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Existing learning-based cortical surface reconstruction approaches heavily rely on the supervision of pseudo ground truth (pGT) cortical surfaces for training. Such pGT surfaces are generated by traditional neuroimage processing pipelines, which are time consuming and difficult to generalize well to low-resolution brain MRI, e.g., from fetuses and neonates. In this work, we present CoSeg, a learning-based cortical surface reconstruction framework weakly supervised by brain segmentations without the need for pGT surfaces. CoSeg introduces temporal attention networks to learn time-varying velocity fields from brain MRI for diffeomorphic surface deformations, which fit an initial surface to target cortical surfaces within only 0.11 seconds for each brain hemisphere. A weakly supervised loss is designed to reconstruct pial surfaces by inflating the white surface along the normal direction towards the boundary of the cortical gray matter segmentation. This alleviates partial volume effects and encourages the pial surface to deform into deep and challenging cortical sulci. We evaluate CoSeg on 1,113 adult brain MRI at 1mm and 2mm resolution. CoSeg achieves superior geometric and morphological accuracy compared to existing learning-based approaches. We also verify that CoSeg can extract high-quality cortical surfaces from fetal brain MRI on which traditional pipelines fail to produce acceptable results., Comment: Accepted by the 27th International Conference on Medical Image Computing and Computer Assisted Intervention (MICCAI 2024)

Clinical adoption of deep learning models has been hindered, in part, because the black-box nature of neural networks leads to concerns regarding their trustworthiness and reliability. These concerns are particularly relevant in the field of neuroimaging due to the complex brain phenotypes and inter-subject heterogeneity often encountered. The challenge can be addressed by interpretable deep learning (iDL) methods that enable the visualisation and interpretation of the inner workings of deep learning models. This study systematically reviewed the literature on neuroimaging applications of iDL methods and critically analysed how iDL explanation properties were evaluated. Seventy-five studies were included, and ten categories of iDL methods were identified. We also reviewed five properties of iDL explanations that were analysed in the included studies: biological validity, robustness, continuity, selectivity, and downstream task performance. We found that the most popular iDL approaches used in the literature may be sub-optimal for neuroimaging data, and we discussed possible future directions for the field.

The Developing Human Connectome Project (dHCP) aims to explore developmental patterns of the human brain during the perinatal period. An automated processing pipeline has been developed to extract high-quality cortical surfaces from structural brain magnetic resonance (MR) images for the dHCP neonatal dataset. However, the current implementation of the pipeline requires more than 6.5 hours to process a single MRI scan, making it expensive for large-scale neuroimaging studies. In this paper, we propose a fast deep learning (DL) based pipeline for dHCP neonatal cortical surface reconstruction, incorporating DL-based brain extraction, cortical surface reconstruction and spherical projection, as well as GPU-accelerated cortical surface inflation and cortical feature estimation. We introduce a multiscale deformation network to learn diffeomorphic cortical surface reconstruction end-to-end from T2-weighted brain MRI. A fast unsupervised spherical mapping approach is integrated to minimize metric distortions between cortical surfaces and projected spheres. The entire workflow of our DL-based dHCP pipeline completes within only 24 seconds on a modern GPU, which is nearly 1000 times faster than the original dHCP pipeline. The qualitative assessment demonstrates that for 82.5% of the test samples, the cortical surfaces reconstructed by our DL-based pipeline achieve superior (54.2%) or equal (28.3%) surface quality compared to the original dHCP pipeline., Comment: Accepted by Medical Image Analysis

Cortical surface analysis has gained increased prominence, given its potential implications for neurological and developmental disorders. Traditional vision diffusion models, while effective in generating natural images, present limitations in capturing intricate development patterns in neuroimaging due to limited datasets. This is particularly true for generating cortical surfaces where individual variability in cortical morphology is high, leading to an urgent need for better methods to model brain development and diverse variability inherent across different individuals. In this work, we proposed a novel diffusion model for the generation of cortical surface metrics, using modified surface vision transformers as the principal architecture. We validate our method in the developing Human Connectome Project (dHCP), the results suggest our model demonstrates superior performance in capturing the intricate details of evolving cortical surfaces. Furthermore, our model can generate high-quality realistic samples of cortical surfaces conditioned on postmenstrual age(PMA) at scan., Comment: 4 pages

