On Continuous-Domain Inverse Problems with Sparse Superpositions of Decaying Sinusoids as Solutions

We study a family of inverse problems in which a continuous-domain object is reconstructed from a finite number of noisy linear measurements. We study regularization methods for solving these problems in which the regularizers promote sparsity in the frequency domain. We show that sparse superpositions of decaying sinusoids are solutions to these continuous-domain linear inverse problems, where the number of terms in the superposition is upper bounded by the number of measurements. This results in new forms of regularization for sparse reconstruction that are different from classical techniques. We numerically illustrate the efficacy of these new regularization techniques in the problem of image reconstruction.