This paper describes a compact perceptual image model intended for morphological representation of the visual information contained in natural images. We explain why the total variation can be a criterion to split the information between the two main visual structures, which are the sketch and the microtextures. We deduce a morphological decomposition scheme, based on a segmentation where the borders of the regions correspond to the location of the topological singularities of a topographic map. This leads to propose a new and morphological definition of edges. The sketch is computed by approximating the image with a piecewise smooth non-oscillating function, using a Lipshitz interpolant given as the solution of a PDE. The data needed to reconstruct the sketch image are very compact, so that an immediate outcome of this image model is the design of a progressive, and artifact-free, image compression scheme.