We generalise results by Lamberti and Lanza de Cristoforis (2005) concerning the continuity of projections onto eigenspaces of self-adjoint differential operators with compact inverses as the (spatial) domain of the functions is perturbed in $\mathbb{R}^2$. Our main case of interest is the Dirichlet Laplacian. We extend these results from bounds from $H_0^1$ to $H_0^1$ to bounds from $L^p$ to $L^p$, under the assumption that $(-\Delta^{-1}-z)^{-1}$ is $L^p$ bounded when $z$ lies outside of the spectrum of $-\Delta^{-1}$. We show that this assumption is met if the initial domain is a square or a rectangle., Comment: 27 pages, 2 figures

This paper introduces GeoMorph, a novel geometric deep-learning framework designed for image registration of cortical surfaces. The registration process consists of two main steps. First, independent feature extraction is performed on each input surface using graph convolutions, generating low-dimensional feature representations that capture important cortical surface characteristics. Subsequently, features are registered in a deep-discrete manner to optimize the overlap of common structures across surfaces by learning displacements of a set of control points. To ensure smooth and biologically plausible deformations, we implement regularization through a deep conditional random field implemented with a recurrent neural network. Experimental results demonstrate that GeoMorph surpasses existing deep-learning methods by achieving improved alignment with smoother deformations. Furthermore, GeoMorph exhibits competitive performance compared to classical frameworks. Such versatility and robustness suggest strong potential for various neuroscience applications.

The development of robust and generalisable models for encoding the spatio-temporal dynamics of human brain activity is crucial for advancing neuroscientific discoveries. However, significant individual variation in the organisation of the human cerebral cortex makes it difficult to identify population-level trends in these signals. Recently, Surface Vision Transformers (SiTs) have emerged as a promising approach for modelling cortical signals, yet they face some limitations in low-data scenarios due to the lack of inductive biases in their architecture. To address these challenges, this paper proposes the surface Masked AutoEncoder (sMAE) and video surface Masked AutoEncoder (vsMAE) - for multivariate and spatio-temporal pre-training of cortical signals over regular icosahedral grids. These models are trained to reconstruct cortical feature maps from masked versions of the input by learning strong latent representations of cortical structure and function. Such representations translate into better modelling of individual phenotypes and enhanced performance in downstream tasks. The proposed approach was evaluated on cortical phenotype regression using data from the young adult Human Connectome Project (HCP) and developing HCP (dHCP). Results show that (v)sMAE pre-trained models improve phenotyping prediction performance on multiple tasks by $\ge 26\%$, and offer faster convergence relative to models trained from scratch. Finally, we show that pre-training vision transformers on large datasets, such as the UK Biobank (UKB), supports transfer learning to low-data regimes. Our code and pre-trained models are publicly available at https://github.com/metrics-lab/surface-masked-autoencoders ., Comment: Accepted for publications for MIDL 2024; 20 figures; 7 figures

Cortical surface reconstruction plays a fundamental role in modeling the rapid brain development during the perinatal period. In this work, we propose Conditional Temporal Attention Network (CoTAN), a fast end-to-end framework for diffeomorphic neonatal cortical surface reconstruction. CoTAN predicts multi-resolution stationary velocity fields (SVF) from neonatal brain magnetic resonance images (MRI). Instead of integrating multiple SVFs, CoTAN introduces attention mechanisms to learn a conditional time-varying velocity field (CTVF) by computing the weighted sum of all SVFs at each integration step. The importance of each SVF, which is estimated by learned attention maps, is conditioned on the age of the neonates and varies with the time step of integration. The proposed CTVF defines a diffeomorphic surface deformation, which reduces mesh self-intersection errors effectively. It only requires 0.21 seconds to deform an initial template mesh to cortical white matter and pial surfaces for each brain hemisphere. CoTAN is validated on the Developing Human Connectome Project (dHCP) dataset with 877 3D brain MR images acquired from preterm and term born neonates. Compared to state-of-the-art baselines, CoTAN achieves superior performance with only 0.12mm geometric error and 0.07% self-intersecting faces. The visualization of our attention maps illustrates that CoTAN indeed learns coarse-to-fine surface deformations automatically without intermediate supervision., Comment: Accepted by the 26th International Conference on Medical Image Computing and Computer Assisted Intervention, MICCAI 2023

