  Let $G$ be a complex reductive group and let $G^\vee$ be its Langlands dual.
Let us choose a triangular decomposition $\mathfrak g^\vee=\mathfrak
n^\vee_-\oplus\mathfrak h^\vee\oplus\mathfrak n^\vee_+$ of the Lie algebra
$G^\vee$. Braverman, Finkelberg and Gaitsgory show that the set of all
Mirkovi\'c-Vilonen cycles in the affine grassmannian $\mathscr G=G\bigl(\mathbb
C((t))\bigr)/G\bigl(\mathbb C[[t]]\bigr)$ is a crystal isomorphic to the
crystal of the canonical basis of $U(\mathfrak n^\vee_+)$. Starting from the
string parameter of an element of the canonical basis, we give an explicit
description of a dense subset of the associated MV cycle. As a corollary, we
show that any MV cycle can be obtained as the closure of one of the varieties
involved in Lusztig's algebraic-geometric parametrization of the canonical
basis. In addition, we prove that the bijection between LS paths and MV cycles
constructed by Gaussent and Littelmann is an isomorphism of crystals.
