In mathematics, a super-resolution problem can be formulated as acquiring high-frequency data from low-frequency measurements. This extrapolation problem in the frequency domain is well-known to be unstable. We propose the model-based super-resolution framework (Model-SR) to address this ill-posedness. Within this framework, we can recover the signal by solving a nonlinear least square problem and achieve the super-resolution. Theoretically, the resolution-enhancing map is proved to have Lipschitz continuity under mild conditions, leading to a stable solution to the super-resolution problem. We apply the general theory to three concrete models and give the stability estimates for each model. Numerical experiments are conducted to show the super-resolution behavior of the proposed framework. The model-based mathematical framework can be extended to problems with similar structures.

翻译：在数学上，超分辨率问题可表述为从低频测量数据中获取高频信息。这一频域外推问题的不稳定性是众所周知的。我们提出基于模型的超分辨率框架（Model-SR）以解决这一不适定性问题。在该框架下，我们可通过求解非线性最小二乘问题来恢复信号并实现超分辨率。理论上，我们证明了在温和条件下分辨率增强映射具有Lipschitz连续性，从而为超分辨率问题提供了稳定解。我们将该通用理论应用于三个具体模型，并给出了各模型的稳定性估计。数值实验展示了所提框架的超分辨率特性。这种基于模型的数学框架可推广至具有类似结构的问题中。