The paper proposes a new multiscale image decomposition based on platelets. Platelets are localized functions at various scales, locations, and orientations that produce piecewise linear image approximations. For smoothness measured in certain Holder classes, the error of m-term platelet approximations can decay significantly faster than that of m-term approximations in terms of sinusoids, wavelets, or wedgelets. Platelet representations are especially well-suited for the analysis of Poisson data, unlike most other multiscale image representations, and they can be rapidly computed. We propose a platelet-based maximum penalized likelihood criterion that encompasses denoising, deblurring, and tomographic reconstruction.