bundle block adjustment with 3d natural cubic splines

Fourier transform has strength in signal and data compression, filtering and smoothing.However, Fourier transform has little strength in the representation of a curveaccurately. More terms to represent a free-form line exactly increase the time andthe computation complexity.2.3.3 Implicit polynomialsImplicit polynomials obtained by the computation of the least squares problemfor the coefficients of the polynomial have the mathematical form:X(t) = a 0 + a 1 t + a 2 t 2 + · · · + a n t nY (t) = b 0 + b 1 t + b 2 t 2 + · · · + b n t n (2.26)Z(t) = c 0 + c 1 t + c 2 t 2 + · · · + c n t nwhere a 0 , a 1 , · · · a n , b 0 , · · · b n , c 0 , · · · c n ∈ R 3 and the value of n is the degree of thepolynomial R n .The coefficients a 0 , b 0 , c 0 characterize a point in the object spaceR 3 and remaining parameters represent the direction.The number of parametersrequired to represent the n degree polynomial R n is 3(n + 1). Implicit polynomial ina degree of 3 is analogous to a natural cubic spline.A free-form line is formulated in a simple form geometrically independent accordingto a polynomial fit. However, a quality of the results is implemented in case ofhigh degrees. A prediction of the outcome of fitting with implicit polynomials is difficultsince coefficients are not geometrically correlated. The best solution is optimizedby the iterative try and error method. In the worst case, the implementation seemsto never end with a tight threshold. The number of trials to fit successful coefficientsdepends on the restriction of the search space. Implicit polynomials are originatedthrough constraints and various minimization criteria.32