A theoretical framework for widefield structured illumination microscopy (SIM) reconstruction from fewer than the commonly used nine raw frame acquisitions is introduced and applied in silico and in vitro. The proposed scheme avoids the recording of redundant spatial frequency components, which was necessary in previous SIM algorithms. This allows for gentler superresolution imaging at faster speeds. A doubling of frame rates is possible solely via changes in the computational reconstruction procedure. Furthermore, we explore numerically the effect of the sample movement on the reconstruction quality and the number of raw frames recordable. Our results show that there exists a limit above which deconvolution microscopy becomes superior to SIM.