In many applications of Image Processing it is crucial to dispose of efficient tools for extraction, analysis and representation of geometrical contents in natural images. These latter can be modelled by classes of bivariate functions, regular on a finite number of regions separated by smooth boundaries. It is by now a well-established fact that the usual two-dimensional tensor product wavelet bases are not optimal for approximating such classes. In the last ten years, several methods have been suggested as a remedy. Among them, wedgelets representations over quadtree structures represent a contour-based approach which allows an efficient digital implementation while capturing mainly geometric features of natural images. We discuss some algorithmic aspects due to the discrete nature of the method, leading to a fast computation of optimal solutions. As a possible application we present a new scheme for digital image compression based on these methods. The main ingredient for the design of an efficient coding scheme is to consider spatial redundancies between neighbouring atoms of the representation, relatively to the properties of the target regularity class. Joint work with Mattia Fedrigo, Felix Friedrich and Hartmut Führ.