  Let $K$ be an algebraically closed field of characteristic $0$ and $f$ be a
homogeneous polynomial. We provide an explicit algorithm, which, given the
invariants of a generic $f$ under the action of $\mathrm{GL}_n(K)$ returns a
polynomial in the orbit of $f$. We derive a specific algorithm for the
reconstruction of generic non-hyperelliptic curve of genus 4 from their
invariants, as well as a direct reconstruction algorithm for generic
non-hyperelliptic curves of genus 3 from their Dixmier-Ohno invariants.
