Representing a LiDAR surface using the 3D Delaunay Triangulation as basis?

I would like to represent a surface, with 3D Delaunay Triangulation. The vertices must be my original input data, a LiDAR point cloud from an urban area. So, the surface must adjust/adapt the input information.

Actually, what I need to do is the following:

I have a 3D point cloud (x, y, z) from an urban area;
I need to represent the surface of this area;
I would like to do a 3D delaunay triangulation (I did with CGAL and I got the tetrahedrons) and identify only the triangles that represent the surface (with CGAL I have 4 vertices and I can't define which ones represent the surface triangle);
Since these surface triangles are known, I need to give a point and get the triangle that contains this given point.

I would like to know which function I need. I saw the "3D surface mesh generation", "3D Delaunay Triangulation", "Surface Reconstruction from point sets", "3D Mesh generation" and "3D Alpha Shapes". But I am not sure which one fits better to my work/need. What function could give me the surface triangles (represented by points from my initial dataset) as output.

Does anyone have a suggestion?

Solution

You can use Fade 2.5D, it uses triangles to mesh surface points:

http://www.geom.at/fade2d/html/ enter image description here

The library is free for scientific use, you can find a code example in Example7 (I'm the author).