Mapping Topographic Structure in White Matter Pathways with Level Set Trees

Fiber tractography on diffusion imaging data offers rich potential for describing white matter pathways in the human brain, but characterizing the spatial organization in these large and complex data sets remains a challenge. We show that level set trees---which provide a concise representation of the hierarchical mode structure of probability density functions---offer a statistically-principled framework for visualizing and analyzing topography in fiber streamlines. Using diffusion spectrum imaging data collected on neurologically healthy controls (N=30), we mapped white matter pathways from the cortex into the striatum using a deterministic tractography algorithm that estimates fiber bundles as dimensionless streamlines. Level set trees were used for interactive exploration of patterns in the endpoint distributions of the mapped fiber tracks and an efficient segmentation of the tracks that has empirical accuracy comparable to standard nonparametric clustering methods. We show that level set trees can also be generalized to model pseudo-density functions in order to analyze a broader array of data types, including entire fiber streamlines. Finally, resampling methods show the reliability of the level set tree as a descriptive measure of topographic structure, illustrating its potential as a statistical descriptor in brain imaging analysis. These results highlight the broad applicability of level set trees for visualizing and analyzing high-dimensional data like fiber tractography output

Topological finiteness properties of monoids. Part 1: Foundations

We initiate the study of higher dimensional topological finiteness properties of monoids. This is done by developing the theory of monoids acting on CW complexes. For this we establish the foundations of $M$-equivariant homotopy theory where $M$ is a discrete monoid. For projective $M$-CW complexes we prove several fundamental results such as the homotopy extension and lifting property, which we use to prove the $M$-equivariant Whitehead theorems. We define a left equivariant classifying space as a contractible projective $M$-CW complex. We prove that such a space is unique up to $M$-homotopy equivalence and give a canonical model for such a space via the nerve of the right Cayley graph category of the monoid. The topological finiteness conditions left-$\mathrm{F}_n$ and left geometric dimension are then defined for monoids in terms of existence of a left equivariant classifying space satisfying appropriate finiteness properties. We also introduce the bilateral notion of $M$-equivariant classifying space, proving uniqueness and giving a canonical model via the nerve of the two-sided Cayley graph category, and we define the associated finiteness properties bi-$\mathrm{F}_n$ and geometric dimension. We explore the connections between all of the these topological finiteness properties and several well-studied homological finiteness properties of monoids which are important in the theory of string rewriting systems, including $\mathrm{FP}_n$, cohomological dimension, and Hochschild cohomological dimension. We also develop the corresponding theory of $M$-equivariant collapsing schemes (that is, $M$-equivariant discrete Morse theory), and among other things apply it to give topological proofs of results of Anick, Squier and Kobayashi that monoids which admit presentations by complete rewriting systems are left-, right- and bi-$\mathrm{FP}_\infty$.Comment: 59 pages, 1 figur

SubCMap: subject and condition specific effect maps

Current methods for statistical analysis of neuroimaging data identify condition related structural alterations in the human brain by detecting group differences. They construct detailed maps showing population-wide changes due to a condition of interest. Although extremely useful, methods do not provide information on the subject-specific structural alterations and they have limited diagnostic value because group assignments for each subject are required for the analysis. In this article, we propose SubCMap, a novel method to detect subject and condition specific structural alterations. SubCMap is designed to work without the group assignment information in order to provide diagnostic value. Unlike outlier detection methods, SubCMap detections are condition-specific and can be used to study the effects of various conditions or for diagnosing diseases. The method combines techniques from classification, generalization error estimation and image restoration to the identify the condition-related alterations. Experimental evaluation is performed on synthetically generated data as well as data from the Alzheimer's Disease Neuroimaging Initiative (ADNI) database. Results on synthetic data demonstrate the advantages of SubCMap compared to population-wide techniques and higher detection accuracy compared to outlier detection. Analysis with the ADNI dataset show that SubCMap detections on cortical thickness data well correlate with non-imaging markers of Alzheimer's Disease (AD), the Mini Mental State Examination Score and Cerebrospinal Fluid amyloid-β levels, suggesting the proposed method well captures the inter-subject variation of AD effects