Convolution-based image interpolation

jim wrote:
> "Ron N." wrote:
>
> > A spline won't do what you wanted in your example either.  A low
> > enough degree spline would be the same as linear interpolation, and
> > a high degree spline would show "ripples" before and/or after any
> > step function.
>
> Nope that's bad information. Some will and some won't. It depends on the
> type of spline interpolation.

What type of, say, 4th degree spline will preserve the original
points (the OP's request, not mine), be continuous, and not
show any overshoot or undershoot when interpolating samples
from a unit step function?

What kind of 1st degree spline would preserve all the original
points and be different from linear interpolation?

Or are there non-linear types of splines which are still
commonly attributed with a "degree"?


IMHO. YMMV.
-- 
rhn A.T nicholson d.0.t C-o-M

"Ron N." wrote:

> What type of, say, 4th degree spline will preserve the original
> points (the OP's request, not mine), be continuous, and not
> show any overshoot or undershoot when interpolating samples
> from a unit step function?

It's the same deal as before if you require the spline to pass thru the
original points you have the overshoot and undershoot or ripple problem.
A more relaxed implementation will be able to transition a step
smoothly. There are even spline algorithms that include a tension factor
so that you can control how tightly or loosely it follows the original
points.
	Degree 4 and other even degree splines are often not used to avoid the
half sample shift they introduce.


> 
> What kind of 1st degree spline would preserve all the original
> points and be different from linear interpolation?
> 

To some people "linear" combined with "spline" is an oxymoron  given the
origins of the meaning of the word spline. But I wasn't disagreeing with
those statements. If you have a generalized algorithm for spline
construction that allows you to plug in any degree you want, then if you
plug-in 1 for degree you usually get linear interpolation, that's true.

-jim

> Or are there non-linear types of splines which are still
> commonly attributed with a "degree"?
> 
> IMHO. YMMV.
> --
> rhn A.T nicholson d.0.t C-o-M

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chris_bore@yahoo.co.uk wrote:
> Michel Rouzic wrote:
> > who would want to interpolate images by FIRs anyways?
>
> It is quite common to interpolate images by FIRs. This is especially so
> in digital video where natural images are concerned, as it may give the
> most pleasing visual effect.

Image data is typically not bandlimited in the formal sense.
More likely rectangularly windowed samples from local blurring
due to some combination of the diffraction Airy disk of
the aperature, the MTF distribution due to lens aberations,
the Bayer pre-CCD scattering blur, and such.

I think this means that their will be aliased high frequency
contamination of the samples.  Are there any characterisitics
of this image noise which might help in the design of an
interpolating filter that would be less affected by the high
frequency aliasing already contained in the image samples,
and produce a "more pleasing visual effect"?


IMHO. YMMV.
-- 
rhn A.T nicholson d.0.t C-o-M

Ron N. wrote:
> jim wrote:
> > "Ron N." wrote:
> >
> > > A spline won't do what you wanted in your example either.  A low
> > > enough degree spline would be the same as linear interpolation, and
> > > a high degree spline would show "ripples" before and/or after any
> > > step function.
> >
> > Nope that's bad information. Some will and some won't. It depends on the
> > type of spline interpolation.
>
> What type of, say, 4th degree spline will preserve the original
> points (the OP's request, not mine)

I never requested it, I just stated that the way I imagined it it would
preserve the original points, now that we're talking about that point,
I remember that there are curve interpolations that don't go through
the original points, and thus would fix both the scalloping on a ramp
and the ripple, I guess.

> be continuous, and not
> show any overshoot or undershoot when interpolating samples
> from a unit step function?
>
> What kind of 1st degree spline would preserve all the original
> points and be different from linear interpolation?
>
> Or are there non-linear types of splines which are still
> commonly attributed with a "degree"?
> 
> 
> IMHO. YMMV.
> -- 
> rhn A.T nicholson d.0.t C-o-M