Derivative compressive sampling with application to phase unwrapping

Reconstruction of multidimensional signals from the samples of their partial derivatives is known to be an important problem in imaging sciences, with its fields of application including optics, interferometry, computer vision, and remote sensing, just to name a few. Due to the nature of the derivative operator, the above reconstruction problem is generally regarded as ill-posed, which suggests the necessity of using some a priori constraints to render its solution unique and stable. The ill-posed nature of the problem, however, becomes much more conspicuous when the set of data derivatives occurs to be incomplete. In this case, a plausible solution to the problem seems to be provided by the theory of compressive sampling, which looks for solutions that fit the measurements on one hand, and have the sparsest possible representation in a predefined basis, on the other hand. One of the most important questions to be addressed in such a case would be of how much incomplete the data is allowed to be for the reconstruction to remain useful. With this question in mind, the present note proposes a way to augment the standard constraints of compressive sampling by additional constraints related to some natural properties of the partial derivatives. It is shown that the resulting scheme of derivative compressive sampling (DCS) is capable of reliably recovering the signals of interest from much fewer data samples as compared to the standard CS. As an example application, the problem of phase unwrapping is discussed.