Time that an imaging device needs to produce results
is one of the most crucial factors in medical imaging.
Shorter scanning duration causes fewer artifacts such as those
created by the patient motion. In addition, it increases patient comfort and
in the case of some imaging modalities also decreases exposure to radiation.
There are some possibilities,
hardware-based or software-based, to improve the imaging speed. One way
is to speed up the scanning process by acquiring fewer measurements. A recently
developed mathematical framework called compressed
sensing shows that it is possible to accurately recover undersampled images
provided a suitable measurement matrix is used and the image itself has a
Nevertheless, not only measurements are important but also good reconstruction
models are required. Such models are usually expressed as optimization
In this thesis, we concentrated on the reconstruction of the undersampled
Magnetic Resonance (MR) images.
For this purpose a complex-valued reconstruction model was provided.
Since the reconstruction should be as quick as possible,
fast methods to find the solution for the reconstruction problem are required.
To meet this objective, three popular algorithms FISTA,
Augmented Lagrangian and Non-linear Conjugate Gradient were
adopted to work with our model.
By changing the complex-valued reconstruction model slightly
and dualizing the problem, we obtained an instance of the quadratically
constrained quadratic program where both the objective function and the
constraints are twice differentiable. Hence new model opened doors to two other
methods, the first order method which resembles FISTA and is called in
this thesis Normed Constrained Quadratic FGP, and the second order
method called Truncated Newton Primal Dual Interior Point.
Next, in order to
compare performance of the methods, we set up the experiments and evaluated all
presented methods against the problem of reconstructing undersampled MR images.
In the experiments we used a number of invocations of the Fourier transform
to measure the performance of all algorithms.
As a result of the experiments we found that in the context of the original
model the performance of Augmented Lagrangian is better than the other
two methods. Performance of Non-linear Conjugate Gradient and
FISTA are about the same.
In the context of the extended model Normed Constrained Quadratic FGP
beats the Truncated Newton Primal Dual Interior Point method.