bundle block adjustment with 3d natural cubic splines

function such as the Bézier curve which is advantageous because of its fast functionevaluations. Adaptive Gaussian integrations employ recursive method which startsfrom few samples and add more samples as necessary. Adaptive Gaussian integrationalso uses a table which maps curves or spline parameter values to the arc-lengthvalues.Nasri et al.[48] proposed the arc-length approximation method of circles and piecewisecircular splines generated by control polygons or points using a recursive subdivisionalgorithm. While B-splines have various tangents over the curve depending onarc-length parameterization, circular splines have constant tangents, which tangentvectors are useful in the arc-length computation.The simple approach for the integral approximation is using equal space valuesof x, independent variable, from a to b. If the range from a to b is divided into nintervals, the integral size iss = b − an(3.22)Integration by the trapezium rule calculates the approximation of integral values ass2 (y 0 + 2y 1 + 2y 2 + · · · + 2y n−1 + y n ) (3.23)where y n = f(a + ns). While the trapezium rule uses intervals by its approximationof degree 1, Simpson’s rule uses intervals by its approximation of degree 2 to providea more accurate integral approximation. Simpson’s one-third rule iss3 (y 0 + 4y 1 + 2y 2 + 4y 3 + · · · + 4y n−3 + 2y n−2 + 4y n−1 + y n ) (3.24)and the integral approximation with two intervals iss3 (y 0 + 4y 1 + y 2 ) (3.25)48