  We study two classes of torsion classes which generalize functorially finite
torsion classes, that is, semistable torsion classes and morphism torsion
classes. Semistable torsion classes are parametrized by the elements in the
real Grothendieck group up to TF equivalence. We give a close connection
between TF equivalence classes and the cones given by canonical decompositions
of the spaces of projective presentations due to Derksen-Fei. More strongly,
for $E$-tame algebras and hereditary algebras, we prove that TF equivalence
classes containing lattice points are exactly the cones given by canonical
decompositions. One of the key steps in our proof is a general description of
semistable torsion classes in terms of morphism torsion classes. We also answer
a question by Derksen-Fei negatively by giving examples of algebras which do
not satisfy the ray condition. As an application of our results, we give an
explicit description of TF equivalence classes of preprojective algebras of
type $\widetilde{\mathbb{A}}$.
