Triangulating a 3D point set


I have to create a concave mesh out of an unsorted point cloud.

I know this is a common problem and I already did some research into this topic. Nearly everybody suggests to use the Delaunay triangulation. I tried the 3D Delaunay which gives you a number of tetrahedra and the convex hull of the points.

What I need instead is the concave hull. A mesh that tightly wraps around the outer points of my set. I read that one could extract these triangles by using 3D alpha shapes. Here is where I'm stuck. I think I don't completely understand how to use them.
How do I utilize the resulting alpha value?

Could anyone tell me if this is the right direction or if there is a simpler way to do what I need to do. I really appreciate your help.

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