  Resolving a linear combination of point sources from their band-limited
Fourier data is a fundamental problem in imaging and signal processing. With
the incomplete Fourier data and the inevitable noise in the measurement, there
is a fundamental limit on the separation distance between 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 classic Rayleigh limit. It is well-known that
Rayleigh limit is heuristic and its drawbacks become prominent when dealing
with data that is subjected to delicate processing, as is what modern
computational imaging methods do. Therefore, 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 number detection and support recovery
in one dimension in [arXiv:2003.02917[cs.IT], arXiv:1912.05430[eess.IV]]. In
this paper, we extend the one-dimensional theory to multi-dimensions. More
precisely, we define and quantitatively characterize the "computational
resolution limit" for the number detection and support recovery problems in a
general k-dimensional space. Our results indicate that there exists a phase
transition phenomenon regarding to the super-resolution factor and the
signal-to-noise ratio in each of the two recovery problems. Our main results
are derived using a subspace projection strategy. Finally, to verify the
theory, we proposed deterministic subspace projection based algorithms for the
number detection and support recovery problems in dimension two and three. The
numerical results confirm the phase transition phenomenon predicted by the
theory.
