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I = ¦ÁF + (1 − ¦Á)B. (2)
We call ¦Á the alpha channel model of image I (see
Fig. 2(a)(b) for an example). If we assume both F and B
are locally smooth1, then
I ∗ h = ¦ÁF ∗ h+(1−¦Á)B ∗ h = ¦ÁbF +(1−¦Áb)B, (3)
where ¦Áb = ¦Á ∗ h is ¦Á blurred by kernel h. From Eqn. 3,
we know that ¦Áb is the alpha channel model of Ib (see
Fig. 2(c)(d) for an example). So the LHS of Eqn. 1 is
∇(¦ÁbF + (1 − ¦Áb)B) ¡¤ b = (F − B)∇¦Áb ¡¤ b. (4)


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