interpolation and filtering

"Andor" <andor.bariska@gmail.com> wrote in message 
news:1152784565.872351.298120@b28g2000cwb.googlegroups.com...
>
> ma wrote:
>
>> Hello,
>>
>>     I know that one way of interpolation is low pass filtering. But I 
>> think
>> any other type of interpolation (such as spile or linre ,...) can be 
>> modeled
>> as a low pass filter with different frequency response. Am I wrong? if 
>> no,
>> where can I read more? for example what is the frequency response of a
>> spilne interpolator?
>
> They are commonly known as "spline" interpolators. There is a paper
> somewhere which discusses general polynomial interpolation filters and
> their frequency responses ... <click, click> ... it moved ....  ah,
> here:
>
> http://www.biochem.oulu.fi/~oniemita/dsp/deip.pdf
>
> Regards,
> Andor
>

Thanks Andor,
    This is a very nice paper on 1D interpolation. Is there any paper 
similar to this one but about 2D interpolations?

Best regards

ma wrote:

> "Andor" wrote in message
> news:1152784565.872351.298120@b28g2000cwb.googlegroups.com...
> >
> > ma wrote:
> >
> >> Hello,
> >>
> >>     I know that one way of interpolation is low pass filtering. But I
> >> think
> >> any other type of interpolation (such as spile or linre ,...) can be
> >> modeled
> >> as a low pass filter with different frequency response. Am I wrong? if
> >> no,
> >> where can I read more? for example what is the frequency response of a
> >> spilne interpolator?
> >
> > They are commonly known as "spline" interpolators. There is a paper
> > somewhere which discusses general polynomial interpolation filters and
> > their frequency responses ... <click, click> ... it moved ....  ah,
> > here:
> >
> > http://www.biochem.oulu.fi/~oniemita/dsp/deip.pdf
> >
> > Regards,
> > Andor
> >
>
> Thanks Andor,
>     This is a very nice paper on 1D interpolation. Is there any paper
> similar to this one but about 2D interpolations?

Your wish ...

http://bigwww.epfl.ch/publications/thevenaz0002.pdf

Wilson wrote:
> [...]
> Linear interpolation is the equivalent of taking your data samples
> (spaced by 'T' seconds) and sending them through a continuous-time
> filter which has a triangular impulse response and then re-sampling the
> output at the new (interpolated) sample times.


>
> The impulse response of this continuous-time filter is:
> h(t) = 1 - abs(t)/T   for abs(t) <= T; and h(t) =  0 abs(t) >= T.
>
> At each new sampling instant, the output is a weighted combination of
> only two samples, in accordance to the the linear interpolation
> formula.  For the case where the new sample falls exactly on the time
> index of an old sample, then the new sample equals the old sample (as
> one would hope!).
>
> The frequency response of this procedure is determined by the Fourier
> Transform of h(t).
> The response has a sinc() squared response.
>
> Wilson

Hi Wilson,

Yes, you are correct. I made a mistake.

I guess I had a strange view of linear interpolation in mind. Assuming
we're interpolating by M, then I was thinking of a process that
*began* with a sequence of values at the upsampled rate, then computed
values m*M + 1, m*M + 2, ..., m*M + M - 1 from the points m*M and
(m+1)*M, where m is an arbitrary integer.

Normally the values m*M are the samples from the original sequence,
but in my gedanken they don't have to be. In case they are, you
get the linear interpolation we all know and love.

If, however, you shifted this sequence by 1 (and assuming the
in-between values were zero), you'd get zeros out with this
scheme. This is why I said it wasn't LTI, and thus cannot be
represented as a linear filter.

But I tend to think in strange ways...

--Randy