Hi Charles,
you need to select all polygons, then all edges.
Bevel the edges with Level one.
Switch from polygon to edge mode, now all inner fields are selected.
Call 'linear subdivide' from polygon tool menu twice, then 'Catmull-Clark subdivide' twice.
Now you have the corners rounded.
Invert the polygon selection, call 'Cover' and move up to get the geometry like in your picture.
I can make a GIF animation later if needed, but am quite busy atm.
Edit: If some points are very close, they will not work well when beveled.
Move them apart a little with point slide tool before you start.