Sparse topological data recovery in medical images

For medical image analysis, the test statistic of the measurements is usually constructed at every voxels in space and thresholded to determine the regions of significant signals. This thresholding produces a small patch of regions around the critical values of the test statistic. It is known that the probability of the critical values bigger than a specific threshold can be computed as the expectation of the Euler characteristic of the patch. Motivated by this topological connection, we present a new computational framework of modeling various functional measurements as topological objects. The level set associated with functional measurements can be approximated using a simplicial complex consisting of nodes and links. The existence of links basically determine the underlying topological structure of the signal. The strength of links can be modeled using an underdetermined linear model. By incorporating sparsity into the model, the links can be sparsely obtained making interpretation and visualization of the simplicial complex easier. The main contribution of this paper is showing the relationship between sparse topological structures to the sparse regression framework. We apply this novel framework in constructing a structural brain network model.