Projective geometry for Computer Vision

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Lines in P 2 ◮ A line equation in R 2 is a 1 x 1 + a 2 x 2 + a 3 = 0 ◮ Substituting by homogeneous coordinates x i = X i /X 3 we get a homogeneous linear equation (a 1 , a 2 , a 3 )· (X 1 , X 2 , X 3 ) = 3∑ a i X i = 0, X ∈ P 2 i=1 ◮ A line in P 2 is represented by a homogeneous 3-vector (a 1 , a 2 , a 3 ). ◮ A point on a line: a· X = 0 or a T X = 0 or X T a = 0 ◮ Two points define a line: l = p × q ◮ Two lines define a point: x = l × m Subhashis Banerjee Projective geometry for Computer Vision
The line at infinity ◮ The line at infinity (l ∞ ): is the line of equation X 3 = 0. Thus, the homogeneous representation of l ∞ is (0, 0, 1). ◮ The line (u 1 , u 2 , u 3 ) intersects l ∞ at the point (−u 2 , u 1 , 0). ◮ Points on l ∞ are directions of affine lines in the embedded affine space (can be extended to higher dimensions). Subhashis Banerjee Projective geometry for Computer Vision
Page 1 and 2: Projective geometry for Computer Vi
Page 3 and 4: Computer vision geometry: main prob
Page 5 and 6: 3D reconstruction from pin-hole pro
Page 7 and 8: In terms of projective coordinates
Page 9 and 10: Euclidean and Affine geometries ◮
Page 11 and 12: Spherical geometry ◮ Lines in S 2
Page 13 and 14: Projective geometry The space P 2 c
Page 15: Canonical injection of R n into P n
Page 19 and 20: Conics in P 2 ◮ The line l tangen
Page 21 and 22: Projective basis Projective basis:
Page 23 and 24: Homography The invertible transform
Page 25 and 26: Homography Subhashis Banerjee Proje
Page 27 and 28: Projective mappings of conics ◮ N
Page 29 and 30: Affine transformations preserve the
Page 31 and 32: Affine calibration of a plane Subha
Page 33 and 34: Reconstruction Camera recovery Metr
Page 35 and 36: Modeling of structured scenes Subha

Projective geometry for Computer Vision

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