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The ability to resolve detail in the object that is being imaged, named byresolution, is the core parameter of an imaging system. Super-resolution is aclass of techniques that can enhance the resolution of an imaging system andeven transcend the diffraction limit of systems. Despite huge success in theapplication, super-resolution is not well understood on the theoretical side,especially for any dimension $d \geq 2$. In particular, in order to recover a$k$-sparse signal, all previous results suffer from either/both poly$(k)$samples or running time.  We design robust algorithms for any (constant) dimension under a strong noisemodel based on developing some new techniques in Sparse Fourier transform(Sparse FT), such as inverting a robust linear system, "eggshell" samplingschemes, and partition and voting methods in high dimension. These algorithmsare the first to achieve running time and sample complexity (nearly) linear inthe number of source points and logarithmic in bandwidth for any constantdimension, and we believe the techniques developed in the work can find theirfurther applications on the Super-resolution and Sparse FT problem.

Super-resolution and Robust Sparse Continuous Fourier Transform in Any Constant Dimension: Nearly Linear Time and Sample Complexity