bv_cvxbook_extra_exercises

# M with an ane vector eld t ie f z az b the matrix

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Unformatted text preview: unit distance from it along the principal axis. The point x′ in the ﬁgure is the intersection of the image plane and the line through the camera center and x, and is given by 1 (x − xc ). x′ = xc + T c (x − xc ) Using the deﬁnition of xc we can write f (x) as f (x) = 1 A(x − xc ) = A(x′ − xc ) = Ax′ + b. c T (x − xc ) This shows that the mapping f (x) can be interpreted as a projection of x on the image plane to get x′ , followed by an aﬃne transformation of x′ . We can interpret f (x) as the point x′ expressed in some two-dimensional coordinate system attached to the image plane. In this exercise we consider the problem of determining the position of a point x ∈ R3 from its image in N cameras. Each of the cameras is characterized by a known linear-fractional mapping fk and camera matrix Pk : fk ( x ) = cT x k 1 (Ak x + bk ), + dk Pk = 66 Ak bk c T dk k , k = 1, . . . , N. The image of the point x in camera k is denoted y (k) ∈ R2 . Due to camera imperfections and calibration errors, we do not expect the equations fk (x) = y (...
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## This note was uploaded on 09/10/2013 for the course C 231 taught by Professor F.borrelli during the Fall '13 term at Berkeley.

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