Surface meshes are a favoured domain for representing structural and functional information on the human cortex, but their complex topology and geometry pose significant challenges for deep learning analysis. While Transformers have excelled as domain-agnostic architectures for sequence-to-sequence learning, the quadratic cost of the self-attention operation remains an obstacle for many dense prediction tasks. Inspired by some of the latest advances in hierarchical modelling with vision transformers, we introduce the Multiscale Surface Vision Transformer (MS-SiT) as a backbone architecture for surface deep learning. The self-attention mechanism is applied within local-mesh-windows to allow for high-resolution sampling of the underlying data, while a shifted-window strategy improves the sharing of information between windows. Neighbouring patches are successively merged, allowing the MS-SiT to learn hierarchical representations suitable for any prediction task. Results demonstrate that the MS-SiT outperforms existing surface deep learning methods for neonatal phenotyping prediction tasks using the Developing Human Connectome Project (dHCP) dataset. Furthermore, building the MS-SiT backbone into a U-shaped architecture for surface segmentation demonstrates competitive results on cortical parcellation using the UK Biobank (UKB) and manually-annotated MindBoggle datasets. Code and trained models are publicly available at https://github.com/metrics-lab/surface-vision-transformers., Comment: Accepted for publication at MIDL 2024, 17 pages, 6 figures

The extension of convolutional neural networks (CNNs) to non-Euclidean geometries has led to multiple frameworks for studying manifolds. Many of those methods have shown design limitations resulting in poor modelling of long-range associations, as the generalisation of convolutions to irregular surfaces is non-trivial. Recent state-of-the-art performance of Vision Transformers (ViTs) demonstrates that a general-purpose architecture, which implements self-attention, could replace the local feature learning operations of CNNs. Motivated by the success of attention-modelling in computer vision, we extend ViTs to surfaces by reformulating the task of surface learning as a sequence-to-sequence problem and propose a patching mechanism for surface meshes. We validate the performance of the proposed Surface Vision Transformer (SiT) on two brain age prediction tasks in the developing Human Connectome Project (dHCP) dataset and investigate the impact of pre-training on model performance. Experiments show that the SiT outperforms many surface CNNs, while indicating some evidence of general transformation invariance. Code available at https://github.com/metrics-lab/surface-vision-transformers, Comment: 7 pages, 1 figure, accepted to Transformers for Vision (T4V) workshop at CVPR 2022. arXiv admin note: substantial text overlap with arXiv:2204.03408, arXiv:2203.16414

Recent state-of-the-art performances of Vision Transformers (ViT) in computer vision tasks demonstrate that a general-purpose architecture, which implements long-range self-attention, could replace the local feature learning operations of convolutional neural networks. In this paper, we extend ViTs to surfaces by reformulating the task of surface learning as a sequence-to-sequence learning problem, by proposing patching mechanisms for general surface meshes. Sequences of patches are then processed by a transformer encoder and used for classification or regression. We validate our method on a range of different biomedical surface domains and tasks: brain age prediction in the developing Human Connectome Project (dHCP), fluid intelligence prediction in the Human Connectome Project (HCP), and coronary artery calcium score classification using surfaces from the Scottish Computed Tomography of the Heart (SCOT-HEART) dataset, and investigate the impact of pretraining and data augmentation on model performance. Results suggest that Surface Vision Transformers (SiT) demonstrate consistent improvement over geometric deep learning methods for brain age and fluid intelligence prediction and achieve comparable performance on calcium score classification to standard metrics used in clinical practice. Furthermore, analysis of transformer attention maps offers clear and individualised predictions of the features driving each task. Code is available on Github: https://github.com/metrics-lab/surface-vision-transformers, Comment: 10 pages, 3 figures, Submitted to IEEE Transactions on Medical Imaging

The extension of convolutional neural networks (CNNs) to non-Euclidean geometries has led to multiple frameworks for studying manifolds. Many of those methods have shown design limitations resulting in poor modelling of long-range associations, as the generalisation of convolutions to irregular surfaces is non-trivial. Motivated by the success of attention-modelling in computer vision, we translate convolution-free vision transformer approaches to surface data, to introduce a domain-agnostic architecture to study any surface data projected onto a spherical manifold. Here, surface patching is achieved by representing spherical data as a sequence of triangular patches, extracted from a subdivided icosphere. A transformer model encodes the sequence of patches via successive multi-head self-attention layers while preserving the sequence resolution. We validate the performance of the proposed Surface Vision Transformer (SiT) on the task of phenotype regression from cortical surface metrics derived from the Developing Human Connectome Project (dHCP). Experiments show that the SiT generally outperforms surface CNNs, while performing comparably on registered and unregistered data. Analysis of transformer attention maps offers strong potential to characterise subtle cognitive developmental patterns., Comment: 22 pages, 6 figures, Accepted to MIDL 2022, OpenReview link https://openreview.net/forum?id=mpp843Bsf-

