A trust region method for implicit orthogonal distance regression

Let a family of curves or surfaces be given in implicit form via the model equationf (x,Β)=0, wherex ε ℝd andΒ ε ℝm 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 {zi}in=1 ⊂ ℝdin a least squares sense. Specifically, we seekΒ* and {xi*}in=1 such thatf (xi*,Β*) = 0,i=1,...,n, and ∑i=1n‖zi−xi*‖22 is minimal. The termorthogonal distance regression is used to describe such constrained nonlinear least squares problems.