Data adaptive tight frame methods have been proven a powerful sparse approximation tool in a variety of settings. We introduce a model of a data adaptive representation that also provides a multi-scale structure. Our idea is to design a multi-scale frame representation for a given data set, with scaling properties similar to the ones of a wavelet basis, but without the necessary self-similar structure. The adaptivity provides better sparsity properties, using Besov-like norm structure both induces sparsity and helps in identifying important features. We focus on investigating the efficiency of a weighted l<sup>1</sup> constraint in the context of sparse recovery from noisy data and compare it to the weighted l<sup>0</sup> model alongside. Numerical experiments confirm that the recovered frame vectors assigned lower weights correspond to image elements of larger scale and lower local variation, thus indicating that weighted sparsity in natural images leads to a natural scale separation.
We introduce a PDE-free variational model for multiphase image segmentation that uses a sparse representation basis (wavelets or other) instead of a Fourier basis in a modified diffuse interface context. The segmentation model we present differs from other state-of-the-art models in several ways. The diffusive nature of the method originates from the sparse representations and thus propagates information in a different manner comparing to any existing PDE models, even though it still has such classical features as coarsening and phase separation. The model has a non-local nature, yet with much reduced diffuse interface blur, thus allowing to connect important features and preserve sharp edges in the output. Numerical experiments show that the method is robust to noise and is highly tunable.