In order to get precise local electron density information from chordal interferometric measurements of a tokamak plasma, a self-consistent and reliable inversion method is necessary. In this paper, an asymmetric Abel-inversion method is introduced. This method includes flexible boundary conditions, application to a noncircular geometry, and estimation of the plasma in the scrape-off layer. The advantages of this method are demonstrated by comparison with other methods. This inversion method is applied to a parametric study which includes dependence on the Shafranov shift and elongation of the profile.