Image convolution with pre-multiplied alpha

I'm trying to implement image convolution with a 3x3 matrix, where my colour components (each ranging from 0 to 255) are stored using pre-multiplied alpha. All the tutorials (e.g. http://www.codeproject.com/KB/GDI-plus/csharpfilters.aspx) I can find only describe performing the convolution calculations on the RGB component开发者_如何学运维s and nothing is mentioned about the alpha component.

My current code leaves the alpha component as it is. The filters I have tried look fine when working on images where every pixel already has full alpha set. When I have partially transparent pixels e.g. a boxblur filter looks strange because pixel colors do not propagate into transparent areas when blurring happens.

What calculations do I perform on the alpha component when running the convolution algorithm and how do I deal with pre-multiplied alphas when setting the final pixel value? Also, do I add the filter offset to the alpha component?

I've tried calculating my new alpha component the same way I calculate the RGB components (i.e. adding up the surrounding alpha values for that pixel according to the filter matrix) but I get colored fringes appearing on the edge of transparent areas and semi-transparent pixels start to darken too much. I think I need to change the new RGB components to take into account the new alpha value but I'm not sure what to do.

Thanks.

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I think that the correct way is to first compute just the alpha of the convolution using the standard formulas

alpha = a1*m1 + a2*m2 + a3*m3 +
        a4*m4 + a5*m5 + a6*m6 +
        a7*m7 + a8*m8 + a9*m9;

then you must compute the convolution of the original (non-premultiplied) r/g/b and post-multiply by alpha

red = (r1/a1*m1 + r2/a2*m2 + r3/a3*m3 +
       r4/a4*m4 + r5/a5*m5 + r6/a6*m6 +
       r7/a7*m7 + r8/a8*m8 + r9/a9*m9) * alpha;

with a similar formula for green and blue.

A more efficient way would be first removing premultiplication (i.e. replacing r with r/a, g with g/a and b with b/a) doing the convolution of all components using standard formulas and then re-premultiply (replacing r with r*a, g with g*a and b with b*a).