6.819 / 6.869: Advances in Computer Vision Antonio Torralba and Bill Freeman. Lecture 11 Geometry, Camera Calibration, and Stereo.

Transcription

1 6.819 / 6.869: Advances in Computer Vision Antonio Torralba and Bill Freeman Lecture 11 Geometry, Camera Calibration, and Stereo.

2 2d from 3d; 3d from multiple 2d measurements? 2d 3d?

3 Perspective projection f f (0,0,0) image plane Virtual image plane

4 Perspective projection Y y? (0,0,0) f Virtual image plane Similar triangles: y / f = Y / Z y = f Y/Z Perspective projection: ( X,Y,Z ) f X Z, f Y Z Z

5 Vanishing points graphics/two_point_perspective.html

6 Other projection models: Orthographic projection ( x, y, z) ( x, y)

7 Three camera projections (1) Perspective: (2) Weak perspective: (3) Orthographic: ), ( ),, (, ),, (, ),, ( 0 0 y x z y x z fy z fx z y x z fy z fx z y x 3-d point 2-d image position

8 Three camera projections Perspective projection Parallel (orthographic) projection Weak perspective?

9 Homogeneous coordinates Is the perspective projection a linear transformation? no division by z is nonlinear Trick: add one more coordinate: homogeneous image coordinates homogeneous world coordinates Converting from homogeneous coordinates Slide by Steve Seitz

10 Perspective Projection Projection is a matrix multiply using homogeneous coordinates: This is known as perspective projection The matrix is the projection matrix Slide by Steve Seitz / f 0 x y z 1 = x y z / f f x z, f y z

11 Perspective Projection How does scaling the projection matrix change the transformation? Slide by Steve Seitz / f 0 x y z 1 = x y z / f f x z, f y z f f x y z 1 = fx fy z f x z, f y z

12 Orthographic Projection Special case of perspective projection Image World Also called parallel projection What s the projection matrix?? Slide by Steve Seitz

13 Orthographic Projection Special case of perspective projection Distance from the COP to the PP is infinite Image World Also called parallel projection What s the projection matrix? Slide by Steve Seitz

14 2D Transformations

15 2D Transformations Example: translation = + tx ty

16 2D Transformations Example: translation = + tx ty = 1 0 tx 0 1 ty. 1

17 2D Transformations Example: translation written 3 ways: non-homog homog in, non-h out, homog in, homog out = + tx ty = 1 0 tx 0 1 ty. 1 = 1 0 tx 0 1 ty Now we can chain transformations

18 Translation and rotation, written in each set of coordinates Non-homogeneous coordinates! B p = B R A! p + B! t A A from Homogeneous coordinates! B p = B C A! p A to where B A C = B A R! B A t

19 Camera calibration Use the camera to tell you things about the world: Relationship between coordinates in the world and coordinates in the image: geometric camera calibration, see Szeliski, section 5.2, 5.3 for references (Relationship between intensities in the world and intensities in the image: photometric image formation, see Szeliski, sect. 2.2.)

20 Camera calibration Intrinsic parameters Image coordinates relative to camera!" Pixel coordinates Extrinsic parameters Camera frame 1!" Camera frame 2

21 Camera calibration Intrinsic parameters Extrinsic parameters

22 Intrinsic parameters: from idealized world coordinates to pixel values Forsyth&Ponce Perspective projection u v = = f f x z y z

23 Intrinsic parameters But pixels are in some arbitrary spatial units u v = α = α x z y z

24 Intrinsic parameters Maybe pixels are not square u v = = α β x z y z

25 Intrinsic parameters The origin of our camera pixel coordinates may be somewhere other than under the camera optical axis. u v = = α β x z y z + u + v 0 0

26 Intrinsic parameters 0 0 ) sin( ) cot( v z y v u z y z x u + = + = θ β θ α α May be skew between camera pixel axes (but usually this angle is 90 deg). v θ u vʹ uʹ v u v u u v v ) cot( ) cos( ) sin( θ θ θ = ʹ = ʹ = ʹ

27 ! p = K C! p Intrinsic parameters, homogeneous coordinates 0 0 ) sin( ) cot( v z y v u z y z x u + = + = θ β θ α α u v 1 = α α cot(θ) u 0 0 β sin(θ) v x y z 1 Using homogenous coordinates, we can write this as: or: In camera-based coords In pixels

28 Camera calibration Intrinsic parameters Extrinsic parameters

29 World and camera coordinate systems In the first lecture, we placed the world coordinates in the center of the scene.

30 Extrinsic parameters: translation and rotation of camera frame C!! p = C W R W p +! C W t Non-homogeneous coordinates C! p C = W R! C W t W! p Homogeneous coordinates

