Many imaging techniques acquire information about an underlying image by making indirect linear measurements. For example, in computed tomography we observe line integrals through the image, while in MRI we observe samples of the image's Fourier transform. To acquire an N-pixel image, we will in general need to make at least N measurements. What happens if the number of measurements is (much) less than N (that is, the measurements are incomplete)? We will present theoretical results showing that if the image is sparse, it can be reconstructed from a very limited set of measurements essentially as well as from a full set by solving a certain convex optimization program. By "sparse", we mean that the image can be closely approximated using a small number of elements from a known orthobasis (a wavelet system, for example). Although the reconstruction procedure is nonlinear, it is exceptionally stable in the presence of noise, both in theory and in practice. We will conclude with several practical examples of how the theory can be applied to "real-world" imaging problems.