We address the computation of the line that minimizes the sum
of the least squared distances with respect to a given set of
$n$ points in 3-space. This problem has a well known satisfying
solution by means of PCA. We offer an alternative interpretation
for this optimal line as the center of the screw motion that
minimizes the sum of squared velocities in the given points.
The numerical translation of this viewpoint is
a generalized eigenproblem, where the
total residue of the optimal line appears as the smallest
generalized eigenvalue.
Publié le : 2008-02-15
Classification:
best line fitting,
least squares,
screw center of motion,
generalized eigenvalues,
15A18,
65F15,
68U05
@article{1203692451,
author = {Penne, Rudi},
title = {A mechanical interpretation of least squares fitting in 3D},
journal = {Bull. Belg. Math. Soc. Simon Stevin},
volume = {15},
number = {1},
year = {2008},
pages = { 127-134},
language = {en},
url = {http://dml.mathdoc.fr/item/1203692451}
}
Penne, Rudi. A mechanical interpretation of least squares fitting in 3D. Bull. Belg. Math. Soc. Simon Stevin, Tome 15 (2008) no. 1, pp. 127-134. http://gdmltest.u-ga.fr/item/1203692451/