Connectivity of projected high dimensional data charts on one-dimensional curves

We propose a principal curve tracing algorithm that uses the gradient and the Hessian of a given density estimate. Curve definition requires the local smoothness of data density and is based on the concept of subspace local maxima. Tracing of the curve is handled through the leading eigenvector where fixed-step updates are used. We also propose an image segmentation algorithm based on the original idea and show the effectiveness of the proposed algorithm on a Brainbow dataset. Lastly, we showed a simple approach to define connectivity in complex topologies, by providing a tree representation for the bifurcating synthetic data.