Prev: Need help in image pattern recognition
Next: cannot restore segment prot after reloc
From: Mary on 14 Jun 2005 22:25
Hi,
Could anyone tell me whether there is a ready made function in Matlab
for natural slpine? Do we have to write a program?
From: John D'Errico on 15 Jun 2005 06:44
In article <ef0aaea.-1(a)webx.raydaftYaTP>,
Mary <shi_xinhui2000(a)yahoo.com.cn> wrote:

> Could anyone tell me whether there is a ready made function in Matlab
> for natural slpine? Do we have to write a program?

The logical (one might even say natural) place to look
is in the function spline. However spline does not offer
a natural spline as an option. Its spline uses what are
called "not-a-knot" end conditions, as opposed to the
natural end conditions.

The natural end conditions for a spline are derived from
the origins of splines. If one solves for the deflection
of a thin flexible beam that is forced to pass through a
set of points in the plane, then (for reasonable deflections)
the shape of that beam will be well modeled by a "natural"
cubic spline. The name itself derives from the solution
of the beam system via the calculus of variations, where
for a beam, the appropriate choice of end conditions are
the "natural" ones. I.e., the second derivative of the
beam deflection at its ends is zero. Said differently,
if there is no force at the ends of a beam to bend it,
then the beam will have no curvature at the ends.

The not-a-knot end conditions are different. They force
the spline to be THREE times continuous at the 2nd and
next to last knot points, as opposed to the normally
twice continuous conditions at most knots of the spline.

Is there a difference between these two sets of end
conditions? Its often a subtle one, and there is rarely
any a priori reason why one would truly prefer the
natural end conditions. Just because the natural end
conditions are appropriate for a thin beam, why should
they be the proper choice for a spline model of some
arbitrary system?

So, if you really want a C2 interpolating spline for
your data, then just use spline, or interp1, which
offers spline as one of its options.

Having said all that...

Often one finds that the basic C2 spline, regardless
of end conditions, is not really appropriate for their
use. Splines work well as models of thin flexible beams.
Shipwrights, carpenters, and draftsmen alike have used
the idea for centuries to draw in a smooth curve. Splines
don't always work perfectly for modeling that arbitrary
set of physical data. For example, one often sees ringing
in a spline model when it is forced to go through a
transition from a region of high slope into a region
where it is constant. Thus even if the data itself is
monotone, the spline need not be. While this may be the
behavior one might expect in a flexible beam, it is not
the behavior one expects from a monotone system.

For this reason, one often is better off with a subtle
cousin of the spline family - pchip. It might be called
a shape preserving interpolant. Pchip will interpolate
your data without introducing any extraneous extrema
into the curve. I like to call it a locally monotone
interpolant. It will produce a monotone result on
monotone data. If your data has noise (bumps) in it,
then pchip will reproduce the noise, as will any other
interpolant. You can find pchip in interp1, or as the
function pchip. Pchip does have one cost of its use -
it is only a C1 interpolant. This means the second
derivative of the pchip function will not be continuous
as a traditional spline is. Is this bad? Rarely so.
In fact, in most cases you will hardly even see any
difference between a C1 and a C2 interpolant. In the
rare event where this is a problem, there are also
C2 shape preserving interpolants (of many different
varieties) available in the literature.

Enough already! I see you yawning off to sleep as I've
rambled on about the subtleties of interpolation. The
feigned snores were a bit too much though. ;-)

HTH,
John D'Errico


--
The best material model of a cat is another, or
preferably the same, cat.
A. Rosenblueth, Philosophy of Science, 1945
 | 
Pages: 1
Prev: Need help in image pattern recognition
Next: cannot restore segment prot after reloc