Cortical surface registration is a fundamental tool for neuroimaging analysis that has been shown to improve the alignment of functional regions relative to volumetric approaches. Classically, image registration is performed by optimizing a complex objective similarity function, leading to long run times. This contributes to a convention for aligning all data to a global average reference frame that poorly reflects the underlying cortical heterogeneity. In this paper, we propose a novel unsupervised learning-based framework that converts registration to a multi-label classification problem, where each point in a low-resolution control grid deforms to one of fixed, finite number of endpoints. This is learned using a spherical geometric deep learning architecture, in an end-to-end unsupervised way, with regularization imposed using a deep Conditional Random Field (CRF). Experiments show that our proposed framework performs competitively, in terms of similarity and areal distortion, relative to the most popular classical surface registration algorithms and generates smoother deformations than other learning-based surface registration methods, even in subjects with atypical cortical morphology., Comment: 13 pages

We present CortexODE, a deep learning framework for cortical surface reconstruction. CortexODE leverages neural ordinary differential equations (ODEs) to deform an input surface into a target shape by learning a diffeomorphic flow. The trajectories of the points on the surface are modeled as ODEs, where the derivatives of their coordinates are parameterized via a learnable Lipschitz-continuous deformation network. This provides theoretical guarantees for the prevention of self-intersections. CortexODE can be integrated to an automatic learning-based pipeline, which reconstructs cortical surfaces efficiently in less than 5 seconds. The pipeline utilizes a 3D U-Net to predict a white matter segmentation from brain Magnetic Resonance Imaging (MRI) scans, and further generates a signed distance function that represents an initial surface. Fast topology correction is introduced to guarantee homeomorphism to a sphere. Following the isosurface extraction step, two CortexODE models are trained to deform the initial surface to white matter and pial surfaces respectively. The proposed pipeline is evaluated on large-scale neuroimage datasets in various age groups including neonates (25-45 weeks), young adults (22-36 years) and elderly subjects (55-90 years). Our experiments demonstrate that the CortexODE-based pipeline can achieve less than 0.2mm average geometric error while being orders of magnitude faster compared to conventional processing pipelines., Comment: Accepted by IEEE Transactions on Medical Imaging

Generalized differential forms are used in discussions of metric geometries and Einstein's vacuum field equations. Cartan's structure equations are generalized and applied. In particular flat generalized connections are associated with any metric. Einstein's vacuum field equations and their Lorentzian four-metric solutions are considered from a novel point of view. It is shown how particular flat generalized connections and geometries can encode such Ricci flat metrics., Comment: 21 pages, minor corrections in Section 2

The study of functional brain connectivity (FC) is important for understanding the underlying mechanisms of many psychiatric disorders. Many recent analyses adopt graph convolutional networks, to study non-linear interactions between functionally-correlated states. However, although patterns of brain activation are known to be hierarchically organised in both space and time, many methods have failed to extract powerful spatio-temporal features. To overcome those challenges, and improve understanding of long-range functional dynamics, we translate an approach, from the domain of skeleton-based action recognition, designed to model interactions across space and time. We evaluate this approach using the Human Connectome Project (HCP) dataset on sex classification and fluid intelligence prediction. To account for subject topographic variability of functional organisation, we modelled functional connectomes using multi-resolution dual-regressed (subject-specific) ICA nodes. Results show a prediction accuracy of 94.4% for sex classification (an increase of 6.2% compared to other methods), and an improvement of correlation with fluid intelligence of 0.325 vs 0.144, relative to a baseline model that encodes space and time separately. Results suggest that explicit encoding of spatio-temporal dynamics of brain functional activity may improve the precision with which behavioural and cognitive phenotypes may be predicted in the future., Comment: MLCN 2021

Traditional cortical surface reconstruction is time consuming and limited by the resolution of brain Magnetic Resonance Imaging (MRI). In this work, we introduce Pial Neural Network (PialNN), a 3D deep learning framework for pial surface reconstruction. PialNN is trained end-to-end to deform an initial white matter surface to a target pial surface by a sequence of learned deformation blocks. A local convolutional operation is incorporated in each block to capture the multi-scale MRI information of each vertex and its neighborhood. This is fast and memory-efficient, which allows reconstructing a pial surface mesh with 150k vertices in one second. The performance is evaluated on the Human Connectome Project (HCP) dataset including T1-weighted MRI scans of 300 subjects. The experimental results demonstrate that PialNN reduces the geometric error of the predicted pial surface by 30% compared to state-of-the-art deep learning approaches., Comment: Accepted in The 4th International Workshop on Machine Learning in Clinical Neuroimaging (MLCN2021)

