

35. Two-Frame Structure from Motion (cont). To estimate direction observe that under the ideal noise-free conditions, the essential matrix E is singular, i.e., This singularity shows up as a singular value of 0 when an SVD of E is performed, Structure from Motion35

36. Pure Translation Figure 7.9: Pure translation camera motion results in visual motion where all the points move towards (or away from) a common focus of expansion (FOE). They therefore satisfies the triple product condition (x 0,x 1 *e) = e * (x o ˣ x 1 ) = 0 Structure from Motion36

37. Pure Translation (cont). Known rotation: The resulting essential matrix E is (in the noise-free case) skew symmetric and can estimate more directly by setting e ij = -e ji and e ii = 0. Two-point parallax now suffices to estimate the FOE. Structure from Motion37

38. Pure Translation (cont). A more direct derivation of FOE estimates can be obtained by minimizing the triple product. which is equivalent to finding null space for the set of equations Structure from Motion38

39. Pure Translation (cont). In a situation where large number of points at infinity are available, (when the camera motion is small compared to distant objects, this suggests a strategy. Pick a pair of points to estimate a rotation, hoping that both of the points lie at infinity (very far from camera). Then compute FOE and check whether residual error is small and whether the motions towards or away from the epipoler (FOE) are all in the same direction. Structure from Motion39

40. Pure Rotation This results in a degenerate estimate of the essential matrix E and the translation direction. If we consider that the rotation matrix is known, the estimates for the FOE will be degenerate, since and hence is degenerate. Structure from Motion40

41. Pure Rotation (cont). Before comparing a full essential matrix to first compute a rotation estimate R, potentially with just a small number of points. Then compute the residuals after rotating the points before processing with a full E computation. Structure from Motion41

42. Projective Reconstruction When we try to build 3D model from the photos taken by unknown cameras, we do not know ahead of time the intrinsic calibration parameters associated with input images. Still, we can estimate a two-frame reconstruction, although the true metric structure may not be available. : the basic epipoler constraint. Structure from Motion42

43. Projective Reconstruction (cont.) In the unreliable case, we do not know the calibration matrices K j, so we cannot use the normalized ray directions. We have access to the image coordinate x j, so essential matrix becomes: fundamental matrix: Structure from Motion43

44. Projective Reconstruction (cont.) Its smallest left singular vector indicates the epipole e 1 in the image 1. Its smallest right singular vector is e 0. Structure from Motion44

45. Projective Reconstruction (cont.) To create a projective reconstruction of a scene, we pick up any valid homography that satisfies and hence : singular value matrix with the smallest value replaced by the middle value. Structure from Motion45

46. Self-calibration Auto-calibration is developed for covering a projective reconstruction into a metric one, which is equivalent to recovering the unknown calibration matrix K j associated with each image. In the presence of additional information about scene, different methods can be applied. If there are parallel lines in the scene, three or more vanishing points, which are the images of points at infinity, can be used to establish homography for the plane at infinity, from which focal length and rotation can be recovered. Structure from Motion46

47. Self-calibration (cont). In the absence of external information: consider all sets of camera matrices P j = K j [ R j | t j ] projecting world coordinates p i =(X i,Y i,Z i,W i ) into screen coordinates x ij ~ P j p i. Consider transforming the 3D scene {p i } through an arbitrary 4 4 projective transformation yielding a new model consisting of points Post-multiplying each other P j matrix by still produces the same screen coordinates and a new set of calibration matrices can be computed by applying RQ decomposition to the new camera matrix. Structure from Motion47

48. Self-calibration (cont). A technique that can recover the focal lengths (f 0,f 1 ) of both images from fundamental matrix F in a two- frame reconstruction. Assume that camera has zero skew, a known aspect ratio, and known optical center. Most cameras have square pixels and an optical center near middle of image and are likely to deviate from simple camera model due to radial distortion Problem occurs when images have been cropped off- center. Structure from Motion48

49. Self-calibration (cont). Take left to right singular vectors {u 0,u 1,v 0,v 1 } of fundamental matrix F and their associated singular values and form the equation: two matrices: Structure from Motion49

50. Self-calibration (cont). Encode the unknown focal length. Write numerators and denominators as: Structure from Motion50

51. Application: View Morphing Application of basic two-frame structure from motion. Also known as view interpolation. Used to generate a smooth 3D animation from one view of a 3D scene to another. To create such a transition: smoothly interpolate camera matrices, i.e., camera position, orientation, focal lengths. More effect is obtained by easing in and easing out camera parameters. To generate in-between frames: establish full set of 3D correspondences or 3D models for each reference view. Structure from Motion51

52. Application: View Morphing Triangulate set of matched feature points in each image. As the 3D points are re-projected into their intermediate views, pixels can be mapped from their original source images to their new views using affine projective mapping. The final image then composited using linear blend of the two reference images as with usual morphing. Structure from Motion52

53. Factorization When processing video sequences, we often get extended feature track from which it is possible to recover the structure and motion using a process called factorization. Structure from Motion53

