bundle block adjustment with 3d natural cubic splines

about correspondences between individual points in the 3D object space and theirprojected features in the 2D image space is not required in extended collinearityequations with 3D natural splines. One point on a cubic spline has 19 parameters(X c , Y c , Z c , ω, ϕ, κ, a 0 , a 1 , a 2 , a 3 , b 0 , b 1 , b 2 , b 3 , c 0 , c 1 , c 2 , c 3 , t). The differentials of (3.14)is (3.15).dx p = − f fudu +w w dw, 2dy p = − f wfvdv + dw (3.15)w2 with the differentials of du, dv, dw (3.16)⎡⎢⎣dudvdw⎤⎥⎦ = ∂RT∂ω+ ∂RT∂κ+ R T ⎡⎢⎣⎡+ R T ⎢⎣⎡+ R T ⎢⎣1000t0⎡⎢⎣⎡⎢⎣X i (t) − X CY i (t) − Y CZ i (t) − Z CX i (t) − X CY i (t) − Y CZ i (t) − Z C⎤⎡⎥⎦ da 0 + R T ⎢⎣⎤ ⎡⎥⎦ db 1 + R T ⎢⎣⎤⎡0⎥0 ⎦ dc 2 + R T ⎢⎣t 2⎤⎥⎦ dω + ∂RT∂ϕ⎤ ⎡⎥⎦ dκ − R T ⎢⎣t000t 2 0100⎡⎢⎣⎤⎡⎥⎦ da 1 + R T ⎢⎣⎤ ⎡⎥⎦ db 2 + R T ⎢⎣⎤⎡0⎥0 ⎦ dc 3 + R T ⎢⎣t 3X i (t) − X CY i (t) − Y CZ i (t) − Z C⎤⎡⎥⎦ dX c − R T ⎢⎣t 2 000t 3 0⎤⎥⎦ dϕ010⎤⎡⎥⎦ da 2 + R T ⎢⎣⎤ ⎡⎥⎦ db 3 + R T ⎢⎣⎤⎡⎥⎦ dY c − R T ⎢⎣t 3 00001a 1 + 2a 2 t + 3a 3 t 2b 1 + 2b 2 t + 3b 3 t 2c 1 + 2c 2 t + 3c 3 t 2001⎤⎤⎡⎥⎦ da 3 + R T ⎢⎣⎤⎡⎥⎦ dc 0 + R T ⎢⎣⎤⎥⎦ dt⎥⎦ dZ c01000t⎤⎥⎦ db 0⎤(3.16)⎥⎦ dc 1Substituting du, dv, dw in (3.15) by the expressions found in (3.16) leads to42