# Image Acquisition from a Highly Incomplete Set of Measurements

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.