# Looking for math background to resampling

Started by September 19, 2003
```Im looking for some mathemathical background for resampling (of images in my case)

What I'm after are 3 things specifically: (#2 most important)

1) How can a wider kernel result in a "better" sometimes "sharper" result.
that feals very unintuitive to me. for example sinc 16*16
2) In the all litterature I've seen (Pratt and a lot of articles on the web)
nowhere is it mentioned that the sum off all contributing weights should be 1.0
otherwise a constant function would not be resampled to the same value.
Specifically: where can I find a reference that discusses different resampling
functions from this aspect? Or am I assuming incorrectly.
3) (A bit compicated) how can a positive function like bicubic result in ripples?
I.E : If you use bicubic resampling for resizing of black text in a gray
surrounding you get lighter "halos" around the text.
That is:the presence of black pixels makes the resulting pixel lighter.
This can only happen if the weighting function is negative at some point
,which bicubc isnt!
I must have made some faulty supposition,which?
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pslant wrote:
>
> Im looking for some mathemathical background for resampling (of images in my case)
>
> What I'm after are 3 things specifically: (#2 most important)
>
> 1) How can a wider kernel result in a "better" sometimes "sharper" result.
>   that feals very unintuitive to me. for example sinc 16*16

If you apply the same kernel twice it is equivalent to applying a wider
kernel once - wouldn't you expect that applying a filter twice would have
a greatewr effect.

> 2) In the all litterature I've seen (Pratt and a lot of articles on the web)
>   nowhere is it mentioned that the sum off all contributing weights should be 1.0
>   otherwise a constant function would not be resampled to the same value.
>   Specifically: where can I find a reference that discusses different resampling
>   functions from this aspect? Or am I assuming incorrectly.

If you expect the overall image brightness to remain the same then the
filter should be normalized for DC gain (all weights sum to one). But if
you use negative coefficients in your filter then it is impossible to have
the gain at all frequencies less than or less than 1 without introducing
some sort of non-linear function. So one possible compromise is to allow
DC not equal 1.

> 3) (A bit compicated) how can a positive function like bicubic result in ripples?
>   I.E : If you use bicubic resampling for resizing of black text in a gray
>   surrounding you get lighter "halos" around the text.
>   That is:the presence of black pixels makes the resulting pixel lighter.
>   This can only happen if the weighting function is negative at some point
>   ,which bicubc isnt!

Well if the filter is strictly positive then there shouldn't be any halos.
I.E., the step response will be monotonic.

>   I must have made some faulty supposition,which?
Yes, probably that the filter doesn't have negative weights.

-jim

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