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Going Back a little

Going Back a little. Cameras.ppt. Applications of RANSAC: Solution for affine parameters. Affine transform of [x,y] to [u,v]: Rewrite to solve for transform parameters:. Assignment. Program-1 info -Link Data. Note: You can generate, bring-in, your own images from www, as long as:

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Going Back a little

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  1. Going Back a little • Cameras.ppt Computer Vision : CISC 4/689

  2. Applications of RANSAC: Solution for affine parameters • Affine transform of [x,y] to [u,v]: • Rewrite to solve for transform parameters: Computer Vision : CISC 4/689

  3. Assignment • Program-1 • info-Link • Data Note: You can generate, bring-in, your own images from www, as long as: For n+1 levels, image must be Mr£2n+1 rows by Mc£2n+1 cols Mr and Mc are any +ve integers Sunday 10pm Computer Vision : CISC 4/689

  4. Another app. : Automatic Homography H Estimation • Homographies describe image transformation of... • General scene when camera motion is rotation about camera center • Planar surfaces under general camera motion • How to get correct correspondences without human intervention? from Hartley & Zisserman Computer Vision : CISC 4/689

  5. Computing a Homography Lets Side-track • 8 degrees of freedom in 3 x 3 matrix H, so at least n = 4 pairs of 2-D points are sufficient to determine it • Use same basic algorithm for P (aka Direct Linear Transformation, or DLT) to compute H • Now stacked matrix A is 2n x 9 vs. 2n x 12 for camera matrix P estimation because all points are 2-D • 3 collinear points in either image is a degenerate configuration preventing a unique solution Computer Vision : CISC 4/689

  6. Estimating H: DLT Algorithm • x0i =Hxiis an equation involving homogeneous vectors, so Hxi and x0i need only be in the same direction, not strictly equal • We can specify “same directionality” by using a cross product formulation: • See Hartley & Zisserman, Chapter 3.1-3.1.1 (linked on course page) for details Computer Vision : CISC 4/689

  7. Texture Mapping • Needed for nice display when applying transformations (like a homography H) to a whole image • Simple approach: Iterate over source image coordinates and apply x0 = Hx to get destination pixel location • Problem: Some destination pixels may not be “hit”, leaving holes • Easy solution: Iterate over destination image and apply inverse transform x = H-1x0 • Round off H-1x0 to address “nearest” source pixel value • This ensures every destination pixel is filled in Computer Vision : CISC 4/689

  8. Automatic H Estimation: Feature Extraction • Find features in pair of images using corner detection—e.g., eigenvalue threshold of: Computer Vision : CISC 4/689 from Hartley & Zisserman ~500 features found

  9. Automatic H Estimation: Finding Feature Matches • Best match over threshold within square search window (here §300 pixels) using SSD or normalized cross-correlation Computer Vision : CISC 4/689 from Hartley & Zisserman

  10. Automatic H Estimation: Finding Feature Matches • Best match over threshold within square search window (here §300 pixels) using SSD or normalized cross-correlation Computer Vision : CISC 4/689 from Hartley & Zisserman

  11. Automatic H Estimation: Initial Match Hypotheses from Hartley & Zisserman 268 matched features (over SSD threshold) in left image pointing to locations of corresponding right image features Computer Vision : CISC 4/689

  12. Automatic H Estimation: Applying RANSAC • Sampling • Size: Recall that 4 correspondences suffice to define homography, so sample size s = 4 • Choice • Pick SSD threshold conservatively to minimize bad matches • Disregard degenerate configurations • Ensure points have good spatial distribution over image • Distance measure • Obvious choice is symmetric transfer error: Computer Vision : CISC 4/689

  13. Automatic H Estimation: Outliers & Inliers after RANSAC • 43 samples used with t = 1.25 pixels Computer Vision : CISC 4/689 from Hartley & Zisserman 117 outliers (²=0.44) 151 inliers

  14. A Short Review of Camera Calibration Computer Vision : CISC 4/689

  15. Pinhole Camera Terminology Image plane Optical axis Principal point/ image center Focal length Camera center/ pinhole Camera point Image point Computer Vision : CISC 4/689

  16. Calibration • Slides (calibration.ppt) Computer Vision : CISC 4/689

  17. Calibration and Pose estimation example • Recover intrinsic and extrinsic parameters of camera by using calibration board. • 3D points are given, can find 2D image coordinates for the corresponding 3D points. • Assume world is located at the folded lower corner, principal point is center of the image, fold is 90 degrees, • Total length and width of board is 9in by 9in. Next 8 slides, courtesy UCF. Computer Vision : CISC 4/689

