The Math Behind Photoshop's Unsharp Mask and Gaussian Blur

[Drazick]

Hello,
Has anyone succeeded replicating Photoshop's Gaussian Blur and Unsharp Mask?
When I say replicating, I mean up to a rounding error or something like that.

It looks like Photoshop is doing something beyond the classic and I want to know what exactly to be able to reproduce things I do in Photoshop using ImageMagick.

I wrote about it here:
http://registry.gimp.org/node/32018

Thank You.

[fmw42]

Now, the Unsharp Mask of 'O' is given by 'O+(O-GB)-inv(O+inv(GB))'.
You can do that by 'Apply Image'.

Remembering inv(Layer) = 1 - Layer (I assume image is [0, 1) yields something interesting.
It means, at the end USM(O) = 3O - 2B.

Your conclusion is not correct. The equivalent is not 3*O+2*GB.

it is 2*O + inv(O) - GB - inv(inv(GB))

inv(O) is not the same as -O and inv(GB) is not the same as -GB.

inv(O) = 1-O and inv(GB) is 1-GB for normalized images in the range of 0 to 1

or

inv(O) = 255-O and inv(GB) is 255-GB for normalized images in the range of 0 to 255

Thus

O+(O-GB)-inv(O+inv(GB)) = O+(O-GB)-inv(O)-inv(inv(GB)) = 2*O - GB - (1-O) - (1-(1-GB)) = 2*O -GB -1 +O -GB = -1+3*O-2*GB

Thus, this should be correct(?)

O+(O-GB)-inv(O+inv(GB)) = 3*O -white - 2*GB

Edited to correct earlier mistakes

[Drazick]

Let's try it again.
Can wee agree that inv(L) = White - L?

USM(O) = O + (O - B) - inv(O + inv(B)) = 2O - B - (White - (O + (White - B))) = 2O - B - (White - (O + White - B)) = 2O - B - (White - O - White + B) = 2O - B - White + O + White - B = 3O - 2B.

So I was right whether it is a [0, 1] image or [0 255] image.
It has to be the same.

I think you were wrong in:
inv(O + inv(B)) = inv(O) + inv(inv(B))

Let's see:
inv(O + inv(B)) = White - (O + (White - B)) = White - O - White + B = B - O.

But,
If you do 3O - 2B in Photoshop you won't get the expected result.
I think it has to do with clipping at each step because you only can calculate O and (O - B) and then add twice (O-B) to O.

[fmw42]

I have corrected my computations earlier as I made a sign mistake.

Your notes said

O+(O-GB)-inv(O+inv(GB))

But I had mistakenly scripted:

O+(O-GB)+inv(O+inv(GB))

So correcting the script gives:

Input:

Code: Select all

thresh=9.8
convert lena3.png \
\( -clone 0 -gaussian-blur 0x3 \) \
\( -clone 0 -clone 1 +swap -compose minus -composite \) \
\( -clone 1 -negate \) \
\( -clone 0 -clone 3 -compose plus -composite -negate \) \
-delete 1,3 \
\( -clone 1 -clone 2 +swap -compose minus -composite \) \
-delete 1,2 \
\( -clone 1 -threshold $thresh% -negate \) \
-compose plus -composite lena3_im_us_script3_3gx9p8x1.png

To be compared with PS

PS r=3, amount=100, thresh=25



PS r=7, amount=100, thresh=25



Edit:

I think you were wrong in:
inv(O + inv(B)) = inv(O) + inv(inv(B))

Yes, I think you are right. But I scripted from your original equation and not the simplified one. I think you are correct that clipping is needed between steps of your processing and you cannot do the breakdown by simple math.

[Drazick]

Thank You for the simulations.

Yet again, what's interesting here is why Photoshop does it and why the direct calculation is different from the "Long Calculation"?

[Drazick]

It turns out the magic happens when after each addition / subtraction operation you clip the image into its range.

[Drazick]

You corrected your answer after I wrote that the magic happens due to clipping in each step.

Yet clipping, since it is Non Linear, must be done at each iteration and not like you wrote.

Try it by yourself.

[fmw42]

Turning clamping off, makes the results worse.