54. Factorization (cont.) Figure 7.10: 3D reconstruction of a rotating ping pong ball using factorization (Tomasi and Kanade 1992) : (a) sample image with tracked features overlaid; (b)sub-sampled feature motion stream ; (c) two views of the reconstructed 3D model. Structure from Motion54

55. Factorization (cont.) Consider orthographic and weak perspective projection models. Since the last row is always [0001], there is no perspective division x ij : location of i th point in j th frame : upper 2 4 portion of projection matrix P j = (X i, Y i, Z i,1): augmented 3D point position Structure from Motion55

56. Factorization (cont.) Assume that every point i is visible in every frame j. We can take the centroid (average) of the projected point locations x ij in frame j. : augmented 3D centroid of the point cloud. so that Centroid of 2D points in each frame directly gives us last element of Structure from Motion56

57. Factorization (cont.) Let be the 2D point locations after their image centroid has been subtracted. we can write; M j : upper 2 by 3 portion of the projection matrix P j and p i =(X i,Y i,Z i ) We can concatenate all of these measurements into one large matrix. Structure from Motion57

58. Factorization (cont.) : measurement matrix : motion matrices : structure matrices Structure from Motion58

59. Factorization (cont.) If SVD of directly returns the matrices and ; but it does not. Instead we can write the relationship = UQ and To recover values of the 3 3 matrix Q depends on motion model being used. In the case of orthographic projection, the entries in M j are the first two rows of rotation matrices R j. Structure from Motion59

60. Factorization (cont.) So we have u k : 3 1 rows of matrix U. This gives us a large set of equations for the entries in matrix QQ T from which matrix Q can be recovered using matrix square root. Structure from Motion60

61. Factorization (cont.) A disadvantage is they require a complete set of tracks i.e., each point must be visible in each frame, in order for the factorization approach to work. To deal with this problem, apply factorization to smaller denser subsets and then use known camera (motion) or point (structure) estimates to hallucinate additional missing values, which allows them to incrementally incorporate more features and cameras. Structure from Motion61

62. Perspective and Projective Factorization Factorization disadvantage is that it cannot deal with perspective cameras. Perform an initial affine (e.g.,orthographic) reconstruction and to then correct for the perspective effects in an iterative manner. Observe that object-centered projection model differ from scaled orthographic projection model Structure from Motion62

63. Perspective and Projective Factorization (cont). By inclusion of denominator terms If we knew correct values of and motion parameters R j and p j ; we cross multiply left hand side by denominator and get correct values for which bilinear projection model is exact. Structure from Motion63

64. Perspective and Projective Factorization (cont). Once the n j have been estimated, the feature locations can then be corrected before applying another factorization. Because of the initial depth reversal ambiguity both reconstructions have to be tried while computing n j. Structure from Motion64

65. Perspective and Projective Factorization (cont). Alternative approach which does not assume calibrated cameras (known optical center, square pixels, and zero skew) is to perform fully projective factorization. The inclusion of third row of camera matrix Is equivalent to multiplying each reconstructed measurement x ij =M j p i by its inverse depth Or equivalently multiplying each measured position by its projective depth d ji. Structure from Motion65

66. Perspective and Projective Factorization (cont). Factorization method provides a “closed form” (linear) method to initialize iterative techniques such as bundle adjustment. Structure from Motion66

67. Application: Sparse 3D Model Extraction Create a denser 3D model than the sparse point cloud that structured from motion produces. Structure from Motion 67 (a) (b)

68. Application: Sparse 3D Model Extraction (cont). Structure from Motion68 (c) (d) Figure 7.11: 3D teacup model from a 240-frame video sequence: (a) first frame of video; (b) last frame of video; (c) side view of 3D model; (d) top view of the model.

69. Application: Sparse 3D Model Extraction (cont). To create more realistic model, a texture map can be extracted for each triangle face. The equations to map points on the surface of a 3D triangle to a 2D image are straightforward: just pass local 2D coordinates on the triangle through the 3 4 camera projection matrix to obtain 3 by 3 homography Alternative is to create a separate texture map from each reference camera and to blend between them during rendering, which is known as view- independent texture mapping. Structure from Motion69

70. Application: Sparse 3D Model Extraction (cont). Structure from Motion70 Figure 7.12: A set of chained transforms for projecting a 3D point p i into a 2D measurement x ij through a series of transformation f (k), each of which is controlled by its own set of parameters. The dashed lines which indicate the flow of information as partial derivatives are computed during a backward pass. f RD (x) = (1+ k 1 r 2 + k 2 r 4 )x: formula for radial distortion function.

71. Bundle Adjustment The most accurate way to recover structure from motion is to perform robust nonlinear minimization of the measurement (re-projection) errors, which is known as photogrammetry (in computer vision) communities as bundle adjustment. Our feature location measurement x ij now depends only on the point (track index) i but also on the camera pose index j. x ij = f(p i,R j,c j,K j ) 3D point positions p i are also updated simultaneously Structure from Motion71

72. Bundle Adjustment ( cont.) Figure : The leftmost box performs a robust comparison of the predicted and measured 2D locations and after re-scaling by the measurement noise covariance Operation can be written as Structure from Motion72

73. 73 (a) (b) Figure 7.13: A camera rig and its associated transform chain. (a) As the mobile rig (robot) moves around in the world, its pose with respect to world at time t is captured by (R r t,c r t ). Each camera’s pose with respect to the rig captured by (R r t,c r t ). (b) A 3D point with world coordinates p w i is first transformed into rig coordinates p r i, and then through rest of the camera-specific chain.

