Structured illumination microscopy (SIM) allows non-invasive visualization of nanoscale subcellular structures. However, image acquisition and reconstruction become the bottleneck to further improve the imaging speed. Here, we propose a method to accelerate SIM imaging by combining the spatial re-modulation principle with Fourier domain filtering and using measured illumination patterns. This approach enables high-speed, high-quality imaging of dense subcellular structures using a conventional nine-frame SIM modality without phase estimation of the patterns. In addition, seven-frame SIM reconstruction and additional hardware acceleration further improve the imaging speed using our method. Furthermore, our method is also applicable to other spatially uncorrelated illumination patterns, such as distorted sinusoidal, multifocal, and speckle patterns.