bear (xgl99@mails.tsinghua.edu.cn) wrote:

Assume there are four layers, all of which are in Normal mode and grayscale. for a specific pixel in each layer,
Their intensity and alpha values(mapped to [0,1]) are i1,a1
i2,a2
i3,a3
i4(background)

the merged intensity is ((i1*a1+i2*(1-a1))*a2+i3*(1-a2))*a3+i4*(1-a3) am I correct? Thanks!

Yes. You might want to look for the paper T. Porter & T. Duff - Compositing Digital Images for an explantation and the reasoning behind these rules.

The paper itself is available at e.g. http://keithp.com/~keithp/porterduff/

Bye, Simon

Bear wrote:

Assume there are four layers, all of which are in Normal mode and grayscale. for a specific pixel in each layer,
Their intensity and alpha values(mapped to [0,1]) are i1,a1
i2,a2
i3,a3
i4(background)

the merged intensity is ((i1*a1+i2*(1-a1))*a2+i3*(1-a2))*a3+i4*(1-a3) am I correct? Thanks!

I hope not. The proper expression should be

a1*i1 + (1-a1) * (a2*i2 + (1-a2) * (a3*i3 + (1-a3) * i4)).

That is, you have the expression reversed. With your expression, the background could show through even with opaque layers above it.

The best way to think about it is that the projection at layer n is given by the recursive formula

P(n) = a_n * i_n + (1 - a_n) * P(n-1),

where the background is layer 0, and P(0) = i_0.

Best, -- Bill

______________ ______________ ______________ ______________ Sent via the KillerWebMail system at primate.ucdavis.edu

On Monday 20 September 2004 11:29, William Skaggs wrote:

Bear wrote:

Assume there are four layers, all of which are in Normal mode and grayscale. for a specific pixel in each layer, Their intensity and alpha values(mapped to [0,1]) are i1,a1
i2,a2
i3,a3
i4(background)

the merged intensity is ((i1*a1+i2*(1-a1))*a2+i3*(1-a2))*a3+i4*(1-a3) am I correct? Thanks!

I hope not. The proper expression should be

a1*i1 + (1-a1) * (a2*i2 + (1-a2) * (a3*i3 + (1-a3) * i4)).

Peering into the code, it is actually simpler than that. The GIMP always treat layers in groups of two, starting at the botton It would do:
new_i3 = i3 * a3 + i4 * (1 - a3)
then:
new_i2 = i2 * a2 + new_i3 * (1 - a2)

and so on. The finalr esult is like above - but it is easier to think on it 2 layers at a time.

That is, you have the expression reversed. With your expression, the background could show through even with opaque layers above it.

The best way to think about it is that the projection at layer n is given by the recursive formula

P(n) = a_n * i_n + (1 - a_n) * P(n-1),

where the background is layer 0, and P(0) = i_0.

Best, -- Bill