User:Federhalter/Alpha channel reconstruction

Definitions:

An image is a vector of pixels x_i with i = 1 ... n_pixel with either one grayscale channel x_i = (gr_i) or three (R,G,B) colour component channels x_i = (r_i, g_i, b_i) and (optionally) one alpha transparency channel α_i. All values are from the interval [0, 1].

In computers these are usually represented in integer values:

gr_i = Gr_i / Maxval

r_i = R_i / Maxval

g_i = G_i / Maxval

b_i = B_i / Maxval

α_i = Α_i / Maxval

with X ∈ [0, Maxval] (X = Gr, R, G, B, Α).

Problem

Given an original image x_0,i (without an alpha channel), and a specific background colour x_b, the task of alpha channel reconstruction is the construction of a new image x_1,i with an alpha channel α_1,i, such that:

1. the new image composed over the specified background colour is identical to the original image, and

2. all pixels with a colour identical to the given background colour become fully transparent,

3. all colours which are at the edge of the colourspace and are not the background colour become fully opaque,

4. all remaining colours are associated with an alpha value which is proportional to their distance in the colourspace from the background colour.

The goal of alpha channel reconstruction is to create an image which can easily be composed over a different background while still looking nice.

Solution

First, calculate the relative colourspace distance for each component:

$\delta _{x,i}={\begin{cases}{\frac {x_{b}-x_{0,i}}{x_{b}}},&{\text{for }}x_{0,i}<x_{b},\\0,&{\text{for }}x_{0,i}=x_{b},\\{\frac {x_{0,i}-x_{b}}{1-x_{b}}},&{\text{for }}x_{0,i}>x_{b}.\\\end{cases}}$

The total distance (and thus the alpha value) is the maximum of the above values for each pixel:

$\alpha _{i}=max(\delta _{X,i})$

For grayscale:

$\alpha _{i}=\delta _{gr,i}$

For RGB: $\alpha _{i}=max(\delta _{r},\delta _{g},\delta _{b})_{i}$

From α we can calculate the new image channels:

$x_{1,i}={\begin{cases}{\frac {x_{0,i}-x_{b}}{\alpha _{i}}}+x_{b},&{\text{for }}\alpha _{i}>0,\\x_{b},&{\text{for }}\alpha _{i}=0.\end{cases}}$

In integer representation:

$\Delta _{X,i}={\begin{cases}{\frac {Maxval\cdot (X_{b}-X_{0,i})}{X_{b}}},&{\text{for }}X_{0,i}<X_{b},\\0,&{\text{for }}X_{0,i}=X_{b},\\{\frac {Maxval\cdot (X_{0,i}-X_{b})}{Maxval-X_{b}}},&{\text{for }}X_{0,i}>X_{b}.\end{cases}}$

$\mathrm {A} _{i}=\Delta _{Gr,i}$

For RGB: $\mathrm {A} _{i}=max(\Delta _{R},\Delta _{G},\Delta _{B})_{i}$

$X_{1,i}={\begin{cases}X_{b}+{\frac {Maxval\cdot (X_{0,i}-X_{b})}{\mathrm {A} _{i}}},&{\text{for }}\mathrm {A} _{i}>0,\\X_{b},&{\text{for }}\mathrm {A} _{i}=0.\end{cases}}$

Implementation notes

The method can be implemented with integer arithmetics if the maximum value range is larger or equal to Maxval² + ½Maxval. With the operation \ (integer division),

$\Delta _{X,i}={\begin{cases}(Maxval\cdot (X_{b}-X_{0,i})+X_{b}\backslash 2)\backslash X_{b},&{\text{for }}X_{0,i}<X_{b},\\0,&{\text{for }}X_{0,i}=X_{b},\\(Maxval\cdot (X_{0,i}-X_{b})+(Maxval-X_{b})\backslash 2)\backslash (Maxval-X_{b}),&{\text{for }}X_{0,i}>X_{b}.\end{cases}}$

$X_{1,i}={\begin{cases}X_{b}+(Maxval\cdot (X_{0,i}-X_{b})+\mathrm {A} _{i}\backslash 2)\backslash \mathrm {A} _{i},&{\text{for }}\mathrm {A} _{i}>0,\\X_{b},&{\text{for }}\mathrm {A} _{i}=0.\end{cases}}$

The latter can be implemented using unsigned integer arithmetics as follows:

$X_{1,i}={\begin{cases}X_{b}-(Maxval\cdot (X_{b}-X_{0,i})+\mathrm {A} _{i}\backslash 2)\backslash \mathrm {A} _{i},&{\text{for }}\mathrm {A} _{i}>0\land X_{0,i}<X_{b},\\X_{b}+(Maxval\cdot (X_{0,i}-X_{b})+\mathrm {A} _{i}\backslash 2)\backslash \mathrm {A} _{i},&{\text{for }}\mathrm {A} _{i}>0\land X_{0,i}\geq X_{b},\\X_{b},&{\text{for }}\mathrm {A} _{i}=0.\end{cases}}$

Alternative methods of transparency reconstruction

A method commonly implemented (simple) image editing software is binary transparency, which fullfills points 1-3 of the above, but not point 4:

$x_{1,i}=x_{0,i}$

$\alpha _{x,i}={\begin{cases}1,&{\text{for }}x_{0,i}\neq x_{b},\\0,&{\text{for }}x_{0,i}=x_{b}.\end{cases}}$

In integer representation:

$X_{1,i}=X_{0,i}$

$\mathrm {A} _{X,i}={\begin{cases}Maxval,&{\text{for }}X_{0,i}\neq X_{b},\\0,&{\text{for }}X_{0,i}=X_{b}.\end{cases}}$