Compressive Sensing and its Applications to Magnetic Resonance Imaging

There exists a class of inverse problems in which scientists must reproduce a desired signal by using an incomplete set of measurements to solve a system of equations. In this experiment, the system is undetermined: instead of having a unique solu­tion, it has infinitely many. If one knows beforehand that the signal is sparse, then Compressive Sensing (CS) algorithms can exploit this property to quickly find the exact solution at the expense of few measurements. Therefore, it is possible to use CS to recreate MRI’s and fMRI of the brain with insufficient data, since this problem is equivalent to solving the underdetermined system for the sparse solution. Using the theory, we hope to replace current reconstruction methods that are costly with respect to time.

To tackle this problem, we will implement and test several well known CS algorithms in MATLAB using sets of limited measurements, comparing the accuracy between the recreated and original image. Moreover, we will see how these reconstruction techniques behave when the observed measurements are corrupted with noise. Finally, we will explore and test new CS algorithms to reconstruct images. 

Implementation of certain codes have been met with some success, especially with regards to solving smaller scale systems, but the results are far from conclusive for higher dimensional problems, and need to be addressed before moving on to image reconstruction of the brain. Future aspirations include programming in lower level languages to reduce computational costs. Overall, the theory is young and will benefit from future research.