> 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
Reply by Ron N.●April 27, 20062006-04-27
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
Reply by jim●April 27, 20062006-04-27
"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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Reply by Ron N.●April 27, 20062006-04-27
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
Reply by jim●April 27, 20062006-04-27
"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.
-jim
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Reply by Michel Rouzic●April 27, 20062006-04-27
jim wrote:
> Michel Rouzic wrote:
> >
> > Jerry Avins wrote:
>
> > >
> > > I get
> > > 0 0 0 1/6 1/2 5/6 1 1 1
> ^^^
>
> This is OK as an interpolating scheme but not what was asked for. He had
> the requirement that only the new values between the old be interpolated
> (old values must remain unchanged). Yours fails to meet that
> requirement. His didn't and was done correctly. It was (by design) made
> to operate only on the new points locations since operations on the old
> points locations is a waste of time.
> But trying to preserve all the old values *and* using only positive
> coefficients makes a fairly ugly interpolating scheme.
I use only positive coefficients because that's all I'll be using. But
somebody pointed out that using a bell-shaped curve would scallop a
ramp, and because of this I have no other choice but to use spline
interpolation if i don't want to meet this problem. makes things harder
that I wanted but i've met bigger difficulties and anyways it was only
a matter of time before I had to deal with splines I guess.
Reply by Ron N.●April 27, 20062006-04-27
Michel Rouzic wrote:
> Ron N. wrote:
> > Michel Rouzic wrote:
> > > Ron N. wrote:
> > > > Michel Rouzic wrote:
> > > > > Ron N. wrote:
> > > > > > Michel Rouzic wrote:
> > > > > > > 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. It can also be very suitable where
> > > > > > > > interpolation by non-simple integer ratios is needed, as polyphase FIR
> > > > > > > > filters can be used and are often available as hardware blocks.
> > > > > > >
> > > > > > > oh, I didn't know. my main problem with FIRs in what I want to do is
> > > > > > > that it introduces ripples where I don't want any...
> > > > > >
> > > > > > Why don't you want ripples? How do you know they don't belong
> > > > > > in the image?
> > > > >
> > > > > by the way, I'm like a hungry shark only asking to be thrown pieces of
> > > > > meat at, and all you people is doing is discussing why I would want
> > > > > that. That's cool, but if you could also throw some names of suitable
> > > > > bell-shaped functions for image interpolation, I'd appreciate that.
> > > > >
> > > > > I'm sure that in the end I'll have explained you all why some method
> > > > > won't work and why I want what i claim to want, but I'm not sure to
> > > > > even be told the name of a single bell-shaped self-overlapping function
> > > > > as it's really all I'm asking for.
> > > >
> > > > Inverted Cosine. One cycle.
> > >
> > > oh, wow, ok, sounds good. i'll give it a try, thanks alot.
> >
> > Don't thank me. It won't do what you want. Try it on a ramp signal
> > ( 0, 1, 2, 3, 4, 5, 6, 7 ) . Using a small bell-shaped kernel will
> > scallop your interpolation. A windowed-Sinc FIR will give you a
> > much smoother interpolation.
>
> oh yeah! I hadn't thought about it... I only thought about the [0 0 0 1
> 1 1] cases. Well, since I don't want an FIR interpolation, widowed-sinc
> or not, the only choice left is spline interpolation it seems...
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. If you want an interpolation that has none of the
above, start inventing.
IMHO. YMMV.
--
rhn A.T nicholson d.0.t C-o-M
Reply by Jerry Avins●April 27, 20062006-04-27
Michel Rouzic wrote:
> Jerry Avins wrote:
...
>>>0 0 0 0 0 0.167 0 0.5 1 0.833 1 1 1 0.833 1
>>
>> ^ ???
>>I get
>> 0 0 0 1/6 1/2 5/6 1 1 1
>>
>>Where did that zero come from?
>
>
> how can you have 9 values when I had 8 at the beginning and that I
> interpolated to twice its length. That zero is the last of the original
> 0's
I only wrote out the interesting part. My assumptions were different
from yours, and so my expansion may not be what you want. Where you
stuff zeros to double, I stuff Xs. Then the four points that are acted
on by the kernel are 4 _original_ points.
0 X 0 X 0 X 0 X 1 X 1 X 1 ....; a unit step. All the
initial Xs will be zero and the final Xs will be one. The interesting
region is the transition. For four points, I use the row of Pascal's
triangle [1 3 3 1]/8.
v
0 0 0 0 0 X 0 X 1 X 1 X 1 The indicated X becomes 1/8.
