  Piecewise-linear (PL) Morse theory and discrete Morse theory are used in
shape analysis tasks to investigate the topological features of discretized
spaces. In spite of their common origin in smooth Morse theory, various notions
of critical points have been given in the literature for the discrete setting,
making a clear understanding of the relationships occurring between them not
obvious. This paper aims at providing equivalence results about critical points
of the two discretized Morse theories. First of all, we prove the equivalence
of the existing notions of PL critical points. Next, under an optimality
condition called relative perfectness, we show a dimension agnostic
correspondence between the set of PL critical points and that of discrete
critical simplices of the combinatorial approach. Finally, we show how a
relatively perfect discrete gradient vector field can be algorithmically built
up to dimension 3. This way, we guarantee a formal and operative connection
between critical sets in the PL and discrete theories.
