  We propose an improved density integration methodology for Background
Oriented Schlieren (BOS) measurements that overcomes the noise sensitivity of
the commonly used Poisson solver. The method employs a weighted least-squares
(WLS) optimization of the 2D integration of the density gradient field by
solving an over-determined system of equations. Weights are assigned to the
grid points based on density gradient uncertainties to ensure that a less
reliable measurement point has less effect on the integration procedure.
Synthetic image analysis with a Gaussian density field shows that WLS
constrains the propagation of random error and reduces it by 80% in comparison
to Poisson for the highest noise level. Using WLS with experimental BOS
measurements of flow induced by a spark plasma discharge show a 30% reduction
in density uncertainty in comparison to Poisson, thereby increasing the overall
precision of the BOS density measurements.
