Linearity and shift invariance (LSI) characteristics of magnetic particle imaging (MPI) are important properties for quantitative medical diagnosis applications. The MPI image equations have been theoretically shown to exhibit LSI; however, in practice, the necessary filtering action removes the first harmonic information, which destroys the LSI characteristics. This lost information can be constant in the x-space reconstruction method. Available recovery algorithms, which are based on signal matching of multiple partial field of views (pFOVs), require much processing time and a priori information at the start of imaging. In this paper, a fast analytical recovery algorithm is proposed to restore the LSI properties of the x-space MPI images, representable as an image of discrete concentrations of magnetic material. The method utilizes the one-dimensional (1D) x-space imaging kernel and properties of the image and lost image equations. The approach does not require overlapping of pFOVs, and its complexity depends only on a small-sized system of linear equations; therefore, it can reduce the processing time. Moreover, the algorithm only needs a priori information which can be obtained at one imaging process. Considering different particle distributions, several simulations are conducted, and results of 1D and 2D imaging demonstrate the effectiveness of the proposed approach.