Multidirectional and Topography-based Dynamic-scale Varifold Representations with Application to Matching Developing Cortical Surfaces

The human cerebral cortex is marked by great complexity as well as substantial dynamic changes during early postnatal development. To obtain a fairly comprehensive picture of its age-induced and/or disorder-related cortical changes, one needs to match cortical surfaces to one another, while maximizing their anatomical alignment. Methods that geodesically shoot surfaces into one another as currents (a distribution of oriented normals) and varifolds (a distribution of non-oriented normals) provide an elegant Riemannian framework for generic surface matching and reliable statistical analysis. However, both conventional current and varifold matching methods have two key limitations. First, they only use the normals of the surface to measure its geometry and guide the warping process, which overlooks the importance of the orientations of the inherently convoluted cortical sulcal and gyral folds. Second, the ‘conversion’ of a surface into a current or a varifold operates at a fixed scale under which geometric surface details will be neglected, which ignores the dynamic scales of cortical foldings. To overcome these limitations and improve varifold-based cortical surface registration, we propose two different strategies. The first strategy decomposes each cortical surface into its normal and tangent varifold representations, by integrating principal curvature direction field into the varifold matching framework, thus providing rich information of the orientation of cortical folding and better characterization of the complex cortical geometry. The second strategy explores the informative cortical geometric features to perform a dynamic-scale measurement of the cortical surface that depends on the local surface topography (e.g., principal curvature), thereby we introduce the concept of a topography-based dynamic-scale varifold. We tested the proposed varifold variants for registering 12 pairs of dynamically developing cortical surfaces from 0 to 6 months of age. Both variants improved the matching accuracy in terms of closeness to the target surface and the goodness of alignment with regional anatomical boundaries, when compared with three state-of-the-art methods: (1) diffeomorphic spectral matching, (2) conventional current-based surface matching, and (3) conventional varifold-based surface matching

Predicting infant cortical surface development using a 4D varifold-based learning framework and local topography-based shape morphing

Longitudinal neuroimaging analysis methods have remarkably advanced our understanding of early postnatal brain development. However, learning predictive models to trace forth the evolution trajectories of both normal and abnormal cortical shapes remains broadly absent. To fill this critical gap, we pioneered the first prediction model for longitudinal developing cortical surfaces in infants using a spatiotemporal current-based learning framework solely from the baseline cortical surface. In this paper, we detail this prediction model and even further improve its performance by introducing two key variants. First, we use the varifold metric to overcome the limitations of the current metric for surface registration that was used in our preliminary study. We also extend the conventional varifold-based surface registration model for pairwise registration to a spatiotemporal surface regression model. Second, we propose a morphing process of the baseline surface using its topographic attributes such as normal direction and principal curvature sign. Specifically, our method learns from longitudinal data both the geometric (vertices positions) and dynamic (temporal evolution trajectories) features of the infant cortical surface, comprising a training stage and a prediction stage. In the training stage, we use the proposed varifold-based shape regression model to estimate geodesic cortical shape evolution trajectories for each training subject. We then build an empirical mean spatiotemporal surface atlas. In the prediction stage, given an infant, we select the best learnt features from training subjects to simultaneously predict the cortical surface shapes at all later timepoints, based on similarity metrics between this baseline surface and the learnt baseline population average surface atlas. We used a leave-one-out cross validation method to predict the inner cortical surface shape at 3, 6, 9 and 12 months of age from the baseline cortical surface shape at birth. Our method attained a higher prediction accuracy and better captured the spatiotemporal dynamic change of the highly folded cortical surface than the previous proposed prediction method

MS FT-2-2 7 Orthogonal polynomials and quadrature: Theory, computation, and applications

Quadrature rules find many applications in science and engineering. Their analysis is a classical area of applied mathematics and continues to attract considerable attention. This seminar brings together speakers with expertise in a large variety of quadrature rules. It is the aim of the seminar to provide an overview of recent developments in the analysis of quadrature rules. The computation of error estimates and novel applications also are described

Generalized averaged Gaussian quadrature and applications

A simple numerical method for constructing the optimal generalized averaged Gaussian quadrature formulas will be presented. These formulas exist in many cases in which real positive GaussKronrod formulas do not exist, and can be used as an adequate alternative in order to estimate the error of a Gaussian rule. We also investigate the conditions under which the optimal averaged Gaussian quadrature formulas and their truncated variants are internal