A mathematical theory of resolution limits for super-resolution of positive sources

A priori information on the positivity of source intensities is ubiquitous in imaging fields and is also important for a multitude of super-resolution and deconvolution algorithms. However, the fundamental resolution limit of positive sources is still unknown, and research in this field is very limited indeed. In this work, we analyze the super-resolving capacity for number and location recoveries in the super-resolution of positive sources and aim to answer the resolution limit problem in a rigorous manner. Specifically, we introduce the computational resolution limit for respectively the number detection and location recovery in the one-dimensional super-resolution problem and quantitatively characterize their dependency on the cutoff frequency, signal-to-noise ratio, and the sparsity of the sources. As a direct consequence, we show that targeting at the sparest positive solution in the super-resolution already provides the optimal resolution order. These results are generalized to multi-dimensional spaces. Our estimates indicate that there exist phase transitions in the corresponding reconstructions, which are confirmed by numerical experiments. On the other hand, despite the fact that positivity plays important roles in improving the resolution of certain super-resolution algorithms, our theory has made several different but significant discoveries: i) The a priori information of positivity cannot further improve the order of the resolution limit; ii) The positivity of the source sometimes deteriorates the resolution limit instead of enhancing it. In particular, under certain signal-to-noise ratio, two point sources with different phases actually have a better resolution limit than those with the same one.