bundle block adjustment with 3d natural cubic splines

to avoid numerical problems as⎡⎢⎣⎤n 1⎥n 2n 3⎦ =⎡⎢⎣sin θ cos ϕsin θ sin ϕcos ϕn 1 (X − X o ) + n 2 (Y − Y o ) + n 3 (Z − Z o ) = 0n 1 X + n 2 Y + n 3 Z = D⎤⎥⎦(2.17)with θ angle from XY plane, ϕ angle around Z axis, n unit vector of plane normal andD the distance between the plane and the origin. Five relative orientation parametersand three planar parameters were obtained by using the homography mapping systemwhich searched the conjugate point in an image corresponding to a point in the otherimage.Lin[40] proposed the method of the autonomous recovery of exterior orientationparameters by the extension of the traditional point-based Modified Iterated HoughTransform (MIHT) to the 3D free-form linear feature based MIHT. Straight polylineswere generalized for matching primitives in the pose estimation since the mathematicalrepresentation of straight lines are much clearer than the algebraic expression ofconic sections and splines.Gruen and Akca[21] matched 3D curves whose forms were defined by a cubicspline using the least squares matching. Subpixels were localized by the least squaresmatching and the quality of the localization was decided by the geometry of imagepatches. Two free-form lines were defined as (2.18).f(u) = [x(u) y(u) z(u)] T = a 0 + a 1 u + a 2 u 2 + a 3 u 3g(u) = [x ′ (u) y ′ (u) z ′ (u)] T = b 0 + b 1 u + b 2 u 2 + b 3 u 3 (2.18)where u ∈ [0, 1], a 0 , a 1 , a 2 , a 3 , b 0 , b 1 , b 2 , b 3 variables and f(u), g(u) ∈ R 3Taylor expansion was employed to adopt Gauss-Markov model as (2.19).22