Diffusion Tensor Estimation, Regularization and Classification

Abstract

In this chapter, we explore diffusion tensor estimation, regularization and classification. To this end, we introduce a variational method for joint estimation and regularization of diffusion tensor fields from noisy raw data as well as a Support Vector Machine (SVM) based classification framework.

In order to simultaneously estimate and regularize diffusion tensor fields from noisy observations, we integrate the classic quadratic data fidelity term derived from the Stejskal-Tanner equation with a new smoothness term leading to a convex objective function. The regularization term is based on the assumption that the signal can be reconstructed using a weighted average of observations on a local neighborhood. The weights measure the similarity between tensors and are computed directly from the diffusion images. We preserve the positive semi-definiteness constraint using a projected gradient descent.

The classification framework we consider in this chapter allows linear as well as non linear separation of diffusion tensors using kernels defined on the space of symmetric positive definite matrices. The kernels are derived from their counterparts on the statistical manifold of multivariate Gaussian distributions with zero mean or from distance substitution in the Gaussian Radial Basis Function (RBF) kernel. Experimental results on diffusion tensor images of the human skeletal muscle (calf) show the potential of our algorithms both in denoising and SVM-driven Markov random field segmentation.