Two complementary solution strategies to the least-squares migration problem with sparseness- & continuity constraints are proposed. The applied formalism explores the sparseness of curvelets on the reflectivity and their invariance under the demigration migration operator. Sparseness is enhanced by (approximately) minimizing a (weighted) l1-norm on the curvelet coefficients. Continuity along imaged reflectors is brought out by minimizing the anisotropic difussion or total variation norm which penalizes variations along and in between reflectors. A brief sketch of the theory is provided as well as a number of synthetic examples. Technical details on the implementation of the optimization strategies are deferred to an accompanying paper: implementation.