New Developments in the Visualization and Processing of Tensor Fields

Description and estimation of local spatial structure has a long history and numerous analysis tools have been developed. A concept that is widely recognized as fundamental in the analysis is the structure tensor. It has, however, a fairly broad and unspecific meaning. This chapter is intended to provide a framework for displaying the differences and similarities of existing structure estimation approaches. A new method for structure tensor estimation, which is a generalization of many of it’s predecessors, is presented. The method uses pairs of filter sets having Fourier directional responses in the form of monomials, one odd order set and one even order set. It is shown that such filter sets allow for a particularly simple way of attaining phase invariant, positive semi-definite, local structure tensor estimates. In addition, we show that the chosen filter sets directly links order, scale and the gradient operator. We continue to compare a number of known structure tensor algorithms by formulating them in monomial filter set terms. In conclusion we show how higher order tensors can be estimated using a generalization of the same simple formulation.

High angular resolution diffusion imaging (HARDI) captures the angular diffusion pattern of water molecules more accurately than diffusion tensor imaging (DTI). This is of importance mainly in areas of complex intra-voxel fiber configurations. However, the extra complexity of HARDI models has many disadvantages that make it unattractive for clinical applications. One of the main drawbacks is the long post-processing time for calculating the diffusion models. Also intuitive and fast visualization is not possible, and the memory requirements are far from modest. Separating the data into anisotropic-Gaussian (i.e., modeled by DTI) and non-Gaussian areas can alleviate some of the above mentioned issues, by using complex HARDI models only when necessary. This work presents a study of DTI and HARDI anisotropy measures applied as classification criteria for detecting non-Gaussian diffusion profiles. We quantify the classification power of these measures using a statistical test of receiver operation characteristic (ROC) curves applied on ex-vivo ground truth crossing phantoms. We show that some of the existing DTI and HARDI measures in the literature can be successfully applied for data classification to the diffusion tensor or different HARDI models respectively. The chosen measures provide fast data classification that can enable data simplification. We also show that increasing the b-value and number of diffusion measurements above clinically accepted settings does not significantly improve the classification power of the measures. Moreover, we show that a denoising pre-processing step improves the classification. This denoising enables better quality classifications even with low b-values and low sampling schemes. Finally, the findings of this study are qualitatively illustrated on real diffusion data under different acquisition schemes.

The segmentation of anatomical structures within the white matter of the brain from DTI is an important task for white matter analysis, and has therefore received considerable attention in the literature during the last few years. Any segmentation method relies on the choice of a tensor dissimilarity measure, which should be small between tensors belonging to the same region and large between tensors belonging to different structures. Many different tensor distances have been proposed in the literature (Frobenius, Kullback-Leibler, Geodesic, Log-Euclidean, Hybrid…) for segmentation or other purposes, and there exist reasons (either theoretical or empirical) to justify the choice of any of them. Thus, determining which is the most appropriate tensor distance for a specific segmentation problem has become an extremely difficult decision. In this chapter we present a study on different tensor dissimilarity measures and their performance for white matter segmentation. The study is based on the use of two different DTI atlases of human brain, which provide a ground truth upon which the distances can be fairly compared. In order for the comparison to be independent of the segmentation method employed, it has been performed in terms of the separability of the different clases. Results show the Hybrid distance to perform better than other traditional tensor dissimilarity measures in terms of separability between classes, while the Frobenius, Kullback-Leibler, Geodesic and Log-Euclidean distances perform similarly.