My problem is about understandig (of physical interpretation) of image function $g$ in the Mumford-Shah functional. Let $\Omega \subset \mathbb R^2$ be an open domain, and $g \in L^\infty (\Omega)$ the image function. What is the interpretation of the function $g$?
My own idea is this: If, for example we are deal with a black and white image, then is $g(x,y)$ may determine the level of blackness in the image. (For example $g $ equals to 1 for pure black, 0 for pure white, and $ 0<g<1 $ for other (i.e., gray) points). If this a right interpretation of $g$, we must have $\|g\|_{L^\infty} \leq 1$, but I think this may not be true because I don't find such an interpretation anywhere (I see some problems in which $\|g\|_{L^\infty} >1$ actually).
With $\Omega $ and $g$ as above, Mumford-Shah functional is defined by $ E(u,K) = \int_{\Omega \setminus K} (u-g)^2+ \int_{\Omega \setminus K}|\nabla u|^2+ \mathcal H^1(K),$ and the problem is to find aclosed one-dimensional subset $K \subset \Omega$, and a function $u \in W^{1,2} (\Omega \setminus K)$ which minimize $E(u,K)$ among all such suitable candidate pairs.
I think your interpretation is basically correct, but there's no need to require any particular scale for $g$; for example you could expect $g$ to take values between $0$ and $255$ in the case of an 8-bit image. The effect of changing the scale is the same as that of adjusting the weights in the energy: if we let $$E_{a,b,c}(u,K) = a\int(u-g)^2 + b\int|\nabla u|^2 + c \mathcal H^1(K)$$ and $E_{a,b,c}(u_*,K_*)$ is optimal for $g,$ then I think $E_{a,b,c\lambda^2}(\lambda u_*,K_*)$ will be optimal for $\lambda g.$