In this talk, I first discuss estimates for the topological degree of maps from sphere into itself. Second, I present characterizations of Sobolev spaces based on the pointwise convergence or the Gamma-convergence of a sequence of nonlocal, nonconvex functionals related to these estimates. If time permits, I will also discuss the  connection between these functionals  with various filters in the denoising problem. The talk is based on joint works with Jean Bourgain and Haim Brezis.