I have to use a graph cut to create a binary image from a grayscale image. I can easily compute both energy functions $E_{data}$ and $E_{smooth}$. But after that, I don't know what is the next step.
What I see from wikipedia is $Pr(x|S) = K^{-E}$ where $E=E_{data} +E_{smooth}$ and $S=\left\{ 0\ for\ background, 1\ for\ foreground \right\}$ but I don't know what is $K$.
From this site, there are several methods but I don't know which one is suitable to my problem.
My energy functions are: $$ \left\{ \begin{array}{ll} E_{data}(x_v = 1) = D(v) \\ E_{data}(x_v = 0) = 1−D(v) \end{array} \right. $$ where $D(v)$ is already computed. Labels can be 1 for foreground and 0 for background $$ E_{smooth}(xu, xv) = \left\{ \begin{array}{ll} λ(1.0−\frac{tan^{−1}(||C_u−C_v||)} {π/2} ),\ if\ x_u\ \ne\ x_v, \\ 0,\ if\ x_u\ =\ x_v, \end{array} \right. $$
$\lambda$ is a constant and $C_u$ is a color vector of the pixel $u$ and $x_u$ is the label of the pixel $u$.
One more thing, for the $E_{data}$, if the pixel is not yet labelled what would be the value ? Would it be 0 ?
EDIT: my image is not any grayscale image, it is already computed and most of pixels are 0 (or close) or 1 (or close).
The methods here are to give fast approximate solutions in the case where there are more than $2$ labels. Since in your case there are only $2$ labels, $0$ and $1$, you can use the exact max-flow min-cut algorithm described in Greig, Porteous, and Seheult (also here.)
The problem is to find labels for which the energy function is minimized. The energy function has no defined value for unlabelled pixels.
The solution should have minimum energy (or maximum probability), so which $K>1$ you choose in your equation above does not matter, as it does not affect the relative ordering of the probabilities.
Addendum: Fox and Nicholls 2001 (here, here) point out that the minimum-energy solution may not be typical of the a posteriori distribution.