Projective geometry for Computer Vision

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Projective geometry ◮ Let π : S 2 → P 2 be the mapping that sends x to {x, −x}. The π is a two-to-one map of S 2 onto P 2 . ◮ A line of P 2 is a set of the form πl, where l is a line of S 2 . Clearly, πx lies on πl if and only if ξ t x = 0. ◮ Homogeneous coordinates: In general, points of real n-dimensional projective space, P n , are represented by n + 1 component column vectors (x 1 , . . . , x n , x n+1 ) ∈ R n+1 such that at least one x i is non-zero and (x 1 , . . . , x n , x n+1 ) and (λx 1 , . . . , λx n , λx n+1 ) represent the same point of P n for all λ ≠ 0. ◮ (x 1 , . . . , x n , x n+1 ) is the homogeneous representation of a projective point. Subhashis Banerjee Projective geometry for Computer Vision
Canonical injection of R n into P n ◮ Affine space R n can be embedded in P n by (x 1 , . . . , x n ) → (x 1 , . . . , x n , 1) ◮ Affine points can be recovered from projective points with x n+1 ≠ 0 by (x 1 , . . . , x n ) ∼ ( x 1 x n+1 , . . . , x n x n+1 , 1) → ( x 1 x n+1 , . . . , x n x n+1 ) ◮ A projective point with x n+1 = 0 corresponds to a point at infinity. ◮ The ray (x 1 , . . . , x n , 0) can be viewed as an additional ideal point as (x 1 , . . . , x n ) recedes to infinity in a certain direction. For example, in P 2 , lim (X /T , Y /T , 1) = lim (X , Y , T ) = (X , Y , 0) T →0 T →0 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: Projective geometry The space P 2 c
Page 17 and 18: The line at infinity ◮ The line a
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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