31 pixels Camera coordinates C!!! p = K C p p C = W R! C W t W! p Intrinsic World coordinates Extrinsic Combining extrinsic and intrinsic calibration parameters, in homogeneous coordinates Forsyth&Ponce! p = K C W R! C W t!! p = M W p ( ) W! p

32 Other ways to write the same equation pixel coordinates! p = M W! p world coordinates u v = 1 T. m 1 T. m 2 T. m W p x W p y W p z 1 Conversion back from homogeneous coordinates leads to: u v = = m m m m ! P! P! P! P

33 Summary camera parameters A camera is described by several parameters Translation T of the optical center from the origin of world coords Rotation R of the image plane focal length f, principle point (x c, y c ), pixel size (s x, s y ) blue parameters are called extrinsics, red are intrinsics Projection equation sx * X * Y Z * 1 x = sy = * * * * = ΠX s * The projection matrix models the cumulative effect of all parameters Useful to decompose into a series of operations * * fs x 0 x' c R Π = 0 fs y y' c x x 3 1 * * 0 3x1 I 3x 3 T 3x1 0 1x 3 1 intrinsics projection rotation translation identity matrix The definitions of these parameters are not completely standardized especially intrinsics varies from one book to another

34 do we calculate the camera s calibration matrix, or measure?

35 Calibration target Find the position, u i and v i, in pixels, of each calibration object feature point.

36 Camera calibration From before, we had these equations relating image positions, u,v, to points at 3-d positions P (in homogeneous coordinates): So for each feature point, i, we have:

37 Camera calibration Stack all these measurements of i=1 n points into a big matrix (cluttering vector arrows omitted from P and m):

38 In vector form: Camera calibration Showing all the elements:

39 Camera calibration Q m = 0 We want to solve for the unit vector m (the stacked one) that minimizes The minimum eigenvector of the matrix Q T Q gives us that (see Forsyth&Ponce, 3.1), because it is the unit vector x that minimizes x T Q T Q x.

40 Camera calibration Once you have the M matrix, can recover the intrinsic and extrinsic parameters.

41 Vision systems One camera Two cameras N cameras

42 Stereo vision ~6cm ~50cm

43 Depth without objects Random dot stereograms (Bela Julesz) Julesz, 1971

44 Stereo photography and stereo viewers Take two pictures of the same subject from two slightly different viewpoints and display so that each eye sees only one of the images. Image courtesy of fisher-price.com Slide credit: Kristen Grauman

45 Public Library, Stereoscopic Looking Room, Chicago, by Phillips, 1923 Slide credit: Kristen Grauman

46 Anaglyph pinhole camera

47 Autostereograms Exploit disparity as depth cue using single image. (Single image random dot stereogram, Single image stereogram) Images from magiceye.com Slide credit: Kristen Grauman

48 My conundrum regarding stereo displays Real 3d scenes often look to me like thin, flat layers, stacked in depth. Why is that? 48

49 Estimating depth with stereo Stereo: shape from disparities between two views We ll need to consider: Info on camera pose ( calibration ) Image point correspondences scene point optical center image plane Slide credit: Kristen Grauman

50 Geometry for a simple stereo system Assume a simple setting: Two identical cameras parallel optical axes known camera parameters (i.e., calibrated cameras). O l O r

51 World point image point (left) Depth of p image point (right) Focal length optical center (left) baseline optical center (right) Slide credit: Kristen Grauman

52 Geometry for a simple stereo system Assume parallel optical axes, known camera parameters (i.e., calibrated cameras). We can triangulate via: Similar triangles (p l, P, p r ) and (O l, P, O r ): disparity Slide credit: Kristen Grauman

53 Depth from disparity image I(x,y) Disparity map D(x,y) image I (x,y ) (x,y )=(x+d(x,y), y) Slide credit: Kristen Grauman

54 General case, with calibrated cameras The two cameras need not have parallel optical axes. Vs.

55 Stereo correspondence constraints Camera 1 Camera 2 p p? O O If we see a point in camera 1, are there any constraints on where we will find it on camera 2?

56 Epipolar constraint p p? O O Geometry of two views constrains where the corresponding pixel for some image point in the first view must occur in the second view: It must be on the line carved out by a plane connecting the world point and optical centers. Why is this useful?

