The propagation of a partially coherent light beam through a weakly inhomogeneous medium with quadratic refractive index profile is described within the framework of the generalized Fresnel transform. Using this formalism, the general problem of the reconstruction of the correlation function of an input optical field from intensity measurements is discussed. As a particular application, the reconstruction of a field having a Gaussian correlation function is carried out. Moreover, a method is proposed to obtain the parameters of the refractive medium, e.g. a GRIN fiber.