bundle block adjustment with 3d natural cubic splines

Algebraic curves have implicit form having a polynomial equation in two variables.The graphs of a polynomial equation are algebraic curves and conic sections arealgebraic curves of degree 2 called a higher plane curve.2.3.1 SplineThe choice of the right feature model is important to develop the feature basedapproach since the ambiguous representation of features leads to an unstable adjustment.A spline is piecewise polynomial functions in n of vector graphics. A spline iswidely used for data fitting in the computer science because of the simplicity of thecurve reconstruction. Complex figures are approximated well through curve fittingand a spline has strength in the accuracy evaluation, data interpolation and curvesmoothing. One of important properties of a spline is that a spline can easily bemorph. A spline represents a 2D or 3D continuous line with a sequence of pixels andsegmentation. The relationship between pixels and lines is applied to a bundle blockadjustment or a functional representation. A spline of degree 0 is the simplest splineand a linear spline of degree 1, a quadratic spline of degree 2 and a common naturalcubic spline of degree 3 with continuity C 2 . The geometrical meaning of continuityC 2 is that the first and second derivatives are proportional at joint points and theparametric importance of continuity C 2 is that the first and second derivatives areequal at connected points.A spline is defined as piecewise parametric form.F : [a, b] → Ra = x 0 < x 1 < · · · < x n−1 = bF (x) = G 0 (x), x ∈ [x 0 , x 1 ] (2.20)26