Abstract:

We introduce a wavelet analogue of the classical Ginzburg-Landau (WGL) energy, where the H1-seminorm is replaced by the Besov (or ``Besov-type'') semi-norm defined via an arbitrary orthonormal wavelet. We prove that, whenever this wavelet is regular, functionals of this type converge, in the Gamma-sense, to a weighted analogue of the TV functional on characteristic functions of finite-perimeter sets.

We show that in 2D the limiting energy is none other than the surface tension energy in the Wulff problem and its minimizers are represented by corresponding Wulff shapes.

We prove that the problem of WGL minimization can be solved via the gradient descent method; in one and two dimensional cases the minimizers are bounded and infinitely smooth, with sharper transitions between equilibria than those in the classical case. We construct a variational model for multi-purpose image recovery that is based on the WGL energy as a regularizing part and includes fidelity terms specific to each particular application.