Anisotropic Diffusion of Tensor Fields for Fold Shape Analysis on Surfaces

Boucher, Matthieu; Evans, Alan C.; Siddiqi, Kaleem

doi:10.1007/978-3-642-22092-0_23

Cited by 8 publications

(11 citation statements)

References 11 publications

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“…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.…”

Section: Discussionmentioning

confidence: 99%

“…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.…”

Section: Methodsmentioning

confidence: 99%

“…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 .…”

Section: Introductionmentioning

confidence: 99%

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

Xia

Wang

et al. 2018

Medical Image Analysis

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The cortical surface of the human brain expands dynamically and regionally heterogeneously during the first postnatal year. As all primary and secondary cortical folds as well as many tertiary cortical folds are well established at term birth, the cortical surface area expansion during this stage is largely driven by the increase of surface area in two orthogonal orientations in the tangent plane: 1) the expansion parallel to the folding orientation (i.e., increasing the lengths of folds) and 2) the expansion perpendicular to the folding orientation (i.e., increasing the depths of folds). This information would help us better understand the mechanisms of cortical development and provide important insights into neurodevelopmental disorders, but still remains largely unknown due to lack of dedicated computational methods. To address this issue, we propose a novel method for longitudinal mapping of orientation-specific expansion of cortical surface area in these two orthogonal orientations during early infancy. First, to derive the two orientation fields perpendicular and parallel to cortical folds, we propose to adaptively and smoothly fuse the gradient field of sulcal depth and also the maximum principal direction field, by leveraging their region-specific reliability. Specifically, we formulate this task as a discrete labeling problem, in which each vertex is assigned to an orientation label, and solve it by graph cuts. Then, based on the computed longitudinal deformation of the cortical surface, we estimate the Jacobian matrix at each vertex by solving a least-squares problem and derive its corresponding stretch tensor. Finally, to obtain the orientation-specific cortical surface expansion, we project the stretch tensor into the two orthogonal orientations separately. We have applied the proposed method to 30 healthy infants, and for the first time we revealed the orientation-specific longitudinal cortical surface expansion maps during the first postnatal year.

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Section: Discussionmentioning

confidence: 99%

Section: Methodsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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

Xia

Wang

et al. 2018

Medical Image Analysis

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“…Several approaches have also used anisotropic diffusion either on textures or directly on the surface in order to analyze MRI data [3,9]. However, as for denoising and regularization, these methods focus on computing the solutions of the associated anisotropic heat diffusion, without computing the anisotropic LB operator per se.…”

Section: Medical Image Analysismentioning

confidence: 99%

Anisotropic Laplace-Beltrami Operators for Shape Analysis

Andreux

Rodolà

Aubry

et al. 2015

Lecture Notes in Computer Science

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Abstract. This paper introduces an anisotropic Laplace-Beltrami operator for shape analysis. While keeping useful properties of the standard Laplace-Beltrami operator, it introduces variability in the directions of principal curvature, giving rise to a more intuitive and semantically meaningful diffusion process. Although the benefits of anisotropic diffusion have already been noted in the area of mesh processing (e.g. surface regularization), focusing on the Laplacian itself, rather than on the diffusion process it induces, opens the possibility to effectively replace the omnipresent Laplace-Beltrami operator in many shape analysis methods. After providing a mathematical formulation and analysis of this new operator, we derive a practical implementation on discrete meshes. Further, we demonstrate the effectiveness of our new operator when employed in conjunction with different methods for shape segmentation and matching.

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“…However, the conventional varifold matching framework developed in (Charon and Trouvé, 2013; Durrleman et al, 2014) does not consider the principal curvature direction of the deforming surface, whereas this represents a key feature of the convoluted cortical surface as it encodes the local orientation of sulcal and gyral folds that marked previous work on the cortex (Boucher et al, 2009; Li et al, 2010; Boucher et al, 2011). Furthermore, using the conventional varifold metric to measure surfaces and estimate distance between them operates at a fixed scale under which geometric surface details (e.g., bumps) will be overlooked, thus ignoring the (spatially-varying) scales of cortical foldings.…”

Section: Introductionmentioning

confidence: 99%

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

Rekik

Lin

et al. 2016

NeuroImage

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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.

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Anisotropic Diffusion of Tensor Fields for Fold Shape Analysis on Surfaces

Cited by 8 publications

References 11 publications

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

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

Anisotropic Laplace-Beltrami Operators for Shape Analysis

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

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