bundle block adjustment with 3d natural cubic splines

x p (t) = −f u(t)w(t) ,y p(t) = −f v(t)w(t)(3.40)where⎡⎢⎣u(t)v(t)w(t)⎤⎡⎥⎦ = R T ⎢(ω, ϕ, κ) ⎣X i (t) − X CY i (t) − Y CZ i (t) − Z C⎤⎥⎦Differentiating the collinearity equations with respect to parameter t leads to 2Dtangent direction in the image space.x ′ p(t) = −f u′ (t)w(t) − u(t)w ′ (t), y ′w 2 (t)p(t) = −f v′ (t)w(t) − v(t)w ′ (t)w 2 (t)(3.41)⎡⎢⎣u ′ (t)v ′ (t)w ′ (t)⎤⎡⎥⎦ = R T ⎢(ω, ϕ, κ) ⎣X i(t)′ ⎤Y i ′ ⎥(t) ⎦Z i(t)′⎡⎢⎣du ′ ⎤dv ′dw ′⎥⎦ = ∂RT∂ω⎡X ′ ⎤⎡iX ′ ⎤⎡⎢⎣ Y i′ ⎥⎦ dω + ∂RT iX ′ ⎤⎢⎣ Y ′ ⎥Z i′ i ⎦ dϕ + ∂RT i⎢⎣ Y ′ ⎥∂ϕZ i′ i ⎦ dκ∂κZ i′ ⎡ ⎤⎡ ⎤⎡12t3t 2 ⎤⎡+R T ⎢ ⎥⎣ 0 ⎦ da 1 + R T ⎢ ⎥⎣ 0 ⎦ da 2 + R T ⎢ ⎥⎣ 0 ⎦ da 3 + R T ⎢⎣000⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎡+R T ⎢ ⎥⎣ ⎦ db 2 + R T ⎢ ⎥⎣ ⎦ db 3 + R T ⎢ ⎥⎣ ⎦ dc 1 + R T ⎢⎣+R T ⎡⎢⎣02t0⎤0⎥03t 2⎡⎦ dc 3 + R T ⎢⎣03t 202a 2 + 6a 3 t2b 2 + 6b 3 t2c 2 + 6c 3 t⎤001010002t⎤⎥⎦ db 1⎤⎥⎦ dc 2⎥⎦ dt j (3.42)tan(θ t ) = y′ px ′ p= v′ (t)w(t) − v(t)w ′ (t)u ′ (t)w(t) − u(t)w ′ (t)(3.43)where tan(θ t ) the tangent in terms of the angle θ t (0 ≤ θ t ≤ 2π)54