1/8 3/8 3/8 1/8 <- Kernel
v
0 0 0 0 0 1/8 0 X 1 X 1 X 1 The indicated X becomes 1/2.
1/8 3/8 3/8 1/8
v
0 0 0 0 0 1/8 0 1/2 1 X 1 X 1 The indicated X becomes 7/8.
1/8 3/8 3/8 1/8
All subsequent Xs become 1.
If you're not doubling, the details become harder to work out. Inserting
zeros and then filtering is only one way to interpolate. With splines
and binomial smoothing, intermediate points are calculated directly.
Jerry
--
Engineering is the art of making what you want from things you can get.
�����������������������������������������������������������������������
Reply by jim●April 27, 20062006-04-27
Michel Rouzic wrote:
>
> Jerry Avins wrote:
> >
> > I get
> > 0 0 0 1/6 1/2 5/6 1 1 1
^^^
This is OK as an interpolating scheme but not what was asked for. He had
the requirement that only the new values between the old be interpolated
(old values must remain unchanged). Yours fails to meet that
requirement. His didn't and was done correctly. It was (by design) made
to operate only on the new points locations since operations on the old
points locations is a waste of time.
But trying to preserve all the old values *and* using only positive
coefficients makes a fairly ugly interpolating scheme.
-jim
> >
> > Where did that zero come from?
>
> how can you have 9 values when I had 8 at the beginning and that I
> interpolated to twice its length. That zero is the last of the original
> 0's
>
> > > Not really what I want, and I don't see how I could possibly change the
> > > 1/3 and 1/6 weightings in order to get to something that'd be more like
> > >
> > > 0 0 0 0 0 0 0 0.5 1 1 1 1 1 1 1
> > >
> > > well changing them to 0.5 and 0 would do it, but like I said, I don't
> > > want linear interpolation. Besides that, it doesn't give you much
> > > freedom, at least not the freedom of zero-stuffing by any number of
> > > samples you want and convolve
> > >
> > > I persist in asking for a bell-shaped function :-)
> >
> > The rows of Pascal's triangle approximate a Gaussian and are finite in
> > extent. Higher orders can be derived by convolving the sequence [1 1]
> > with itself any number of times. Each row is normalized by a division by
> > a power of 2.
>
> oh alright. well, sounds like just what I need, plus the algorithm to
> get to that is quite simple.
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Reply by Michel Rouzic●April 27, 20062006-04-27
Ron N. wrote:
> Michel Rouzic wrote:
> > Ron N. wrote:
> > > Michel Rouzic wrote:
> > > > Ron N. wrote:
> > > > > Michel Rouzic wrote:
> > > > > > 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. It can also be very suitable where
> > > > > > > interpolation by non-simple integer ratios is needed, as polyphase FIR
> > > > > > > filters can be used and are often available as hardware blocks.
> > > > > >
> > > > > > oh, I didn't know. my main problem with FIRs in what I want to do is
> > > > > > that it introduces ripples where I don't want any...
> > > > >
> > > > > Why don't you want ripples? How do you know they don't belong
> > > > > in the image?
> > > >
> > > > by the way, I'm like a hungry shark only asking to be thrown pieces of
> > > > meat at, and all you people is doing is discussing why I would want
> > > > that. That's cool, but if you could also throw some names of suitable
> > > > bell-shaped functions for image interpolation, I'd appreciate that.
> > > >
> > > > I'm sure that in the end I'll have explained you all why some method
> > > > won't work and why I want what i claim to want, but I'm not sure to
> > > > even be told the name of a single bell-shaped self-overlapping function
> > > > as it's really all I'm asking for.
> > >
> > > Inverted Cosine. One cycle.
> >
> > oh, wow, ok, sounds good. i'll give it a try, thanks alot.
>
> Don't thank me. It won't do what you want. Try it on a ramp signal
> ( 0, 1, 2, 3, 4, 5, 6, 7 ) . Using a small bell-shaped kernel will
> scallop your interpolation. A windowed-Sinc FIR will give you a
> much smoother interpolation.
oh yeah! I hadn't thought about it... I only thought about the [0 0 0 1
1 1] cases. Well, since I don't want an FIR interpolation, widowed-sinc
or not, the only choice left is spline interpolation it seems... thanks
alot for help!