Quantitative Analysis of Open Curves in Brain Imaging: Applications to White Matter Fibers and Sulci

There are about a hundred sulci in the human brain and over a hundred billion whitematter fibers. These two anatomical structures differ in their number, their physicalarrangements and in the function they serve but they do share a common geometricdescription: they are both open continuous curves. This thesis is a study of how thephysical attributes of open curves can be used to advantage in the many varied quan-titative applications of sulci and white matter fibers.Shape, scale, orientation and position, the four physical features associated withopen curves, have different properties so the usual approach has been to design differentmetrics and spaces to treat them individually. We take an alternative approach usinga comprehensive Riemannian framework where joint feature spaces allow for analysisof combinations of features. We can compare curves by using geodesic distances whichquantify their differences.We validate the metrics we use, demonstrate practical uses and apply the tools toimportant clinical problems. To begin, specific tract configurations in the corpus cal-losum are used to showcase clustering results that depend on the Riemannian distancemetric used. This nicely argues for the judicious selection of metrics in various applica-tions, a central premise in our work. The framework also provides tools for computingstatistical summaries of curves. We represent fiber bundles with a mean and variancewhich describes their essential characteristics. This is both a convenient way to workwith a large volume of fibers and is a first step towards statistical analysis. Next, wedesign and implement methods to detect morphological changes which can potentiallytrack progressive white matter disease.With sulci, we address the specific problem of labeling. An evaluation of physicalfeatures and methods such as clustering leads us to a pattern matching solution in whichthe sulcal configuration itself is the best feature.