We present a new numerical method for reconstruction of instantaneous density volume from 3D background-oriented schlieren (3DBOS) measurements, with a validation on a dedicated flexible experimental BOS bench. In contrast to previous works, we use a direct formulation where density is estimated from measured deviation fields without the intermediate step of density gradient reconstruction. Regularization techniques are implemented to deal with the ill-posed problem encountered. The resulting high-dimensional optimization is conducted by conjugate gradient techniques. A parallel algorithm, implemented on graphics processing unit, helps to speed up the calculation. The resulting software is validated on synthetic BOS images of a 3D density field issued from a numerical simulation. Then, we describe a dedicated 3DBOS experimental facility which has been built to study various BOS settings and to assess the performance of the proposed numerical reconstruction process. Results on various datasets illustrate the potential of the method for flow characterization and measurement in real-world conditions.