Theses and Dissertations

Issuing Body

Mississippi State University


Kim, Seongjai

Committee Member

Razzaghi, Mohsen

Committee Member

Lim, Hyeona

Committee Member

Shivaji, Ratnasingham

Committee Member

Smith, Robert

Date of Degree


Document Type

Dissertation - Open Access


Mathematics and Statistics

Degree Name

Doctor of Philosophy


College of Arts and Sciences


Department of Mathematics and Statistics


Most of PDE-based restoration models and their numerical realizations show a common drawback: loss of fine structures. In particular, they often introduce an unnecessary numerical dissipation on regions where the image content changes rapidly such as on edges and textures. This thesis studies the magnitude data/imagery of magnetic resonance imaging (MRI) which follows Rician distribution. It analyzes statistically that the noise in the magnitude MRI data is approximately Gaussian of mean zero and of the same variance as in the frequency-domain measurements. Based on the analysis, we introduce a novel partial differential equation (PDE)-based denoising model which can restore fine structures satisfactorily and simultaneously sharpen edges as needed. For an efficient simulation we adopt an incomplete Crank-Nicolson (CN) time-stepping procedure along with the alternating direction implicit (ADI) method. The algorithm is analyzed for stability. It has been numerically verified that the new model can reduce the noise satisfactorily, outperforming the conventional PDE-based restoration models in 3-4 alternating direction iterations, with the residual (the difference between the original image and the restored image) being nearly edgeree. It has also been verified that the model can perform edge-enhancement effectively during the denoising of the magnitude MRI imagery. Numerical examples are provided to support the claim.



Crank-Nicolson (CN) alternating direction implicit||equalized net diffusion||Rician distribution||Gaussian distribution||Magnetic resonance imaging (MRI)||PDE based denoising models