Image region-based representation I have a homogeneity image that includes two regions $\Omega_1$ and $\Omega_2$. How to represent the image by decomposition of two regions? 

For example, the famous way is representing by the mean feature. Image $I$ can be  represented as follows:

$$I=u_1c_1+u_2c_2$$

$c_1$ is mean of region $\Omega_1$

$c_2$ is mean of region $\Omega_2$

$ u_1 =
  \begin{cases}
   1 &amp; \text{if is $\Omega_1$ }\\
   0       &amp; \text{otherwise}
  \end{cases}
$

$ u_2 =
  \begin{cases}
   1 &amp; \text{if is $\Omega_2$ }\\
   0       &amp; \text{otherwise}
  \end{cases}
$

Do you know another way to represent image $I$ without using mean feature. Thank you so much

As the question, I will show a simple example as following
Your example seems to imply you are talking about mutually exclusive, non-overlapping regions.  

In this case, given N such regions, use an image where each pixel has enough values (depth), to represent N regions. Just set the pixel value to the index of the region in that position.

You could think of the result image as a sum of N binary images, each multiplied by the $i$ subscript of the corresponding $\Omega_i$.

Alternatively, if you are asking about how to segment such an image into 2 (or more) regions, the you should use a connected component algorithm, which is common is most image processing libraries.