Mathematical Methods in Medical Image Processing and Magnetic Resonance Imaging

In this paper,we describe some central mathematical problems in medical imaging.The subject has been undergoing rapid changes driven by better hardware and software.Much of the software is based on novel methods utilizing geometric partial differential equations in conjunction with standard signal/image processing techniques as well as computer graphics facilitating man/machine interactions.As part of this enterprise,researchers have been trying to base biomedical engineering principles on rigorous mathematical foundations for the development of software methods to be integrated into complete therapy delivery systems.These systems support the more effective delivery of many image-guided procedures such as radiation therapy,biopsy,and minimally invasive surgery.We will show how mathematics may impact some of the main problems in this area including image enhancement,registration,and segmentation.This paper[1] describes image processing techniques for Diffusion Tensor Magnetic Resonance.In Diffusion Tensor MRI,a tensor describing local water diffusion is acquired for each voxel. The geometric nature of the diffusion tensors can quantitatively characterize the local structure in tissues such as bone,muscles,and white matter of the brain.The close relationship between local image structure and apparent diffusion makes this image modality very interesting for medical image analysis.We present a decomposition of the diffusion tensor based on its symmetry properties resulting in useful measures describing the geometry of the diffusion ellipsoid. A simple anisotropy measure follows naturally from this analysis.We describe how the geometry,or shape,of the tensor can be visualized using a coloring scheme based on the derived shape measures.We show how filtering of the tensor data of a human brain can provide a description of macrostructural diffusion which can be used for measures of fiber-tract organization.We also describe how tracking of white matter tracts can be implemented using the introduced methods.