A linear distance metric commonly used in computer vision applications because of its simple form and standard matrix based least mean square estimation operations.
If a curve or surface is defined implicitly by (e.g. for a hyperplane) the algebraic distance of a point to the surface is simply .
If can be factored as
for a hyperplane and
for a circle)
then the least square estimation of the parameters is as follows.
(This factoring is straightforward if the components of are polynomials
in the components of .)
. Then we wish to estimate the
The algebraic distance is the same as the Euclidean distance for hyperplanes,
but can have significant error for curved surfaces. As a consequence, Taubin
produced a variant to the algebraic distance that may give parameter
estimates with lower Euclidean distance. This approach is a first order
approximation to the Euclidean distance and minimizes:
G. Taubin, F. Cukierman, S. Sullivan, J. Ponce, D.J. Kriegman. Parameterized families of polynomials for bounded algebraic curve and surface fitting. IEEE. Trans. Pat. Anal. and Mach. Intel., March 1994 (Vol. 16, No. 3) p.287-303.