74. Exploiting Sparsity Large bundle adjustment problems, such as those involving 3D scenes from thousands of Internet photographs can require solving non-linear least square problems with millions of measurements Structure from motion is bipartite problem in structure and motion. Each feature point x ij in a given image depends on one 3D point position p i and 3D camera pose (R j,c j ). Structure from Motion74

75. Exploiting Sparsity (cont). Structure from Motion75 (a)(b) (c) Figure 7.14: (a) Bipartite graph for a toy structure from motion problem and (b) its associated Jacobian J and (c) Hessian A. Numbers indicate cameras. The dashed arcs and light blue squares indicate the fill-in that occurs when the structure (point) variables are eliminated.

76. Structure from motion involves the estimation of so many highly coupled parameters, often with no known “ground truth” components. The estimates produces by structure from motion algorithm can often exhibit large amounts of uncertainty. Example: bas-relief ambiguity, which makes it hard to simultaneously estimate 3D depth of scene and the amount of camera motion. Structure from Motion76 Uncertainty and Ambiguity

77. Uncertainty and Ambiguity (cont). A unique coordinate frame and scale for a reconstructed scene can not be recovered from monocular visual measurements alone. This seven-degrees-of-freedom gauge ambiguity makes it tricky to compute the variance matrix associated with a 3D reconstruction. To compute a convex matrix that ignores gauge freedom is to throw away the seven smallest eigenvalues of the information matrix, whose values are equivalent to the problem Hessian A up to noise scaling. Structure from Motion77

78. Reconstruction from Internet Photos Widely used application of structure from motion: the reconstruction of 3D objects and scenes from video sequences and collection of images. Before structure from motion comparison can begin, it is first necessary to establish sparse correspondences between different pairs of images and to then link such correspondences into feature track, which associates individual 2D image feature with global 3D points. Structure from Motion78

79. Reconstruction from Internet Photos (cont). For the reconstruction process, it is important to select good pair of images and a significant amount of out-of-plane parallax to ensure that a stable reconstruction can be obtained. The EXIF tags associated with photographs can be used to get good initial estimates for camera focal lengths. This is not always strictly necessary, since these parameters are re-adjusted as part of the bundle adjustment process. Structure from Motion79

80. Reconstruction from Internet Photos (cont). Structure from Motion80 Figure 7.15: Incremental structure from motion: Starting with an initial two-frame reconstruction of Trevi Fountain, batches of images are added using pose estimation, and their positions (along with 3D model) are refined using bundle adjustment

81. Reconstruction from Internet Photos (cont). Structure from Motion81 Figure7.16: 3D reconstruction produced by the incremental structure from motion algorithm. (a) cameras and point cloud from Trafalgar Square; (b) cameras and points overlaid on an image from the Great Wall of China.; (c) overhead view of reconstruction of Old Town Square in Prague registered to an aerial photograph.

82. Reconstruction from Internet Photos (cont). Structure from Motion82 Figure 7.17: Large scale structure from motion using skeletal sets: (a) original match graph for 784 images; (b) skeletal set containing 101 images; (c) top-down view of scene (Pantheon) reconstructed from the skeletal set; (d) reconstruction after adding in the remaining images using pose estimation; (e) final bundle adjusted reconstruction, which is almost identical.

83. Constrained Structure and Motion If the object of interest is rotating around a fixed but unknown axis, specialized techniques can be used to recover this motion. In other situation, the camera itself may be moving in a fixed arc around some center of rotation. Specialized capture steps, such as mobile stereo camera rings or moving vehicles equipped with multiple fixed cameras, can also take advantage of the knowledge that individual cameras are mostly fixed with respect to the capture rig. Structure from Motion83

84. Constrained Structure and Motion (cont). Line-based technique: Pairwise epipolar geometry cannot be recovered from line matches alone, even if the cameras are calibrated. Consider projecting the set of lines in each image into a set of 3D planes in space. You can move the two cameras around into any configuration and still obtain a valid reconstruction for 3D lines. Structure from Motion84

85. Constrained Structure and Motion (cont). When lines are visible in three or more views, the trifocal tensor can be used to transfer lines from one pair of image to another. The trifocal tensor can also be computed on the basis line matches alone. For triples of images, the trifocal tensor is used to verify that the lines are in geometric correspondence before evaluating the correlations between line segments. Structure from Motion85

86. Constrained Structure and Motion (cont). Structure from Motion86 Figure 7.18: Two images of toy house along with their matched 3D line segments.

87. Constrained Structure and Motion (cont). Plane-based technique: Better approach is to hallucinate virtual point correspondences within the area from which each homography was computed and to feed them into a standard structure from motion algorithm. Structure from Motion87

88. End Structure from Motion88