57 Epipolar constraint This is useful because it reduces the correspondence problem to a 1D search along an epipolar line. Image from Andrew Zisserman Slide credit: Kristen Grauman

58 Epipolar geometry Epipolar Line Epipolar Plane Epipolar Line Epipole Baseline Epipole Epipolar plane: plane containing baseline and world point Epipole: point of intersection of baseline with the image plane Epipolar line: intersection of epipolar plane with the image plane Baseline: line joining the camera centers All epipolar lines intersect at the epipole An epipolar plane intersects the left and right image planes in epipolar lines Slide credit: Kristen Grauman

59 Example Slide credit: Kristen Grauman

60 Example: parallel cameras Where are the epipoles? Figure from Hartley & Zisserman Slide credit: Kristen Grauman

61 Example: converging cameras Figure from Hartley & Zisserman Slide credit: Kristen Grauman

62 So far, we have the explanation in terms of geometry. Now, how to express the epipolar constraints algebraically? Slide credit: Kristen Grauman

63 Stereo geometry, with calibrated cameras Main idea Slide credit: Kristen Grauman

64 Stereo geometry, with calibrated cameras If the stereo rig is calibrated, we know : how to rotate and translate camera reference frame 1 to get to camera reference frame 2. Rotation: 3 x 3 matrix R; translation: 3 vector T. Slide credit: Kristen Grauman

65 Stereo geometry, with calibrated cameras If the stereo rig is calibrated, we know : how to rotate and translate camera reference frame 1 to get to camera reference frame 2. Slide credit: Kristen Grauman

66 From geometry to algebra X' = RX + T T Xʹ = T RX + T T Normal to the plane = T RX ( T Xʹ) = Xʹ ( T RX) X ʹ = 0 Slide credit: Kristen Grauman

67 From geometry to algebra X' = RX + T T Xʹ = T RX + T T Normal to the plane = T RX ( T Xʹ) = Xʹ ( T RX) X ʹ = 0 Slide credit: Kristen Grauman

68 Aside: cross product Vector cross product takes two vectors and returns a third vector that s perpendicular to both inputs. So here, c is perpendicular to both a and b, which means the dot product = 0. Slide credit: Kristen Grauman

69 Matrix form of cross product Can be expressed as a matrix multiplication. c b b b a a a a a a b a!!! = = [ ] = a a a a a a a x Slide credit: Kristen Grauman

70 Let Xʹ Xʹ E = ( T RX) = 0 ( T ) x RX = 0 T x R Xʹ T EX = 0 Essential matrix E is called the essential matrix, and it relates corresponding image points between both cameras, given the rotation and translation. If we observe a point in one image, its position in the other image is constrained to lie on line defined by above.

71 x and x are scaled versions of X and X

72 Let pts x and x in the image planes are scaled versions of X and X. E is called the essential matrix, and it relates corresponding image points between both cameras, given the rotation and translation. If we observe a point in one image, its position in the other image is constrained to lie on line defined by above. Note: these points are in camera coordinate systems.

73 Essential matrix example: parallel cameras d 0 d 0 For the parallel cameras, image of any point must lie on same horizontal line in each image plane. Slide credit: Kristen Grauman

74 image I(x,y) Disparity map D(x,y) image I (x,y ) (x,y )=(x+d(x,y),y) What about when cameras optical axes are not parallel? Slide credit: Kristen Grauman

75 Stereo image rectification: example Source: Alyosha Efros

76 Stereo image rectification In practice, it is convenient if image scanlines (rows) are the epipolar lines. Reproject image planes onto a common plane parallel to the line between optical centers Pixel motion is horizontal after this transformation Two homographies (3x3 transforms), one for each input image reprojection See Szeliski book, Sect , Fig. 2.12, and Mapping from one camera to another p. 56: Adapted from Li Zhang Slide credit: Kristen Grauman

77 Your basic stereo algorithm For each epipolar line For each pixel in the left image compare with every pixel on same epipolar line in right image pick pixel with minimum match cost Improvement: match windows CSE 576, Slide Spring credit: 2008 Rick Szeliski Stereo matching 77

78 Image block matching How do we determine correspondences? block matching or SSD (sum squared differences) d is the disparity (horizontal motion) How big should the neighborhood be? CSE 576, Spring 2008 Slide credit: Rick Szeliski Stereo matching 78

79 Neighborhood size Smaller neighborhood: more details Larger neighborhood: fewer isolated mistakes w = 3 w = 20 CSE 576, Slide Spring credit: 2008 Rick Szeliski Stereo matching 79

80 Matching criteria Raw pixel values (correlation) Band-pass filtered images [Jones & Malik 92] Corner like features [Zhang, ] Edges [many people ] Gradients [Seitz 89; Scharstein 94] Rank statistics [Zabih & Woodfill 94] CSE 576, Slide Spring credit: 2008 Rick Szeliski Stereo matching 80

