Re Mexican hat

Re: Mexican hat

On Sep 25, 3:36=A0pm, Anicet Franken 
wrote:
> Hello,
>
> who knows the following kernel : mexican hat?

Normally this is the negative of the Lapacian of a Gaussian. As
always, try wikipedia yourself before you bother people here:

http://en.wikipedia.org/wiki/Mexican_hat_wavelet

illywhacker;

Re: Mexican hat

On Sep 25, 10:44=A0am, illywhacker  wrote:
> On Sep 25, 3:36=A0pm, Anicet Franken 
> wrote:
>
> > Hello,
>
> > who knows the following kernel : mexican hat?
>
> Normally this is the negative of the Lapacian of a Gaussian. As
> always, try wikipedia yourself before you bother people here:
>
> http://en.wikipedia.org/wiki/Mexican_hat_wavelet
>
> illywhacker;

Anicet,

As a kernel, the Mexican Hat or Sombrero function is the (2D) Fourier
transform of the top-hat function. The form of the Mexican Hat is
2*J1(ar)/(ar) (where J1 is the Bessel function of the first kind and
first order). The top-hat has the form circ(r) =3D 1 for r < 1, zero
otherwise.  BTW, there  is also a Chinese hat function which is
obtained from the auto-correlation of the top-hat function.

HTH

Jomar

Re: Mexican hat

On Sep 25, 10:44=A0am, illywhacker  wrote:
> On Sep 25, 3:36=A0pm, Anicet Franken 
> wrote:
>
> > Hello,
>
> > who knows the following kernel : mexican hat?
>
> Normally this is the negative of the Lapacian of a Gaussian. As
> always, try wikipedia yourself before you bother people here:
>
> http://en.wikipedia.org/wiki/Mexican_hat_wavelet
>
> illywhacker;

illywhacker:
I didn't know that - learn something new every day.  I don't use
wavelets but have a background in optics and imaging and there the
Mexican Hat "normally" refers to the Bessel function, like Jomar
said.  Just as often, or maybe more or less, it's also called the
"Sombrero function."  You get this from the diffraction pattern of a
circular aperture - analogous to the sinc function which you get from
diffraction of a rectangular aperture.  You can convolve your image
with these to simulate blurring by imaging through apertures shaped
like that.
Regards,
ImageAnalyst

Re: Mexican hat

On Sep 26, 1:23=A0am, ImageAnalyst  wrote:
> On Sep 25, 10:44=A0am, illywhacker  wrote:
>
> > On Sep 25, 3:36=A0pm, Anicet Franken 
> > wrote:
>
> > > Hello,
>
> > > who knows the following kernel : mexican hat?
>
> > Normally this is the negative of the Lapacian of a Gaussian. As
> > always, try wikipedia yourself before you bother people here:
>
> >http://en.wikipedia.org/wiki/Mexican_hat_wavelet
>
> > illywhacker;
>
> illywhacker:
> I didn't know that - learn something new every day. =A0I don't use
> wavelets but have a background in optics and imaging and there the
> Mexican Hat "normally" refers to the Bessel function, like Jomar
> said. =A0Just as often, or maybe more or less, it's also called the
> "Sombrero function." =A0You get this from the diffraction pattern of a
> circular aperture - analogous to the sinc function which you get from
> diffraction of a rectangular aperture. =A0You can convolve your image
> with these to simulate blurring by imaging through apertures shaped
> like that.

Yes, thanks: I see that. I guess the truth is that it could be used to
refer to any function of radius with a positive centre and negative
surround, tending to zero at infinity. The Bessel function has
multiple zeros, which is unlike any sombrero I have ever seen, but
then sombreros do not extend to infinity either. Calling it a kernel
or not is irrelevant though. The Laplacian of a Gaussian is also a
kernel.

illywhacker;