Gaussian kernel in image processing

[PainterGuy]

I was reading the following webpage on Gaussian kernel but couldn't understand few details: https://www.imageeprocessing.com/2014/04/gaussian-filter-without-using-matlab.html . Would really appreciate it if you could guide me. Thanks in advance!

Here you can find the high-res screenshot of the webpage: https://imagizer.imageshack.com/img922/8473/QIx8L3.jpg

Question 1:
Where do the matrices X and Y come from?

Question 2:
I could see how the value "94.9296" is calculated but I cannot see how other values in the matrix are calculated?

 

[BvU]

Question 1:
Where do the matrices X and Y come from?

I had no clear idea either, but it is really easy nowadays to find some explanation
https://setosa.io/ev/image-kernels/
If still unclear:

Consult a textbook​

If still unclear:​

Ask a specific question on PF​

Still thinking on answer 2 (but I'm at least as lazy as I think you are )

----

Re first link: it's actually more an example to show why you shouldn't filter like that !
(other comment: for a club that faetures image processing, they do rather a poor job ! And the web page isn't much of a confidence building advert either !)
Re second link: not that high res !

Here you can find the high-res screenshot of the webpage:

I can't and after a few months probably no one can. No loss.

No need to think long about

Question 2:
I could see how the value "94.9296" is calculated but I cannot see how other values in the matrix are calculated?

You didn't even try to move the red box one column to the left and repeat the matrix multiplication ? Or one row down ? Or one column to the left and one row down ?

That should give you three more values. For more, you need more

$\$

 

[PainterGuy]

Thanks a lot for the reply!

I'd say my Question 1 was quite specific. Where do X and Y come from?

Suppose both X and Y have 5x5 dimensions instead of 3x3. I don't think I can get the kernel below. I've looked up around and can't see how the following kernel is derived using the Gaussian equation

.

Source: https://en.wikipedia.org/wiki/Kernel_(image_processing)

Thanks for the help, in advance!

 

[phyzguy]

The X and Y matrices are just the distances in the x and y directions from the center pixel:
If they were 5x5 matrices, X and Y would be $$X = \begin{matrix} -2&-1&0&1&2\\-2&-1&0&1&2\\-2&-1&0&1&2\\-2&-1&0&1&2\\-2&-1&0&1&2\\ \end{matrix} \,\,\,\,Y = \begin{matrix} -2&-2&-2&-2&-2\\-1&-1&-1&-1&-1\\0&0&0&0&0\\1&1&1&1&1\\2&2&2&2&2\\ \end{matrix}$$ Do you see the pattern?

 

[BvU]

I'd say my Question 1 was quite specific. Where do X and Y come from?

It was. I just didn't interpret it the way you intended.

x is the step in the x-direction
[edit]never mind ! Thanks @phyzguy !

 

[PainterGuy]

Suppose both X and Y have 5x5 dimensions instead of 3x3. I don't think I can get the kernel below. I've looked up around and can't see how the following kernel is derived using the Gaussian equation View attachment 277892.

View attachment 277891
Source: https://en.wikipedia.org/wiki/Kernel_(image_processing)

The X and Y matrices are just the distances in the x and y directions from the center pixel:
If they were 5x5 matrices, X and Y would be $$X = \begin{matrix} -2&-1&0&1&2\\-2&-1&0&1&2\\-2&-1&0&1&2\\-2&-1&0&1&2\\-2&-1&0&1&2\\ \end{matrix} \,\,\,\,Y = \begin{matrix} -2&-2&-2&-2&-2\\-1&-1&-1&-1&-1\\0&0&0&0&0\\1&1&1&1&1\\2&2&2&2&2\\ \end{matrix}$$ Do you see the pattern?

Thank you!

In my last message I made a mistake. I wanted to ask about 5x5 Gaussian kernel as shown below instead of 3x3.

I did the calculation but my values for 5x5 kernel are different. Where am I going wrong? Could you please help me?

 

[BvU]

Where am I going wrong?

Perhaps in the interpretation of the word 'approximation'?

 

[PainterGuy]

Perhaps in the interpretation of the word 'approximation'?

But my middle number is 40.742 and in the standard 5x5 kernel it's 36. Isn't the difference too much for an approximation? Thank you.

 

[BvU]

You'll be hard pressed to see any difference between the two !

 

[phyzguy]

Ideally you want the kernel to sum to 1.0, so you are only moving signal around, not gaining or losing signal. The approximation in Post #4 does that, but your calculation doesn't. If you continue the Gaussian out far enough, it will sum to 1.0, but since you truncated it, it doesn't.

 

[PainterGuy]

The X and Y matrices are just the distances in the x and y directions from the center pixel:
If they were 5x5 matrices, X and Y would be $$X = \begin{matrix} -2&-1&0&1&2\\-2&-1&0&1&2\\-2&-1&0&1&2\\-2&-1&0&1&2\\-2&-1&0&1&2\\ \end{matrix} \,\,\,\,Y = \begin{matrix} -2&-2&-2&-2&-2\\-1&-1&-1&-1&-1\\0&0&0&0&0\\1&1&1&1&1\\2&2&2&2&2\\ \end{matrix}$$ Do you see the pattern?

This is a minor point but I wanted to clarify.

Normally the values increases toward the top of y-axis and become more negative toward the bottom. In the shown case, the values become negative toward the top and increase toward the bottom, as shown below.

I don't think it affects the end result but for consistency, shouldn't the X matrix be written as shown below?

Ideally you want the kernel to sum to 1.0, so you are only moving signal around, not gaining or losing signal. The approximation in Post #4 does that, but your calculation doesn't. If you continue the Gaussian out far enough, it will sum to 1.0, but since you truncated it, it doesn't.

I'm sorry but I didn't exactly get your point. Do you mean that if the dimensions of X and Y were, say, 1000x1000 then it would sum to 1.0?

Thanks for the help, in advance!

 

[phyzguy]

Which direction is + or - is just a convention, so you can do it either way. But since the function squares x and y, it is symmetric in x and y, so it makes no difference which way you define it. On the other point, the $\frac {1}{2 \pi \sigma^2}$ normalizes the Gaussian function so that it integrates to 1. To do it properly, instead of each pixel (for example x=1, y=2) having the value $\frac {1}{2 \pi \sigma^2} \exp (-(1^2+2^2)/2 \sigma^2)$, it should have the value $\frac {1}{2 \pi \sigma^2}\int_{1.5}^{2.5} \int_{0.5}^{1.5} \exp (-(x^2+y^2)/2 \sigma^2) dx dy$. Then if you did that and the matrices are large enough (even 10x10 should be enough) then the matrix values should sum to 1.0. You might try evaluating these and see if it comes closer to the integer values in your earlier post.