A trust region method for implicit orthogonal distance regression

Abstract

Let a family of curves or surfaces be given in implicit form via the model equationf (x,Β)=0, wherex ε ℝ^d andΒ ε ℝ^m is a parameter vector. We present a trust region algorithm for solving the problem:find a parameter vector Β ^* such that the contour f(x,Β ^*)=0is a best fit to given data {z_i} ⁿ_i =1 ⊂ ℝ^d in a least squares sense. Specifically, we seekΒ ^* and {x ^* _i } ⁿ _i =1 such thatf (x ^*_i ,Β ^*) = 0,i=1,...,n, and ∑ ⁿ _i=1 ‖z _i −x ^* _i ‖ ²₂ is minimal. The termorthogonal distance regression is used to describe such constrained nonlinear least squares problems.