Abel inversion from central-projection background oriented schlieren observations for reconstruction of axisymmetric refractive media

The background oriented schlieren (BOS) technique provides quantitative measurements of integrated light deflection caused by gradients of refractive index in an optical medium. For an axisymmetric medium and parallel rays orthogonal to the axis of symmetry, Abel inversion enables fast, direct, and non-iterative reconstruction of the refractive index distribution. In this work, we relax the parallel ray hypothesis, demonstrating that Abel inversion can be effectively applied to observations obtained by central projection with a standard entocentric lens, with a simple correction for measured deflections. This correction derives from an original second-order approximation of ray deflection under the paraxial approximation. The accuracy of the method is demonstrated using a synthetic low-density jet and a cone of vision aperture of approximately 40°. Its practical relevance is further illustrated by reconstructing the temporal mean density fields of two experimental cases: a heated high-subsonic dual-stream jet and an under-expanded supersonic jet characterized by significant shock-induced discontinuities.