This paper describes a Fast Factorized Back-Projection (FFBP) formulation that includes a fully integrated autofocus algorithm, i.e. the Factorized Geometrical Autofocus (FGA) algorithm. The base-two factorization is executed in a horizontal plane, using a Merging (M) and a Range History Preserving (RHP) transform. Six parameters are adopted for each sub-aperture pair, i.e. to establish the geometry stage-by-stage via triangles in 3-dimensional space. If the parameters are derived from navigation data, the algorithm is used as a conventional processing chain. If the parameters on the other hand are varied from a certain factorization step and forward, the algorithm is used as a joint image formation and autofocus strategy. By regulating the geometry at multiple resolution levels, challenging defocusing effects, e.g. residual space-variant Range Cell Migration (RCM), can be corrected. The new formulation also serves another important purpose, i.e. as a parameter characterization scheme. By using the FGA algorithm and its inverse, relations between two arbitrary geometries can be studied, in consequence, this makes it feasible to analyze how errors in navigation data, and topography, affect image focus. The versatility of the factorization procedure is demonstrated successfully on simulated Synthetic Aperture Radar (SAR) data. This is achieved by introducing different GPS/IMU errors and Focus Target Plane (FTP) deviations prior to processing. The characterization scheme is then employed to evaluate the sensitivity, to determine at what step the autofocus function should be activated, and to decide the number of necessary parameters at each step. Resulting FGA images are also compared to a reference image (processed without errors and autofocus) and to a defocused image (processed without autofocus), i.e. to validate the novel approach further.