Multiresolution triangulation meshes are widely used in computer graphics for 3-d modelling of shapes. We propose an image representation and processing framework using a multiscale triangulation of the grayscale function. Triangles have the potential of approximating edges better than the blocky structures of tensor-product wavelets. Among the many possible triangulation schemes, normal meshes are natural for efficiently representing singularities in image data thanks to their adaptivity to the smoothness of the modeled image. Our non-linear, multiscale image decomposition algorithm, based on this subdivision scheme, takes edges into account in a way that is closely related to wedgelets and curvelets. The highly adaptive property of the normal meshes construction provides a very efficient representation of images, which potentially outperforms standard wavelet transforms. We demonstrate the approximation performance of the normal mesh representation through mathematical analyses for simple functions and simulations for real images.