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Orthogonal distance fitting


To overcome the problems with the algebraic distances, it is natural to replace them by the orthogonal distances which are invariant to transformations in Euclidean space and which do not exhibit the high curvature bias.

The orthogonal distance tex2html_wrap_inline2815 between a point tex2html_wrap_inline2845 and a conic Q(x,y) is the Euclidean distance between tex2html_wrap_inline2849 and the point tex2html_wrap_inline2851 in the conic whose tangent is orthogonal to the line joining tex2html_wrap_inline2849 and tex2html_wrap_inline2855 (see fig:orth-dis).

Figure 2: Orthogonal distance of a point to a conic

Given n points tex2html_wrap_inline2849 ( tex2html_wrap_inline2549 ), the orthogonal distance fitting is to estimate the conic Q by minimizing the following function


However, as the expression of tex2html_wrap_inline2815 is very complicated (see below), an iterative optimization procedure must be carried out. Many techniques are readily available, including Gauss-Newton algorithm, Steepest Gradient Descent, Levenberg-Marquardt procedure, and simplex method. A software ODRPACK (written in Fortran) for weighted orthogonal distance regression is domain public and is available from NETLIB ( Initial guess of the conic parameters must be supplied, which can be obtained using the techniques described in the last section.

Let us now proceed to compute the orthogonal distance tex2html_wrap_inline2815 . The subscript i will be omitted for clarity. Refer again to fig:orth-dis. The conic is assumed to be described by


Point tex2html_wrap_inline2871 must satisfy the following two equations:


Equation (9) merely says the point tex2html_wrap_inline2855 is on the conic, while (10) says that the tangent at tex2html_wrap_inline2855 is orthogonal to the vector tex2html_wrap_inline2877 .

Let tex2html_wrap_inline2879 and tex2html_wrap_inline2881 . From (9),


where tex2html_wrap_inline2883 . From (10),


Substituting the value of tex2html_wrap_inline2885 (11) in the above equation leads to the following equation:




Squaring the above equation, we have


Rearranging the terms, we obtain an equation of degree four in tex2html_wrap_inline2887 :




The two or four real roots of (13) can be found in closed form. For one solution tex2html_wrap_inline2887 , we can obtain tex2html_wrap_inline2891 from (12), i.e.:


Thus, tex2html_wrap_inline2885 is computed from (11). Eventually comes the orthogonal distance d, which is given by


Note that we possibly have four solutions. Only the one which gives the smallest distance is the one we are seeking for.

next up previous contents
Next: Gradient Weighted Least-Squares Fitting Up: Least-Squares Fitting Based on Previous: Why are algebraic distances

Zhengyou Zhang
Thu Feb 8 11:42:20 MET 1996