  For each integer $k\geq 4$ we describe diagrammatically a positively graded
Koszul algebra $\mathbb{D}_k$ such that the category of finite dimensional
$\mathbb{D}_k$-modules is equivalent to the category of perverse sheaves on the
isotropic Grassmannian of type ${\rm D}_k$ or ${\rm B}_{k-1}$, constructible
with respect to the Schubert stratification. The algebra is obtained by a
(non-trivial) ``folding'' procedure from a generalized Khovanov arc algebra.
Properties like graded cellularity and explicit closed formulas for graded
decomposition numbers are established by elementary tools.
