Lefschetz decompositions and Categorical resolutions of singularities

Let $Y$ be a singular algebraic variety and let $\TY$ be a resolution of singularities of $Y$. Assume that the exceptional locus of $\TY$ over $Y$ is an irreducible divisor $\TZ$ in $\TY$ such that the image $Z$ of $\TZ$ in $Y$ is smooth and $\TZ$ is smooth over $Z$. For every relative Lefschetz decomposition of $\TZ$ over $Z$ we construct a triangulated subcategory $\TD \subset \D^b(\TY)$ which gives a desingularization of $\D^b(Y)$. If the Lefschetz decomposition is generated by a vector bundle tilting over $Y$ then $\TD$ is a noncommutative resolution, and if the Lefschetz decomposition is rectangular, then $\TD$ is a crepant resolution.