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Bicubic Interpolation

 
 
durgesh29@gmail.com
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      03-03-2006
Hi all,
can we do Bicubic Interpolation with5x5 pixels of image
instead of 4x4

 
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Paul Heslop
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      03-03-2006
http://www.velocityreviews.com/forums/(E-Mail Removed) wrote:
>
> Hi all,
> can we do Bicubic Interpolation with5x5 pixels of image
> instead of 4x4


I can't even do long division! (or short for that matter) :O)

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Bart van der Wolf
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      03-03-2006

<(E-Mail Removed)> wrote in message
news:(E-Mail Removed) oups.com...
> Hi all,
> can we do Bicubic Interpolation with5x5 pixels of
> image instead of 4x4


http://astronomy.swin.edu.au/~pbourk...ion/index.html

Bart

 
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Dave Martindale
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      03-04-2006
(E-Mail Removed) writes:
>Hi all,
> can we do Bicubic Interpolation with5x5 pixels of image
>instead of 4x4


No, yes, and no, depending on how deeply you look.

If you used 5x5 pixels, you'd have enough data to use 4th-degree
polynomials instead of 3rd degree, so it would be "biquartic"
interpolation instead of bicubic. Thus your terminology is not quite
right.

You *could* build an interpolation method that used 5 points and 5
4th-degree polynomial segments in each direction. You just wouldn't
want to.

Polynomial interpolation is always done with odd-degree polynomials and
an even number of points, because that allows building an interpolating
function that goes through the points (0, 1), (-1, 0), (1, 0), (-2, 0),
(2, 0), and so on. In other words the function is 1 at X=0 and zero at
all integer offsets from that. This, in turn, has the nice property
that interpolating an image to exactly the same size without any shift
gives the original image back.

With an even-degree polynomial and an odd number of input points, the
"joints" in the polynomial are at X=0.5, -0.5, 1.5, -1.5, etc. You
can't create an interpolating function that goes through (0, 1) and has
zero crossings at other integer offsets (while keeping the integral of
the function equal to 1) so using even-degree polynomial interpolation
never gives the original image even at the same size. There's always
some blurring.

So in practice, interpolation methods jump from linear interpolation
(which is the trivial case of polynomial interpolation, using a 1st
degree polynomial and 2 points) to cubic using 3rd-degree polynomial and
4 points, to 5th, 7th, etc degrees using 6, 8, ... points.

Dave
 
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Matt Ion
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      03-04-2006
Uhhh, you hurt my brain!

Dave Martindale wrote:
> (E-Mail Removed) writes:
>
>>Hi all,
>> can we do Bicubic Interpolation with5x5 pixels of image
>>instead of 4x4

>
>
> No, yes, and no, depending on how deeply you look.
>
> If you used 5x5 pixels, you'd have enough data to use 4th-degree
> polynomials instead of 3rd degree, so it would be "biquartic"
> interpolation instead of bicubic. Thus your terminology is not quite
> right.
>
> You *could* build an interpolation method that used 5 points and 5
> 4th-degree polynomial segments in each direction. You just wouldn't
> want to.
>
> Polynomial interpolation is always done with odd-degree polynomials and
> an even number of points, because that allows building an interpolating
> function that goes through the points (0, 1), (-1, 0), (1, 0), (-2, 0),
> (2, 0), and so on. In other words the function is 1 at X=0 and zero at
> all integer offsets from that. This, in turn, has the nice property
> that interpolating an image to exactly the same size without any shift
> gives the original image back.
>
> With an even-degree polynomial and an odd number of input points, the
> "joints" in the polynomial are at X=0.5, -0.5, 1.5, -1.5, etc. You
> can't create an interpolating function that goes through (0, 1) and has
> zero crossings at other integer offsets (while keeping the integral of
> the function equal to 1) so using even-degree polynomial interpolation
> never gives the original image even at the same size. There's always
> some blurring.
>
> So in practice, interpolation methods jump from linear interpolation
> (which is the trivial case of polynomial interpolation, using a 1st
> degree polynomial and 2 points) to cubic using 3rd-degree polynomial and
> 4 points, to 5th, 7th, etc degrees using 6, 8, ... points.
>
> Dave



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