  18. Matlab code • Matlab • fx = 1.5031 • fy =1.2773 • Rc = -0.0201 -0.2000 -0.9796 0.2198 0.9588 -0.1797 0.9752 -0.2189 0.0247 • Tc = (29.0725, -2.8850, 53.4196) • camera position:( -51.0289, 19.7118 26.6985) Computer Vision : CISC 4/689

  19. 3D World Points • Camera Centers • Camera Orientations Multi-View Geometry Relates Computer Vision : CISC 4/689

  20. 3D World Points • Camera Centers • Camera Intrinsic Parameters • Image Points Multi-View Geometry Relates • Camera Orientations Computer Vision : CISC 4/689

  21. Stereo scene point image plane optical center Computer Vision : CISC 4/689

  22. Stereo • Basic Principle: Triangulation • Gives reconstruction as intersection of two rays • Requires • calibration • point correspondence Computer Vision : CISC 4/689

  23. Stereo Constraints p’ ? p Given p in left image, where can the corresponding point p’in right image be? Computer Vision : CISC 4/689

  24. Epipolar Line p’ Y2 X2 Z2 O2 Epipole Stereo Constraints M Image plane Y1 p O1 Z1 X1 Focal plane Computer Vision : CISC 4/689

  25. Stereo • The geometric information that relates two different viewpoints of the same scene is entirely contained in a mathematical construct known as fundamental matrix. • The geometry of two different images of the same scene is called the epipolar geometry. Computer Vision : CISC 4/689

  26. Stereo/Two-View Geometry • The relationship of two views of a scene taken from different camera positions to one another • Interpretations • “Stereo vision” generally means two synchronized cameras or eyes capturing images • Could also be two sequential views from the same camera in motion • Assuming a static scene http://www-sop.inria.fr/robotvis/personnel/sbougnou/Meta3DViewer/EpipolarGeo Computer Vision : CISC 4/689

  27. 3D from two-views There are two ways of extracting 3D from a pair of images. • Classical method, called Calibrated route, we need to calibrate both cameras (or viewpoints) w.r.t some world coordinate system. i.e, calculate the so-called epipolar geometry by extracting the essential matrix of the system. • Second method, called uncalibrated route, a quantity known as fundamental matrix is calculated from image correspondences, and this is then used to determine the 3D. Either way, principle of binocular vision is triangulation. Given a single image, the 3D location of any visible object point must lie on the straight line that passes through COP and image point (see fig.). Intersection of two such lines from two views is triangulation. Computer Vision : CISC 4/689

  28. Mapping Points between Images • What is the relationship between the images x, x’ of the scene point X in two views? • Intuitively, it depends on: • The rigid transformation between cameras (derivable from the camera matrices P, P’) • The scene structure (i.e., the depth of X) • Parallax: Closer points appear to move more Computer Vision : CISC 4/689

  29. x3 x’3 x’2 x2 x1 x’1 Example: Two-View Geometry courtesy of F. Dellaert Is there a transformation relating the points xi tox’i ? Computer Vision : CISC 4/689

  30. Epipolar Geometry • Baseline: Line joining camera centers C, C’ • Epipolar plane ¦: Defined by baseline and scene point X Computer Vision : CISC 4/689 baseline from Hartley & Zisserman

  31. Epipolar Lines • Epipolar lines l, l’: Intersection of epipolar plane ¦ with image planes • Epipoles e, e’: Where baseline intersects image planes • Equivalently, the image in one view of the other camera center. C’ C Computer Vision : CISC 4/689 from Hartley & Zisserman

  32. Epipolar Pencil • As position of X varies, epipolar planes “rotate” about the baseline (like a book with pages) • This set of planes is called the epipolar pencil • Epipolar lines “radiate” from epipole—this is the pencil of epipolar lines Computer Vision : CISC 4/689 from Hartley & Zisserman

  33. Epipolar Constraint • Camera center C and image point define ray in 3-D space that projects to epipolar line l’ in other view (since it’s on the epipolar plane) • 3-D point X on this ray, so image of X in other view x’ must be on l’ • In other words, the epipolar geometry defines a mapping x !l’, of points in one image to lines in the other x’ C’ C Computer Vision : CISC 4/689 from Hartley & Zisserman

  34. Example: Epipolar Lines for Converging Cameras Left view Right view from Hartley & Zisserman Intersection of epipolar lines = Epipole ! Indicates direction of other camera Computer Vision : CISC 4/689

  35. Special Case: Translation Parallel to Image Plane Note that epipolar lines are parallel and corresponding points lie on correspond- ing epipolar lines (the latter is true for all kinds of camera motions) Computer Vision : CISC 4/689

  36. P p p’ O’ O From Geometry to Algebra Courtesy, UCF Computer Vision : CISC 4/689

  37. P p p’ O’ O From Geometry to Algebra Computer Vision : CISC 4/689

  38. Linear Constraint:Should be able to express as matrix multiplication. Computer Vision : CISC 4/689

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