81 Local evidence framework For every disparity, compute raw matching costs Why use a robust function? occlusions, other outliers Can also use alternative match criteria CSE 576, Slide Spring credit: 2008 Rick Szeliski Stereo matching 81

82 Local evidence framework Aggregate costs spatially Here, we are using a box filter (efficient moving average implementation) Can also use weighted average, [non-linear] diffusion CSE 576, Slide Spring credit: 2008 Rick Szeliski Stereo matching 82

83 Local evidence framework Choose winning disparity at each pixel Interpolate to sub-pixel accuracy E(d) d* d CSE 576, Slide Spring credit: 2008 Rick Szeliski Stereo matching 83

84 Active stereo with structured light Li Zhang s one-shot stereo camera 1 camera 1 projector projector camera 2 Project structured light patterns onto the object simplifies the correspondence problem Li Zhang, Brian Curless, and Steven M. Seitz. Rapid Shape Acquisition Using Color Structured Light and Multi-pass Dynamic Programming. In Proceedings of the 1st International Symposium on 3D Data Processing, Visualization, and Transmission (3DPVT), Padova, Italy, June 19-21, 2002, pp CSE 576, Slide Spring credit: 2008 Rick Szeliski Stereo matching 84

85 CSE 576, Spring 2008 Li Zhang, Brian Curless, and Steven M. Seitz Stereo matching 85

86 CSE 576, Spring 2008 Stereo matching 86

87 CSE 576, Spring 2008 Stereo matching 87

88 CSE 576, Spring 2008 Stereo matching 88

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92 Bibliography D. Scharstein and R. Szeliski. A taxonomy and evaluation of dense twoframe stereo correspondence algorithms. International Journal of Computer Vision, 47(1):7-42, May R. Szeliski. Stereo algorithms and representations for image-based rendering. In British Machine Vision Conference (BMVC'99), volume 2, pages , Nottingham, England, September G. M. Nielson, Scattered Data Modeling, IEEE Computer Graphics and Applications, 13(1), January 1993, pp S. B. Kang, R. Szeliski, and J. Chai. Handling occlusions in dense multiview stereo. In CVPR'2001, vol. I, pages , December Y. Boykov, O. Veksler, and Ramin Zabih, Fast Approximate Energy Minimization via Graph Cuts, Unpublished manuscript, A.F. Bobick and S.S. Intille. Large occlusion stereo. International Journal of Computer Vision, 33(3), September pp D. Scharstein and R. Szeliski. Stereo matching with nonlinear diffusion. International Journal of Computer Vision, 28(2): , July 1998 CSE 576, Slide Spring credit: 2008 Rick Szeliski Stereo matching 92

93 Bibliography Volume Intersection Martin & Aggarwal, Volumetric description of objects from multiple views, Trans. Pattern Analysis and Machine Intelligence, 5(2), 1991, pp Szeliski, Rapid Octree Construction from Image Sequences, Computer Vision, Graphics, and Image Processing: Image Understanding, 58(1), 1993, pp Voxel Coloring and Space Carving Seitz & Dyer, Photorealistic Scene Reconstruction by Voxel Coloring, Proc. Computer Vision and Pattern Recognition (CVPR), 1997, pp Seitz & Kutulakos, Plenoptic Image Editing, Proc. Int. Conf. on Computer Vision (ICCV), 1998, pp Kutulakos & Seitz, A Theory of Shape by Space Carving, Proc. ICCV, 1998, pp CSE 576, Slide Spring credit: 2008 Rick Szeliski Stereo matching 93

94 Bibliography Related References Bolles, Baker, and Marimont, Epipolar-Plane Image Analysis: An Approach to Determining Structure from Motion, International Journal of Computer Vision, vol 1, no 1, 1987, pp Faugeras & Keriven, Variational principles, surface evolution, PDE's, level set methods and the stereo problem", IEEE Trans. on Image Processing, 7(3), 1998, pp Szeliski & Golland, Stereo Matching with Transparency and Matting, Proc. Int. Conf. on Computer Vision (ICCV), 1998, Roy & Cox, A Maximum-Flow Formulation of the N-camera Stereo Correspondence Problem, Proc. ICCV, 1998, pp Fua & Leclerc, Object-centered surface reconstruction: Combining multi-image stereo and shading", International Journal of Computer Vision, 16, 1995, pp Narayanan, Rander, & Kanade, Constructing Virtual Worlds Using Dense Stereo, Proc. ICCV, 1998, pp CSE 576, Slide Spring credit: 2008 Rick Szeliski Stereo matching 94