We use ITK’s automatic threshold computation algorithms options. Their documentation is available here.
Smoothing pipeline consists of a number of VTK filters. See summary here and source code here.
Vertebrae are very easy to separate on high-resolution CT scans, but on typical patient scans have much lower resolution.
@amine gave an excellent answer on how to approach this. In addition to his suggestions, you may also try masked grow from seeds. See this segmentation recipe for details and this short demo video:
