计算RGBA以匹配RGB颜色

Question

If I have an RGB color, with 100% opacity.

I want that same color (or close to it) with a transparent alpha channel. I will paint the transparent color over a white background.

How do I compute the RGBA color?

I guess what I am asking is the opposite of this question .

Answer 1

You mean you want the RGBA color with maximum transparency which, when drawn on top of a white background, gives the original RGB color?

Let R ₀ , G ₀ and B ₀ be the components of the original color, each ranging from 0.0 to 1.0, and let R , G , B and A be the components of the new RGBA color (with A = 1 denoting 100% opacity). We know that the colors must satisfy:

R ₀ = A · R + (1 − A )
G ₀ = A · G + (1 − A )
B ₀ = A · B + (1 − A )

which, if we knew A , we could easily solve for R , G and B :

R = ( R ₀ − 1 + A ) / A = 1 − (1 − R ₀ ) / A
G = ( G ₀ − 1 + A ) / A = 1 − (1 − G ₀ ) / A
B = ( B ₀ − 1 + A ) / A = 1 − (1 − B ₀ ) / A

Since we require that R ≥ 0, G ≥ 0 and B ≥ 0, it follows that 1 − R ₀ ≥ A, 1 − G ₀ ≥ A and 1 − B ₀ ≥ A, and therefore the smallest possible value for A is:

A = max( 1 − R ₀ , 1 − G ₀ , 1 − B ₀ ) = 1 − min( R ₀ , G ₀ , B ₀ )

Thus, the color we want is:

A = 1 − min( R ₀ , G ₀ , B ₀ )
R = 1 − (1 − R ₀ ) / A
G = 1 − (1 − G ₀ ) / A
B = 1 − (1 − B ₀ ) / A

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Ps. For a black background, the same formulas would be even simpler:

A = max( R ₀ , G ₀ , B ₀ )
R = R ₀ / A
G = G ₀ / A
B = B ₀ / A

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Pps. Just to clarify, all the formulas above are for non -premultiplied RGBA colors. For premultiplied alpha, just multiply R , G and B as calculated above by A , giving:

R = A · ( 1 − (1 − R ₀ ) / A ) = R ₀ − (1 − A )
G = A · ( 1 − (1 − G ₀ ) / A ) = G ₀ − (1 − A )
B = A · ( 1 − (1 − B ₀ ) / A ) = B ₀ − (1 − A )

(or, for a black background, simply R = R ₀ , G = G ₀ and B = B ₀ .)