Super-resolution structured illumination microscopy (SIM) has become a widely used method for biological imaging. Standard reconstruction algorithms, however, are prone to generate noise-specific artifacts that limit their applicability for lower signal-to-noise data. Here we present a physically realistic noise model that explains the structured noise artifact, which we then use to motivate new complementary reconstruction approaches. True-Wiener-filtered SIM optimizes contrast given the available signal-to-noise ratio, and flat-noise SIM fully overcomes the structured noise artifact while maintaining resolving power. Both methods eliminate ad hoc user-adjustable reconstruction parameters in favor of physical parameters, enhancing objectivity. The new reconstructions point to a trade-off between contrast and a natural noise appearance. This trade-off can be partly overcome by further notch filtering but at the expense of a decrease in signal-to-noise ratio. The benefits of the proposed approaches are demonstrated on focal adhesion and tubulin samples in two and three dimensions, and on nanofabricated fluorescent test patterns.