Introduction to homography
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This document introduces homography and projective geometry concepts. It discusses that homography is a projective transformation represented by a 3x3 matrix that maps points from one projective plane to another. The document outlines key homography topics like the line at infinity, mapping between image planes using homography, and the Direct Linear Transform (DLT) algorithm to estimate homography from point correspondences between images. It also provides an overview of homography functions in the OpenCV library.
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Introduction to homography
- 1. Introduction to Homography 2010.01.19
- 2. Coordinate System view coordinate system object coordinate system x y z world coordinate system
- 3. History P->G: 以 E 為眼點從平面 G 到平面 W 的 透視對應
- 4. Outline Projective Geometry Projective Transformation Estimation Document Perspective Distortion Direct Linear Transform (DLT) Algorithm OpenCV Implementation
- 5. Homo. representation (1) Lines This equivalence relationship is known as a homogeneous vector representative
- 6. Homo. representation (2) Points Homo. representation
- 7. Degree of freedom (DoF) Point Line
- 8. 2 lines Assume Intersection of lines (1)
- 9. Intersection of lines (2) Ex: determining the intersection of the lines x = 1 and y = 1 y = 1 x = 1
- 10. Line joining points The line passing through two points x and x’ Assume
- 11. Line at infinity (1) Consider 2 lines Intersection
- 12. Line at infinity (2) Inhomogeneous representation No sense ! Suggest the point has infinitely large coordinates Homogeneous coordinates (x, y, 0) T do not correspond to any finite point in R 2
- 13. Line at infinity (3) Ex: intersection point of 2 lines x=1 and x=2 The point at infinity in the direction of the y-axis
- 14. Line at infinity (4) Projective Space P 2 : augment R 2 by adding points with last coordinate x 3 = 0 to all homogeneous 3-vectors The points with last coordinate x 3 = 0 are known as ideal points , or points at infinity The set of all ideal points may be written (x 1 ,x 2 , 0) T . The set lies on a single line, the line at infinity , denoted by the vector (0,0,1) T => (x 1 ,x 2 ,0)(0,0,1) T =0
- 15. Concept of points at infinity Simplify the intersection properties of points and lines In P 2 , one may state without qualification (not true in the R 2 ) 2 distinct lines meet in a single point 2 distinct points lie on a single line
- 16. A model for the projective plane Points and Lines of P 2 are represented by Rays and Planes , respectively, through the origin in R 3 Lines lying in the x 1 x 2 -plane represent ideal points , and the x 1 x 2 -plane represents l ∞
- 17. Outline Projective Geometry Projective Transformation Estimation Document Perspective Distortion Direct Linear Transform (DLT) Algorithm OpenCV Implementation
- 19. Definition (1) A projectivity is an invertible mapping h from P 2 to itself such that 3 points x 1 , x 2 and x 3 lie on the same line if and only if h(x 1 ) , h(x 2 ) and h(x 3 ) do Synonymous Collineation Homography
- 20. Definition (2) A mapping h from P 2 to P 2 There exists a non-singular 3 x 3 matrix H such that for any point in P 2 represented by a vector x it is true that
- 21. Definition (3) H is a homogeneous matrix , since as in the homogeneous representation of a point, only the ratio of the matrix elements is significant (Dof = 8)
- 22. A hierarchy of transformations (1)
- 23. A hierarchy of transformations (2)
- 24. Outline Projective Geometry Projective Transformation Estimation Document Perspective Distortion Direct Linear Transform (DLT) Algorithm OpenCV Implementation
- 26. Correction
- 27. 2D Homography x i in P 2 x j in P 2 In a practical situation, the points x i and x j are points in 2 images (or the same image), each image being considered as a projective plane P 2
- 29. DLT Algorithm (2) Since each point correspondence provides 2 equations, 4 correspondences are sufficient to solve for the 8 degrees of freedom of H Use more than 4 correspondences to ensure a more robust solution (The problem then becomes to solve for a vector h that minimizes a suitable cost function )
- 30. Outline Projective Geometry Projective Transformation Estimation Document Perspective Distortion Direct Linear Transform (DLT) Algorithm OpenCV Implementation
- 31. OpenCV Implementation Mat findHomography ( const Mat & srcPoints, const Mat & dstPoints, int method=0, double ransacReprojThreshold=3); void warpPerspective ( const Mat & src, CV_OUT Mat & dst, const Mat & M, Size dsize, int flags=INTER_LINEAR, int borderMode=BORDER_CONSTANT, const Scalar& borderValue=Scalar()); Homography Estimation Projective Transform
- 32. Reference Multiple View Geometry in Computer Vision , Cambridge University Press, 2000. Projective geometry and homogeneous coordinates [Video] - http://www.youtube.com/watch?v=q3turHmOWq4