A novel method for inverting time-resolved line-integrated interferometric plasma density measurements is described. The method uses singular value decomposition of local density profiles from Thomson scattering measurements obtained at low sampling rates in the same or equivalent plasmas to determine a set of orthogonal spatial basis functions which is well adapted to the physical processes under investigation. The sought-for density profile is expanded into a limited series of these functions and a solution is calculated by using a simple least-squares fit method. The new method overcomes the difficulties encountered with other methods, such as regularization methods, which smoothen gradients and depend on the availability of accurate measurements in the plasma edge region. The small number of computations required provides for a fast algorithm. This method, which combines the high bandwidth of interferometer systems with the spatial accuracy of Thomson scattering, is applied to invert interferometer measurements in a wide variety of operational regimes in the Tokamak à Configuration Variable and Joint European Torus (JET) tokamaks. In particular, the collisionality dependence of density peaking observed in ASDEX Upgrade is confirmed in JET H-modes using this method.