Projective geometry for Computer Vision

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Euclidean and Affine geometries ◮ Given a coordinate system, n-dimensional real affine space is the set of all points parameterized by x = (x 1 , . . . , x n ) t ∈ R n . ◮ An affine transformation is expressed as x ′ = Ax + b where A is a n × n (usually) non-singular matrix and b is a n × 1 vector representing a translation. ◮ By SVD A = UΣV T = (UV T )(VΣV T ) = R(θ)R(−φ)ΣR(φ) where where Σ = [ ] λ1 0 0 λ 2 Subhashis Banerjee Projective geometry for Computer Vision
Euclidean and Affine geometries ◮ In the special case of when A is a rotation (i.e., AA t = A t A = I, then the transformation is Euclidean. ◮ An affine transformation preserves parallelism and ratios of lengths along parallel directions. ◮ An Euclidean transformation, in addition to the above, also preserves lengths and angles. ◮ Since an affine (or Euclidean) transformation preserves parallelism it cannot be used to describe a pinhole projection. 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: In terms of projective coordinates
Page 11 and 12: Spherical geometry ◮ Lines in S 2
Page 13 and 14: Projective geometry The space P 2 c
Page 15 and 16: Canonical injection of R n into P n
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

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