Noncommutative Bell polynomials, quasideterminants and incidence Hopf algebras

Bell polynomials appear in several combinatorial constructions throughout mathematics. Perhaps most naturally in the combinatorics of set partitions, but also when studying compositions of diffeomorphisms on vector spaces and manifolds, and in the study of cumulants and moments in probability theory. We construct commutative and noncommutative Bell polynomials and explain how they give rise to Fa\`a di Bruno Hopf algebras. We use the language of incidence Hopf algebras, and along the way provide a new description of antipodes in noncommutative incidence Hopf algebras, involving quasideterminants. We also discuss M\"obius inversion in certain Hopf algebras built from Bell polynomials.Comment: 37 pages, final version, to appear in IJA

Pulmonary Nodule Classification in Lung Cancer from 3D Thoracic CT Scans Using fastai and MONAI

We construct a convolutional neural network to classify pulmonary nodules as malignant or benign in the context of lung cancer. To construct and train our model, we use our novel extension of the fastai deep learning framework to 3D medical imaging tasks, combined with the MONAI deep learning library. We train and evaluate the model using a large, openly available data set of annotated thoracic CT scans. Our model achieves a nodule classification accuracy of 92.4% and a ROC AUC of 97% when compared to a “ground truth” based on multiple human raters subjective assessment of malignancy. We further evaluate our approach by predicting patient-level diagnoses of cancer, achieving a test set accuracy of 75%. This is higher than the 70% obtained by aggregating the human raters assessments. Class activation maps are applied to investigate the features used by our classifier, enabling a rudimentary level of explainability for what is otherwise close to “black box” predictions. As the classification of structures in chest CT scans is useful across a variety of diagnostic and prognostic tasks in radiology, our approach has broad applicability. As we aimed to construct a fully reproducible system that can be compared to new proposed methods and easily be adapted and extended, the full source code of our work is available at https://github.com/MMIV-ML/Lung-CT-fastai-2020

On the Lie enveloping algebra of a post-Lie algebra

We consider pairs of Lie algebras $g$ and $\bar{g}$, defined over a common vector space, where the Lie brackets of $g$ and $\bar{g}$ are related via a post-Lie algebra structure. The latter can be extended to the Lie enveloping algebra $U(g)$. This permits us to define another associative product on $U(g)$, which gives rise to a Hopf algebra isomorphism between $U(\bar{g})$ and a new Hopf algebra assembled from $U(g)$ with the new product. For the free post-Lie algebra these constructions provide a refined understanding of a fundamental Hopf algebra appearing in the theory of numerical integration methods for differential equations on manifolds. In the pre-Lie setting, the algebraic point of view developed here also provides a concise way to develop Butcher's order theory for Runge--Kutta methods.Comment: 25 page

Correlations between measures of executive attention and cortical thickness of left posterior middle frontal gyrus - a dichotic listening study

Background: The frontal lobe has been associated to a wide range of cognitive control functions and is also vulnerable to degeneration in old age. A recent study by Thomsen and colleagues showed a difference between a young and old sample in grey matter density and activation in the left middle frontal cortex (MFC) and performance on a dichotic listening task. The present study investigated this brain behaviour association within a sample of healthy older individuals, and predicted a positive correlation between performance in a condition requiring executive attention and measures of grey matter structure of the posterior left MFC. Methods: A dichotic listening forced attention paradigm was used to measure attention control functions. Subjects were instructed to report only the left or the right ear syllable of a dichotically presented consonant-vowel syllable pair. A conflict situation appears when subjects are instructed to report the left ear stimulus, caused by the conflict with the bottom-up, stimulus-driven right ear advantage. Overcoming this processing conflict was used as a measure of executive attention. Thickness and volumes of frontal lobe regions were derived from automated segmentation of 3D magnetic resonance image acquisitions. Results: The results revealed a statistically significant positive correlation between the thickness measure of the left posterior MFC and performance on the dichotic listening measures of executive attention. Follow-up analyses showed that this correlation was only statistically significant in the subgroup that showed the typical bottom-up, stimulus-driven right ear advantage. Conclusion: The results suggest that the left MFC is a part of an executive attention network, and that the dichotic listening forced attention paradigm may be a feasible tool for assessing subtle attentional dysfunctions in older adults

2D and 3D U-Nets for skull stripping in a large and heterogeneous set of head MRI using fastai

Skull stripping in brain imaging is the removal of the parts of images corresponding to non-brain tissue. Fast and accurate skull stripping is a crucial step for numerous medical brain imaging applications, e.g. registration, segmentation and feature extraction, as it eases subsequent image processing steps. In this work, we propose and compare two novel skull stripping methods based on 2D and 3D convolutional neural networks trained on a large, heterogeneous collection of 2777 clinical 3D T1-weighted MRI images from 1681 healthy subjects. We investigated the performance of the models by testing them on 927 images from 324 subjects set aside from our collection of data, in addition to images from an independent, large brain imaging study: the IXI dataset (n = 556). Our models achieved mean Dice scores higher than 0:978 and Jaccard indices higher than 0:957 on all tests sets, making predictions on new unseen brain MR images in approximately 1.4s for the 3D model and 12.4s for the 2D model. A preliminary exploration of the models’ robustness to variation in the input data showed favourable results when compared to a traditional, well-established skull stripping method. With further research aimed at increasing the models’ robustness, such accurate

