Our previous works in image analysis dealing with image representations by means of adjacency or boundary graphs, led us to the need for a coherent representation model. In fact, classical approaches to this problem seem to be unsuffcient or even uncoherent; for example they are unclear with the well-known connectivity paradox or with the border description of regions. Diierent works like those of Kovalevsky, Herman and Malandain pointed out the advantages of cellular complex based topologies [Kovalevsky89, Herman90, Malandain93]. But no one of them suggested a formalism that can be applied to any type of image. In this work we propose a topological representation for any type of image, colour or grey-level of whatever dimension. It is based on convex complexes, and looks closely at the elements realizing the connectivity within complexes and later within regions. Futhermore it remains coherent with pixel and voxel only based representations. An important feature is still maintaining a direct correspondance with the classical IR n topology. Finally, we suggest a characterization of regions, their borders and boundaries which is useful as a basic tool for segmentation.