Home The Basics Gaussian Blur and the High-Pass Filter - What gives?
Gaussian Blur and the High-Pass Filter - What gives?

Last update:  12-31-1969

Submitted by byRo

As we say in Brazil - they're just the two sides of the same coin. (PS 7.0 - but applies everywhere)

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You may find this rather boring (now that is a good way to start a tutorial, isn't it? ). However, the basics put down here will be built on later in other exciting, breath-taking .... tutorials.

So, there are three ways to get through this:

1) Skip to the end, and see the conclusions;
2) Read through the tutorial, just looking at the pictures;
3) Work through the examples yourself.

Your choice, OK?

Let's have a look at these two images..........

 

onthe left we have a Gaussian blur at radius 10 and to the right is a high-pass filter also at radius 10. Although they look completely different we will see here that they are very closely related.

 

 

Once again we are looking at our young friend Maria. As a first step we have, of course, unclipped the colors [see tutorial], and from  now we will be using just the luminosity information. Everything here fully applies to colored images, but it's just that much easier to understand looking at it in grayscale.

So, make a layer with just the Luminosity and call it "Original".

[hint]
Quick way to get luminosity (*4). Duplicate the layer <ctrl><J>. Edit>Fill Black, blending Colour

[end of hint]

Make a duplicate copy (<ctrl><J>) of the "Original" luminosity image and call it "GB 10", do a Gaussian blur at radius 10 on this layer.


We are now going to subtract this Gaussian blur from the original. How?


Photoshop does not have a direct blending option for  subtraction (the blending options "Difference" and "Exclusion" do not work this way). Although there is a subtraction option in Image>Calculations which will do the job directly, we'll stick with layers and do it in two steps.

We do have a way  to addition with the blending option "Linear Light" (*1),  so we can invert (*2) <ctrl><I> the "GB 10" layer and then add this to the Original using the blending option "Linear Light" at opacity 50%. This will end up doing the subtraction we wanted.

 

 

 

 

Hey, that looks a little familiar........

 

Let's check this out. Make a new duplicate of the "Original" and put it on top. Change the name to "HP 10" and apply Filters>Other>High Pass... with radius 10 (you'd already figured that out, I know). Click on the eye-ball of the "HP 10" layer to turn it on and off. If it looks like it's just the same - that's because it is!

For the more skeptical - change the blending of the HP 10 layer to Difference and have a look at the histogram (Image>Histogram), give or take one or two  pixel values (rouding errors) the are exactly the same thing.

Great, so what?
Well, writing some
really intricate maths, this is what we just did:

Original - GB(r) = HP(r)

The (r), beside being fancy maths notation, means that (as long as they are both the same) it will work for any radius, not just 10. (a sly generalization, OK, but it is true)

Now, going back to school, we can rearrange that to get:

Original = GB(r)  + HP(r)

This means that we can split up any image into a Gaussian blur part and a High-pass part, or looking at it another way, these two are exact complements of each other.

To go a little further, we could think about the strange mix-up of these names. Surely the complement of a high-pass would be a low-pass, and the complement of blurring would be sharpening. Sure is!

Original = Low Pass + High Pass, or
Original = Blurred image+ Sharpened image

Let's test this out too......


Set this up with the layers: Invert <ctrl><I> the GB layer back to the normal looking image and set blending to Normal at 100%. Change the blending of the High-pass layer to Linear Light with opacity 50%. Put the Original layer on top now.

Should look like the palette to the left.

Toggling the original layer on and off we can see that the GB and HP have now added up to get back to the original.

OK, works over - let's play
(Leave the Original off now.)
- Play around with
the opacity slider of  the HP layer:

up = more sharpened,

down = more blurred.

 

Congratulations!! You have constructed a new Photoshop tool - the blur / sharpen slider thingy tool.

Conclusions:
- An image can be thought of as being made up of two parts, one corresponding to the Gaussian blur and one to the High-pass filter.
- The Linear Light blending option can be used to add two layers together.
- As we change the mix of these two components the original image can be either blurred or sharpened.

Next up:
choosing the radius, without the guesswork
sharpening, the true story


Notes, observations and nit-picking:
*1 - in this type of addition mid-gray (*3) is taken as the zero point, lighter than mid-gray is positive and darker is negative;

*2
- a pixel can't have a negative value, we are in fact inverting around the mid-gray (*3) value;
*3 -  mid-gray is 127.5, not 127. Funny thing is, you can't actually put this value in your channel. Try this: Fill a layer with the standard PS 50% gray , which is 128 in all channels, and invert it - you'll now have 127!

*4 - thanks, LisaG for pointing out my errors here, and Stroker for the real cute luminosity trick!