Tomographic imaging from limited projections: Theory and algorithms

Justin K. Romberg, Applied Mathematics Department, California Institute of Technology.

Reconstructing an object of interest from its projections is a fundamental task in imaging. Mathematically, the measurement process can be represented as a linear operator A applied to a unknown vector x. Recovering x can be reduced to the most classical problem in linear algebra: given measurements y=Ax, solve a linear system of equations What happens if the number of projections is severely limited? That is, what if our measurement matrix A has many fewer rows than columns? The linear inverse problem is now severely ill-posed. There are many more unknowns than constraints, and recovering a general signal x becomes impossible. However, if x is sparse (in that it consists of a few "significant" components, and many "insignificant" components), we will show that it can still be recovered by solving a certain type of convex optimization problem. The recovery procedure, despite being nonlinear, is also exceptionally stable in the presence of noise. We will close by discussing some of the algorithmic challenges for the recovery procedure in the context of tomographic imaging.

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