convolution frequency relation

Re: convolution-frequency relation

On Oct 17, 5:50=A0pm, mark_h...@sbcglobal.net wrote:
> 17-OCT-2008
>
> Hi all --
>
> I'm using two Gaussian kernels (5x5) and (15x15) to convolve
> duplicates of an image, which I'm then differencing (small blur minus
> big blur); the kernel coefficients are as follows:
>
> 5x5
>
> 1 =A01 =A02 =A01 =A01
> 1 =A02 =A04 =A02 =A01
> 2 =A04 =A08 =A04 =A02
> 1 =A02 =A04 =A02 =A01
> 1 =A01 =A02 =A01 =A01
>
> 15x15
>
> 2 2 =A03 =A04 =A05 =A05 =A06 =A06 =A06 =A05 =A05 =A04 =A03 2 2
> 2 3 =A04 =A05 =A07 =A07 =A08 =A08 =A08 =A07 =A07 =A05 =A04 3 2
> 3 4 =A06 =A07 =A09 10 10 11 10 10 =A09 =A07 =A06 4 3
> 4 5 =A07 =A09 10 12 13 13 13 12 10 =A09 =A07 5 4
> 5 7 =A09 11 13 14 15 16 15 14 13 11 =A09 7 5
> 5 7 10 12 14 16 17 18 17 16 14 12 10 7 5
> 6 8 10 13 15 17 19 19 19 17 15 13 10 8 6
> 6 8 11 13 16 18 19 20 19 18 16 13 11 8 6
> 6 8 10 13 15 17 19 19 19 17 15 13 10 8 6
> 5 7 10 12 14 16 17 18 17 16 14 12 10 7 5
> 5 7 =A09 11 13 14 15 16 15 14 13 11 =A09 7 5
> 4 5 =A07 =A09 10 12 13 13 13 12 10 =A09 =A07 5 4
> 3 4 =A06 =A07 =A09 10 10 11 10 10 =A09 =A07 =A06 4 3
> 2 3 =A04 =A05 =A07 =A07 =A08 =A08 =A08 =A07 =A07 =A05 =A04 3 2
> 2 2 =A03 =A04 =A05 =A05 =A06 =A06 =A06 =A05 =A05 =A04 =A03 2 2
>
> These are the kernels that come canned with NIH-Image. =A0I have two
> questions:
>
> 1. =A0I don't know how these particular kernels are computed in order to
> result in such relatively small integer coefficients. =A0What is a
> simple method to obtain identical or similar results for arbitrary
> size matrices?

Inaccurately if they are meant to be a 2D Gaussian. The centre line of
the first 5x5 matrix should look more like
1 5 8 5 1, 2 8 13 8 2 or even 1 4 6 4 1 and the extreme corner
elements should round to 0. BTW For the larger one I think a peak
height of 18 gives smaller residual error from rounding ona 15x15
grid.
The lines of Pascals triangle are not a bad integer coefficient
approximation to a Gaussian.
Or

int( 0.5+A*exp( -B*(x^2 + y^2))

For suitable choices of A and B.

> 2. Is there a reasonably straightforward means of quantifying the
> spread in spatial frequencies defined by these (or any pair of)
> Gaussians?
>
> Many thanks in advance for any help.

Their Fourier transform shows you how they attentaute various
frequencies.

Regards,
Martin Brown