Making a rectangle map

So you start with a random map:

Then connect vertexes on the map with straight lines instead of irregular ones, but make sure every region has at least four sides.

Move vertices so every line that is closest to horizontal becomes totally horizontal, same with lines that are closest to vertical, and the rest make 45 degree diagonals. 

Imagine the diagonals are the hypotenuse of a right triangle, replace the diagonal with the other two sides of the triangle...

Move vertical and horizontal lines into as few columns and rows as you can while preserving the borders...

This still has the same border information but everything is on a grid....

Can every map be put into grid form? I don't know...