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

Abstract

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.

A computational method for longitudinal mapping of orientation-specific expansion of cortical surface in infants

“…There are some possible solutions to estimate the cortical orientation fields. On one hand, several methods have proposed to leverage the maximum principal direction field (Boucher et al, 2009, 2011; Li et al, 2009; Rekik et al, 2016a), which is perpendicular to cortical folds on the highly-bended regions (e.g., gyral crests and sulcal bottoms). However, the maximum principal direction field is inherently am­biguous and sensitive to noise on other regions, e.g., sulcal walls as well as flat regions at gyral crests and sulcal bottoms, thus leading to unreliable orientation fields.…”

“…Several methods have been proposed to use the maximum principal direction (the direction corresponding to the maximum principal curvature) to approximate p e (Boucher et al, 2009, 2011; Li et al, 2009; Rekik et al, 2016a). However, this approximation indeed brings ambiguity in certain regions.…”

“…However, this is a challenging task due to the remarkable complexity, irregularity and inter-subject variability of cortical folds. In the literature, several methods used the maximum principal direction field to approximate p e (Boucher et al, 2009, 2011; Li et al, 2009; Rekik et al, 2016a). However, this strategy is only valid on the highly-bended regions at gyral crests and sulcal bottoms, where the maximum principal direction is perpendicular to the folding orientation, consistent with p e .…”

A General Framework for Curve and Surface Comparison and Registration with Oriented Varifolds

“…Later on, [11,10] introduced the alternative but orientation-invariant representation known as varifolds before the higher order model of normal cycles was recently investigated in [31]. Since then, all these different frameworks have found numerous applications to the morphological analysis of cortical surfaces [26,30], brain sulci [31], white matter fiber bundles [13,18] or registration of lung vessels [29]...…”

Riemannian metric optimization on surfaces (RMOS) for intrinsic brain mapping in the Laplace–Beltrami embedding space

“…Using popular volume-based registration methods such as the Demons (Vercauteren et al, 2009; Thirion, 1998), and large deformation diffeomorphic metric mapping (LDDMM) (Beg et al, 2005), nonlinear maps can be computed between image volumes and thus obtain correspondences across different brain structures, but such methods may not accurately align the surface geometry. With LDDMM, sophisticated surface matching methods (Vaillant and Glaunès, 2005; Vaillant et al, 2007; Durrleman et al, 2009; Qiu et al, 2009; Charon and Trouve, 2013; Rekik et al, 2016) were further developed by deforming the ambient space, i.e., the 3D volume. In this approach, the surface correspondences are not optimized directly but rather obtained once the deformation is completed.…”