bundle block adjustment with 3d natural cubic splines

=√∫ √√√ ()ti+1−f u′ (t)w(t) − u(t)w ′ 2 ()(t)+ −f v′ (t)w(t) − v(t)w ′ 2(t)dtt i w 2 (t)w 2 (t)where f is the focal length and⎡ ⎤⎡u(t)⎢ ⎥⎣ v(t) ⎦ = R T ⎢(ω, ϕ, κ) ⎣w(t)⎡u ′ ⎤⎡(t)⎢⎣ v ′ ⎥(t) ⎦ = R T ⎢(ω, ϕ, κ) ⎣w ′ (t)a 0 + a 1 t + a 2 t 2 + a 3 t 3 − X Cb 0 + b 1 t + b 2 t 2 + b 3 t 3 − Y Cc 0 + c 1 t + c 2 t 2 + c 3 t 3 − Z Ca 1 + 2a 2 t + 3a 3 t 2b 1 + 2b 2 t + 3b 3 t 2c 1 + 2c 2 t + 3c 3 t 2⎤⎥⎦⎤⎥⎦Since neither the problem of the arc-length parameterization of splines has an analyticalsolution, several numerical approximations of reparameterization techniquesfor splines or other curve representations have been developed. While most curvesare not parameterized for the arc-length, the arc-length of a B-spline can be reparameterizedby adjusting the knots of a B-spline. Wang et al.[74] approximated theparameterized arc-length of spline curves by generating a new curve which accuratelyapproximated the original spline curve to reduce the computation complexity of thearc-length parameterization. They showed that the approximation of the arc-lengthparameterization works well in a variety of real time applications including a drivingsimulation.Guenter and Parent[22] employed the hierarchical approach algorithm of the linearsearch of the arc-length subdivision table for parameterized curves to reduce the arclengthcomputation time. A table of the correspondence between parameter t and thearc-length can be established to accelerate the arc-length computation. After dividingthe parameter range into intervals, the arc-length of each interval is computed formapping parameters to the arc-length. A table is the reference of the arc-length forvarious intervals. Another method of the arc-length approximation is using explicit47