2D and 3D U-Nets for skull stripping in a large and heterogeneous set of head MRI using fastai

Skull stripping in brain imaging is the removal of the parts of images corresponding to non-brain tissue. Fast and accurate skull stripping is a crucial step for numerous medical brain imaging applications, e.g. registration, segmentation and feature extraction, as it eases subsequent image processing steps. In this work, we propose and compare two novel skull stripping methods based on 2D and 3D convolutional neural networks trained on a large, heterogeneous collection of 2777 clinical 3D T1-weighted MRI images from 1681 healthy subjects. We investigated the performance of the models by testing them on 927 images from 324 subjects set aside from our collection of data, in addition to images from an independent, large brain imaging study: the IXI dataset (n = 556). Our models achieved mean Dice scores higher than 0:978 and Jaccard indices higher than 0:957 on all tests sets, making predictions on new unseen brain MR images in approximately 1.4s for the 3D model and 12.4s for the 2D model. A preliminary exploration of the models’ robustness to variation in the input data showed favourable results when compared to a traditional, well-established skull stripping method. With further research aimed at increasing the models’ robustness, such accurate and fast skull stripping methods can potentially form a useful component of brain MRI analysis pipelines.publishedVersio

Representative factor generation for the interactive visual analysis of high-dimensional data

Datasets with a large number of dimensions per data item (hundreds or more) are challenging both for computational and visual analysis. Moreover, these dimensions have different characteristics and relations that result in sub-groups and/or hierarchies over the set of dimensions. Such structures lead to heterogeneity within the dimensions. Although the consideration of these structures is crucial for the analysis, most of the available analysis methods discard the heterogeneous relations among the dimensions. In this paper, we introduce the construction and utilization of representative factors for the interactive visual analysis of structures in high-dimensional datasets. First, we present a selection of methods to investigate the sub-groups in the dimension set and associate representative factors with those groups of dimensions. Second, we introduce how these factors are included in the interactive visual analysis cycle together with the original dimensions. We then provide the steps of an analytical procedure that iteratively analyzes the datasets through the use of representative factors. We discuss how our methods improve the reliability and interpretability of the analysis process by enabling more informed selections of computational tools. Finally, we demonstrate our techniques on the analysis of brain imaging study results that are performed over a large group of subjects

Post-Lie Algebras and Isospectral Flows

In this paper we explore the Lie enveloping algebra of a post-Lie algebra derived from a classical $R$-matrix. An explicit exponential solution of the corresponding Lie bracket flow is presented. It is based on the solution of a post-Lie Magnus-type differential equation

On algebraic structures of numerical integration on vector spaces and manifolds

Numerical analysis of time-integration algorithms has been applying advanced algebraic techniques for more than fourty years. An explicit description of the group of characters in the Butcher-Connes-Kreimer Hopf algebra first appeared in Butcher's work on composition of integration methods in 1972. In more recent years, the analysis of structure preserving algorithms, geometric integration techniques and integration algorithms on manifolds have motivated the incorporation of other algebraic structures in numerical analysis. In this paper we will survey structures that have found applications within these areas. This includes pre-Lie structures for the geometry of flat and torsion free connections appearing in the analysis of numerical flows on vector spaces. The much more recent post-Lie and D-algebras appear in the analysis of flows on manifolds with flat connections with constant torsion. Dynkin and Eulerian idempotents appear in the analysis of non-autonomous flows and in backward error analysis. Non-commutative Bell polynomials and a non-commutative Fa\`a di Bruno Hopf algebra are other examples of structures appearing naturally in the numerical analysis of integration on manifolds.Comment: 42 pages, final versio

Fractional anisotropy shows differential reduction in frontal-subcortical fiber bundles - A longitudinal MRI study of 76 middle-aged and older adults

Motivated by the frontal- and white matter (WM) retrogenesis hypotheses and the assumptions that fronto-striatal circuits are especially vulnerable in normal aging, the goal of the present study was to identify fiber bundles connecting subcortical nuclei and frontal areas and obtain site-specific information about age related fractional anisotropy (FA) changes. Multimodal magnetic resonance image acquisitions [3D T1-weighted and diffusion weighted imaging (DWI)] were obtained from healthy older adults (N = 76, range 49–80 years at inclusion) at two time points, 3 years apart. A subset of the participants (N = 24) was included at a third time-point. In addition to the frontal-subcortical fibers, the anterior callosal fiber (ACF) and the corticospinal tract (CST) was investigated by its mean FA together with tract parameterization analysis. Our results demonstrated fronto-striatal structural connectivity decline (reduced FA) in normal aging with substantial inter-individual differences. The tract parameterization analysis showed that the along tract FA profiles were characterized by piece-wise differential changes along their extension rather than being uniformly affected. To the best of our knowledge, this is the first longitudinal study detecting age-related changes in frontal-subcortical WM connections in normal aging.publishedVersio