Biomechanical modeling of tissue deformation can be used to simulate different scenarios of longitudinal brain evolution. In this work,we present a deep learning framework for hyper-elastic strain modelling of brain atrophy, during healthy ageing and in Alzheimer's Disease. The framework directly models the effects of age, disease status, and scan interval to regress regional patterns of atrophy, from which a strain-based model estimates deformations. This model is trained and validated using 3D structural magnetic resonance imaging data from the ADNI cohort. Results show that the framework can estimate realistic deformations, following the known course of Alzheimer's disease, that clearly differentiate between healthy and demented patterns of ageing. This suggests the framework has potential to be incorporated into explainable models of disease, for the exploration of interventions and counterfactual examples., Comment: MLCN 2021

We consider a problem relating to magnetic confinement devices known as stellarators. Plasma is confined by magnetic fields generated by current-carrying coils, and here we investigate how closely to the plasma they need to be positioned. Current-carrying coils are represented as singularities within the magnetic field and therefore this problem can be modelled mathematically as finding how far we can harmonically extend a vector field from the boundary of a domain. For this paper we consider two-dimensional domains with real analytic boundary, and prove that a harmonic extension exists if and only if the boundary data satisfies a combined compatibility and regularity condition. Our method of proof uses a generalisation of a result of Hadamard on the Cauchy problem for the Laplacian. We then provide a lower bound on how far we can harmonically extend the vector field from the boundary via the Cauchy--Kovalevskaya Theorem., Comment: 30 pages, 4 figures

We identify explicitly the fractional power spaces for the $L^2$ Dirichlet Laplacian and Dirichlet Stokes operators using the theory of real interpolation. The results are not new, but we hope that our arguments are relatively accessible., Comment: 10 pages. Most of this previously appeared as part of arXiv:1904.03337v1, which was subsequently split for publication

Fetal ultrasound screening during pregnancy plays a vital role in the early detection of fetal malformations which have potential long-term health impacts. The level of skill required to diagnose such malformations from live ultrasound during examination is high and resources for screening are often limited. We present an interpretable, atlas-learning segmentation method for automatic diagnosis of Hypo-plastic Left Heart Syndrome (HLHS) from a single `4 Chamber Heart' view image. We propose to extend the recently introduced Image-and-Spatial Transformer Networks (Atlas-ISTN) into a framework that enables sensitising atlas generation to disease. In this framework we can jointly learn image segmentation, registration, atlas construction and disease prediction while providing a maximum level of clinical interpretability compared to direct image classification methods. As a result our segmentation allows diagnoses competitive with expert-derived manual diagnosis and yields an AUC-ROC of 0.978 (1043 cases for training, 260 for validation and 325 for testing)., Comment: MICCAI'21 Main Conference

Lorentzian 4-metrics are expressed in spinorial coordinates. In these coordinates the metric components can be factorized into a product of complex conjugate quantities. The linearized theory and Einstein's vacuum field equations are studied using these coordinates. The relationship between Lorentzian and complex 4-metrics is discussed., Comment: Equations added. Final section re-written

An important goal of medical imaging is to be able to precisely detect patterns of disease specific to individual scans; however, this is challenged in brain imaging by the degree of heterogeneity of shape and appearance. Traditional methods, based on image registration to a global template, historically fail to detect variable features of disease, as they utilise population-based analyses, suited primarily to studying group-average effects. In this paper we therefore take advantage of recent developments in generative deep learning to develop a method for simultaneous classification, or regression, and feature attribution (FA). Specifically, we explore the use of a VAE-GAN translation network called ICAM, to explicitly disentangle class relevant features from background confounds for improved interpretability and regression of neurological phenotypes. We validate our method on the tasks of Mini-Mental State Examination (MMSE) cognitive test score prediction for the Alzheimer's Disease Neuroimaging Initiative (ADNI) cohort, as well as brain age prediction, for both neurodevelopment and neurodegeneration, using the developing Human Connectome Project (dHCP) and UK Biobank datasets. We show that the generated FA maps can be used to explain outlier predictions and demonstrate that the inclusion of a regression module improves the disentanglement of the latent space. Our code is freely available on Github https://github.com/CherBass/ICAM.