Clamp On (by default)
thresh=9.8
im6937hdri convert lena3.png \
\( -clone 0 -gaussian-blur 0x3 \) \
\( -clone 0 -clone 1 +swap -compose minus -composite \) \
\( -clone 1 -negate \) \
\( -clone 0 -clone 3 -compose plus -composite -negate \) \
-delete 1,3 \
\( -clone 1 -clone 2 +swap -compose minus -composite \) \
-delete 1,2 \
\( -clone 1 -threshold $thresh% -negate \) \
-compose plus -composite lena3_im_us_script3_3gx9p8x1a.png


Clamp Off:
thresh=9.8
im6937hdri convert lena3.png \
\( -clone 0 -gaussian-blur 0x3 \) \
\( -clone 0 -clone 1 +swap -define compose:clamp=off -compose minus -composite \) \
\( -clone 1 -negate \) \
\( -clone 0 -clone 3 -define compose:clamp=off -compose plus -composite -negate \) \
-delete 1,3 \
\( -clone 1 -clone 2 +swap -define compose:clamp=off -compose minus -composite \) \
-delete 1,2 \
\( -clone 1 -threshold $thresh% -negate \) \
-define compose:clamp=off -compose plus -composite lena3_im_us_script3_3gx9p8x1b.png


compare -metric rmse lena3_im_us_script3_3gx9p8x1a.png lena3_im_us_script3_3gx9p8x1b.png null:
3940.39 (0.0601265)

[fmw42]

If you remove the thresholding and clamp (clip) in HDRI, you get


im6937hdri convert lena3.png \
\( -clone 0 -gaussian-blur 0x3 \) \
\( -clone 0 -clone 1 +swap -compose minus -composite \) \
\( -clone 1 -negate \) \
\( -clone 0 -clone 3 -compose plus -composite -negate \) \
-delete 1,3 \
\( -clone 1 -clone 2 +swap -compose minus -composite \) \
-delete 1,2 \
-compose plus -composite lena3_im_us_script3_3gx9p8x1c.png



IM internal unsharp:
im6937hdri convert lena3.png -unsharp 0x3 lena3_im_uns_0x3.png



PS r=3, amount=100, thresh=25



PS r=7, amount=100, thresh=25


Differences may be in the quality or argument interpetation of the gaussian blur(?) It is also possible that PS adds some slight smoothing of some kind to the result.

[Casper]

Here's the image manually applying the operations in Photoshop, no threshold, using PS Gaussian Blur with radius 3.
It's strange that this image is not identical to the one made with ImageMagic.

Actually we need a sample from IM _without_ any threshold to be absolutely sure, but it already looks like
they're still going to be different..which is a bit strange. Perhaps Gaussian Blur is not calculated the same
on PS and IM, or the radius value is used differently(?)

[Casper]

IM internal unsharp:
im6937hdri convert lena3.png -unsharp 0x3 lena3_im_uns_0x3.png

Interesting..this seems to be the best match to the image I posted above, but it's still not as good as the PS version.

[fmw42]

I do not see you posted image or its PS argument values.

Edit: Sorry, we must have posted at about the same time. I see your image above.

[Casper]

Sorry. Not sure why it's not showing up for you. This is the URL:
http://i.imgur.com/ZKwoD4L.png

Can you open that one?

I did these steps in PS:
O+(O-GB)-inv(O+inv(GB))

For GB I used radius 3.

[fmw42]

One of the issues is interpreting the PS arguments. It has 3 and IM has 4. The main question is what is radius for photoshop and how does that map to IM radiusxsigma. Is the PS radius the radius or the sigma value or some combination. This affects the gaussian blur significantly. With IM you can use 0xsigma and it will compute the appropriate radius. So is radius 3 in PS the same IM radius or IM sigma?

[fmw42]

Sorry. Not sure why it's not showing up for you. This is the URL:
http://i.imgur.com/ZKwoD4L.png

Can you open that one?

I did these steps in PS:
O+(O-GB)-inv(O+inv(GB))

For GB I used radius 3.

Can you reproduce that using PS unsharp mask and if so what values are you using. Presumably the above formula would be for amount=100, and threshold=0. But I do not know what radius 3 means in terms of a gaussian blur the uses radius x sigma values.