We consider variational problems which are defined over scalar valued functions and that are non-convex in both the variable function and its gradient. We propose two methods for finding minimizers of the associated relaxation: i) a semi-analytical method based on Control Systems Theory, ii) and a numerical method that combines a convex splitting scheme together with a modified version of the Split Bregman algorithm. These solutions are then used to deduce the Young-measure-valued minimizers of the non-convex problem and to reconstruct the location and length scale of microstructures.