A new generation of optical devices that generate images covering a larger part of the field of view than conventional cameras, namely catadioptric cameras, is slowly emerging. These omnidirectional images will most probably deeply impact computer vision in the forthcoming years, provided that the necessary algorithmic background stands strong. In this paper, we propose a general framework that helps define various computer vision primitives. We show that geometry, which plays a central role in the formation of omnidirectional images, must be carefully taken into account while performing such simple tasks as smoothing or edge detection. Partial differential equations (PDEs) offer a very versatile tool that is well suited to cope with geometrical constraints. We derive new energy functionals and PDEs for segmenting images obtained from catadioptric cameras and show that they can be implemented robustly using classical finite difference schemes. Various experimental results illustrate the potential of these new methods on both synthetic and natural images.