For thousands of years, doctors had to face the daunting task of diagnosing and treating all sorts of medical ailments without the ability to view the insides of their patients. It was not until the 1970's that CT and MRI technology enabled doctors to develop cross-sectional images of internal anatomy. This work discusses the application of sparse approximation theory and the closely related field compressive sensing medical image processing. We discuss one related theoretical problem and two major practical applications.

Orthogonal Matching Pursuit (IMP) is a fast and efficient greedy algorithm that is well known in the sparse approximation community. We prove restricted isometry conditions that guarantee its correctness and establish theoretical error bounds on its performance. Then we prove stronger results for variations of this algorithm where multiple items are allowed to be selected per iteration.

The orthogonalities matching pursuit algorithms are then applied to the problem of recovering sparse gradient images from a small number of Fourier samples. In MRI, this translates into reducing patient scan time by eliminating the need to sample the entire spectrum of an image at the Nyquist rate. A novel algorithm called Gradient Matching Pursuit is introduced that uses some variation of IMP to recover an image in the edge domain and then use one of several proposed inverse-filtering techniques to obtain a final reconstruction. Gradient Matching Pursuit is analyzed theoretically and is empirically shown to perform as accurately, but more efficiently, than conventional total-variation minimization routines.

The last part of this work will describe how sparse approximation methods can be utilized to correct imperfections in MRI transmission coils. In the general case of an MRI scanner with multiple transmission coils, the MRI Parallel Excitation problem can be recast into a parallel sparse approximation problem, which is basically an interpolation between sparse and simultaneous sparse approximation. An efficient algorithm called Parallel Orthogonal Matching Pursuit is proposed to solve the MRI Parallel Excitation Problem as well as other similar problems.