Image region-based representation
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.