Converting height-map to 3D without a plane floor.

So, I have a height-map as shown in the picture below. I use this to determine the y-axis height of each vertex in a plane. The problem with this is that the final 3D mesh has a surrounding floor where y-axis = 0.

I want to only generate the areas where the height map isn’t 0 but I do not know how to approach this specific case. Thanks in advance!

Don’t produce triangles where all the verts are at or below the floor you don’t want to see.