A mathematical theory of the computational resolution limit in one dimension

Abstract

Given an image generated by the convolution of point sources with a band-limited function, the deconvolution problem involves reconstructing the source number, positions, and amplitudes. This problem is related to many important applications in imaging and signal processing. It is well known that it is impossible to resolve the sources when they are sufficiently close in practice. Rayleigh investigated this problem and formulated a resolution limit known as the Rayleigh limit for the case of two sources with identical amplitudes. However, many numerical experiments have demonstrated that stable recovery of the sources is possible even if the sources are separated below the Rayleigh limit. In this study, a mathematical theory for the deconvolution problem in one dimension is developed. The theory addresses the problem when the source number can be recovered exactly from noisy data. The key component is a new concept called the “computational resolution limit,” which is defined as the minimum separation distance between the sources such that exact recovery of the source number is possible. This new resolution limit is determined by the signal-to-noise ratio and the sparsity of sources, as well as the cutoff frequency of the image. Quantitative bounds for this limit are derived, and they demonstrate the importance of the sparsity and signal-to-noise ratio for the recovery problem. The stability of recovering the source positions is also analyzed under a condition on their separation distances. Moreover, a singular value thresholding algorithm is proposed to recover the source number for a cluster of closely spaced point sources and to verify our theoretical results regarding the computational resolution limit. The results are based on a multipole expansion method and a nonlinear approximation theory in Vandermonde space.