Enclosed geometric shape from Delaunay triangulation

Hello?

I am working on making some enclosed geometry from Delaunay triangulation whose vertices are from two different sources, base plane vertices and additional nodes.  
I have made some sloppy definition that works if the second chunk of points are not on the perimeter or at the corner of the base geometry.  

However, if more than one point’s projection towards the base geometry is at the same age or at the corner, the whole shape has become non-enclosed.

It could be seen from the image below (Red parts supposed to be the complimentary surfaces to the white surfaces). The desired outcome should be some kind of enclosed type of continuous triangular surfaces without its base geometry.

In this case, only 4 points are considered, and I am going to increase the number of the additional points for the future investigation.  

I have tried different way to solve this issue but couldn’t make the perfect answer yet. And I would greatly appreciate for any suggestion.

Kind regards.
Amraa.