Least Squares Fitting or Cubic Splines?

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2 total messages Started by Louis G. Stern Thu, 15 Oct 1992 16:23

#640

Author: Louis G. Stern
Date: Thu, 15 Oct 1992 16:23

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I'm posting this problem for a friend, and I don't have easy access
to usenet, so please reply by e-mail.  I will post a summary of the
replies.

The problem:  I have some reasonably good data, 2 pi periodic, which
indicates where, on the inside of a tube, there are holes and bends.
While it is perfectly obvious visually, I'd like to fit the data with
a fairly smooth curve that does *not* incorporate the holes, but *does*
capture most of the bends.  I have tried one approach:  a least-squares
fit of the data to the first 5 cos/sin frequencies, then weighting the
data inversely as their distance from the fit and trying again.  After
several iterations most of the outliers are not counted in the fit.  This
doesn't work so well when the pipe is bent, though, because the cross
section of a bent pipe incorporates a lot of high frequency information!

The other idea is this:  is there some way to pick a 'sufficient' number
of control points in the data automatically, without including the outliers
(such as might arise from pits in the pipe), and generate a cubic or
exponential spline that fits well with the rest of the data?  Has anyone
come across some combination of splines + least-squared fitting before?

I would appreciate references to articles, textbooks, et al.

L.G. 'Ted' Stern
lstern@orville.nas.nasa.gov

Re: Least Squares Fitting or Cubic Splines?

#647

Author: W.A.B.Evans
Date: Fri, 16 Oct 1992 21:49

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In article <LSTERN.92Oct15162328@orville.nas.nasa.gov> lstern@orville.nas.nasa.gov (Louis G. Stern) writes:
>	Stuff Deleted
>
> Has anyone
>come across some combination of splines + least-squared fitting before?
>
	In developing a plotter algorithm for my PLOT program, I implemented
a piecewise least squares fit of cubic polynomials to the "local" data. The
usual least squares algorithm was subject to 2 constraints i.e. the cubic
had to be continuous in value and gradient with the previous cubic - as
indeed are cubic splines. These constraints are coded by means of Lagrange
multipliers - so that to obtain the next cubic, I had to invert a 6x6 matrix
(instead of the usual 4x4). The first 4 elements of the 6-vector are the
cubic coeffs one desires and the last two are the Lagrange multipliers that
are of no further use. To start it off, I do a normal unconstrained cubic fit
to the leftmost local data - so that this technique is not as nicely symmetric
as a normal cubic spline. I also used a "Lorentzian" weighting function to
weight the "local" points the width of which I could choose. For narrow widths
the piecewise polynomial fit curve makes valiant efforts to pass very close
to every point (like cubic splines) but for larger widths it produces a
smoother curve that ignores high frequency (noise?) in the data.

	Although this technique is fairly "obvious" I have never seen any
textbook refer to it - but I dare say someone out there may know better.

				Regards,	W. Alan B. Evans
						[ wabe@ukc.ac.uk ]

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