Noise Removal with Tissue Boundary Preservation Using Fourth-Order Partial Differential Equations
Marius Lysaker, Arvid Lundervold, Xue‐Cheng Tai, M. F. M. De Bock, Lothar Rudi Schad · 2001
Introduction A major source of MR image degradation, with impairment of diagnostic information, is random thermal noise entering the MR data in the time domain [1]. Methods for noise suppression are thus an important issue in many MRI applications. To overcome the deficiencies of acquisition-based noise reduction methods, such as increased acquisition time (i.e. time averaging over repeated measurements) or decreased spatial resolution (i.e. enlarging voxel volume), several postprocessing methods have been proposed. The ultimate goal of these methods is to obtain piecewise constant, or slowly varying signals in homogeneous tissue regions while preserving the tissue boundaries. The most powerful and promising mathematical approaches towards this goal are PDE-based methods (e.g. [2],[3],[4]) and wavelet-based methods (e.g. [5]). Methods PDE model: We use a partial differential equation of 4th order to remove noise from MR images. It is assumed that the noise level is approximate known. We are trying to get a new image u that has the same noise level but with a smallest norm for the second order derivatives, i.e. we are trying to minimize the norm of the second order derivatives under the noise level constraint. The minimization problem is solved by a Lagrangian technique in connection with finite difference approximations. More specifically, we assume u_d is the image to be restored. Then we update u^k as in the following with an appropriate u_0.