A mathematical theory of computational resolution limit in multi-dimensional spaces * *Hai Zhang was partially supported by Hong Kong RGC grant GRF 16305419.

Abstract

Resolving a linear combination of point sources from their Fourier data in a bounded domain is a fundamental problem in imaging and signal processing. With incomplete Fourier data and inevitable noise in the measurement, there is a fundamental limit on the separation distance between the point sources that can be resolved. This is the so-called resolution limit problem. Characterization of this resolution limit is still a long-standing puzzle despite the prevalent use of the classical Rayleigh limit. It is well-known that the Rayleigh limit is heuristic and its drawbacks become prominent when dealing with data that is subjected to elaborated processing, as is what modern computational imaging methods do. Therefore, a more precise characterization of the resolution limit becomes increasingly necessary with the development of data processing methods. For this purpose, we developed a theory of ‘computational resolution limit’ for both the number detection problem and the support recovery problem in one dimension in (Liu and Zhang 2019 arXiv:1912.05430; Liu and Zhang 2021 IEEE Trans. Inf. Theory 67 4812–27). In this paper, we extend the theory from dimension one to multiple dimensions. More precisely, we define and quantitatively characterize the ‘computational resolution limit’ for the number detection problem and the support recovery problem respectively in general multi-dimensional spaces. Our results indicate that there exists a phase transition phenomenon regarding the super-resolution factor and the signal-to-noise ratio in each of the two recovery problems. Our main results are obtained by using a projection strategy. Finally, to verify the theory, we propose deterministic projection-based algorithms for the number detection problem and the support recovery problem respectively. The numerical results in dimensions two and three confirm the phase transition phenomenons.